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@@ -0,0 +1,1542 @@
+1
+00:00:21,600 --> 00:00:29,560
+الـ .. في المحاضرة السابقة بدأنا التعرف على Cauchy
+
+2
+00:00:29,560 --> 00:00:35,020
+sequences فأخذنا تعريف الـ Cauchy sequence و أثبتنا 
+
+3
+00:00:35,020 --> 00:00:41,000
+أنه كل convergent sequence is Cauchy و أعتقد كمان
+
+4
+00:00:41,000 --> 00:00:48,510
+أثبتنا أنه كل Cauchy sequence is bounded صحيح؟ اليوم
+
+5
+00:00:48,510 --> 00:00:54,970
+هنثبت العكس و هو أن كل كوشي sequence is convergent
+
+6
+00:00:54,970 --> 00:01:00,630
+فنستعيد بس نستذكر مع بعض تعريف الكوشي sequence
+
+7
+00:01:00,630 --> 00:01:07,450
+definition a
+
+8
+00:01:07,450 --> 00:01:14,010
+sequence of real numbers xn is
+
+9
+00:01:14,010 --> 00:01:14,710
+Cauchy
+
+10
+00:01:18,570 --> 00:01:26,170
+إذا تحقق الشرط التالي لكل epsilon أكبر من الصفر
+
+11
+00:01:26,170 --> 00:01:32,270
+يوجد capital N depends on epsilon natural number
+
+12
+00:01:32,270 --> 00:01:41,660
+such that لو كان n و m bigger than or equal N this
+
+13
+00:01:41,660 --> 00:01:48,840
+implies أن absolute value ل xn minus xm أصغر من 
+
+14
+00:01:48,840 --> 00:01:53,960
+epsilon وشوفنا
+
+15
+00:01:53,960 --> 00:02:03,260
+المرة اللي فاتت أو برهنا lemma 2 و 21 every
+
+16
+00:02:03,260 --> 00:02:06,820
+convergent
+
+17
+00:02:11,450 --> 00:02:17,190
+sequence is Cauchy
+
+18
+00:02:17,190 --> 00:02:27,510
+ثم برهنا another lemma lemma 2 و 22 بتقول
+
+19
+00:02:27,510 --> 00:02:34,250
+اللمّة هذه أن every Cauchy
+
+20
+00:02:34,250 --> 00:02:35,010
+sequence
+
+21
+00:02:40,290 --> 00:02:49,630
+is bounded اليوم
+
+22
+00:02:49,630 --> 00:02:59,890
+هنثبت نظرية مهمة نظرية 2 و 33 وهذه
+
+23
+00:02:59,890 --> 00:03:07,310
+النظرية هي كوشي كوشي
+
+24
+00:03:07,310 --> 00:03:08,150
+criterion
+
+25
+00:03:11,820 --> 00:03:18,680
+أو معيار كوشي معيار 
+
+26
+00:03:18,680 --> 00:03:25,580
+كوشي للتقارب النظرية
+
+27
+00:03:25,580 --> 00:03:35,800
+بتنص على أن a sequence xn contained in R is
+
+28
+00:03:35,800 --> 00:03:36,700
+convergent
+
+29
+00:03:39,150 --> 00:03:55,130
+is convergent if and only if it is Cauchy any
+
+30
+00:03:55,130 --> 00:04:00,610
+sequence of real numbers بتكون convergent if and
+
+31
+00:04:00,610 --> 00:04:04,750
+only if it is Cauchy البرهان
+
+32
+00:04:09,110 --> 00:04:15,430
+this part اللي هو الـ only if part هذا هو نفسه لمّة
+
+33
+00:04:15,430 --> 00:04:29,890
+21 if xn is convergent then
+
+34
+00:04:29,890 --> 00:04:32,990
+by
+
+35
+00:04:32,990 --> 00:04:37,210
+لمّة 21
+
+36
+00:04:40,710 --> 00:04:46,370
+it is Cauchy it is Cauchy
+
+37
+00:04:46,370 --> 00:04:51,850
+لأن هذا جزء برهناه في المحاضرة السابقة على صورة
+
+38
+00:04:51,850 --> 00:05:00,710
+لمّة 21 الـ .. الـ if part هنبرهنه اليوم
+
+39
+00:05:00,710 --> 00:05:09,520
+هنشوف مع بعض assume العكس assume أن الـ sequence xn
+
+40
+00:05:09,520 --> 00:05:16,100
+in is Cauchy وبدنا
+
+41
+00:05:16,100 --> 00:05:25,280
+نثبت إنها convergent طيب بما إنها Cauchy then
+
+42
+00:05:25,280 --> 00:05:31,540
+by لمّا 22 تطلع bounded
+
+43
+00:05:36,760 --> 00:05:43,280
+إذا by لمّة 22 الـ sequence xn is
+
+44
+00:05:43,280 --> 00:05:52,560
+bounded باستخدام 
+
+45
+00:05:52,560 --> 00:05:55,600
+Bolzano-Weierstrass theorem
+
+46
+00:06:05,480 --> 00:06:09,240
+اللي أخدناها المحاضرة السابقة أو اللي قبلها هذا
+
+47
+00:06:09,240 --> 00:06:15,960
+اختصار بولزانو ويرشتراس بولزانو ويرشتراس هنا بتقول
+
+48
+00:06:15,960 --> 00:06:18,900
+أن كل bounded sequence has a convergent
+
+49
+00:06:18,900 --> 00:06:27,720
+subsequence فهي عندي bounded sequence sequence xn
+
+50
+00:06:27,720 --> 00:06:33,480
+in has a
+
+51
+00:06:33,480 --> 00:06:34,560
+convergent
+
+52
+00:06:39,950 --> 00:06:44,370
+sub-sequence xn
+
+53
+00:06:44,370 --> 00:06:57,970
+nk وها دي converges to x* تنتمي إلى R طبعا؟
+
+54
+00:06:57,970 --> 00:07:03,550
+إذن هذه sub-sequence من xn وconvergent to some x
+
+55
+00:07:03,550 --> 00:07:05,350
+* تنتمي إلى R
+
+56
+00:07:08,910 --> 00:07:14,610
+طيب احنا عايزين نثبت claim عايزين 
+
+57
+00:07:14,610 --> 00:07:24,210
+احنا نثبت أن الـ sequence xn converges إلى العدد 
+
+58
+00:07:24,210 --> 00:07:32,530
+x* وبالتالي هيك بنكمل برهان النظرية صح؟ فلبرهان
+
+59
+00:07:32,530 --> 00:07:33,090
+ذلك
+
+60
+00:07:36,470 --> 00:07:44,330
+نستخدم تعريف epsilon capital N للـ limit فبنبدأ بـ
+
+61
+00:07:44,330 --> 00:07:47,790
+epsilon أكبر من الصفر عشوائية let epsilon أكبر من
+
+62
+00:07:47,790 --> 00:07:57,240
+الصفر be given نحتاج أن نشهر أن هناك كابتل N كمية
+
+63
+00:07:57,240 --> 00:07:59,060
+عامة تعتمد على إبسلون كمية عامة تعتمد على إبسلون
+
+64
+00:07:59,060 --> 00:08:00,120
+كمية عامة تعتمد على إبسلون كمية عامة تعتمد على
+
+65
+00:08:00,120 --> 00:08:02,780
+إبسلون كمية عامة تعتمد على إبسلون كمية عامة تعتمد
+
+66
+00:08:02,780 --> 00:08:05,280
+على إبسلون كمية عامة تعتمد على إبسلون كمية عامة
+
+67
+00:08:05,280 --> 00:08:08,080
+تعتمد على إبسلون كمية عامة تعتمد على إبسلون كمية
+
+68
+00:08:08,080 --> 00:08:09,960
+عامة تعتمد على إبسلون كمية عامة تعتمد على إبسلون
+
+69
+00:08:09,960 --> 00:08:14,020
+كمية عامة تعتمد على إبسلون 
+
+70
+00:08:14,020 --> 00:08:21,080
+كمية عامة تعتمد على إبسلون
+
+71
+00:08:23,720 --> 00:08:27,960
+وهي epsilon given، إذا by definition of Cauchy
+
+72
+00:08:27,960 --> 00:08:33,920
+sequence there exists capital N depends on epsilon
+
+73
+00:08:33,920 --> 00:08:44,200
+natural number such that لكل n و m أكبر من أو يساوي
+
+74
+00:08:44,200 --> 00:08:50,400
+capital N، this implies an absolute xn minus xm
+
+75
+00:08:52,000 --> 00:09:00,760
+less than epsilon at null نسمي
+
+76
+00:09:00,760 --> 00:09:06,380
+الـ implication هيا دي (*) طيب 
+
+77
+00:09:06,380 --> 00:09:12,900
+احنا حصلنا على أن الـ sequence xn أو الـ subsequence
+
+78
+00:09:12,900 --> 00:09:18,080
+xnk converges to x*
+
+79
+00:09:20,890 --> 00:09:25,870
+إذا لنفس الـ epsilon و epsilon هي نفس الـ epsilon 
+
+80
+00:09:25,870 --> 00:09:34,130
+given فمن تعريف الـ convergence for 
+
+81
+00:09:34,130 --> 00:09:39,950
+same epsilon أكبر من الصفر نفس الـ epsilon اللي
+
+82
+00:09:39,950 --> 00:09:47,070
+هناك نقدر نلاقي يوجد capital K عدد طبيعي capital K
+
+83
+00:09:49,980 --> 00:09:55,920
+وهذا العدد .. هذا عبارة عن عدد طبيعي وهذا العدد
+
+84
+00:09:55,920 --> 00:10:01,240
+الطبيعي هو واحد من مؤشرات الـ subsequence اللي هم 
+
+85
+00:10:01,240 --> 00:10:10,300
+n1, n2, n3 و
+
+86
+00:10:10,300 --> 00:10:17,500
+هكذا إذا يوجد كابتل K اللي هو واحد عدد طبيعي وهذا
+
+87
+00:10:17,500 --> 00:10:23,080
+واحد من مؤشرات الـ subsequence ممكن أختاره هذا
+
+88
+00:10:23,080 --> 00:10:32,400
+كابتل K أكبر من أو يساوي كابتل N بحيث
+
+89
+00:10:32,400 --> 00:10:41,800
+أن الـ absolute value لـ Xcapital K minus X*
+
+90
+00:10:41,800 --> 00:10:50,000
+أصغر من epsilon على 2 كمان
+
+91
+00:10:50,000 --> 00:10:54,280
+مرة الـ subsequence هي هتconverge لـ X* إذا في
+
+92
+00:10:54,280 --> 00:11:02,780
+capital K natural number و هو واحد من large واحد
+
+93
+00:11:02,780 --> 00:11:10,220
+من الـ indices وطبعا كبير هو ممكن نختاره أكبر من أو 
+
+94
+00:11:10,220 --> 00:11:15,720
+يساوي capital N بحيث المسافة بين X كابتل K و X* أصلا
+
+95
+00:11:15,720 --> 00:11:21,480
+أصغر من epsilon على 2 هو ممكن أن أنا يعني هذا أحط هنا K
+
+96
+00:11:21,480 --> 00:11:26,160
+و أقول أن هذا أصغر من epsilon على 2 لكل K أكبر من أو
+
+97
+00:11:26,160 --> 00:11:33,030
+يساوي كابتل Kصح؟ مش هيك تعريف الـ convergence لكن
+
+98
+00:11:33,030 --> 00:11:39,730
+أنا بدي اخد K بساوي كابتل K وبالتالي اخد بس X
+
+99
+00:11:39,730 --> 00:11:45,270
+كابتل K المسافة بينها و بين X* أصغر من epsilon على 2
+
+100
+00:11:45,270 --> 00:11:53,290
+نسمي المتباينة هذه (**) الآن
+
+101
+00:11:53,290 --> 00:11:53,950
+now
+
+102
+00:11:59,240 --> 00:12:08,140
+أنا عندي كابتل K أكبر من أو يساوي كابتل N so
+
+103
+00:12:08,140 --> 00:12:14,320
+by (*) by
+
+104
+00:12:14,320 --> 00:12:25,600
+(*) with m بساوي كابتل K we
+
+105
+00:12:25,600 --> 00:12:26,720
+have لدينا
+
+106
+00:12:30,820 --> 00:12:40,660
+absolute xn minus x capital k أصغر من epsilon على 2 نسمي
+
+107
+00:12:40,660 --> 00:12:49,900
+هذه المتباينة (***) (***) كمان مرة الـ k هذه
+
+108
+00:12:49,900 --> 00:12:59,980
+اختارناها أكبر منها و يساوي n و من (*) إذا كانت
+
+109
+00:12:59,980 --> 00:13:05,100
+الـ K .. إذا خدت m بساوي كابتل K و هذه أكبر من أو 
+
+110
+00:13:05,100 --> 00:13:11,400
+يساوي N فبتصير المتباينة هذه absolute xn minus xk
+
+111
+00:13:11,400 --> 00:13:17,260
+أصغر من epsilon على 2 و الـ n هذه لازم تكون أكبر
+
+112
+00:13:17,260 --> 00:13:22,780
+من أو يساوي m، إذن هذا صحيح لكل small m أكبر من أو
+
+113
+00:13:22,780 --> 00:13:24,260
+يساوي كابتل N
+
+114
+00:13:29,670 --> 00:13:35,670
+تمام hence by 
+
+115
+00:13:35,670 --> 00:13:44,050
+(**) الآن من (**) and (***)
+
+116
+00:13:48,930 --> 00:13:57,270
+لدينا we have لو كان n أكبر من أو يساوي capital N
+
+117
+00:13:57,270 --> 00:14:11,330
+فهذا بيقدي أن الـ absolute xn minus x* طبعا
+
+118
+00:14:11,330 --> 00:14:18,530
+هنا هترح x capital K و هرجعها 
+
+119
+00:14:28,690 --> 00:14:38,730
+إذا I subtracted xk and get it back باخد هدول
+
+120
+00:14:38,730 --> 00:14:43,640
+الأثنين مع بعض والتحدين هدول مع بعض الـ absolute
+
+121
+00:14:43,640 --> 00:14:49,100
+value بالترانجل inequality بالترانجل inequality
+
+122
+00:14:49,100 --> 00:14:54,380
+هذا أصغر من absolute الحد الأول اللي هو xn
+
+123
+00:14:54,380 --> 00:15:01,400
+minus xk زائد absolute الحد الثاني اللي هو xk
+
+124
+00:15:01,400 --> 00:15:08,500
+minus x* الآن
+
+125
+00:15:08,500 --> 00:15:16,750
+باستخدام (***) من المتباينة هذه هاي عندي أنا 
+
+126
+00:15:16,750 --> 00:15:23,170
+xn أول شي الـ n small n أكبر من أو يساوي capital N
+
+127
+00:15:23,170 --> 00:15:28,590
+هاي small n أكبر من أو يساوي capital N وبالتالي الـ
+
+128
+00:15:28,590 --> 00:15:34,870
+absolute value هذه أصغر من epsilon على 2 زائد و
+
+129
+00:15:34,870 --> 00:15:42,360
+من (**) من (**) هي عندي absolute xk
+
+130
+00:15:42,360 --> 00:15:47,640
+minus x* أصغر من epsilon على 2 المجموع بتطلع
+
+131
+00:15:47,640 --> 00:15:53,460
+epsilon since
+
+132
+00:15:53,460 --> 00:16:00,000
+epsilon أكبر من الصفر was arbitrarily
+
+133
+00:16:03,850 --> 00:16:09,870
+نحن لدينا من مفهوم الـ convergence أنه هيك منكون
+
+134
+00:16:09,870 --> 00:16:16,690
+أثبتنا أن الـ limit xn as n tends to infinity equals
+
+135
+00:16:16,690 --> 00:16:26,790
+x* وهذا بكمل برهان الـ claim و النظرية تمام؟
+
+136
+00:16:26,790 --> 00:16:32,160
+هاي لاحظوا أن احنا بنثبت أننا ندعي أن الـ sequence 
+
+137
+00:16:32,160 --> 00:16:35,660
+xn هي الـ convergent لـ x* حسب تعريف epsilon
+
+138
+00:16:35,660 --> 00:16:40,360
+capital N للـ limits بدأنا بـ epsilon given عشوائية 
+
+139
+00:16:40,360 --> 00:16:46,360
+عدد موجب أثبتنا هي يوجد capital N يعتمد على 
+
+140
+00:16:46,360 --> 00:16:51,660
+epsilon natural number بحيث أن لكل n أكبر من أو يساوي 
+
+141
+00:16:51,660 --> 00:16:59,400
+capital N طلع absolute |xn - x*| <
+
+142
+00:16:59,400 --> 00:17:07,300
+ε لما إن هذا الكلام صحيح لكل ε إذا by 
+
+143
+00:17:07,300 --> 00:17:10,880
+definition limit xn = x*، إذا ال sequence
+
+144
+00:17:10,880 --> 00:17:14,240
+convergent إذا هذا بيكمل البرهان إن لو كانت ال
+
+145
+00:17:14,240 --> 00:17:18,800
+sequence كوشي then it is convergent تمام واضح
+
+146
+00:17:18,800 --> 00:17:23,960
+البرهان؟ okay حلو إذا نعم
+
+147
+00:17:28,410 --> 00:17:32,690
+مش احنا حاكينا أن x<sub>n</sub> is bounded؟ صحيح طيب الحين
+
+148
+00:17:32,690 --> 00:17:37,530
+في عندنا بالنظام و بالسترس في عندنا x<sub>n</sub> في عندنا
+
+149
+00:17:37,530 --> 00:17:41,470
+convergent subsequence صح هذا هي صح convergent ل x
+
+150
+00:17:41,470 --> 00:17:46,590
+and to some x* احنا أخذنا نظرية إذا كانت ال
+
+151
+00:17:46,590 --> 00:17:51,110
+convergent subsequence converge to x* و x to r ف x
+
+152
+00:17:51,110 --> 00:17:55,890
+* and تكون converge ل x* لا ماأخذنا نظرية زيك أنت مش
+
+153
+00:17:55,890 --> 00:17:57,050
+خارق النظرية صح
+
+154
+00:18:00,740 --> 00:18:05,020
+لأ النظرية ما بتحكيش هيك معلش النظرية هذه بتقول لو
+
+155
+00:18:05,020 --> 00:18:09,700
+أنا في عندي bounded sequence و لو كل convergent
+
+156
+00:18:09,700 --> 00:18:13,940
+subsequence من ال sequence هذه convergent لعدد x*
+
+157
+00:18:13,940 --> 00:18:19,160
+فلازم ال sequence نفسها تكون convergent ل x* أنا 
+
+158
+00:18:19,160 --> 00:18:22,900
+عندي بس subsequence واحدة converged ل x*
+
+159
+00:18:22,900 --> 00:18:26,860
+أصلاً وليس every convergent subsequence converged
+
+160
+00:18:26,860 --> 00:18:31,790
+ل x* أصلاً فالفرض الثاني تبع النظرية اللي بتحكي عنها
+
+161
+00:18:31,790 --> 00:18:38,270
+مش متحقق وبالتالي لا أستطيع تطبيق النظرية تمام؟ في
+
+162
+00:18:38,270 --> 00:18:45,290
+أي سؤال ثاني؟ okay ده سؤال كثير يعني مهم و .. و ..
+
+163
+00:18:45,290 --> 00:18:51,790
+و جيد و يا ريت يعني أي حد عنده تساؤل زي هذا يعني
+
+164
+00:18:51,790 --> 00:18:58,410
+يسأله هل في أي شيء في القرآن مش واضح؟ واضح أكثر من 
+
+165
+00:18:58,410 --> 00:19:02,990
+هيك؟ Okay أعتقد أن البرهان واضح يعني لو قرأته
+
+166
+00:19:02,990 --> 00:19:10,710
+بتمعن هتجد أنه يعني سهل و بسيط طيب نأخذ أمثلة على
+
+167
+00:19:10,710 --> 00:19:17,430
+كيف نستخدم تعريف ال Cauchy sequence في إثبات أنه
+
+168
+00:19:17,430 --> 00:19:25,730
+given sequence is Cauchy باستخدام التعريف مباشرة و
+
+169
+00:19:25,730 --> 00:19:28,410
+ليس باستخدام اللي هو Cauchy criterion
+
+170
+00:19:31,950 --> 00:19:47,490
+إذا نأخذ هنا بعض الأمثلة examples
+
+171
+00:19:47,490 --> 00:19:52,930
+الأمثلة
+
+172
+00:19:52,930 --> 00:20:00,250
+دي أنا أعطيها الرقم 224 أول مثال show
+
+173
+00:20:04,710 --> 00:20:13,310
+directly show direct that
+
+174
+00:20:13,310 --> 00:20:20,390
+ال sequence ال
+
+175
+00:20:20,390 --> 00:20:25,130
+sequence 1/n is Cauchy
+
+176
+00:20:35,150 --> 00:20:39,630
+لما أقول show directly أن ال sequence معينة is
+
+177
+00:20:39,630 --> 00:20:44,450
+Cauchy معناها ده بدي أستخدم التعريف بدي أستخدم 
+
+178
+00:20:44,450 --> 00:20:50,450
+التعريف تبع Cauchy sequence فنشوف
+
+179
+00:20:50,450 --> 00:20:56,210
+مع بعض طبعاً
+
+180
+00:20:56,210 --> 00:21:02,230
+البرهان باستخدام التعريف هنبدأ بـ ε > 
+
+181
+00:21:02,230 --> 00:21:07,510
+الصفر ونرد عليها بـ capital N بتخلي ال implication
+
+182
+00:21:07,510 --> 00:21:13,890
+هي دي تتحقق بالظبط زي .. يعني قريب يعني بالظبط زي
+
+183
+00:21:13,890 --> 00:21:18,370
+ما عملنا في إثبات أن ال sequence is convergent و
+
+184
+00:21:18,370 --> 00:21:27,670
+هنستخدم ال Archimedean property نشوف مع بعض let
+
+185
+00:21:29,570 --> 00:21:37,110
+بالمناسبة .. بالمناسبة يعني احنا كيف نحدد ال
+
+186
+00:21:37,110 --> 00:21:40,010
+capital N for any given ε؟
+
+187
+00:21:48,010 --> 00:21:54,270
+أنا يعني هي عندي |x<sub>n</sub> - x<sub>m</sub>| لو في عندي
+
+188
+00:21:54,270 --> 00:21:59,310
+ε given ε موجبة given فمن الآخر أنا
+
+189
+00:21:59,310 --> 00:22:05,190
+عايز أثبت أنه هذا أصغر من ε، مظبوط؟ طب ما هذا
+
+190
+00:22:05,190 --> 00:22:11,530
+عبارة عن |1/n - 1/m| وهذا
+
+191
+00:22:11,530 --> 00:22:16,250
+أصغر من أو يساوي |1/n| + |1/m| مظبوط وهذه أعداد موجبة فهذا 1/n
+
+192
+00:22:16,250 --> 00:22:23,490
++ 1/m مظبوط وهذه أعداد موجبة فهذا 1/n 
+
+193
+00:22:23,490 --> 00:22:28,290
++ 1/m طيب
+
+194
+00:22:28,290 --> 00:22:33,950
+أنا عايز أجيب capital N بحيث
+
+195
+00:22:33,950 --> 00:22:39,050
+أنه لو كانت ال n و ال m أكبر من أو يساوي capital N
+
+196
+00:22:39,050 --> 00:22:44,670
+فبدنا هذا يؤدي إلى ال absolute value هذه أصغر من
+
+197
+00:22:44,670 --> 00:22:49,460
+ε إذن ال n و ال m هدول لازم يكونوا أكبر من
+
+198
+00:22:49,460 --> 00:22:53,880
+capital N اللي أنا مش عارف إيش هي، بدي أجيبها، إذن
+
+199
+00:22:53,880 --> 00:23:02,480
+و بالتالي من هنا هذا بيقود إلى أن 1/n  و كذلك
+
+200
+00:23:02,480 --> 00:23:10,060
+1/m أصغر من أو يساوي 1/capital N، صح؟ إذا
+
+201
+00:23:10,060 --> 00:23:14,720
+كانت n أكبر من أو يساوي capital N فـ 1/n هتصير
+
+202
+00:23:14,720 --> 00:23:19,100
+أصغر من أو يساوي 1/capital N وكذلك بالنسبة ل
+
+203
+00:23:19,100 --> 00:23:25,620
+m، مظبوط؟ إذاً هذا هيصير أصغر من أو يساوي 1/
+
+204
+00:23:25,620 --> 00:23:30,620
+capital N وهذا أصغر من 1/capital N بيساوي 2/
+
+205
+00:23:30,620 --> 00:23:34,980
+capital N الآن بدي أخلي هذا، متى بيكون هذا أصغر من
+
+206
+00:23:34,980 --> 00:23:45,300
+ε؟ أه، إذا هأخذ n أصغر من ε/2 أو 1
+
+207
+00:23:45,300 --> 00:23:51,080
+على n أصغر من ε/2 إذا هذا أصغر من
+
+208
+00:23:51,080 --> 00:23:56,720
+ε عندما 1/n أصغر من ε/2 طيب،
+
+209
+00:23:56,720 --> 00:24:03,640
+أنا لو بدأت بـ ε عدد موجب فـ ε/2 بيطلع
+
+210
+00:24:03,640 --> 00:24:08,540
+عدد موجب و by Archimedean property لأي عدد موجب زي
+
+211
+00:24:08,540 --> 00:24:13,740
+هذا يوجد capital N عدد طبيعي بحيث مقلوبه وأصغر من
+
+212
+00:24:13,740 --> 00:24:18,440
+ε على اتنين إذا capital N اللي بتعتمد على ال
+
+213
+00:24:18,440 --> 00:24:22,380
+given ε لازم تكون مقلوبها أصغر من ε على
+
+214
+00:24:22,380 --> 00:24:26,800
+اتنين عشان يطلع هذا أصغر من ε شفتوا كيف
+
+215
+00:24:26,800 --> 00:24:31,240
+منطلق ال capital N و ال Archimedean property طبعاً
+
+216
+00:24:31,240 --> 00:24:38,930
+تضمن لي وجود مثل هالعدد capital N تمام؟ إذا بآجي
+
+217
+00:24:38,930 --> 00:24:42,790
+بقول هنا let ε الكلام هذا طبعاً بعمله في
+
+218
+00:24:42,790 --> 00:24:47,690
+الهامش بعدين بآجي برتبه بقول let ε أكبر من
+
+219
+00:24:47,690 --> 00:24:56,610
+الصفر be given إذاً
+
+220
+00:24:56,610 --> 00:25:00,930
+it choose by
+
+221
+00:25:00,930 --> 00:25:04,110
+Archimedean property
+
+222
+00:25:08,750 --> 00:25:19,250
+نختار capital N عدد طبيعي بحيث أن مقلوب ال N أصغر
+
+223
+00:25:19,250 --> 00:25:24,910
+من ε على اتنين إذا هنا أثبتت يوجد capital N
+
+224
+00:25:24,910 --> 00:25:29,190
+وهي اعتمدت على ε هي مرتبطة بـ ε
+
+225
+00:25:35,760 --> 00:25:41,740
+هذا هيعطينا ال implication تبع ال Cauchy sequence
+
+226
+00:25:41,740 --> 00:25:46,240
+then
+
+227
+00:25:46,240 --> 00:25:54,280
+لو أخذت n و m أكبر من أو يساوي ال capital N هذه
+
+228
+00:25:54,280 --> 00:26:04,420
+فبالتأكيد هذا هيقود إلى أن 1/n  و كذلك 1/m
+
+229
+00:26:04,420 --> 00:26:09,300
+كلهما أصغر من أو يساوي 1/capital N وهذا
+
+230
+00:26:09,300 --> 00:26:15,120
+بدوره بيقود إلى أن |1/n - 1/m|
+
+231
+00:26:15,120 --> 00:26:26,020
+m طبعاً هذه x<sub>m</sub> وهذه x<sub>n</sub> فشفنا أن هذا أصغر من أو
+
+232
+00:26:26,020 --> 00:26:29,940
+يساوي |1/n| باستخدام ال triangle
+
+233
+00:26:29,940 --> 00:26:36,140
+inequality زائد |-1/m| اللي هو
+
+234
+00:26:36,140 --> 00:26:44,520
+|1/m| طيب هذا بيساوي 1/n +
+
+235
+00:26:44,520 --> 00:26:49,800
+1/m لأن  أعداد موجبة وقول إن هذا أصغر من
+
+236
+00:26:49,800 --> 00:26:55,830
+أو يساوي 1/n + 1/m وهذا بيساوي
+
+237
+00:26:55,830 --> 00:27:05,510
+2/capital N وهذا من الاختيار تبعنا لـ capital N by
+
+238
+00:27:05,510 --> 00:27:15,990
+2/capital N أصغر من ε طب
+
+239
+00:27:15,990 --> 00:27:22,830
+ما هذا .. هذا هو شرط Cauchy صح؟ هذا هو شرط Cauchy إذا
+
+240
+00:27:22,830 --> 00:27:28,310
+by definition بما أن هذا صحيح لكل ε since
+
+241
+00:27:28,310 --> 00:27:39,990
+ε أكبر من الصفر was arbitrary by
+
+242
+00:27:39,990 --> 00:27:45,830
+definition of Cauchy sequence ال sequence x<sub>n</sub> is
+
+243
+00:27:45,830 --> 00:27:50,990
+اللي هي 1/n اللي الحد العام تبعها 1/
+
+244
+00:27:50,990 --> 00:27:57,610
+n is Cauchy تمام
+
+245
+00:27:57,610 --> 00:28:04,230
+هنا أثبتنا أن ال sequence Cauchy مباشرة باستخدام
+
+246
+00:28:04,230 --> 00:28:12,050
+التعريف طبعاً في برهان ثاني ممكن نستخدم Cauchy
+
+247
+00:28:12,050 --> 00:28:18,290
+criterion احنا ممكن نثبت أن ال sequence هذي
+
+248
+00:28:18,290 --> 00:28:24,620
+convergent وأثبتنا هذا الكلام قبل كده صح؟ و حسب
+
+249
+00:28:24,620 --> 00:28:28,640
+Cauchy criterion بما أنه ال sequence convergent
+
+250
+00:28:28,640 --> 00:28:32,760
+then it is Cauchy صح؟ هذا برهان ثاني لكن إذا كنا
+
+251
+00:28:32,760 --> 00:28:38,260
+لكم برهنيها directly يعني استخدم التعريف لازم
+
+252
+00:28:38,260 --> 00:28:45,600
+البرهان هذا هو اللي إيه تكتبوه واضح تمام؟ في أي
+
+253
+00:28:45,600 --> 00:28:46,400
+استفسار؟
+
+254
+00:28:50,060 --> 00:28:51,700
+نأخذ مثال ثاني
+
+255
+00:29:18,730 --> 00:29:27,750
+مثال رقم 2 consider .. consider
+
+256
+00:29:27,750 --> 00:29:36,370
+ال sequence defined
+
+257
+00:29:36,370 --> 00:29:38,710
+inductively
+
+258
+00:29:48,750 --> 00:29:52,770
+إذا في عندي sequence معرفة بطريقة استقرائية
+
+259
+00:29:52,770 --> 00:30:03,070
+كالتالي كما يلي هناخد x<sub>1</sub> = 1 و x<sub>2</sub> = 
+
+260
+00:30:03,070 --> 00:30:12,710
+2 طب و x<sub>n</sub> n ≥ 3 هناخده =
+
+261
+00:30:12,710 --> 00:30:24,340
+1/2 في x<sub>n-2</sub> + x<sub>n-1</sub> طبعاً هذا
+
+262
+00:30:24,340 --> 00:30:30,740
+لكل n عدد طبيعي أكبر من أو يساوي 3 إذا هنا في
+
+263
+00:30:30,740 --> 00:30:35,160
+inductive sequence معرفة بطريقة استقرائية أول حدين اللي
+
+264
+00:30:35,160 --> 00:30:41,200
+هم قيم معينة الحد الثالث وانت طالع معرف بدلالة
+
+265
+00:30:41,200 --> 00:30:47,520
+الحدين اللي قبله مباشرة هذا طبعاً بيعطينا
+
+266
+00:30:47,520 --> 00:30:53,720
+sequence المطلوب عايزين نثبت show أن ال sequence x<sub>n</sub>
+
+267
+00:30:53,720 --> 00:31:04,020
+is convergent و converges to the number 5/
+
+268
+00:31:04,020 --> 00:31:12,220
+3 البرهان
+
+269
+00:31:17,020 --> 00:31:24,000
+هنثبت we first show
+
+270
+00:31:24,000 --> 00:31:37,700
+that sequence x<sub>n</sub> converges by
+
+271
+00:31:37,700 --> 00:31:41,700
+showing
+
+272
+00:31:41,700 --> 00:31:46,540
+بإثبات أنه
+
+273
+00:31:51,510 --> 00:32:00,170
+إنها Cauchy thanks
+
+274
+00:32:00,170 --> 00:32:07,610
+to Cauchy criterion
+
+275
+00:32:07,610 --> 00:32:17,390
+طبعاً هذا بفضل معيار كوشي أو Cauchy criterion هنثبت
+
+276
+00:32:17,390 --> 00:32:23,510
+الأول أن ال sequence هي to convergent بإثبات إنه
+
+277
+00:32:23,510 --> 00:32:28,970
+Cauchy وهذا طبعاً حسب Cauchy criterion إذا أثبتنا إن ال
+
+278
+00:32:28,970 --> 00:32:35,730
+sequence Cauchy بتكون convergent تمام فنشوف كيف ممكن
+
+279
+00:32:35,730 --> 00:32:40,150
+نثبت الكلام هذا فأول شيء بدي أثبت إن ال sequence
+
+280
+00:32:40,150 --> 00:32:44,750
+bounded إذن هنا الإدعاء 
+
+281
+00:32:44,750 --> 00:32:51,710
+الأول أو claim number one السيكونس xn الحد العام
+
+282
+00:32:51,710 --> 00:32:56,890
+تبعها أكبر من أو يساوي الواحد أصغر من أو يساوي اثنين
+
+283
+00:32:56,890 --> 00:33:05,050
+لكل n في N لبرهان
+
+284
+00:33:05,050 --> 00:33:11,810
+ذلك to see this use
+
+285
+00:33:11,810 --> 00:33:14,310
+induction
+
+286
+00:33:19,650 --> 00:33:27,010
+on n so I will leave it for you to prove claim one
+
+287
+00:33:27,010 --> 00:33:33,250
+by induction on n فالحالة
+
+288
+00:33:33,250 --> 00:33:38,010
+لو بنشوف بقرا ال statement هذا when n equals one
+
+289
+00:33:38,010 --> 00:33:44,090
+هذا معناه أن المتباينة هذه هتكون x one أكبر من أو
+
+290
+00:33:44,090 --> 00:33:50,360
+يساوي الواحد أصغر من أو يساوي اثنين وهذا true وهذه
+
+291
+00:33:50,360 --> 00:33:56,880
+صحيحة لأن هاي x واحد بساوي واحد والواحد أكبر من
+
+292
+00:33:56,880 --> 00:34:01,620
+أو يساوي الواحد هو less than or equal to إذن ال
+
+293
+00:34:01,620 --> 00:34:06,000
+statement هذا is true for n يساوي one assume it is
+
+294
+00:34:06,000 --> 00:34:09,620
+true for n يساوي k وprove it for n يساوي k زائد
+
+295
+00:34:09,620 --> 00:34:13,500
+واحد فيعني
+
+296
+00:34:13,500 --> 00:34:16,200
+هسيبكم أنتم تكملوا البرهان البرهان سهل
+
+297
+00:34:19,520 --> 00:34:28,600
+So this is claim one الآن by claim one
+
+298
+00:34:28,600 --> 00:34:36,400
+By claim one the
+
+299
+00:34:36,400 --> 00:34:43,020
+sequence xn is bounded حسب
+
+300
+00:34:43,020 --> 00:34:50,140
+claim one لأن claim one أثبتنا فيه أو هتثبتوا فيه
+
+301
+00:34:50,140 --> 00:34:53,880
+أن الـ xn ال sequence xn كل حدود ال sequence
+
+302
+00:34:53,880 --> 00:34:57,740
+محصورة بين واحد واثنين وبالتالي bounded below by
+
+303
+00:34:57,740 --> 00:35:02,680
+one bound above by two وبالتالي bounded okay إذا
+
+304
+00:35:02,680 --> 00:35:15,440
+ال sequence bounded الآن لو كتبنا writing
+
+305
+00:35:15,440 --> 00:35:16,220
+out
+
+306
+00:35:21,120 --> 00:35:29,260
+الأول مرات... المرات
+
+307
+00:35:29,260 --> 00:35:32,100
+الأول مرات... المرات الأول مرات... المرات الأول
+
+308
+00:35:32,100 --> 00:35:32,120
+المرات الأول مرات... المرات الأول مرات الأول مرات
+
+309
+00:35:32,120 --> 00:35:33,160
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+310
+00:35:33,160 --> 00:35:33,480
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+311
+00:35:33,480 --> 00:35:34,040
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+312
+00:35:34,040 --> 00:35:34,600
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+313
+00:35:34,600 --> 00:35:35,980
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+314
+00:35:35,980 --> 00:35:39,620
+الأول مرات الأول
+
+315
+00:35:39,620 --> 00:35:46,440
+مرات الأول
+
+316
+00:35:46,440 --> 00:35:47,720
+مرات
+
+317
+00:35:49,600 --> 00:35:56,440
+is not monotone لو
+
+318
+00:35:56,440 --> 00:36:02,300
+كتبنا أول أربع خمس ست حدود من ال sequence هذه
+
+319
+00:36:02,300 --> 00:36:08,060
+فبلاحظ أنها ليست monotone ليست increasing neither
+
+320
+00:36:08,060 --> 00:36:14,100
+increasing nor decreasing وبالتالي نقدر نستخدم الـ
+
+321
+00:36:14,100 --> 00:36:23,600
+monotone convergence theorem so we can't we can't
+
+322
+00:36:23,600 --> 00:36:31,120
+use the monotone convergence theorem we
+
+323
+00:36:31,120 --> 00:36:31,940
+can't
+
+324
+00:36:35,030 --> 00:36:42,910
+we can't use monotone convergence theorem الـ
+
+325
+00:36:42,910 --> 00:36:46,570
+sequence bounded عشان نستخدم الـ monotone
+
+326
+00:36:46,570 --> 00:36:49,530
+convergence theorem لازم تكون monotone increasing
+
+327
+00:36:49,530 --> 00:36:53,730
+أو monotone decreasing ف it is not monotone
+
+328
+00:36:53,730 --> 00:36:56,610
+فما أقدرش أستخدم ال monotone convergence theorem
+
+329
+00:36:56,610 --> 00:37:03,210
+عشان أفحص ال convergence ال sequence لازم أبحث عن
+
+330
+00:37:03,210 --> 00:37:09,890
+طريقة ثانية غير الـ monotone convergence فيها طيب
+
+331
+00:37:09,890 --> 00:37:16,530
+هنثبت claim 2 claim
+
+332
+00:37:16,530 --> 00:37:24,230
+2 ادعاء ثاني وهو أن ال sequence xn بتحقق المعادلة
+
+333
+00:37:24,230 --> 00:37:30,290
+absolute xn minus xn زائد واحد يساوي واحد على
+
+334
+00:37:30,290 --> 00:37:37,450
+اثنين أس n ناقص واحد وهذا الكلام صحيح for every n
+
+335
+00:37:37,450 --> 00:37:41,950
+في N to
+
+336
+00:37:41,950 --> 00:37:50,510
+see
+
+337
+00:37:50,510 --> 00:37:54,790
+this لبرهان ذلك use induction
+
+338
+00:37:57,680 --> 00:38:03,880
+use induction on n برضه ممكن برهان المعادلة هذه by
+
+339
+00:38:03,880 --> 00:38:09,460
+induction on n هينبرهن
+
+340
+00:38:09,460 --> 00:38:13,540
+البرهان if
+
+341
+00:38:13,540 --> 00:38:24,670
+n يساوي واحد ف absolute x واحد minus x اثنين يساوي
+
+342
+00:38:24,670 --> 00:38:30,010
+absolute واحد ناقص اثنين يساوي absolute واحد يساوي
+
+343
+00:38:30,010 --> 00:38:35,810
+واحد هذا الطرف اليمين والطرف اليسار واحد على
+
+344
+00:38:35,810 --> 00:38:41,370
+اثنين أس n ناقص واحد يساوي واحد على اثنين زائد
+
+345
+00:38:41,370 --> 00:38:49,110
+صفر يساوي واحد واحد يساوي واحد إذا
+
+346
+00:38:49,110 --> 00:38:54,930
+المعادلة true for n يساوي واحد طيب assume ال
+
+347
+00:38:54,930 --> 00:39:06,670
+induction hypothesis الفرض طبع ال induction assume
+
+348
+00:39:06,670 --> 00:39:12,710
+أنه ال...
+
+349
+00:39:12,710 --> 00:39:25,640
+ال claim is true for n يساوي k و k طبعا أكبر من أو يساوي واحد هذا
+
+350
+00:39:25,640 --> 00:39:30,920
+معناه أن absolute xk minus xk زائد واحد يساوي
+
+351
+00:39:30,920 --> 00:39:37,460
+واحد على اثنين أس k ناقص واحد، صح؟ هذه العبارة
+
+352
+00:39:37,460 --> 00:39:44,020
+صحيحة and k
+
+353
+00:39:44,020 --> 00:39:45,600
+أكبر من أو يساوي واحد
+354
+00:39:49,840 --> 00:39:54,580
+الآن تعال نثبت صحة العبارة عند n يساوي k زائد
+
+355
+00:39:54,580 --> 00:39:59,420
+واحد ناخذ الطرف الشمال عندما n يساوي k زائد واحد
+
+356
+00:39:59,420 --> 00:40:06,600
+هذا عبارة عن x k زائد واحد ناقص x k زائد اثنين
+
+357
+00:40:06,600 --> 00:40:14,020
+بدنا نثبت أن هذا يساوي واحد على اثنين أس k صح؟ طب
+
+358
+00:40:14,020 --> 00:40:21,460
+تعال نشوف هي absolute xk زائد واحد ناقص الآن xk
+
+359
+00:40:21,460 --> 00:40:26,760
+زائد اثنين من ال definition تبع ال sequence بدل n
+
+360
+00:40:26,760 --> 00:40:38,320
+بدل n ب k زائد اثنين فبيطلع نص في xk زائد xk زائد
+
+361
+00:40:38,320 --> 00:40:38,760
+واحد
+
+362
+00:40:49,170 --> 00:41:04,590
+وهذا يساوي وهذا يساوي نص في absolute x x
+
+363
+00:41:04,590 --> 00:41:09,430
+k ناقص x k زائد واحد
+
+364
+00:41:16,590 --> 00:41:19,730
+بعد ما نطرح بيطلع عنده نص عامل مشترك و absolute
+
+365
+00:41:19,730 --> 00:41:26,890
+الآن by induction hypothesis من الفرض تبع ال
+
+366
+00:41:26,890 --> 00:41:33,130
+induction ال absolute value هذه أيها ايش يساوي
+
+367
+00:41:33,130 --> 00:41:39,210
+عوض عنها أي نص ضرب one over two to k ناقص one
+
+368
+00:41:39,210 --> 00:41:43,550
+ويساوي واحد على
+
+369
+00:42:09,140 --> 00:42:09,700
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+370
+00:42:09,700 --> 00:42:09,720
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+371
+00:42:09,720 --> 00:42:09,820
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+372
+00:42:09,820 --> 00:42:10,040
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+373
+00:42:10,040 --> 00:42:10,480
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+374
+00:42:10,480 --> 00:42:10,960
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+375
+00:42:15,820 --> 00:42:23,080
+الآن باستخدام ال claim الثاني ممكن نثبت شغلة مهمة
+
+376
+00:42:23,080 --> 00:42:36,380
+في البرهان إذا
+
+377
+00:42:36,380 --> 00:42:43,650
+خليها هادي للمرة الجاية بس بدي أكتبها خليكم أنتم 
+
+378
+00:42:43,650 --> 00:42:53,390
+تفكروا فيها... خليكم أنتم تفكروا فيها Now
+
+379
+00:42:53,390 --> 00:43:11,210
+using a claim to verify... verify that...
+
+380
+00:43:14,770 --> 00:43:23,190
+Fm أكبر من N فهذا
+
+381
+00:43:23,190 --> 00:43:33,530
+بيودي أن absolute Xn ناقص Xm أصغر من واحد على
+
+382
+00:43:33,530 --> 00:43:39,170
+اثنين أس M ناقص اثنين
+
+383
+00:43:45,950 --> 00:43:54,290
+إذاً هذا ممكن إثباته by the triangle inequality و
+
+384
+00:43:54,290 --> 00:44:06,850
+claim اثنين فبنوقف
+
+385
+00:44:06,850 --> 00:44:14,460
+هنا وبنكمل ال... بنكمل إن شاء الله البرهان في
+
+386
+00:44:14,460 --> 00:44:19,680
+المحاضرة الجاية، في حد عنده أي سؤال أو استفسار؟ 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3eaeb3f4fe659f198485c7c67531dac2c402b39b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/-3lwzxWH7r8_raw.srt
@@ -0,0 +1,1556 @@
+1
+00:00:21,600 --> 00:00:29,560
+ال .. في المحاضرة السابقة بدأنا التعرف على cushy
+
+2
+00:00:29,560 --> 00:00:35,020
+sequences فأخدنا تعريف ال cushy sequence و أثبتنا
+
+3
+00:00:35,020 --> 00:00:41,000
+أنه كل convergence sequence is cushy و أعتقد كمان
+
+4
+00:00:41,000 --> 00:00:48,510
+أثبتنا أنه كل cushy sequence is bounded صحيح؟اليوم
+
+5
+00:00:48,510 --> 00:00:54,970
+هنثبت العكس و هو ان كل كوشي sequence is convergent
+
+6
+00:00:54,970 --> 00:01:00,630
+فنستعيد بس نستذكر مع بعض تعريف الكوشي sequence
+
+7
+00:01:00,630 --> 00:01:07,450
+definition a
+
+8
+00:01:07,450 --> 00:01:14,010
+sequence of real numbers xn is
+
+9
+00:01:14,010 --> 00:01:14,710
+cauchy
+
+10
+00:01:18,570 --> 00:01:26,170
+إذا تحقق الشرط التالي لكل epsilon أكبر من الصفر
+
+11
+00:01:26,170 --> 00:01:32,270
+يوجد capital N depends on epsilon natural number
+
+12
+00:01:32,270 --> 00:01:41,660
+such that لو كان N و N bigger than or equal Nthis
+
+13
+00:01:41,660 --> 00:01:48,840
+implies أن absolute value ل xn minus xm أصغر من
+
+14
+00:01:48,840 --> 00:01:53,960
+إيصال وشوفنا
+
+15
+00:01:53,960 --> 00:02:03,260
+المرة اللي فاتت أو برهننا limit 2 و 21 every
+
+16
+00:02:03,260 --> 00:02:06,820
+convergent
+
+17
+00:02:11,450 --> 00:02:17,190
+sequence is cauchy
+
+18
+00:02:17,190 --> 00:02:27,510
+ثم برهنة another لمبة لمبة اتنين و عشرين بتقول
+
+19
+00:02:27,510 --> 00:02:34,250
+اللمبة هذه ان every cauchy
+
+20
+00:02:34,250 --> 00:02:35,010
+sequence
+
+21
+00:02:40,290 --> 00:02:49,630
+is bounded اليوم
+
+22
+00:02:49,630 --> 00:02:59,890
+هنثبت نظرية مهمة نظرية اتنين تلاتة وعشرين وهذه
+
+23
+00:02:59,890 --> 00:03:07,310
+النظرية هي كوشي كوشي
+
+24
+00:03:07,310 --> 00:03:08,150
+criterion
+
+25
+00:03:11,820 --> 00:03:18,680
+أو معيار كوشي معيار
+
+26
+00:03:18,680 --> 00:03:25,580
+كوشي للتقارب النظرية
+
+27
+00:03:25,580 --> 00:03:35,800
+بتنص على أن a sequence x in contained in R is
+
+28
+00:03:35,800 --> 00:03:36,700
+convergent
+
+29
+00:03:39,150 --> 00:03:55,130
+is convergent if and only if it is cauchy any
+
+30
+00:03:55,130 --> 00:04:00,610
+sequence of real numbers بتكون convergent if and
+
+31
+00:04:00,610 --> 00:04:04,750
+only if it is cauchy البرهان
+
+32
+00:04:09,110 --> 00:04:15,430
+this part اللي هو ال only if part هذا هو نفسه لمّة
+
+33
+00:04:15,430 --> 00:04:29,890
+واحدة عشرين if x in is convergent then
+
+34
+00:04:29,890 --> 00:04:32,990
+by
+
+35
+00:04:32,990 --> 00:04:37,210
+لمّة واحدة عشرين
+
+36
+00:04:40,710 --> 00:04:46,370
+it is cushy it is cushy
+
+37
+00:04:46,370 --> 00:04:51,850
+لأن هذا جزء برهناه في المحاضرة السابقة على صورة
+
+38
+00:04:51,850 --> 00:05:00,710
+لمة واحد وعشرين ال .. ال if part هنبرهنه اليوم
+
+39
+00:05:00,710 --> 00:05:09,520
+هنشوف مع بعض assume العكسassume أن الـ sequence x
+
+40
+00:05:09,520 --> 00:05:16,100
+in is Cauchy وبدنا
+
+41
+00:05:16,100 --> 00:05:25,280
+نثبت إنها convergent طيب بما إنها Cauchy then
+
+42
+00:05:25,280 --> 00:05:31,540
+by لمّا اتنين و عشرين تطلع bounded
+
+43
+00:05:36,760 --> 00:05:43,280
+إذا by لمبة إتنين و عشرين ال sequence x in is
+
+44
+00:05:43,280 --> 00:05:52,560
+bounded بستخدام
+
+45
+00:05:52,560 --> 00:05:55,600
+Bolzano-Weierstrass theorem
+
+46
+00:06:05,480 --> 00:06:09,240
+اللي أخدناها المحاضرة السابقة أو اللي قبلها هذا
+
+47
+00:06:09,240 --> 00:06:15,960
+اختصار بولزانو ويرشتراس بولزانو ويرشتراس هنا بتقول
+
+48
+00:06:15,960 --> 00:06:18,900
+انه كل bounded sequence has a convergent
+
+49
+00:06:18,900 --> 00:06:27,720
+subsequence فهي عندي bounded sequence sequence x
+
+50
+00:06:27,720 --> 00:06:33,480
+in has a
+
+51
+00:06:33,480 --> 00:06:34,560
+convergent
+
+52
+00:06:39,950 --> 00:06:44,370
+sub-sequence x
+
+53
+00:06:44,370 --> 00:06:57,970
+in k وها دي converges to x star تنتمي إلى R طبعا؟
+
+54
+00:06:57,970 --> 00:07:03,550
+إذن هذه sub-sequence من x in وconvergent to some x
+
+55
+00:07:03,550 --> 00:07:05,350
+star تنتمي إلى R
+
+56
+00:07:08,910 --> 00:07:14,610
+طيب احنا عايزين نثبت claim عايزين
+
+57
+00:07:14,610 --> 00:07:24,210
+احنا نثبت ان ال sequence xn converges الى العدد
+
+58
+00:07:24,210 --> 00:07:32,530
+x star وبالتالي هيك بنكمل برهان انظرية صح؟ فلبرهان
+
+59
+00:07:32,530 --> 00:07:33,090
+ذلك
+
+60
+00:07:36,470 --> 00:07:44,330
+نستخدم تعريف epsilon capital N للـ limit فبنبدأ بـ
+
+61
+00:07:44,330 --> 00:07:47,790
+epsilon أكبر من السفر عشوائية let epsilon أكبر من
+
+62
+00:07:47,790 --> 00:07:57,240
+السفر be givenنحتاج أن نشهر أن هناك كابتل N كمية
+
+63
+00:07:57,240 --> 00:07:59,060
+عامة تعتمد على إبسلون كمية عامة تعتمد على إبسلون
+
+64
+00:07:59,060 --> 00:08:00,120
+كمية عامة تعتمد على إبسلون كمية عامة تعتمد على
+
+65
+00:08:00,120 --> 00:08:02,780
+إبسلون كمية عامة تعتمد على إبسلون كمية عامة تعتمد
+
+66
+00:08:02,780 --> 00:08:05,280
+على إبسلون كمية عامة تعتمد على إبسلون كمية عامة
+
+67
+00:08:05,280 --> 00:08:08,080
+تعتمد على إبسلون كمية عامة تعتمد على إبسلون كمية
+
+68
+00:08:08,080 --> 00:08:09,960
+عامة تعتمد على إبسلون كمية عامة تعتمد على إبسلون
+
+69
+00:08:09,960 --> 00:08:14,020
+كمية عامة تعتمد على إبسلون
+
+70
+00:08:14,020 --> 00:08:21,080
+كمية عامة تعتمد على إبسلون
+
+71
+00:08:23,720 --> 00:08:27,960
+وهي إبسلون given، إذا by definition of Cauchy
+
+72
+00:08:27,960 --> 00:08:33,920
+sequence there exists capital N depends on إبسلون
+
+73
+00:08:33,920 --> 00:08:44,200
+natural number such that لكل N و M أكبر من أو ساوي
+
+74
+00:08:44,200 --> 00:08:50,400
+capital N، this implies an absolute X N minus X M
+
+75
+00:08:52,000 --> 00:09:00,760
+less than epsilon at null نسمي
+
+76
+00:09:00,760 --> 00:09:06,380
+ال implication هيا دي star طيب
+
+77
+00:09:06,380 --> 00:09:12,900
+احنا حصلنا على انه ال sequence x أو ال subsequence
+
+78
+00:09:12,900 --> 00:09:18,080
+x in k converges to x star
+
+79
+00:09:20,890 --> 00:09:25,870
+إذا لنفس الـ Epsilon و Epsilon هي نفس الـ Epsilon
+
+80
+00:09:25,870 --> 00:09:34,130
+given فمن تعريف ال convergence for
+
+81
+00:09:34,130 --> 00:09:39,950
+same Epsilon أكبر من الصفر نفس الـ Epsilon اللي
+
+82
+00:09:39,950 --> 00:09:47,070
+هناك نقدر نلاقي يوجد capital K عدد طبيعي capital K
+
+83
+00:09:49,980 --> 00:09:55,920
+وهذا العدد .. هذا عبارة عن عدد طبيعي وهذا العدد
+
+84
+00:09:55,920 --> 00:10:01,240
+الطبيعي هو واحد من مؤشرات الـ subsequence اللي هم
+
+85
+00:10:01,240 --> 00:10:10,300
+n واحد, n اتنين, n تلاتة و
+
+86
+00:10:10,300 --> 00:10:17,500
+هكذاإذا يوجد كابتل K اللي هو واحد عدد طبيعي وهذا
+
+87
+00:10:17,500 --> 00:10:23,080
+واحد من مؤشرات ال subsequence ممكن أختاره هذا
+
+88
+00:10:23,080 --> 00:10:32,400
+كابتل K أكبر من أو ساوي كابتل N بحيث
+
+89
+00:10:32,400 --> 00:10:41,800
+أن ال absolute value ل Xcapital K minus X star
+
+90
+00:10:41,800 --> 00:10:50,000
+أصغر من إبسلون على اتنين كمان
+
+91
+00:10:50,000 --> 00:10:54,280
+مرة السب سيكوينس هي هتconverge ل X star إذا في
+
+92
+00:10:54,280 --> 00:11:02,780
+capital K natural number و هو واحد من large واحد
+
+93
+00:11:02,780 --> 00:11:10,220
+من ال indices و طبعا كبيرهو ممكن نختاره أكبر من أو
+
+94
+00:11:10,220 --> 00:11:15,720
+ساوي capital N بحيث المسافة بين X كابتل K و X أصلا
+
+95
+00:11:15,720 --> 00:11:21,480
+أصغر من ي على 2 هو ممكن أن أنا يعني هذا أحط هنا K
+
+96
+00:11:21,480 --> 00:11:26,160
+و أقول أن هذا أصغر من ي على 2 لكل K أكبر من أو
+
+97
+00:11:26,160 --> 00:11:33,030
+ساوي كابتل Kصح؟ مش هيك تعريف ال convergence لكن
+
+98
+00:11:33,030 --> 00:11:39,730
+انا بدي اخد K بساوي كابتل K وبالتالي اخد بس X
+
+99
+00:11:39,730 --> 00:11:45,270
+كابتل K المسافة بينها و بين X star أصغر من Y على 2
+
+100
+00:11:45,270 --> 00:11:53,290
+نسمي المتباينة هذه double star الان
+
+101
+00:11:53,290 --> 00:11:53,950
+now
+
+102
+00:11:59,240 --> 00:12:08,140
+أنا عندي كابتل كأكبر من أو ساوي كابتل N so
+
+103
+00:12:08,140 --> 00:12:14,320
+by star by
+
+104
+00:12:14,320 --> 00:12:25,600
+star with M بساوي كابتل K we
+
+105
+00:12:25,600 --> 00:12:26,720
+have لدينا
+
+106
+00:12:30,820 --> 00:12:40,660
+absolute xn minus x capital k أصغر من y ع 2 نسمي
+
+107
+00:12:40,660 --> 00:12:49,900
+هذه المتباينة triple triple star كمان مرة ال k هذه
+
+108
+00:12:49,900 --> 00:12:59,980
+اختارناها أكبر منها و يساوي n و من starإذا كانت
+
+109
+00:12:59,980 --> 00:13:05,100
+الـ K .. إذا خدت M بساوي كابتال K و هذه أكبر من أو
+
+110
+00:13:05,100 --> 00:13:11,400
+ساوي N فبتصير المتباينة هذه absolute XN minus XK
+
+111
+00:13:11,400 --> 00:13:17,260
+أزرع من إبسط على اتنين و الـ N هذه لازم تكون أكبر
+
+112
+00:13:17,260 --> 00:13:22,780
+من أو ساوي M، إذن هذا صحيح لكل small M أكبر من أو
+
+113
+00:13:22,780 --> 00:13:24,260
+ساوي كابتال M
+
+114
+00:13:29,670 --> 00:13:35,670
+تمام hence by
+
+115
+00:13:35,670 --> 00:13:44,050
+double star الآن من double star and triple star
+
+116
+00:13:48,930 --> 00:13:57,270
+لدينا we have لو كان n أكبر من أو ساوي capital N
+
+117
+00:13:57,270 --> 00:14:11,330
+فهذا بيقدي أنه absolute xn minus x star طبعا
+
+118
+00:14:11,330 --> 00:14:18,530
+هنا هترح x capital K و هرجعها
+
+119
+00:14:28,690 --> 00:14:38,730
+إذا I subtracted XK and get it back باخد هدوع
+
+120
+00:14:38,730 --> 00:14:43,640
+الأثنين مع بعض و التحدين هدوع مع بعضالـ absolute
+
+121
+00:14:43,640 --> 00:14:49,100
+value بالترانجل inequality بالترانجل الانيقواليتي
+
+122
+00:14:49,100 --> 00:14:54,380
+هذا أصغر من أسابع absolute الحد الأول اللي هو xn
+
+123
+00:14:54,380 --> 00:15:01,400
+minus xk زائد absolute الحد التاني اللي هو xk
+
+124
+00:15:01,400 --> 00:15:08,500
+minus x star الآن
+
+125
+00:15:08,500 --> 00:15:16,750
+باستخدام triple starمن المتباينة هذه هاي عندي انا
+
+126
+00:15:16,750 --> 00:15:23,170
+x اول شي ال n small n أكبر من أو ساوي capital N
+
+127
+00:15:23,170 --> 00:15:28,590
+هاي small n أكبر من أو ساوي capital N وبالتالي ال
+
+128
+00:15:28,590 --> 00:15:34,870
+absolute value هذه أصغر من epsilon على اتنين زاد و
+
+129
+00:15:34,870 --> 00:15:42,360
+من double star من double starهي عندي absolute x K
+
+130
+00:15:42,360 --> 00:15:47,640
+minus x star أصغر من إبسمن على اتنين المجموع بتطلع
+
+131
+00:15:47,640 --> 00:15:53,460
+إبسمن since
+
+132
+00:15:53,460 --> 00:16:00,000
+إبسمن أكبر من السفر was arbitrarily
+
+133
+00:16:03,850 --> 00:16:09,870
+نحن لدينا من مفهوم الـ convergence أنه هيك منكون
+
+134
+00:16:09,870 --> 00:16:16,690
+أثبتنا أنه limit xn as n tends to infinity equals
+
+135
+00:16:16,690 --> 00:16:26,790
+x star وهذا بكمل برهان ال claim و النظرية تمام؟
+
+136
+00:16:26,790 --> 00:16:32,160
+هاي لاحظوا أن احنابنثبت أننا ندعي أن الـSequence
+
+137
+00:16:32,160 --> 00:16:35,660
+Xn هي الـConversion لـX الصار حسب تعريف epsilon
+
+138
+00:16:35,660 --> 00:16:40,360
+capital N للـLimits بدأنا بـepsilon given عشوائية
+
+139
+00:16:40,360 --> 00:16:46,360
+عدد موجب أثبتنا هي يوجد capital N يعتمد على
+
+140
+00:16:46,360 --> 00:16:51,660
+epsilon أشرر numberبحيث انه لكل N أكبر من او ساوي
+
+141
+00:16:51,660 --> 00:16:59,400
+capital N طلع absolute xn minus x star less than
+
+142
+00:16:59,400 --> 00:17:07,300
+epsilon لما ان هذا الكلام صحيح لكل epsilonإذا by
+
+143
+00:17:07,300 --> 00:17:10,880
+definition limit xn ساوي x أسطورة، إذا ال sequence
+
+144
+00:17:10,880 --> 00:17:14,240
+convergent إذا هذا بيكمل البرهان إن لو كانت ال
+
+145
+00:17:14,240 --> 00:17:18,800
+sequence كوشي then it is convergent تمام واضح
+
+146
+00:17:18,800 --> 00:17:23,960
+البرهان؟ okay حلو إذا نعم
+
+147
+00:17:28,410 --> 00:17:32,690
+مش احنا حاكينا ان x and is bounded؟ صحيح طيب الحين
+
+148
+00:17:32,690 --> 00:17:37,530
+في عند قلب بالنزام و بالسترس في عند x and في عند
+
+149
+00:17:37,530 --> 00:17:41,470
+conversion subsequence صح هدا هي صح conversion ل x
+
+150
+00:17:41,470 --> 00:17:46,590
+and to some x star احنا أخدنا نظرية إذا كانت ال
+
+151
+00:17:46,590 --> 00:17:51,110
+conversion subsequence converge to x و x to r ف x
+
+152
+00:17:51,110 --> 00:17:55,890
+and تكون converge ل x لا مااخدنا نظرية زيك انت مش
+
+153
+00:17:55,890 --> 00:17:57,050
+خارق النظرية صح
+
+154
+00:18:00,740 --> 00:18:05,020
+لأ النظرية مابتحكيش هيك معلش النظرية هذه بتقول لو
+
+155
+00:18:05,020 --> 00:18:09,700
+أنا في عندي bounded sequence و لو كل convergent
+
+156
+00:18:09,700 --> 00:18:13,940
+sequence من ال sequence هذه convergent لعدد X
+
+157
+00:18:13,940 --> 00:18:19,160
+فلازم ال sequence نفسها تكون convergent ل X أنا
+
+158
+00:18:19,160 --> 00:18:22,900
+عندي بس sequence sub sequence واحدة converged ل X
+
+159
+00:18:22,900 --> 00:18:26,860
+أصلا و ليس every convergent subsequence converged
+
+160
+00:18:26,860 --> 00:18:31,790
+ل X أصلافالفرض التاني تبع النظرية اللي بتحكي عنها
+
+161
+00:18:31,790 --> 00:18:38,270
+مش متحقق وبالتالي لا استطيع تطبيق النظرية تمام؟ في
+
+162
+00:18:38,270 --> 00:18:45,290
+اي سؤال تاني؟ okay ده سؤال كتير يعني مهم و .. و ..
+
+163
+00:18:45,290 --> 00:18:51,790
+و جيد و ياريت يعني اي حد عنده تساؤل زي هذا يعني
+
+164
+00:18:51,790 --> 00:18:58,410
+يسأله هل في اي شي في القرآن مش واضح؟ واضح اكتر من
+
+165
+00:18:58,410 --> 00:19:02,990
+هيك؟Okay أعتقد أن البرهان واضح يعني لو قرأته
+
+166
+00:19:02,990 --> 00:19:10,710
+بتماعه هتجد أنه يعني سهل و بسيط طيب ناخد أمثلة على
+
+167
+00:19:10,710 --> 00:19:17,430
+كيف نستخدم تعريف ال koshi sequence في إثبات أنه
+
+168
+00:19:17,430 --> 00:19:25,730
+given sequence is koshi باستخدام التعريف مباشرة و
+
+169
+00:19:25,730 --> 00:19:28,410
+ليس باستخدام اللي هو koshi criterion
+
+170
+00:19:31,950 --> 00:19:47,490
+إذا ناخد هنا بعض الأمثلة examples
+
+171
+00:19:47,490 --> 00:19:52,930
+الأمثلة
+
+172
+00:19:52,930 --> 00:20:00,250
+دي أنا أعطيها الرقم 224 أول مثال show
+
+173
+00:20:04,710 --> 00:20:13,310
+directly show direct that
+
+174
+00:20:13,310 --> 00:20:20,390
+ال sequence ال
+
+175
+00:20:20,390 --> 00:20:25,130
+sequence واحد على ان is Cauchy
+
+176
+00:20:35,150 --> 00:20:39,630
+لما اقول show directly ان ال sequence معينة is
+
+177
+00:20:39,630 --> 00:20:44,450
+Cauchy معناها ده بدي استخدم التعريف بدي استخدم
+
+178
+00:20:44,450 --> 00:20:50,450
+التعريف تبع Cauchy sequence فنشوف
+
+179
+00:20:50,450 --> 00:20:56,210
+مع بعض طبعا
+
+180
+00:20:56,210 --> 00:21:02,230
+البرهانباستخدام التعريف هنبدأ بأبسلون أكبر من
+
+181
+00:21:02,230 --> 00:21:07,510
+السفر ونرد عليها بcapital N بتخلي ال implication
+
+182
+00:21:07,510 --> 00:21:13,890
+هي دي تتحقق بالظبط زي .. يعني قريب يعني بالظبط زي
+
+183
+00:21:13,890 --> 00:21:18,370
+ما عملنا في اثبات ان ال sequence is convergent و
+
+184
+00:21:18,370 --> 00:21:27,670
+هنستخدم الarchimedean property نشوف مع بعض let
+
+185
+00:21:29,570 --> 00:21:37,110
+بالمناسبة .. بالمناسبة يعني احنا كيف نحدد ال
+
+186
+00:21:37,110 --> 00:21:40,010
+capital N for any given epsilon؟
+
+187
+00:21:48,010 --> 00:21:54,270
+أنا يعني هي عندي absolute xn minus xm لو في عندي
+
+188
+00:21:54,270 --> 00:21:59,310
+epsilon given epsilon موجة given فمن الآخر أنا
+
+189
+00:21:59,310 --> 00:22:05,190
+عايز أثبت أنه هذا أصغر من إمسنان، مظبوط؟ طب ما هذا
+
+190
+00:22:05,190 --> 00:22:11,530
+عبارة عن absolute واحد على n minus واحد على m وهذا
+
+191
+00:22:11,530 --> 00:22:16,250
+أصغر من أو ساوي absolute واحد على n زائد absolute
+
+192
+00:22:16,250 --> 00:22:23,490
+واحد على mبصبوط وهذه أعداد موجبة فهذا واحد على M
+
+193
+00:22:23,490 --> 00:22:28,290
+زائد واحد على M طيب
+
+194
+00:22:28,290 --> 00:22:33,950
+أنا عايز أجيب capital M بحيث
+
+195
+00:22:33,950 --> 00:22:39,050
+أنه لو كانت ال N و ال M أكبر من أو ساوي capital N
+
+196
+00:22:39,050 --> 00:22:44,670
+فبدنا هذا يؤدي إلى ال absolute value هذه أصغر من
+
+197
+00:22:44,670 --> 00:22:49,460
+إبسلونإذن ال N و ال M هدول لازم يكونوا أكبر من
+
+198
+00:22:49,460 --> 00:22:53,880
+capital N اللي أنا مش عارف إيش هي، بدي أجيبها، إذن
+
+199
+00:22:53,880 --> 00:23:02,480
+و بالتالي من هنا هذا بيقدي إن واحد على N و كذلك
+
+200
+00:23:02,480 --> 00:23:10,060
+واحد على M أصغر من أوسع واحد على capital N، صح؟إذا
+
+201
+00:23:10,060 --> 00:23:14,720
+كانت n أكبر من أو يساوي capital N ف1 على n هتصير
+
+202
+00:23:14,720 --> 00:23:19,100
+أصغر من أو يساوي 1 على capital N وكذلك بالنسبة ل
+
+203
+00:23:19,100 --> 00:23:25,620
+M، مظبوط؟ إذاً هذا هيصير أصغر من أو يساوي 1 على
+
+204
+00:23:25,620 --> 00:23:30,620
+capital N وهذا أصغر من 1 على capital N بساوي 2 على
+
+205
+00:23:30,620 --> 00:23:34,980
+capital M الآن بدي أخلي هذا، متى بيكون هذا أصغر من
+
+206
+00:23:34,980 --> 00:23:45,300
+epsilon؟أه، إذا هاخد n أصغر من epsilon على 2 أو 1
+
+207
+00:23:45,300 --> 00:23:51,080
+على n أصغر من epsilon على 2 إذا هذا أصغر من
+
+208
+00:23:51,080 --> 00:23:56,720
+epsilon عندما 1 على n أصغر من epsilon على 2 طيب،
+
+209
+00:23:56,720 --> 00:24:03,640
+أنا لو بدأت بepsilon عدد موجب فepsilon على 2 بطلع
+
+210
+00:24:03,640 --> 00:24:08,540
+عدد موجب و by Archimedean propertyلأي عدد موجب زي
+
+211
+00:24:08,540 --> 00:24:13,740
+هذا يوجد capital N عدد طبيعي بحيث مقلوب و أصغر من
+
+212
+00:24:13,740 --> 00:24:18,440
+epsilon ع اتنين اذا capital N اللي بتعتمد ع ال
+
+213
+00:24:18,440 --> 00:24:22,380
+given epsilon لازم تكون مقلوبها أصغر من epsilon ع
+
+214
+00:24:22,380 --> 00:24:26,800
+اتنين عشان يطلع هذا أصغر من epsilon شوفتوا كيف
+
+215
+00:24:26,800 --> 00:24:31,240
+منطلق ال capital N و ال Archimedean property طبعا
+
+216
+00:24:31,240 --> 00:24:38,930
+تضمنلي وجود مثل هالعدد capital Nتمام؟ إذا باجي
+
+217
+00:24:38,930 --> 00:24:42,790
+بقول هنا let epsilon الكلام هذا طبعا بعمله في
+
+218
+00:24:42,790 --> 00:24:47,690
+الهامش بعدين باجي برتبه بقول let epsilon أكبر من
+
+219
+00:24:47,690 --> 00:24:56,610
+السفر be given إذا
+
+220
+00:24:56,610 --> 00:25:00,930
+it choose by
+
+221
+00:25:00,930 --> 00:25:04,110
+Archimedean property
+
+222
+00:25:08,750 --> 00:25:19,250
+نختار capital N عدد طبيعي بحيث انه مقلوب ال N أصغر
+
+223
+00:25:19,250 --> 00:25:24,910
+من إبسلون على اتنين إذا هان أثبتت يوجد capital N
+
+224
+00:25:24,910 --> 00:25:29,190
+وهي اعتمد على إبسلون هي مرتبطة بإبسلون
+
+225
+00:25:35,760 --> 00:25:41,740
+هذا هيعطينا ال implication تبع الكوشي sequence
+
+226
+00:25:41,740 --> 00:25:46,240
+then
+
+227
+00:25:46,240 --> 00:25:54,280
+لو أخدت N و M أكبر من أوسع ال capital N هذه
+
+228
+00:25:54,280 --> 00:26:04,420
+فبالتأكيد هذا هيقدر ال 1 على N و كذلكواحد على M
+
+229
+00:26:04,420 --> 00:26:09,300
+كلهما أصغر من أو ساوي واحد على capital N وهذا
+
+230
+00:26:09,300 --> 00:26:15,120
+بدوره بيقدي أنه absolute واحد على M minus واحد على
+
+231
+00:26:15,120 --> 00:26:26,020
+M طبعا هذه XM وهذه XM فشوفنا أن هذا أصغر من أو
+
+232
+00:26:26,020 --> 00:26:29,940
+ساوي absolute واحد على Mباستخدام ال triangle
+
+233
+00:26:29,940 --> 00:26:36,140
+inequality زائد absolute سالب واحد على M اللي هو
+
+234
+00:26:36,140 --> 00:26:44,520
+absolute واحد على M طيب هذا بساوي واحد على M زائد
+
+235
+00:26:44,520 --> 00:26:49,800
+واحد على M لإن رد عداد موجبة وقول إن هذا أصغر من
+
+236
+00:26:49,800 --> 00:26:55,830
+أو يساوي واحد على Mزايد واحد على N وهذا بيساوي
+
+237
+00:26:55,830 --> 00:27:05,510
+اتنين على N وهذا من الاختيار تبعنا ل capital N by
+
+238
+00:27:05,510 --> 00:27:15,990
+star اتنين على N أصغر من epsilon طب
+
+239
+00:27:15,990 --> 00:27:22,830
+ما هذه .. هذا هو شرط Koshi صح؟ هذا هو شرط Koshiإذا
+
+240
+00:27:22,830 --> 00:27:28,310
+by definition بما أن هذا صحيح لكل epsilon since
+
+241
+00:27:28,310 --> 00:27:39,990
+epsilon أكبر من الصفر was arbitrary by
+
+242
+00:27:39,990 --> 00:27:45,830
+definition of Cauchy sequence ال sequence xn is
+
+243
+00:27:45,830 --> 00:27:50,990
+اللي هي واحد على n اللي الحد العام تبعها واحد على
+
+244
+00:27:50,990 --> 00:27:57,610
+nis Cauchy تمام
+
+245
+00:27:57,610 --> 00:28:04,230
+هنا أثبتنا إن ال sequence Cauchy مباشرة باستخدام
+
+246
+00:28:04,230 --> 00:28:12,050
+التعريف طبعا في برهان تاني ممكن نستخدم Cauchy
+
+247
+00:28:12,050 --> 00:28:18,290
+criterion احنا ممكن نثبت إن ال sequence هذي
+
+248
+00:28:18,290 --> 00:28:24,620
+convergentو أثبتنا هذا الكلام جبليك صح؟ و حسب
+
+249
+00:28:24,620 --> 00:28:28,640
+cushy criterion بما أنه ال sequence convergent
+
+250
+00:28:28,640 --> 00:28:32,760
+then it is cushy صح؟ هذا برهان تاني لكن إذا كنا
+
+251
+00:28:32,760 --> 00:28:38,260
+لكم برهنيها directly يعني استخدم التعريف لازم
+
+252
+00:28:38,260 --> 00:28:45,600
+البرهان هذا هو اللي إيه تكتبوه واضح تمام؟ في أي
+
+253
+00:28:45,600 --> 00:28:46,400
+استفسار؟
+
+254
+00:28:50,060 --> 00:28:51,700
+ناخد مثال تاني
+
+255
+00:29:18,730 --> 00:29:27,750
+مثال تقم اتنين consider .. consider
+
+256
+00:29:27,750 --> 00:29:36,370
+ال sequence defined
+
+257
+00:29:36,370 --> 00:29:38,710
+inductively
+
+258
+00:29:48,750 --> 00:29:52,770
+إذا في عندي sequence معرفة بطريقة استقرائية
+
+259
+00:29:52,770 --> 00:30:03,070
+كالتالي كما هي ليه هناخد x1 بساوي واحد و x2 بساوي
+
+260
+00:30:03,070 --> 00:30:12,710
+اتنين طب و xn-n أكبر من أو ساوي تلاتة هناخده بساوي
+
+261
+00:30:12,710 --> 00:30:24,340
+نص فيxn سالب اتنين زائد xn negative one طبعا هذا
+
+262
+00:30:24,340 --> 00:30:30,740
+لكل n أدب طبيعي أكبر من أو ساوى تلاتة اذا هنا في
+
+263
+00:30:30,740 --> 00:30:35,160
+اندي سيكوانس معرفة بطريقة استقرائية اول حدين اللي
+
+264
+00:30:35,160 --> 00:30:41,200
+هم قيم معينة الحد التالت وانت طالع معرف بدلالة
+
+265
+00:30:41,200 --> 00:30:47,520
+الحد اللي حدين اللي جابله مباشرةهذا طبعا بيعطينا
+
+266
+00:30:47,520 --> 00:30:53,720
+sequence المطلوب عايزين نثبت show ان ال sequence x
+
+267
+00:30:53,720 --> 00:31:04,020
+in is convergent و converges to the number 5 over
+
+268
+00:31:04,020 --> 00:31:12,220
+3 البرهان
+
+269
+00:31:17,020 --> 00:31:24,000
+هنثبت we first show
+
+270
+00:31:24,000 --> 00:31:37,700
+that sequence xn converges by
+
+271
+00:31:37,700 --> 00:31:41,700
+showing
+
+272
+00:31:41,700 --> 00:31:46,540
+بإثبات أنه
+
+273
+00:31:51,510 --> 00:32:00,170
+إنها كوشي thanks
+
+274
+00:32:00,170 --> 00:32:07,610
+to koshi criterion
+
+275
+00:32:07,610 --> 00:32:17,390
+طبعا هذا بفضل معيار كوشي أو كوشي criterion هنثبت
+
+276
+00:32:17,390 --> 00:32:23,510
+الأول أن ال sequence هي to convergentبإثبات إنه
+
+277
+00:32:23,510 --> 00:32:28,970
+كوشي وهذا طبعا حسب كوشي criterion إذا أثبتنا إن ال
+
+278
+00:32:28,970 --> 00:32:35,730
+sequence كوشي بتكون convergent تمام فنشوف كيف ممكن
+
+279
+00:32:35,730 --> 00:32:40,150
+نثبت الكلام هذا فأول شيء بدي أثبت إن ال sequence
+
+280
+00:32:40,150 --> 00:32:44,750
+bounded إذن هنا الإدعاء
+
+281
+00:32:44,750 --> 00:32:51,710
+الأول أو claim number oneالسيكونس xn الحد العام
+
+282
+00:32:51,710 --> 00:32:56,890
+تبعها أكبر من أو ساوي الواحد أصغر من أو ساوي اتنين
+
+283
+00:32:56,890 --> 00:33:05,050
+لكل n في n لبرهان
+
+284
+00:33:05,050 --> 00:33:11,810
+ذلك to see this use
+
+285
+00:33:11,810 --> 00:33:14,310
+induction
+
+286
+00:33:19,650 --> 00:33:27,010
+on n so I will leave it for you to prove claim one
+
+287
+00:33:27,010 --> 00:33:33,250
+by induction on n فالحالة
+
+288
+00:33:33,250 --> 00:33:38,010
+لو بنشوف بقرا ال statement هذا when n equals one
+
+289
+00:33:38,010 --> 00:33:44,090
+هذا معناه ان المتباين هذه هتكون x one أكبر من أو
+
+290
+00:33:44,090 --> 00:33:50,360
+ساوى الواحد أصغر من أو ساوى اتنين وهذا trueو هذه
+
+291
+00:33:50,360 --> 00:33:56,880
+صحيحة لأن هاي x واحد بساوي واحد و الواحد أكبر من
+
+292
+00:33:56,880 --> 00:34:01,620
+أو ساوي الواحد هو less than or equal to لذن ال
+
+293
+00:34:01,620 --> 00:34:06,000
+statement هذا is true for n بساوي one assume it is
+
+294
+00:34:06,000 --> 00:34:09,620
+true for n بساوي k و prove it for n بساوي k زاد
+
+295
+00:34:09,620 --> 00:34:13,500
+واحد فيعني
+
+296
+00:34:13,500 --> 00:34:16,200
+هسيبكم أنتم تكملوا البرهان البرهان سهل
+
+297
+00:34:19,520 --> 00:34:28,600
+So this is claim one الان by claim one
+
+298
+00:34:28,600 --> 00:34:36,400
+By claim one The
+
+299
+00:34:36,400 --> 00:34:43,020
+sequence x in is bounded حسب
+
+300
+00:34:43,020 --> 00:34:50,140
+claim one لأن claim oneأثبتنا فيه أو هتثبتوا فيه
+
+301
+00:34:50,140 --> 00:34:53,880
+ان الـ x in ال sequence x كل حدود ال sequence
+
+302
+00:34:53,880 --> 00:34:57,740
+محصورة بين واحد واتنين وبالتالي bounded below by
+
+303
+00:34:57,740 --> 00:35:02,680
+one bound above by two وبالتالي bounded okay إذا
+
+304
+00:35:02,680 --> 00:35:15,440
+ال sequence bounded الآن لو كتبنا writing
+
+305
+00:35:15,440 --> 00:35:16,220
+out
+
+306
+00:35:21,120 --> 00:35:29,260
+الأول مرات .. المرات
+
+307
+00:35:29,260 --> 00:35:32,100
+الأول مرات .. المرات الأول مرات .. المرات الأول
+
+308
+00:35:32,100 --> 00:35:32,100
+مرات .. المرات الأول مرات .. المرات الأول مرات ..
+
+309
+00:35:32,100 --> 00:35:32,120
+المرات الأول مرات .. المرات الأول مرات الأول مرات
+
+310
+00:35:32,120 --> 00:35:32,120
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+311
+00:35:32,120 --> 00:35:33,160
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+312
+00:35:33,160 --> 00:35:33,480
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+313
+00:35:33,480 --> 00:35:34,040
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+314
+00:35:34,040 --> 00:35:34,600
+الأول مرات الأول مرات الأول مرات الأول مرات الأول
+
+315
+00:35:34,600 --> 00:35:35,980
+مرات الأول مرات الأول مرات الأول مرات الأول مرات
+
+316
+00:35:35,980 --> 00:35:39,620
+الأول مرات الأول
+
+317
+00:35:39,620 --> 00:35:46,440
+مرات الأول
+
+318
+00:35:46,440 --> 00:35:47,720
+مرات
+
+319
+00:35:49,600 --> 00:35:56,440
+is not monotone لو
+
+320
+00:35:56,440 --> 00:36:02,300
+كتبنا أول أربع خمس ست حدود من ال sequence هذه
+
+321
+00:36:02,300 --> 00:36:08,060
+فبلاحظ أنها ليست monotone ليست increasing neither
+
+322
+00:36:08,060 --> 00:36:14,100
+increasing nor decreasingوبالتالي نقدر نستخدم الـ
+
+323
+00:36:14,100 --> 00:36:23,600
+monotone convergence theorem so we can't we can't
+
+324
+00:36:23,600 --> 00:36:31,120
+use ال monotone convergence theorem we
+
+325
+00:36:31,120 --> 00:36:31,940
+can't
+
+326
+00:36:35,030 --> 00:36:42,910
+we can't use monotone convergence theorem الـ
+
+327
+00:36:42,910 --> 00:36:46,570
+sequence bounded عشان استخدم الـ monotone
+
+328
+00:36:46,570 --> 00:36:49,530
+convergence theorem لازم تكون monotone increasing
+
+329
+00:36:49,530 --> 00:36:53,730
+او monotone decreasing ف it is not monotone
+
+330
+00:36:53,730 --> 00:36:56,610
+فماقدرش استخدم ال monotone convergence theorem
+
+331
+00:36:56,610 --> 00:37:03,210
+عشان افحص ال convergenceالـ sequence لازم ابحث عن
+
+332
+00:37:03,210 --> 00:37:09,890
+طريقه تانية غير الـ monotone convergence فيها طيب
+
+333
+00:37:09,890 --> 00:37:16,530
+هنثبت claim 2 claim
+
+334
+00:37:16,530 --> 00:37:24,230
+2 ادعاء تاني وهو ان ال sequence xn بتحقق المعادلة
+
+335
+00:37:24,230 --> 00:37:30,290
+absolute xn minus xn زيادة واحدبساوي واحد على
+
+336
+00:37:30,290 --> 00:37:37,450
+اتنين قص ان نجاتف وان وهذا الكلام صحيح for every n
+
+337
+00:37:37,450 --> 00:37:41,950
+في n to
+
+338
+00:37:41,950 --> 00:37:50,510
+see
+
+339
+00:37:50,510 --> 00:37:54,790
+this لبرهان ذلك use induction
+
+340
+00:37:57,680 --> 00:38:03,880
+use induction on n برضه ممكن برهان المعادلة هذه by
+
+341
+00:38:03,880 --> 00:38:09,460
+induction on n هينبرهن
+
+342
+00:38:09,460 --> 00:38:13,540
+البرهان if
+
+343
+00:38:13,540 --> 00:38:24,670
+n بسبب واحد ف absolute x واحد minus xأتنين بساوي
+
+344
+00:38:24,670 --> 00:38:30,010
+absolute واحد سالي اتنين بساوي absolute واحد بساوي
+
+345
+00:38:30,010 --> 00:38:35,810
+واحد هذا الطرف اليمين و الطرف اليسار واحد على
+
+346
+00:38:35,810 --> 00:38:41,370
+اتنين plus n minus واحد بساوي واحد على اتنين plus
+
+347
+00:38:41,370 --> 00:38:49,110
+سفر بساوي واحدة واحد بساوي واحد اذا
+
+348
+00:38:49,110 --> 00:38:54,930
+المعادلة true for n بساوي واحدطيب assume ال
+
+349
+00:38:54,930 --> 00:39:06,670
+induction hypothesis الفرض طبع ال induction assume
+
+350
+00:39:06,670 --> 00:39:12,710
+أنه ال ..
+
+351
+00:39:12,710 --> 00:39:25,640
+ال claim is true for n بساوة kو k طبعا أكبر من أول
+
+352
+00:39:25,640 --> 00:39:30,920
+سالة من واحد هذا
+
+353
+00:39:30,920 --> 00:39:37,460
+معناه أن absolute xk minus xk زيادة واحد بسالة
+
+354
+00:39:37,460 --> 00:39:44,020
+واحد على اتنين أس كسالب واحد، صح؟ هذه الأدارة
+
+355
+00:39:44,020 --> 00:39:45,600
+صحيحة and k
+
+356
+00:39:49,840 --> 00:39:54,580
+الان تعالى نثبت صحة العبارة عند n بساوي k زايد
+
+357
+00:39:54,580 --> 00:39:59,420
+واحد ناخد الطرف الشمال عندما n بساوي k زايد واحد
+
+358
+00:39:59,420 --> 00:40:06,600
+هذا عبارة عن x k زايد واحد سالد x k زايد اتنين
+
+359
+00:40:06,600 --> 00:40:14,020
+بدنا نثبت ان هذا بساوي واحد على اتنين اص k صح؟ طب
+
+360
+00:40:14,020 --> 00:40:21,460
+تعالى نشوفهي absolute xk plus one minus الان xk
+
+361
+00:40:21,460 --> 00:40:26,760
+زائد اتنين من ال definition تبع ال sequence بدل n
+
+362
+00:40:26,760 --> 00:40:38,320
+بدل n بk زائد اتنين فبطلع نص في xk زائد xk زائد
+
+363
+00:40:38,320 --> 00:40:38,760
+واحد
+
+364
+00:40:49,170 --> 00:41:04,590
+وهذا بيساوي و هذا بيساوي نص في absolute x x
+
+365
+00:41:04,590 --> 00:41:09,430
+k negative x k plus one
+
+366
+00:41:16,590 --> 00:41:19,730
+بعد ما نطرح بطلع عنده نص عامل مشترك و absolute
+
+367
+00:41:19,730 --> 00:41:26,890
+الان by induction hypothesis من الفرض تبع ال
+
+368
+00:41:26,890 --> 00:41:33,130
+induction ال absolute value هذه أيها ايش بيساوي
+
+369
+00:41:33,130 --> 00:41:39,210
+عوض عنها اي نص ضرب one over two to k negative one
+
+370
+00:41:39,210 --> 00:41:43,550
+ويساوي واحد على
+
+371
+00:42:09,140 --> 00:42:09,700
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+372
+00:42:09,700 --> 00:42:09,720
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+373
+00:42:09,720 --> 00:42:09,820
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+374
+00:42:09,820 --> 00:42:10,040
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+375
+00:42:10,040 --> 00:42:10,480
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+376
+00:42:10,480 --> 00:42:10,480
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+377
+00:42:10,480 --> 00:42:10,960
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+378
+00:42:15,820 --> 00:42:23,080
+الان باستخدام ال claim التاني ممكن نثبت شغلة مهمة
+
+379
+00:42:23,080 --> 00:42:36,380
+في البرهان اذا
+
+380
+00:42:36,380 --> 00:42:43,650
+خليها هادى للمرة الجاية بس بدى اكتبهاخليكم أنتوا
+
+381
+00:42:43,650 --> 00:42:53,390
+تفكروا فيها .. خليكم أنتوا تفكروا فيها Now
+
+382
+00:42:53,390 --> 00:43:11,210
+using a claim to verify .. verify that ..
+
+383
+00:43:14,770 --> 00:43:23,190
+F M أكبر من N فهذا
+
+384
+00:43:23,190 --> 00:43:33,530
+بيقدي أن absolute X N minus X M أصغر من واحد على
+
+385
+00:43:33,530 --> 00:43:39,170
+اتنين قص M نجاتي باتنين
+
+386
+00:43:45,950 --> 00:43:54,290
+إذاً هذا ممكن إثباته by ال triangle ال equality و
+
+387
+00:43:54,290 --> 00:44:06,850
+claim اثنين فبنوقف
+
+388
+00:44:06,850 --> 00:44:14,460
+هنا و بنكمل ال .. بنكمل ان شاء اللهالبرهان في
+
+389
+00:44:14,460 --> 00:44:19,680
+المحاضرة الجاية، في حد عنده أي سؤال أو استفسار؟
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM.srt
new file mode 100644
index 0000000000000000000000000000000000000000..61e94d27abb6355474039310c0b87a7274a69aff
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM.srt
@@ -0,0 +1,1395 @@
+1
+00:00:23,230 --> 00:00:28,870
+بسم الله الرحمن الرحيم في الساعة هذه طبعا هيكون
+
+2
+00:00:28,870 --> 00:00:34,770
+فيانا مناقشة نشوف 
+
+3
+00:00:34,770 --> 00:00:39,790
+الـ section الأخيرة في chapter تلاتة نبدأ section
+
+4
+00:00:39,790 --> 00:00:43,610
+تلاتة ستة فيانكم أي سؤال في section تلاتة ستة؟
+
+5
+00:00:51,050 --> 00:00:56,790
+التالي هذا نقشناه المرة اللي فاتت طيب
+
+6
+00:00:56,790 --> 00:01:02,630
+في section تلاتة سبعة في عندكم أي أسئلة في section
+
+7
+00:01:02,630 --> 00:01:08,970
+تلاتة سبعة سؤال رقم احداشر نعم رقم احداشر
+
+8
+00:01:22,550 --> 00:01:35,490
+بس الرقم 11 تلاتة سابعة if 
+
+9
+00:01:35,490 --> 00:01:42,950
+the series sigma a n with
+
+10
+00:01:42,950 --> 00:01:46,270
+a 
+
+11
+00:01:46,270 --> 00:01:51,070
+n أكبر من الصفر is convergent
+
+12
+00:01:54,210 --> 00:02:01,230
+is convergent then
+
+13
+00:02:01,230 --> 00:02:14,890
+is the series sigma للجذر التربيعي ولا 
+
+14
+00:02:14,890 --> 00:02:15,410
+لأ؟
+
+15
+00:02:24,900 --> 00:02:29,340
+is the series and
+
+16
+00:02:29,340 --> 00:02:39,240
+if and
+
+17
+00:02:39,240 --> 00:02:52,300
+if BN BN بيساوي A واحد زائد إلى AN كل هذا مجسوم على
+
+18
+00:02:52,300 --> 00:02:52,720
+N
+
+19
+00:02:55,990 --> 00:03:03,350
+مع الـ n يشبه الـ n ثم 
+
+20
+00:03:03,350 --> 00:03:08,310
+اظهر .. اظهر
+
+21
+00:03:08,310 --> 00:03:15,590
+ان السيريز سيجما bn دائما
+
+22
+00:03:15,590 --> 00:03:19,510
+.. دائما
+
+23
+00:03:19,510 --> 00:03:21,290
+متحرر
+
+24
+00:03:33,740 --> 00:03:34,160
+Okay
+
+25
+00:03:51,610 --> 00:03:56,550
+بنثبت ان لو كانت ال series هذه حدودها كلها موجبة و
+
+26
+00:03:56,550 --> 00:04:02,670
+convergent وعرفنا Pn على ان ال average لمجموعة أو
+
+27
+00:04:02,670 --> 00:04:09,750
+ال average لأول n من حدود ال series An فبنثبت ان
+
+28
+00:04:09,750 --> 00:04:12,790
+ال series هذه بتطلع دائما divergent
+
+29
+00:04:18,290 --> 00:04:21,610
+وارجي ال unbounded ال series لما تكون unbounded
+
+30
+00:04:21,610 --> 00:04:25,710
+تتطير مين هي ال unbounded؟ الأسئلة ال sequence of
+
+31
+00:04:25,710 --> 00:04:36,990
+partial sums صحيح يعني
+
+32
+00:04:36,990 --> 00:04:42,710
+أنا عندي أول شي not
+
+33
+00:04:42,710 --> 00:04:43,290
+first
+
+34
+00:04:47,630 --> 00:05:00,050
+رحزي أولا أنه لكل K ينتمي إلى N EK
+
+35
+00:05:00,050 --> 00:05:12,590
+اللي هو بيساوي A1 زايد EK على N على K هذا
+
+36
+00:05:12,590 --> 00:05:15,870
+بيكون دايما أكبر من أو يساوي
+
+37
+00:05:20,590 --> 00:05:25,570
+A1 على K لأن
+
+38
+00:05:25,570 --> 00:05:32,410
+ال .. ال sum اللي هنا أكبر من A1 لأن الأعداد هنا
+
+39
+00:05:32,410 --> 00:05:37,150
+اللي في ال sum كل أعداد موجبة فال sum اللي هنا
+
+40
+00:05:37,150 --> 00:05:40,930
+أكبر من ال sum اللي هناك وبالتالي هذا دايما صحيح
+
+41
+00:05:40,930 --> 00:05:45,650
+لكل K في N hence
+
+42
+00:05:45,650 --> 00:05:46,790
+وبالتالي
+
+43
+00:05:48,890 --> 00:05:57,350
+لو أخدت الـ nth partial sum للسيريز سيجما BN
+
+44
+00:06:05,920 --> 00:06:10,120
+إذن هذا عبارة عن الـ nth partial sum لل series sigma
+
+45
+00:06:10,120 --> 00:06:17,480
+bn الآن عندي bk أكبر من أو يساوي هاي summation من
+
+46
+00:06:17,480 --> 00:06:24,380
+k بساوي واحد إلى n و ال bk هادي أكبر من أو يساوي a
+
+47
+00:06:24,380 --> 00:06:29,640
+واحد على k ال a واحد ثابت بالنسبة ل k ده تمليش على
+
+48
+00:06:29,640 --> 00:06:36,860
+k فبطلّه برا هاي a واحد ضربSummation من K بيساوي
+
+49
+00:06:36,860 --> 00:06:44,400
+واحد إلى N لواحد على K واحنا
+
+50
+00:06:44,400 --> 00:06:50,140
+أثبتنا قبل هيك أنه ال sequence of partial sums لل
+
+51
+00:06:50,140 --> 00:06:57,620
+harmonic series is unbounded
+
+52
+00:06:57,620 --> 00:07:03,380
+في كان مثال سابق بيقول إنه
+
+53
+00:07:07,390 --> 00:07:14,970
+إن الـ sequence هذه من n بساوي واحد to infinity is
+
+54
+00:07:14,970 --> 00:07:18,590
+unbounded
+
+55
+00:07:18,590 --> 00:07:25,010
+is unbounded حسب
+
+56
+00:07:25,010 --> 00:07:31,030
+مثال سألت إذا لما أضربها ال sequence هذه لما أضرب
+
+57
+00:07:31,030 --> 00:07:35,350
+حدودها أو أضربها في ثابت موجب تبقى unbounded
+
+58
+00:07:39,070 --> 00:07:48,770
+وبالتالي إذا SM هذا بيقدي ان ال sequence SM is
+
+59
+00:07:48,770 --> 00:07:52,610
+unbounded
+
+60
+00:07:52,610 --> 00:07:59,870
+therefore ال
+
+61
+00:07:59,870 --> 00:08:08,950
+limit ل SM لما انتقل ل infinity does not exist and
+
+62
+00:08:08,950 --> 00:08:16,510
+therefore the series sigma dn diverges لان احنا
+
+63
+00:08:16,510 --> 00:08:19,970
+قلنا قبلك ان اي infinite series بتكون convergent
+
+64
+00:08:19,970 --> 00:08:24,570
+if and only if the sequence of partial sums is
+
+65
+00:08:24,570 --> 00:08:32,870
+convergent لان هذا هو الحل okay تمام في
+
+66
+00:08:32,870 --> 00:08:35,730
+أي أسئلة تانية في section تلاتة سبعة
+
+67
+00:08:53,340 --> 00:08:58,320
+مفهوم الحل؟ في 
+
+68
+00:08:58,320 --> 00:09:03,800
+أسئلة تانية في ال section هذا أو أي section سابق؟
+
+69
+00:09:03,800 --> 00:09:11,180
+فسؤال سبعة هذا
+
+70
+00:09:11,180 --> 00:09:16,210
+المماثل بيشبه مثال تلاتة سبعة ستة فاقرأي المثال
+
+71
+00:09:16,210 --> 00:09:22,330
+حاولي تطبقي نفس الطريقة مشروحليك في المثال فحاولي
+
+72
+00:09:22,330 --> 00:09:28,710
+اتجلدي المثال في اي اسئلة تانية؟
+
+73
+00:09:28,710 --> 00:09:35,950
+مان
+
+74
+00:09:35,950 --> 00:09:37,170
+لديها سؤال؟
+
+75
+00:09:56,850 --> 00:10:11,850
+في عندكم أسرة طيب
+
+76
+00:10:11,850 --> 00:10:14,790
+لما تفكروا في أسرة بدي أنا بارهنكم Cauchy
+
+77
+00:10:14,790 --> 00:10:21,390
+condensation test لأن هذا في عليه أسرة ومهم
+
+78
+00:10:38,680 --> 00:10:56,660
+سؤال اتماشي section تلاتة .. سابعة Cauchy
+
+79
+00:10:56,660 --> 00:11:00,760
+condensation
+
+80
+00:11:00,760 --> 00:11:01,180
+test
+
+81
+00:11:13,290 --> 00:11:19,130
+فال test هذا بيقول let sigma
+
+82
+00:11:19,130 --> 00:11:29,970
+an be a series .. a series of
+
+83
+00:11:29,970 --> 00:11:42,270
+monotone .. of monotone decreasing positive
+
+84
+00:11:45,320 --> 00:11:54,260
+مجموعات اثنين اثنين
+
+85
+00:11:54,260 --> 00:11:54,340
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+86
+00:11:54,340 --> 00:11:58,160
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+87
+00:11:58,160 --> 00:11:59,760
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+88
+00:11:59,760 --> 00:12:03,980
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+89
+00:12:03,980 --> 00:12:05,320
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+90
+00:12:05,320 --> 00:12:08,420
+اثنين اثنين اثنين
+
+91
+00:12:08,420 --> 00:12:14,380
+اثنين
+
+92
+00:12:14,380 --> 00:12:14,820
+اثن
+
+93
+00:12:42,930 --> 00:12:48,630
+وهي البرهان أولا
+
+94
+00:12:48,630 --> 00:13:02,350
+خلّينا نلاحظ note that لاحظي انه لو أخدت نص في
+
+95
+00:13:02,350 --> 00:13:12,530
+summation من k بساوي zero to infinity ل two أُس k
+
+96
+00:13:12,530 --> 00:13:18,930
+في a two to k هذا
+
+97
+00:13:18,930 --> 00:13:20,830
+بيطلع بساوي نص
+
+98
+00:13:23,540 --> 00:13:33,720
+في A1 اول حد لما كدا ساوى سفر فبطلع نص A1 الحد
+
+99
+00:13:33,720 --> 00:13:43,940
+اللي بعده هيطلع A2 اللي بعده اتنين A4 واللي بعده
+
+100
+00:13:43,940 --> 00:13:52,020
+اربعة في A8 وهكذا نستمر على هذا النمط إلى
+
+101
+00:13:55,470 --> 00:14:00,650
+أتنين خلّينا ناخد المجموعة من K بساوي سفر إلى M
+
+102
+00:14:00,650 --> 00:14:08,290
+حيث M عدد طبيعي ما فأخر حد هيكون اتنين أس M سالب
+
+103
+00:14:08,290 --> 00:14:15,670
+واحد في A اتنين أس M الآن
+
+104
+00:14:15,670 --> 00:14:21,770
+هذا المجموع أصغر من A واحد نص A واحد بالتأكيد أصغر
+
+105
+00:14:21,770 --> 00:14:32,160
+من A واحد وطبعا ال .. ال .. الأعداد هذه كلها موجبة
+
+106
+00:14:32,160 --> 00:14:38,600
+و بتكون decrease in sequence فنص a1 أصغر من a1 و
+
+107
+00:14:38,600 --> 00:14:59,570
+a2 بساوي a2 و 2 a4 أصغر من a3 زائد a4 صح؟A4 أصغر
+
+108
+00:14:59,570 --> 00:15:07,110
+من A3 لأن ال sequence A N decreasing فعندي A4 زائد
+
+109
+00:15:07,110 --> 00:15:17,350
+A4 أصغر من A3 زائد A4 و هكذا برضه عندي A8 أصغر من
+
+110
+00:15:17,350 --> 00:15:25,190
+A5 و أصغر من A6 و أصغر من A7وبالتالي هدا هيكون
+
+111
+00:15:25,190 --> 00:15:30,470
+اربعة A8 اصغر من مجموعة اربعة حدود اللي هم a
+
+112
+00:15:30,470 --> 00:15:42,270
+خمسة زائد a ستة زائد a سبعة زائد a تمانية و هكذا
+
+113
+00:15:42,270 --> 00:15:48,830
+استمر على هذا النمط الى ان نصل لاخر
+
+114
+00:15:51,230 --> 00:15:58,170
+هدول الحدود هيكون اصغر من .. او لحد هذا الأخير
+
+115
+00:15:58,170 --> 00:16:04,530
+اصغر من المجموعة اللي هو a اتنين أُس ام سالب واحد
+
+116
+00:16:04,530 --> 00:16:11,430
+زائد واحد زائد a
+
+117
+00:16:11,430 --> 00:16:18,920
+اتنين أُس ام سالب واحد زائد اتنين زائد و هكذابقت
+
+118
+00:16:18,920 --> 00:16:24,700
+أصغر من مجموعة كل ال series لأن هذه كلها حدود
+
+119
+00:16:24,700 --> 00:16:29,680
+موجبة، أعداد موجبة وهذا
+
+120
+00:16:29,680 --> 00:16:38,540
+الكلام صحيح لكل M، لكل M عدد طبيعي أكبر
+
+121
+00:16:38,540 --> 00:16:47,240
+من أو يساوي، يعني عدد طبيعي وبالتالي
+
+122
+00:16:47,240 --> 00:16:48,120
+and so
+
+123
+00:16:50,890 --> 00:17:01,690
+وبالتالي نضرب sum من k بساوي سفر إلى m لتو أس ك
+
+124
+00:17:01,690 --> 00:17:10,490
+بإتنين أُس ك ده هيطلع أصغر من أو ساوي نضرب الطرفين
+
+125
+00:17:10,490 --> 00:17:15,470
+في اتنين عشان نتخلص من النصف بصير المجموع هذا أصغر
+
+126
+00:17:15,470 --> 00:17:21,410
+من أو يساوي اتنين في summation من n equals zero to
+
+127
+00:17:21,410 --> 00:17:27,750
+infinity ل a n تمام؟
+
+128
+00:17:27,750 --> 00:17:34,330
+وهذا
+
+129
+00:17:34,330 --> 00:17:39,650
+صحيح لكل m belonging to N
+
+130
+00:17:44,360 --> 00:18:02,120
+بنسمي ال quality هذه واحد طيب
+
+131
+00:18:02,120 --> 00:18:05,680
+now next
+
+132
+00:18:09,650 --> 00:18:21,350
+given any m أكبر من أو سوى الواحد choose using
+
+133
+00:18:21,350 --> 00:18:34,050
+Archimedean property choose
+
+134
+00:18:34,050 --> 00:18:44,810
+k بحيث أنه two to K أكبر من M لأي عدد طبيعي ممكن
+
+135
+00:18:44,810 --> 00:18:58,530
+ألاقي عدد طبيعي بحياتي two to K أكبر من M then ال
+
+136
+00:18:58,530 --> 00:19:06,690
+summation from N equals zero to M لان هذا بيطلع
+
+137
+00:19:06,690 --> 00:19:12,630
+أصغر من a0
+
+138
+00:19:12,630 --> 00:19:19,710
+زائد a1 زائد a2
+
+139
+00:19:19,710 --> 00:19:31,150
+زائد a3 زائد a4 زائد a5 زائد a6 زائد a7 زائد a8
+
+140
+00:19:35,390 --> 00:19:47,670
+مع بعض زائد و هكذا إلى اتنين 
+
+141
+00:19:47,670 --> 00:19:55,190
+أس 2 زائد 2 أس 2 زائد 1 زائد وهكذا إلى 
+
+142
+00:19:55,190 --> 00:19:59,330
+2 
+
+143
+00:19:59,330 --> 00:20:03,950
+أس 2 زائد 1 سالب 1
+
+144
+00:20:12,500 --> 00:20:17,840
+أنا عند الـ M هذا الـ M أصغر من 2 أس K في آخر 
+
+145
+00:20:17,840 --> 00:20:26,460
+حد اللي هو A<sub>M</sub> هيكون أصغر من A رقم 2 أس K أو
+
+146
+00:20:26,460 --> 00:20:34,180
+أصغر من أو يساوي 2 رقم A أس 2K زي واحد 
+
+147
+00:20:34,180 --> 00:20:35,780
+ناقص 1
+
+148
+00:20:43,450 --> 00:20:52,190
+والمجموع هذا .. هذا المجموع أصغر من أو يساوي a<sub>0</sub>
+
+149
+00:20:52,190 --> 00:20:57,490
+زائد a<sub>1</sub> زائد
+
+150
+00:20:57,490 --> 00:21:06,710
+2 a<sub>2</sub> لأن a<sub>3</sub> أصغر من a<sub>2</sub> صح؟ عشان الـ sequence a<sub>n</sub> is
+
+151
+00:21:06,710 --> 00:21:13,030
+decreasing وهذا المجموع أصغر من 4 a
+
+152
+00:21:14,740 --> 00:21:26,420
+4 صح وهكذا إلى المجموع هذا هيكون أصغر من 2
+
+153
+00:21:26,420 --> 00:21:37,280
+أس K هذول عدد الحدود في a 2 أس K يعني هذول
+
+154
+00:21:37,280 --> 00:21:41,860
+عدد الحدود عددهم 2 أس K وكل واحد منهم
+
+155
+00:21:45,050 --> 00:21:55,350
+أصغر من 2 أول واحد اللي هو 2 أس 2K وهذا 
+
+156
+00:21:55,350 --> 00:22:01,830
+بدوره أصغر من 2 أس 2K زائد summation من K
+
+157
+00:22:01,830 --> 00:22:09,730
+بساوي 0 to infinity لـ 2 أس K في 2 أس 
+
+158
+00:22:09,730 --> 00:22:17,540
+K هاي أول حد 2 أس K لما K بيساوي 0 بيطلع
+
+159
+00:22:17,540 --> 00:22:25,640
+1 واحد وبعدين اللي بعده بيطلع 2 2 لما K 
+
+160
+00:22:25,640 --> 00:22:33,480
+بيساوي 1 واللي بعده 4 4 وهكذا طبعا
+
+161
+00:22:33,480 --> 00:22:37,400
+هذا بوقف المجموعة هذا finite هذا أصغر من المجموعة
+
+162
+00:22:37,400 --> 00:22:41,400
+من K بيساوي 0 إلى ما لا نهاية هذا طبعا في حدود 
+
+163
+00:22:41,400 --> 00:22:41,820
+أكثر
+
+164
+00:22:44,960 --> 00:22:53,040
+تمام؟ وبالتالي إذا نستنتج and so نستنتج 
+
+165
+00:22:53,040 --> 00:23:02,980
+إنه المجموعة ∑ from n equal 0 to infinity لـ
+
+166
+00:23:02,980 --> 00:23:12,050
+a<sub>n</sub> بطلع أصغر من أو يساوي a<sub>0</sub> زائد ∑ from k equals
+
+167
+00:23:12,050 --> 00:23:20,790
+0 to infinity لـ 2<sup>k</sup> a<sub>2<sup>k</sup></sub> لأن
+
+168
+00:23:20,790 --> 00:23:26,810
+هذا صحيح لكل M أكبر من أو يساوي الـ 1 لأن هذا
+
+169
+00:23:26,810 --> 00:23:33,330
+عبارة عن هذا عبارة عن upper bound هذا العدد أو هذا
+
+170
+00:23:33,330 --> 00:23:39,530
+العدد upper bound للـ sequence of partial sums هنا
+
+171
+00:23:39,530 --> 00:23:44,190
+فما 
+
+172
+00:23:44,190 --> 00:23:47,210
+هذه الـ sequence of partial sums is increasing
+
+173
+00:23:47,210 --> 00:23:50,750
+متزايدة
+
+174
+00:23:50,750 --> 00:23:55,110
+و bounded above by this number إذا الـ limit تبعت
+
+175
+00:23:55,110 --> 00:23:58,650
+الـ sequence of partial sums exist وبالساوي
+
+176
+00:23:58,650 --> 00:24:04,990
+supremum للـ sequence of partial sums الـ supremum
+
+177
+00:24:04,990 --> 00:24:11,150
+للـ sequence of partial sums أقل من الـ upper bound
+
+178
+00:24:11,150 --> 00:24:13,670
+هذا upper bound للـ sequence of partial sums الـ 
+
+179
+00:24:13,670 --> 00:24:17,050
+supremum أصغر upper bound وبالتالي إذا الـ supremum
+
+180
+00:24:17,050 --> 00:24:21,690
+للـ sequence of partial sums هو عبارة عن limit للـ
+
+181
+00:24:21,690 --> 00:24:23,730
+sequence of partial sums اللي هو مجموعة الـ
+
+182
+00:24:23,730 --> 00:24:29,190
+infinite series أصغر من أو يساوي الـ upper bound by
+
+183
+00:24:29,190 --> 00:24:34,290
+monotone convergence theorem السيريز 
+
+184
+00:24:34,290 --> 00:24:39,610
+هذي convergence ومجموعة بساوي limit للـ sequence of 
+
+185
+00:24:39,610 --> 00:24:44,710
+partial sums اللي هي أصغر من أو ساوي عددها okay
+
+186
+00:24:44,710 --> 00:24:54,170
+إذا نسمي المتباينة هذه 2 إذا من المتباينة 1 
+
+187
+00:24:54,170 --> 00:24:54,870
+و2
+
+188
+00:25:11,870 --> 00:25:19,130
+الآن بمقارنة مباشرة الاختلافات
+
+189
+00:25:19,130 --> 00:25:30,640
+المتباينات 1 و 2 بيقدوا السيريز ∑ a<sub>n</sub>
+
+190
+00:25:30,640 --> 00:25:39,720
+converges if and only if السيريز ∑ 2<sup>2</sup>
+
+191
+00:25:39,720 --> 00:25:47,820
+a<sub>2<sup>2</sup></sub> converges تعالى 
+
+192
+00:25:47,820 --> 00:25:54,680
+نشوف لو كانت السيريز هذه convergent فالسيريز
+
+193
+00:25:54,680 --> 00:25:55,780
+هذه convergent
+
+194
+00:25:58,080 --> 00:26:03,500
+وبالتالي طبعا أن هذا صحيح لكل M بالمناسبة بقدر أن
+
+195
+00:26:03,500 --> 00:26:08,880
+هذه أيضا sequence of partial sums هذه الـ limit
+
+196
+00:26:08,880 --> 00:26:19,460
+تبعتها exist وبالتالي الـ infinite series هذه إذا
+
+197
+00:26:19,460 --> 00:26:27,250
+أن ال ممكن نقول أن هذا الكلام صحيح الآن لو كانت الـ
+
+198
+00:26:27,250 --> 00:26:31,430
+series هادي convergent فنضربها في ثابت 2 تطلع
+
+199
+00:26:31,430 --> 00:26:35,270
+convergent وبالتالي الـ series هادي convergent by 
+
+200
+00:26:35,270 --> 00:26:40,170
+direct comparison test العكس لو كانت الـ series
+
+201
+00:26:40,170 --> 00:26:41,670
+هادي convergent
+
+202
+00:26:44,460 --> 00:26:50,840
+فلما أضفلها حد عدد موجب بيبقى conversion وبالتالي
+
+203
+00:26:50,840 --> 00:26:54,080
+by direct comparison test الـ series الأصغر بتطلع
+
+204
+00:26:54,080 --> 00:26:58,160
+conversion okay تمام؟ لأن هذا بثبت Cauchy
+
+205
+00:26:58,160 --> 00:27:03,600
+condensation test هذا الـ test قوي كتير وله فوائد
+
+206
+00:27:03,600 --> 00:27:13,000
+كتيرة فمن الفوائد تبعتها يعني
+
+207
+00:27:13,000 --> 00:27:13,800
+هذه مثال
+
+208
+00:27:22,170 --> 00:27:37,410
+ممكن نستنتج الـ P-series test مثال، 
+
+209
+00:27:37,410 --> 00:27:46,290
+أنا موجود في إحدى التمارين التمرين 13
+
+210
+00:27:53,040 --> 00:28:05,440
+تعملين تلتاش سيكشن 3 7 ايش بيقول هذا if if
+
+211
+00:28:05,440 --> 00:28:16,600
+P أكبر من الـ 0 is a real number discuss
+
+212
+00:28:16,600 --> 00:28:20,940
+the 
+
+213
+00:28:20,940 --> 00:28:21,680
+convergence
+
+214
+00:28:42,640 --> 00:28:44,720
+تعالوا نفحص
+
+215
+00:28:49,400 --> 00:28:58,120
+∑ from n equals 1 to infinity لـ 2 أس
+
+216
+00:28:58,120 --> 00:29:08,700
+n في 1 على هاي أو خليني أقول 2 أس n في a
+
+217
+00:29:08,700 --> 00:29:16,120
+and a 2 أس m ايش بيساوي هذا طبعا هاي عندي a<sub>n</sub>
+
+218
+00:29:16,120 --> 00:29:24,230
+هذا هو عبارة عن a<sub>m</sub> الحد العام للـ series فان بيساوي 
+
+219
+00:29:24,230 --> 00:29:30,290
+1 على n<sup>p</sup> فبتبحث هل الـ series هذي convergent أو
+
+220
+00:29:30,290 --> 00:29:33,990
+متى بتكون هذي الـ series convergent وبالتالي بقدر
+
+221
+00:29:33,990 --> 00:29:37,890
+أطبق اللي هو Cauchy condensation test فهذه عبارة
+
+222
+00:29:37,890 --> 00:29:43,970
+عن ∑ from n equals 1 to infinity الآن ايه
+
+223
+00:29:43,970 --> 00:29:53,550
+2 أس n بطلع 1 على 2 أس n الكل أس P
+
+224
+00:29:53,550 --> 00:30:03,810
+تمام؟ وهذا بيساوي ∑ from n equals 1 to
+
+225
+00:30:03,810 --> 00:30:18,940
+infinity لـ 2 أس 1−P الكل أس n وهدي 
+
+226
+00:30:18,940 --> 00:30:27,020
+is a geometric series is a geometric series
+
+227
+00:30:27,020 --> 00:30:33,680
+وبالتالي
+
+228
+00:30:33,680 --> 00:30:38,320
+مظبوط هذا عبارة عن geometric series لو بدى أكتب
+
+229
+00:30:38,320 --> 00:30:40,620
+حدود تبعتها
+
+230
+00:30:43,100 --> 00:30:52,660
+فأول حد عبارة عن 2 أس 1−P الحد الثاني
+
+231
+00:30:52,660 --> 00:31:00,940
+2 أس 1−P الكل تربيع وهكذا فالحد 
+
+232
+00:31:00,940 --> 00:31:05,480
+الأول 2 أس 1−P الحد الثاني 2 أس
+
+233
+00:31:05,480 --> 00:31:09,980
+1−P وهكذا with ratio
+
+234
+00:31:14,090 --> 00:31:28,710
+with ratio with 
+
+235
+00:31:28,710 --> 00:31:34,830
+ratio R
+
+236
+00:31:34,830 --> 00:31:41,790
+بيساوي 2 أس 1−P 
+
+237
+00:31:48,590 --> 00:31:58,790
+So by geometric series test it converges if
+
+238
+00:31:58,790 --> 00:32:06,450
+and only if |R| بيساوي 2 أس 1−P
+
+239
+00:32:06,450 --> 00:32:16,670
+أصغر من 1 وهذا بتحقق 2 أس 1−P أصغر
+
+240
+00:32:16,670 --> 00:32:25,590
+من 1 فنقول if 1−P إذا 
+
+241
+00:32:25,590 --> 00:32:36,910
+كان 1−P أصغر من الـ 0 سالب لأن لو كان
+
+242
+00:32:36,910 --> 00:32:41,430
+1−P موجب فـ 2 أس أي عدد موجب عمره ما
+
+243
+00:32:41,430 --> 00:32:47,440
+بيكون أصغر من 1 نصفوت لكن لو كان الأس سالب فبيصير
+
+244
+00:32:47,440 --> 00:32:52,620
+هذا 1 على 2 أس وموجب فبيصير أصغر من 1 إذا
+
+245
+00:32:52,620 --> 00:32:57,020
+هذا صحيح if and only if الأس تابع الـ 2 اللي هو
+
+246
+00:32:57,020 --> 00:33:06,240
+1−P أصغر من 0 if and only if 1 أصغر 
+
+247
+00:33:06,240 --> 00:33:12,920
+من P أو P أكبر من 1 okay تمام وهذا هو الـ P
+
+248
+00:33:12,920 --> 00:33:19,120
+series test لأن احنا استنتجنا الـ P series test من
+
+249
+00:33:19,120 --> 00:33:26,200
+Cauchy Condensation test فاكرين الـ P series هذي أو
+
+250
+00:33:26,200 --> 00:33:29,840
+الـ P series test اثبتنا أن Convergent if and only 
+
+251
+00:33:29,840 --> 00:33:35,200
+if P أكبر من 1 وDivergent إذا كانت P أصغر منها
+
+252
+00:33:35,200 --> 00:33:35,960
+وسائل 1
+
+253
+00:33:42,110 --> 00:33:51,730
+Okay إذا الـ .. هذا المعنى So by Cauchy Cauchy's
+
+254
+00:33:51,730 --> 00:34:01,910
+Condensation Test The series ∑
+
+255
+00:34:01,910 --> 00:34:07,830
+from N equals 1 to infinity الـ 1 over N<sup>P</sup>
+
+256
+00:34:08,830 --> 00:34:16,530
+convergence if and only if P أكبر من 1 وهذا هو
+
+257
+00:34:16,530 --> 00:34:23,030
+الـ P-series test إذن هذا بورجينا قوة Cauchy 
+
+258
+00:34:23,030 --> 00:34:29,530
+Condensation Test okay تمام؟ في طبعا أسئلة أخرى 
+
+259
+00:34:29,530 --> 00:34:33,430
+على Cauchy Condensation Test وأنا طالب منكم تحلوها
+
+260
+00:34:33,430 --> 00:34:41,340
+زي السؤال 14 و15 صح؟ ففي أي شيء في الأسئلة دي أو
+
+261
+00:34:41,340 --> 00:34:47,160
+أسئلة ثانية؟
+
+262
+00:34:47,160 --> 00:34:56,500
+في
+
+263
+00:34:56,500 --> 00:34:58,180
+عندكم أي أسئلة؟
+
+264
+00:35:13,000 --> 00:35:19,560
+إذا سيكشن 1 3 7 في أي سؤال ثاني عندكم في
+
+265
+00:35:19,560 --> 00:35:25,540
+الأسئلة هذه أو
+
+266
+00:35:25,540 --> 00:35:31,860
+السيكاشن السابقة أو سيكشن 4 1 إذا بتحبه
+
+267
+00:35:31,860 --> 00:35:35,560
+سيكشن 4 1
+
+268
+00:36:06,090 --> 00:36:13,070
+مافيش أسئلة؟ طيب الـ .. مدام مافيش أسئلة نواصل ..
+
+269
+00:36:13,070 --> 00:36:16,190
+نكمل
+
+270
+00:36:16,190 --> 00:36:17,490
+المحاضرة في السابقة
+
+271
+00:36:49,090 --> 00:36:53,250
+المرة الأخرى تحدثنا عن الـ two-sided limits وعن
+
+272
+00:36:53,250 --> 00:37:00,350
+الـ one-sided limits وأخذنا بعض النظريات وقلنا إن
+
+273
+00:37:00,350 --> 00:37:05,090
+جميع النظريات اللي برهناها هو one-sided limit
+
+274
+00:37:05,090 --> 00:37:12,990
+صحيحة للـ two-sided limits أو النظريات الصحيحة لـ 
+
+275
+00:37:12,990 --> 00:37:17,070
+two-sided limits بتكون أيضا صحيحة لـ one-sided
+
+276
+00:37:17,070 --> 00:37:26,650
+limit فناخد 
+
+277
+00:37:26,650 --> 00:37:31,350
+مثال show
+
+278
+00:37:31,350 --> 00:37:31,950
+that
+
+279
+00:37:35,020 --> 00:37:55,100
+Limit لـ Signum X لإن X تقول لـ 0 لا يوجد فنلاحظ
+
+280
+00:37:55,100 --> 00:37:59,600
+أن Limit لأول شيء Signum X
+
+281
+00:38:03,790 --> 00:38:11,230
+بساوي x على absolute x لكل x لا يساوي صفر لما 
+
+282
+00:38:11,230 --> 00:38:15,010
+أعرفنا الدالة هذه قلت لها بس هي نفسها x على
+
+283
+00:38:15,010 --> 00:38:20,690
+absolute x لو كان x بساوي صفر الآن ال limit ل
+
+284
+00:38:20,690 --> 00:38:30,890
+sigma x لما x تقول إلى صفر من اليمين بساوي ال
+
+285
+00:38:30,890 --> 00:38:31,310
+limit
+
+286
+00:38:35,810 --> 00:38:41,530
+لما x تقول إلى صفر من اليمين لما x تقول إلى صفر من
+
+287
+00:38:41,530 --> 00:38:50,190
+اليمين لما x تقول إلى صفر من اليمين لما x تقول إلى 
+
+288
+00:38:50,190 --> 00:38:55,650
+صفر من اليمين لما x تقول إلى صفر من اليمين لما x
+
+289
+00:38:55,650 --> 00:38:57,330
+تقول إلى صفر من اليمين لما x تقول إلى صفر من
+
+290
+00:38:57,330 --> 00:39:02,560
+اليمين لما x تقول إلى صفر من اليمين لما x تقول أصغر
+
+291
+00:39:02,560 --> 00:39:21,640
+من اليسار لما x أصغر من صفر لما
+
+292
+00:39:21,640 --> 00:39:28,560
+x أصغر من صفر لما x أصغر من صفر لما x أصغر من صفر
+
+293
+00:39:28,560 --> 00:39:33,950
+لما x أصغر من صفر، السالب واحد بيطلع السالب واحد إن
+
+294
+00:39:33,950 --> 00:39:37,670
+أنا عندي ال limit من اليمين يساوي واحد، ال limit
+
+295
+00:39:37,670 --> 00:39:44,230
+من اليسار يساوي سالب واحد، مش متساويين الاثنين، so by
+
+296
+00:39:44,230 --> 00:39:50,150
+theorem، حسب النظرية اللي أخدناها theorem أربعة
+
+297
+00:39:50,150 --> 00:39:55,630
+ثلاثة، بيطلع
+
+298
+00:39:55,630 --> 00:40:01,080
+عندي ال limit أو ال two sided limit للـ signal
+
+299
+00:40:01,080 --> 00:40:09,560
+function لما x تقول إلى الصفر does not exist تمام؟
+
+300
+00:40:09,560 --> 00:40:22,280
+طيب خلّيني أنا آخد show
+
+301
+00:40:22,280 --> 00:40:27,380
+that ال
+
+302
+00:40:27,380 --> 00:40:32,350
+limit لل function e والواحد على x لما x تقول إلى
+
+303
+00:40:32,350 --> 00:40:40,550
+صفر من اليمين does not exist and
+
+304
+00:40:40,550 --> 00:40:43,910
+من
+
+305
+00:40:43,910 --> 00:40:51,170
+ال limit لنفس ال function e to واحد على x لما x
+
+306
+00:40:51,170 --> 00:40:58,710
+تقول إلى صفر من اليسار تطلع موجودة و بساوي صفر
+
+307
+00:41:23,830 --> 00:41:31,010
+طيب ال ...
+
+308
+00:41:31,010 --> 00:41:34,050
+نحاول نبرهن الجزء الأول
+
+309
+00:41:56,420 --> 00:42:03,380
+بناخد الجزء الأول let
+
+310
+00:42:03,380 --> 00:42:13,540
+z of x بساوي e to 1 على x، حفة x لا تساوي 0، وبدنا
+
+311
+00:42:13,540 --> 00:42:19,260
+نثبت to
+
+312
+00:42:19,260 --> 00:42:28,130
+show إن ال limit لـ g of x لما x تقول لصفر من
+
+313
+00:42:28,130 --> 00:42:38,750
+اليمين does not exist، it suffices to
+
+314
+00:42:38,750 --> 00:42:42,710
+show يكفي
+
+315
+00:42:42,710 --> 00:42:52,650
+إثبات أن ال function g of x is not bounded on
+
+316
+00:42:56,170 --> 00:43:05,850
+on a right ... on a right neighborhood
+
+317
+00:43:05,850 --> 00:43:13,670
+... on a right neighborhood اللي هو صفر دلتا of
+
+318
+00:43:13,670 --> 00:43:15,230
+zero
+
+319
+00:43:25,230 --> 00:43:28,670
+أخذنا قبل ذلك نظرية بتقول إيه؟ ده عشان أثبت أنه ال
+
+320
+00:43:28,670 --> 00:43:35,710
+limit ل function عن نقطة معينة مش موجودة يكفي أثبت
+
+321
+00:43:35,710 --> 00:43:43,910
+أنه أنه الدالة unbounded عند أي unbounded
+
+322
+00:43:43,910 --> 00:43:48,650
+عند أي neighborhood
+
+323
+00:43:48,650 --> 00:43:56,210
+للنقطة الآن بالنسبة لل one-sided limit عشان أقول إن
+
+324
+00:43:56,210 --> 00:44:02,230
+ال limit ل function زي هذه g of x لما x تقول إلى
+
+325
+00:44:02,230 --> 00:44:09,430
+صفر من اليمين does not exist فهي
+
+326
+00:44:09,430 --> 00:44:16,390
+الصفر و ال x تقول لسفر من اليمين فبدل ما أخد delta
+
+327
+00:44:16,390 --> 00:44:20,290
+neighborhood للصفر
+
+328
+00:44:20,290 --> 00:44:29,960
+فباخد right neighborhood right neighborhood للصفر
+
+329
+00:44:29,960 --> 00:44:35,960
+فيكفي إن ال function هذه ماهياش bounded عن كل
+
+330
+00:44:35,960 --> 00:44:41,780
+right neighborhood يعني جوار من اليمين للصفر لأن
+
+331
+00:44:41,780 --> 00:44:46,000
+أنا بتعامل مع نهاية من اليمين لكن لما كنت اتعامل
+
+332
+00:44:46,000 --> 00:44:51,240
+مع نهاية من الطرفين فكنت آخد delta neighborhood
+
+333
+00:44:51,240 --> 00:44:56,840
+كامل، ولو أثبتت إن الـ function هذه ماهياش bounded
+
+334
+00:44:56,840 --> 00:45:01,280
+عند أي right neighborhood للصفر على الصورة هذه
+
+335
+00:45:01,280 --> 00:45:06,220
+فحسب نظرية سابقة الدالة مش ممكن يكون لها limit من
+
+336
+00:45:06,220 --> 00:45:09,960
+اليمين عند الصفر لأن لو كان لها limit عند الصفر من
+
+337
+00:45:09,960 --> 00:45:15,820
+اليمين فلازم تكون bounded على some neighborhood ...
+
+338
+00:45:15,820 --> 00:45:25,650
+right neighborhood للصفر Okay تمام و لإثبات ذلك to
+
+339
+00:45:25,650 --> 00:45:29,270
+see
+
+340
+00:45:29,270 --> 00:45:39,290
+this we use ال inequality التالية وهي T أكبر من
+
+341
+00:45:39,290 --> 00:45:47,010
+صفر دايما أصغر من E أس T for all T أكبر من صفر هذه
+
+342
+00:45:47,010 --> 00:45:54,200
+المتباينة هذه المتباينة موجودة
+
+343
+00:45:54,200 --> 00:46:01,780
+برهانها في Chapter 8 برهانها
+
+344
+00:46:01,780 --> 00:46:07,600
+موجودة في Chapter 8 اللي هتاخدوه لاحقا فهنستخدم
+
+345
+00:46:07,600 --> 00:46:11,460
+اللي هو المتباينة هذه في إثبات إن ال function
+
+346
+00:46:11,460 --> 00:46:17,420
+ماهياش bounded على neighborhood أو right
+
+347
+00:46:17,420 --> 00:46:25,130
+neighborhood للصفر Okay عشان الوجد خلص بنوقف و
+
+348
+00:46:25,130 --> 00:46:29,590
+بناخد خمس دقايق break وبعدين بنكمل إن شاء الله
+
+349
+00:46:29,590 --> 00:46:35,550
+البرهان فحنوقف ونكمل في الجزء التالي من المحاضرة
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..7b35f4db4c8799590adb9d52f6a722114ed6362e
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_postprocess.srt
@@ -0,0 +1,1396 @@
+1
+00:00:23,230 --> 00:00:28,870
+بسم الله الرحمن الرحيم في الساعة هذه طبعا هيكون
+
+2
+00:00:28,870 --> 00:00:34,770
+فيانا مناقشة نشوف
+
+3
+00:00:34,770 --> 00:00:39,790
+ال section الأخيرة في chapter تلاتة نبدأ section
+
+4
+00:00:39,790 --> 00:00:43,610
+تلاتة ستة فيانكم أي سؤال في section تلاتة ستة؟
+
+5
+00:00:51,050 --> 00:00:56,790
+التالي هذا نقشناه المرة اللي فاتت طيب
+
+6
+00:00:56,790 --> 00:01:02,630
+في section تلاتة سبعة في عندكم أي أسئلة في section
+
+7
+00:01:02,630 --> 00:01:08,970
+تلاتة سبعة سؤال رقم احداشر نعم رقم احداشر
+
+8
+00:01:22,550 --> 00:01:35,490
+بس الرقم 11 تلاتة سابعة if
+
+9
+00:01:35,490 --> 00:01:42,950
+the series sigma a n with
+
+10
+00:01:42,950 --> 00:01:46,270
+a
+
+11
+00:01:46,270 --> 00:01:51,070
+n أكبر من الصفر is convergent
+
+12
+00:01:54,210 --> 00:02:01,230
+is convergent then
+
+13
+00:02:01,230 --> 00:02:14,890
+is the series sigma للجدر التربيهي ولا
+
+14
+00:02:14,890 --> 00:02:15,410
+لأ؟
+
+15
+00:02:24,900 --> 00:02:29,340
+is the series and
+
+16
+00:02:29,340 --> 00:02:39,240
+if and
+
+17
+00:02:39,240 --> 00:02:52,300
+if BN BN بساوي A واحد زائد إلى AN كل هذا مجسوم على
+
+18
+00:02:52,300 --> 00:02:52,720
+N
+
+19
+00:02:55,990 --> 00:03:03,350
+مع الـ n يشبه الـ n ثم
+
+20
+00:03:03,350 --> 00:03:08,310
+اظهر .. اظهر
+
+21
+00:03:08,310 --> 00:03:15,590
+ان السيريز سيجما bn دائما
+
+22
+00:03:15,590 --> 00:03:19,510
+.. دائما
+
+23
+00:03:19,510 --> 00:03:21,290
+متحرر
+
+24
+00:03:33,740 --> 00:03:34,160
+Okay
+
+25
+00:03:51,610 --> 00:03:56,550
+بنثبت ان لو كانت ال series هذه حدودها كلها موجبة و
+
+26
+00:03:56,550 --> 00:04:02,670
+convergent وعرفنا Pn على ان ال average لمجموعة او
+
+27
+00:04:02,670 --> 00:04:09,750
+ال average لأول n من حدود ال series An فبنثبت ان
+
+28
+00:04:09,750 --> 00:04:12,790
+ال series هذه بتطلع دائما divergent
+
+29
+00:04:18,290 --> 00:04:21,610
+وارجي ال unbounded ال series لما تكون unbounded
+
+30
+00:04:21,610 --> 00:04:25,710
+تتطير مين هي ال unbounded؟ الأسئلة ال sequence of
+
+31
+00:04:25,710 --> 00:04:36,990
+partial sums صحيح يعني
+
+32
+00:04:36,990 --> 00:04:42,710
+أنا عندي أول شي not
+
+33
+00:04:42,710 --> 00:04:43,290
+first
+
+34
+00:04:47,630 --> 00:05:00,050
+رحزي أولا أنه لكل K ينتمي إلى N EK
+
+35
+00:05:00,050 --> 00:05:12,590
+اللي هو بيساوي A1 زايد EK على N على K هذا
+
+36
+00:05:12,590 --> 00:05:15,870
+بيكون دايما أكبر من أو يساوي
+
+37
+00:05:20,590 --> 00:05:25,570
+A1 على K لأن
+
+38
+00:05:25,570 --> 00:05:32,410
+ال .. ال bus اللي هنا أكبر من A1 لأن الأعداد هنا
+
+39
+00:05:32,410 --> 00:05:37,150
+اللي في ال bus كل أعداد موجبة فال bus اللي هنا
+
+40
+00:05:37,150 --> 00:05:40,930
+أكبر من ال bus اللي هناك وبالتالي هذا دايما صحيح
+
+41
+00:05:40,930 --> 00:05:45,650
+لكل K في N hence
+
+42
+00:05:45,650 --> 00:05:46,790
+وبالتالي
+
+43
+00:05:48,890 --> 00:05:57,350
+لو أخدت الـ SIN الانف بارشيل سام للسيريز سيجما BN
+
+44
+00:06:05,920 --> 00:06:10,120
+إذن هذا عبارة عن ال F partial sum لل series sigma
+
+45
+00:06:10,120 --> 00:06:17,480
+bn الآن عندي bk أكبر من أو يساوي هاي summation من
+
+46
+00:06:17,480 --> 00:06:24,380
+k بساوي واحد إلى n و ال bk هادي أكبر من أو يساوي a
+
+47
+00:06:24,380 --> 00:06:29,640
+واحد على k ال a واحد ثابت بالنسبة ل k ده تمليش على
+
+48
+00:06:29,640 --> 00:06:36,860
+k فبطلّه برا هاي a واحد ضربSummation من K بيسار
+
+49
+00:06:36,860 --> 00:06:44,400
+واحد إلى N لواحد على K واحنا
+
+50
+00:06:44,400 --> 00:06:50,140
+أثبتنا قبل هيك أنه ال sequence of partial sums لل
+
+51
+00:06:50,140 --> 00:06:57,620
+harmonic series is unbounded
+
+52
+00:06:57,620 --> 00:07:03,380
+في كان مثال سابق بيقول إنه
+
+53
+00:07:07,390 --> 00:07:14,970
+إن الـ sequence هذه من n بساوي واحد to infinity is
+
+54
+00:07:14,970 --> 00:07:18,590
+unbounded
+
+55
+00:07:18,590 --> 00:07:25,010
+is unbounded حسب
+
+56
+00:07:25,010 --> 00:07:31,030
+مثال سألت إذا لما أضربها ال sequence هذه لما أضرب
+
+57
+00:07:31,030 --> 00:07:35,350
+حدودها أو أضربها في ثابت موجة تبقى unbounded
+
+58
+00:07:39,070 --> 00:07:48,770
+وبالتالي إذا SM هذا بيقدي ان ال sequence SM is
+
+59
+00:07:48,770 --> 00:07:52,610
+unbounded
+
+60
+00:07:52,610 --> 00:07:59,870
+therefore ال
+
+61
+00:07:59,870 --> 00:08:08,950
+limit ل SM لما انتقل ل infinity does not existand
+
+62
+00:08:08,950 --> 00:08:16,510
+therefore the series sigma dn diverges لان احنا
+
+63
+00:08:16,510 --> 00:08:19,970
+قلنا قبلك ان اي infinite series بتكون convergent
+
+64
+00:08:19,970 --> 00:08:24,570
+if and only if the sequence of partial sums is
+
+65
+00:08:24,570 --> 00:08:32,870
+convergent لان هذا هو الحل okay تمام في
+
+66
+00:08:32,870 --> 00:08:35,730
+أي أسئلة تانية في section تلاتة سبعة
+
+67
+00:08:53,340 --> 00:08:58,320
+مفهوم الحل؟ في
+
+68
+00:08:58,320 --> 00:09:03,800
+أسئلة تانية في ال section هذا أو أي section سابق؟
+
+69
+00:09:03,800 --> 00:09:11,180
+فسؤال سبعة هذا
+
+70
+00:09:11,180 --> 00:09:16,210
+المماثل بيشبه مثال تلاتة سبعة ستةفاقرأي المثال
+
+71
+00:09:16,210 --> 00:09:22,330
+حاولي تطبقي نفس الطريقة مشروحليك مثال فحاولي
+
+72
+00:09:22,330 --> 00:09:28,710
+اتجلدي المثال في اي اسئلة تانية؟
+
+73
+00:09:28,710 --> 00:09:35,950
+مان
+
+74
+00:09:35,950 --> 00:09:37,170
+لديها سؤال؟
+
+75
+00:09:56,850 --> 00:10:11,850
+في عندكم أسرة طيب
+
+76
+00:10:11,850 --> 00:10:14,790
+لما تفكروا في أسرة بدي أنا بارهنكم koshi
+
+77
+00:10:14,790 --> 00:10:21,390
+condensation set test لأن هذا في عليه أسرة ومهم
+
+78
+00:10:38,680 --> 00:10:56,660
+سؤال اتماشي section تلاتة .. سابعة قوشيز
+
+79
+00:10:56,660 --> 00:11:00,760
+condensation
+
+80
+00:11:00,760 --> 00:11:01,180
+test
+
+81
+00:11:13,290 --> 00:11:19,130
+فال test هذا بيقول let sigma
+
+82
+00:11:19,130 --> 00:11:29,970
+an be a series .. a series of
+
+83
+00:11:29,970 --> 00:11:42,270
+monotone .. of monotone decreasing positive
+
+84
+00:11:45,320 --> 00:11:54,260
+مجموعات اثنين اثنين
+
+85
+00:11:54,260 --> 00:11:54,340
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+86
+00:11:54,340 --> 00:11:58,160
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+87
+00:11:58,160 --> 00:11:59,760
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+88
+00:11:59,760 --> 00:12:03,980
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+89
+00:12:03,980 --> 00:12:05,320
+اثنين اثنين اثنين اثنين اثنين اثنين اثنين اثنين
+
+90
+00:12:05,320 --> 00:12:08,420
+اثنين اثنين اثنين
+
+91
+00:12:08,420 --> 00:12:14,380
+اثنين
+
+92
+00:12:14,380 --> 00:12:14,820
+اثن
+
+93
+00:12:42,930 --> 00:12:48,630
+وهي البرهان اولا
+
+94
+00:12:48,630 --> 00:13:02,350
+خلّينا نلاحظnote that لاحظي انه لو أخدت نص في
+
+95
+00:13:02,350 --> 00:13:12,530
+summation من k بساوي zero to infinity ل two أُس k
+
+96
+00:13:12,530 --> 00:13:18,930
+في a two to k هذا
+
+97
+00:13:18,930 --> 00:13:20,830
+بيطلع بساوي نص
+
+98
+00:13:23,540 --> 00:13:33,720
+في A1 اول حد لما كدا ساوى سفر فبطلع نص A1 الحد
+
+99
+00:13:33,720 --> 00:13:43,940
+اللي بعده هيطلع A2 اللي بعده اتنين A4 واللي بعده
+
+100
+00:13:43,940 --> 00:13:52,020
+اربعة في A8 وهكذا نستمر على هذا النمط إلى
+
+101
+00:13:55,470 --> 00:14:00,650
+أتنين خلّينا ناخد المجموعة من K بساوي سفر إلى M
+
+102
+00:14:00,650 --> 00:14:08,290
+حيث M عدد طبيعي ما فأخر حد هيكون اتنين أس M سالب
+
+103
+00:14:08,290 --> 00:14:15,670
+واحد في A اتنين أس M الآن
+
+104
+00:14:15,670 --> 00:14:21,770
+هذا المجموع أصغر من A واحد نص A واحد بالتأكيد أصغر
+
+105
+00:14:21,770 --> 00:14:32,160
+من A واحدو طبعا ال .. ال .. الأعداد هذه كلها موجبة
+
+106
+00:14:32,160 --> 00:14:38,600
+و بتكون decrease in sequence فنص a1 أصغر من a1 و
+
+107
+00:14:38,600 --> 00:14:59,570
+a2 بساوي a2 و 2 a4 أصغر من a3 زائد a4 صح؟A4 أصغر
+
+108
+00:14:59,570 --> 00:15:07,110
+من A3 لأن ال sequence A N decreasing فعندي A4 زاد
+
+109
+00:15:07,110 --> 00:15:17,350
+A4 أصغر من A3 زاد A4 و هكذا برضه عندي A8 أصغر من
+
+110
+00:15:17,350 --> 00:15:25,190
+A5 و أصغر من A6 و أصغر من A7وبالتالي هدا هيكون
+
+111
+00:15:25,190 --> 00:15:30,470
+اربعة ا تمانية اصغر من مجموعة اربعة حدود اللي هم a
+
+112
+00:15:30,470 --> 00:15:42,270
+خمسة زائد a ستة زائد a سبعة زائد a تمانية و هكذا
+
+113
+00:15:42,270 --> 00:15:48,830
+استمر على هذا النمط الى ان نصل لاخر
+
+114
+00:15:51,230 --> 00:15:58,170
+هدول الحدود هيكون اصغر من .. او لحد هذا الأخير
+
+115
+00:15:58,170 --> 00:16:04,530
+اصغر من المجموعة اللي هو a اتنين اص ام سالب واحد
+
+116
+00:16:04,530 --> 00:16:11,430
+زائد واحد زائد a
+
+117
+00:16:11,430 --> 00:16:18,920
+اتنين اص ام سالب واحد زائد اتنين زائد و هكذابقت
+
+118
+00:16:18,920 --> 00:16:24,700
+أصغر من مجموعة كل ال series لأن هذه كلها حدود
+
+119
+00:16:24,700 --> 00:16:29,680
+موجبة، أعداد موجبة وهذا
+
+120
+00:16:29,680 --> 00:16:38,540
+الكلام صحيح لكل M، لكل M عدد طبيعي أكبر
+
+121
+00:16:38,540 --> 00:16:47,240
+من أو يساوي، يعني عدد طبيعي وبالتالي
+
+122
+00:16:47,240 --> 00:16:48,120
+and so
+
+123
+00:16:50,890 --> 00:17:01,690
+وبالتالي نضرب sum من k بساوي سفر إلى m لتو أس ك
+
+124
+00:17:01,690 --> 00:17:10,490
+بإتنين أس ك ده هيطلع أصغر من أو ساوي نضرب الطرفين
+
+125
+00:17:10,490 --> 00:17:15,470
+في اتنين عشان نتخلص من النصفبصير المجموع هذا أصغر
+
+126
+00:17:15,470 --> 00:17:21,410
+من أوسعه اتنين في summation من n equals zero to
+
+127
+00:17:21,410 --> 00:17:27,750
+infinity ل a n تمام؟
+
+128
+00:17:27,750 --> 00:17:34,330
+وهذا
+
+129
+00:17:34,330 --> 00:17:39,650
+صحيح لكل m belonging to N
+
+130
+00:17:44,360 --> 00:18:02,120
+بنسمي ال quality هذه واحد طيب
+
+131
+00:18:02,120 --> 00:18:05,680
+now next
+
+132
+00:18:09,650 --> 00:18:21,350
+given any m أكبر من أو سوى الواحد choose using
+
+133
+00:18:21,350 --> 00:18:34,050
+Archimedean property choose
+
+134
+00:18:34,050 --> 00:18:44,810
+k بحيث أنهtwo to K أكبر من M لأي عدد طبيعي ممكن
+
+135
+00:18:44,810 --> 00:18:58,530
+ألاقي عدد طبيعي بحياتي two to K أكبر من M then ال
+
+136
+00:18:58,530 --> 00:19:06,690
+summation from N equals zero to Mلان هذا بيطلع
+
+137
+00:19:06,690 --> 00:19:12,630
+أصغر من a0
+
+138
+00:19:12,630 --> 00:19:19,710
+زائد a1 زائد a2
+
+139
+00:19:19,710 --> 00:19:31,150
+زائد a3 زائد a4 زائد a5 زائد a6 زائد a7 زائد a8
+
+140
+00:19:35,390 --> 00:19:47,670
+مع بعض زائد و هكذا إلى اتنين
+
+141
+00:19:47,670 --> 00:19:55,190
+أسكت زائد اتنين أسكت زائد واحد زائد و هكذا إلى
+
+142
+00:19:55,190 --> 00:19:59,330
+اتنين
+
+143
+00:19:59,330 --> 00:20:03,950
+أسكت زائد واحد سالب واحد
+
+144
+00:20:12,500 --> 00:20:17,840
+أنا عند ال M هذا ال M أصغر من اتنين أس كي في آخر
+
+145
+00:20:17,840 --> 00:20:26,460
+حد اللي هو AM هيكون أصغر من A رقم اتنين أس كي أو
+
+146
+00:20:26,460 --> 00:20:34,180
+أصغر من أو ساوي اتنين رقم A أس اتنين كي زي واحد
+
+147
+00:20:34,180 --> 00:20:35,780
+minus واحد
+
+148
+00:20:43,450 --> 00:20:52,190
+والمجموع هذا .. هذا المجموع أصغر من او يساوي a0
+
+149
+00:20:52,190 --> 00:20:57,490
+زائد a1 زائد
+
+150
+00:20:57,490 --> 00:21:06,710
+2 a2 لأن a3 أصغر من a2 صح؟ عشان ال sequence an is
+
+151
+00:21:06,710 --> 00:21:13,030
+decreasing و هذا المجموع أصغر من 4 a
+
+152
+00:21:14,740 --> 00:21:26,420
+أربعة صح وهكذا إلى المجموع هذا هيكون أصغر من اتنين
+
+153
+00:21:26,420 --> 00:21:37,280
+أث كيه هذول عدد الحدود في a اتنين أث كيه يعني هذول
+
+154
+00:21:37,280 --> 00:21:41,860
+عدد الحدود عددهم اتنين أث كيه وكل واحد منهم
+
+155
+00:21:45,050 --> 00:21:55,350
+أصغر من ات اول واحد اللي هو ات نين اص كيه وهذا
+
+156
+00:21:55,350 --> 00:22:01,830
+بدوره أصغر من ات نين اص كيه زائد summation من كيه
+
+157
+00:22:01,830 --> 00:22:09,730
+بساوي zero to infinity لاتنين اص كيه في ات نين اص
+
+158
+00:22:09,730 --> 00:22:17,540
+كيه هاي أول حد ات نين اص كيهلما ك بيساوي سفر بيطلع
+
+159
+00:22:17,540 --> 00:22:25,640
+ا واحد و بعدين اللي بعده بيطلع اتنين اتنين لما ك
+
+160
+00:22:25,640 --> 00:22:33,480
+بيساوي واحد و اللي بعده اربعة اربعة و هكذا طبعا
+
+161
+00:22:33,480 --> 00:22:37,400
+هذا بوقف المجموعة هذا finite هذا أصغر من المجموعة
+
+162
+00:22:37,400 --> 00:22:41,400
+من ك بيساوي سفر إلى ملا نهاية هذا طبعا في حدود
+
+163
+00:22:41,400 --> 00:22:41,820
+أكتر
+
+164
+00:22:44,960 --> 00:22:53,040
+تمام؟ وبالتالي إذا نستنتج and so نستنتج
+
+165
+00:22:53,040 --> 00:23:02,980
+إنه المجموعة sigma from n equal zero to infinity ل
+
+166
+00:23:02,980 --> 00:23:12,050
+a nبطلع أصغر من أو ساوي a0 زاد sigma from k equals
+
+167
+00:23:12,050 --> 00:23:20,790
+zero to infinity ل 2 أُس k a2 أُس k لأن
+
+168
+00:23:20,790 --> 00:23:26,810
+هذا صحيح لكل m أكبر من أو ساوي الواحد لأن هذا
+
+169
+00:23:26,810 --> 00:23:33,330
+عبارة عن هذا عبارة عن upper bound هذا العددأو هذا
+
+170
+00:23:33,330 --> 00:23:39,530
+العدد upper bound لل sequence of partial sums هنا
+
+171
+00:23:39,530 --> 00:23:44,190
+فما
+
+172
+00:23:44,190 --> 00:23:47,210
+هذه ال sequence of partial sums is increasing
+
+173
+00:23:47,210 --> 00:23:50,750
+متزايدة
+
+174
+00:23:50,750 --> 00:23:55,110
+و bounded above by this number إذا ال limit تبعت
+
+175
+00:23:55,110 --> 00:23:58,650
+ال sequence of partial sums exist و بالساوية
+
+176
+00:23:58,650 --> 00:24:04,990
+supremumلـ sequence of partial sums الـ supremum
+
+177
+00:24:04,990 --> 00:24:11,150
+لـ sequence of partial sums أقل من ال upper bound
+
+178
+00:24:11,150 --> 00:24:13,670
+هذا upper bound لـ sequence of partial sums ال
+
+179
+00:24:13,670 --> 00:24:17,050
+supremum أصغر upper bound وبالتالي إذا ال supremum
+
+180
+00:24:17,050 --> 00:24:21,690
+لـ sequence of partial sums هو عبارة عن limit لـ
+
+181
+00:24:21,690 --> 00:24:23,730
+sequence of partial sums اللي هو مجموعة ال
+
+182
+00:24:23,730 --> 00:24:29,190
+infinite series أصغر من أو ساوي ال upper boundby
+
+183
+00:24:29,190 --> 00:24:34,290
+monotone convergence theorem السيريز
+
+184
+00:24:34,290 --> 00:24:39,610
+هذي convergence ومجموعة بساول limit ل sequence of
+
+185
+00:24:39,610 --> 00:24:44,710
+partial sums اللي هي أصغر من أو ساول عددها okay
+
+186
+00:24:44,710 --> 00:24:54,170
+إذا نسمي المتباينة هذه اتنين إذا من المتباينة واحد
+
+187
+00:24:54,170 --> 00:24:54,870
+واتنين
+
+188
+00:25:11,870 --> 00:25:19,130
+الان بمقارنة مباشرة الاختلافات
+
+189
+00:25:19,130 --> 00:25:30,640
+المتباينات واحدة و اتنين بيقدواالسيريز sigma a n
+
+190
+00:25:30,640 --> 00:25:39,720
+converges if and only if السيريز sigma اثنين اثنين
+
+191
+00:25:39,720 --> 00:25:47,820
+a اثنين اثنين converges تعالى
+
+192
+00:25:47,820 --> 00:25:54,680
+نشوف لو كانت السيريز هذه convergent فالسيريز
+
+193
+00:25:54,680 --> 00:25:55,780
+هذه convergent
+
+194
+00:25:58,080 --> 00:26:03,500
+وبالتالي طبعا أن هذا صحيح لكل M بالمناسبة بقدر أن
+
+195
+00:26:03,500 --> 00:26:08,880
+هذه أيضا sequence of partial sums هذه ال limit
+
+196
+00:26:08,880 --> 00:26:19,460
+تبعتها exist وبالتالي ال infinite series هذه إذا
+
+197
+00:26:19,460 --> 00:26:27,250
+أن ال ممكن نقول أن هذا الكلام صحيحالان لو كانت ال
+
+198
+00:26:27,250 --> 00:26:31,430
+series هادي convergent فنضربها في ثابت اتنين تطلع
+
+199
+00:26:31,430 --> 00:26:35,270
+convergent وبالتالي ال series هادي convergent by
+
+200
+00:26:35,270 --> 00:26:40,170
+direct comparison test العكس لو كانت ال series
+
+201
+00:26:40,170 --> 00:26:41,670
+هادي convergent
+
+202
+00:26:44,460 --> 00:26:50,840
+فلما أضفلها حد عدد موجب بيبقى conversion وبالتالي
+
+203
+00:26:50,840 --> 00:26:54,080
+by direct comparison test ال series الأصغر بتطلع
+
+204
+00:26:54,080 --> 00:26:58,160
+conversion okay تمام؟ لأن هذا بثبت koshi
+
+205
+00:26:58,160 --> 00:27:03,600
+condensation test هذا ال test قوي كتير ويله فوائد
+
+206
+00:27:03,600 --> 00:27:13,000
+كتيرة فمن الفوائد تبعته يعني
+
+207
+00:27:13,000 --> 00:27:13,800
+هذه مثال
+
+208
+00:27:22,170 --> 00:27:37,410
+ممكن نستنتج ال test P-series مثال،
+
+209
+00:27:37,410 --> 00:27:46,290
+أنا موجود في أحدى التمرين التمرين 13
+
+210
+00:27:53,040 --> 00:28:05,440
+تعملين تلتاش سيكشن تلاتة سبعة ايش بيقول هذا if if
+
+211
+00:28:05,440 --> 00:28:16,600
+P أكبر من السفر is a real number discuss
+
+212
+00:28:16,600 --> 00:28:20,940
+the
+
+213
+00:28:20,940 --> 00:28:21,680
+convergence
+
+214
+00:28:42,640 --> 00:28:44,720
+تعالوا نفحص
+
+215
+00:28:49,400 --> 00:28:58,120
+Summation from n equals one to infinity لإتنين أُس
+
+216
+00:28:58,120 --> 00:29:08,700
+n في واحد على هاي أو خلّيني أقول إتنين أُس n في a
+
+217
+00:29:08,700 --> 00:29:16,120
+and a إتنين أُس m إيش بيساوي هذا طبعا هاي عندي a n
+
+218
+00:29:16,120 --> 00:29:24,230
+هذا هو عبارة عن a mالحد العام لل series فان بساوي
+
+219
+00:29:24,230 --> 00:29:30,290
+1 على n to p فبتبحث هل ال series هذي convergent او
+
+220
+00:29:30,290 --> 00:29:33,990
+متى بتكون هذي ال series convergent وبالتالي بقدر
+
+221
+00:29:33,990 --> 00:29:37,890
+اطبق اللي هو cauchy condensation test فهذه عبارة
+
+222
+00:29:37,890 --> 00:29:43,970
+عن sigma from n equals one to infinityالان ايه
+
+223
+00:29:43,970 --> 00:29:53,550
+اتنين اص ان بطلع واحد على اتنين اص ان الكل اص P
+
+224
+00:29:53,550 --> 00:30:03,810
+تمام؟ وهذا بيساوي summation from n equals one to
+
+225
+00:30:03,810 --> 00:30:18,940
+infinity لاتنين اص واحد minus Pالكل أسئلة وهدي
+
+226
+00:30:18,940 --> 00:30:27,020
+is a geometric series is a geometric series
+
+227
+00:30:27,020 --> 00:30:33,680
+وبالتالي
+
+228
+00:30:33,680 --> 00:30:38,320
+مظبوط هذا عبارة عن geometric series لو بدى أكتب
+
+229
+00:30:38,320 --> 00:30:40,620
+حدود تبعتها
+
+230
+00:30:43,100 --> 00:30:52,660
+فاول حد عبارة عن اتنين اص واحد minus P الحد التاني
+
+231
+00:30:52,660 --> 00:31:00,940
+اتنين اص واحد minus P الكل تربية و هكذا فالحد
+
+232
+00:31:00,940 --> 00:31:05,480
+الاول اتنين اص واحد minus P الحد التاني اتنين اص
+
+233
+00:31:05,480 --> 00:31:09,980
+واحد minus P و هكذا with ratio
+
+234
+00:31:14,090 --> 00:31:28,710
+with ratio with
+
+235
+00:31:28,710 --> 00:31:34,830
+ratio R
+
+236
+00:31:34,830 --> 00:31:41,790
+بساوي اتنين اص واحد minus P
+
+237
+00:31:48,590 --> 00:31:58,790
+So by geometric series test it converges if
+
+238
+00:31:58,790 --> 00:32:06,450
+and all if absolute R بيساوي اتنين أس واحد minus P
+
+239
+00:32:06,450 --> 00:32:16,670
+أصغر من واحد وهذا بتحقق اتنين أس واحد minus P أصغر
+
+240
+00:32:16,670 --> 00:32:25,590
+من واحدفنقول if واحد minus P اذا
+
+241
+00:32:25,590 --> 00:32:36,910
+كان واحد minus P أصغر من السفر سالم لأن لو كان
+
+242
+00:32:36,910 --> 00:32:41,430
+واحد minus P موجب فاتنين أس أي عدد موجب عمره ما
+
+243
+00:32:41,430 --> 00:32:47,440
+بيكون أصغر من واحدنصبوت لكن لو كان الأس سالم فبصير
+
+244
+00:32:47,440 --> 00:32:52,620
+هذا واحد على اتنين أس وموجب فبصير أصغر من واحد اذا
+
+245
+00:32:52,620 --> 00:32:57,020
+هذا صحيح if and only if الأس تابع الأتنين اللي هو
+
+246
+00:32:57,020 --> 00:33:06,240
+واحد minus P أصغر من سفر if and only if واحد أصغر
+
+247
+00:33:06,240 --> 00:33:12,920
+من P أو P أكبر من واحد okay تماموهذا هو ال P
+
+248
+00:33:12,920 --> 00:33:19,120
+Series Test لان احنا استنتجنا ال P Series Test من
+
+249
+00:33:19,120 --> 00:33:26,200
+Koshi Condensation Test فاكرين ال P Series هذي او
+
+250
+00:33:26,200 --> 00:33:29,840
+ال P Series Test اثبتنا ان Convergent if and only
+
+251
+00:33:29,840 --> 00:33:35,200
+if P أكبر من 1 وDivergent اذا كانت P أصغر منها
+
+252
+00:33:35,200 --> 00:33:35,960
+وسائل 1
+
+253
+00:33:42,110 --> 00:33:51,730
+Okay إذا ال .. هذا المعنى So by Cauchy Cauchy's
+
+254
+00:33:51,730 --> 00:34:01,910
+Condensation Test The series Sigma
+
+255
+00:34:01,910 --> 00:34:07,830
+from N equals one to infinity ال one over N to P
+
+256
+00:34:08,830 --> 00:34:16,530
+convergence if and only if P أكبر من واحد وهذا هو
+
+257
+00:34:16,530 --> 00:34:23,030
+ال P-series test إذن هذا بورجينا قوة Koshi
+
+258
+00:34:23,030 --> 00:34:29,530
+Condensation Test okay تمام؟ في طبعا أسئلة أخرى
+
+259
+00:34:29,530 --> 00:34:33,430
+على Koshi Condensation Test وانا طالب منكم تحلوها
+
+260
+00:34:33,430 --> 00:34:41,340
+زي السؤال 14 و15 صح؟ففي أي شيء في الأسئلة دي أو
+
+261
+00:34:41,340 --> 00:34:47,160
+أسئلة تانية؟
+
+262
+00:34:47,160 --> 00:34:56,500
+في
+
+263
+00:34:56,500 --> 00:34:58,180
+عندكم أي أسئلة؟
+
+264
+00:35:13,000 --> 00:35:19,560
+إذا سيكشن واحد تلاتة سبعة في أي سؤال تاني عندكم في
+
+265
+00:35:19,560 --> 00:35:25,540
+الأسئلة هذه أو
+
+266
+00:35:25,540 --> 00:35:31,860
+السيكاشن السابقة أو سيكشن أربعة واحد إذا بتحبه
+
+267
+00:35:31,860 --> 00:35:35,560
+سيكشن أربعة واحد
+
+268
+00:36:06,090 --> 00:36:13,070
+مافيش أسئلة؟ طيب ال .. مدان مافيش أسئلة نواصل ..
+
+269
+00:36:13,070 --> 00:36:16,190
+نكمل
+
+270
+00:36:16,190 --> 00:36:17,490
+المحاضرة في السابقة
+
+271
+00:36:49,090 --> 00:36:53,250
+المرة الأخرى اتحدثنا عن ال two-sided limits و عن
+
+272
+00:36:53,250 --> 00:37:00,350
+ال one-sided limits و أخدنا بعض النظريات و قلنا إن
+
+273
+00:37:00,350 --> 00:37:05,090
+جميع النظريات اللي برهنناها هو one-sided limit
+
+274
+00:37:05,090 --> 00:37:12,990
+صحيحة لل two-sided limitsأو المباريات الصحيحة لـ
+
+275
+00:37:12,990 --> 00:37:17,070
+two-sided limits بتكون أيضا صحيحة لـ one-sided
+
+276
+00:37:17,070 --> 00:37:26,650
+limit فناخد
+
+277
+00:37:26,650 --> 00:37:31,350
+أنفلة show
+
+278
+00:37:31,350 --> 00:37:31,950
+that
+
+279
+00:37:35,020 --> 00:37:55,100
+Limit لـ Signum X لإن X تقول لسفر لا يوجد فنلاحظ
+
+280
+00:37:55,100 --> 00:37:59,600
+أن Limit لأول شئ Signum X
+
+281
+00:38:03,790 --> 00:38:11,230
+بساوي x على absolute x لكل x لا يساوي سفر لما
+
+282
+00:38:11,230 --> 00:38:15,010
+أعرفنا الدالة هذه قلت لها بس هي نفسها x على
+
+283
+00:38:15,010 --> 00:38:20,690
+absolute x لو كان x بساوي سفر الآن ال limit ل
+
+284
+00:38:20,690 --> 00:38:30,890
+sigma x لما x تقول إلى سفر من اليمين بساوي ال
+
+285
+00:38:30,890 --> 00:38:31,310
+limit
+
+286
+00:38:35,810 --> 00:38:41,530
+لما x تقول إلى صفر من اليمين لما x تقول إلى صفر من
+
+287
+00:38:41,530 --> 00:38:50,190
+اليمين لما x تقول إلى صفر من اليمين لما x تقول إلى
+
+288
+00:38:50,190 --> 00:38:55,650
+صفر من اليمين لما x تقول إلى صفر من اليمين لما x
+
+289
+00:38:55,650 --> 00:38:57,330
+تقول إلى صفر من اليمين لما x تقول إلى صفر من
+
+290
+00:38:57,330 --> 00:39:02,560
+اليمين لما x تقول إلى صفر من اليمينلما x تقول أصغر
+
+291
+00:39:02,560 --> 00:39:21,640
+من اليسار لما x أصغر من صفر لما
+
+292
+00:39:21,640 --> 00:39:28,560
+x أصغر من صفر لما x أصغر من صفر لما x أصغر من صفر
+
+293
+00:39:28,560 --> 00:39:33,950
+لما x أصغر من صفرالسالب واحد بيطلع السالب واحد ان
+
+294
+00:39:33,950 --> 00:39:37,670
+انا عندي ال limit من اليامين يساوي واحد ال limit
+
+295
+00:39:37,670 --> 00:39:44,230
+من اليسار يساوي سالب واحد مش متساوي اتين so by
+
+296
+00:39:44,230 --> 00:39:50,150
+theorem حسب النظرية اللي أخدناها theorem اربعة
+
+297
+00:39:50,150 --> 00:39:55,630
+تلاتة تلاتة بيطلع
+
+298
+00:39:55,630 --> 00:40:01,080
+عندي ال limit او ال two sided limitللـ signal
+
+299
+00:40:01,080 --> 00:40:09,560
+function لما x تقول السفر does not exist تمام؟
+
+300
+00:40:09,560 --> 00:40:22,280
+طيب خلّيني انا اخد show
+
+301
+00:40:22,280 --> 00:40:27,380
+that ال
+
+302
+00:40:27,380 --> 00:40:32,350
+limit لل function e والواحد على xلما x تقول إلى
+
+303
+00:40:32,350 --> 00:40:40,550
+سفر من اليمين does not exist and
+
+304
+00:40:40,550 --> 00:40:43,910
+من
+
+305
+00:40:43,910 --> 00:40:51,170
+ال limit لنفس ال function e to واحد على x لما x
+
+306
+00:40:51,170 --> 00:40:58,710
+تقول إلى سفر من اليسار تطلع موجودة و بساوي سفر
+
+307
+00:41:23,830 --> 00:41:31,010
+طيب ال ..
+
+308
+00:41:31,010 --> 00:41:34,050
+نحاول نبرهن الجزء الأول
+
+309
+00:41:56,420 --> 00:42:03,380
+بناخد الجزء الأول let
+
+310
+00:42:03,380 --> 00:42:13,540
+z of x بساوي e to 1 على x حفة x لا تساوي 0 وبدنا
+
+311
+00:42:13,540 --> 00:42:19,260
+نثبت to
+
+312
+00:42:19,260 --> 00:42:28,130
+show ان ال limitلـ g of x لما x تقول لصفر من
+
+313
+00:42:28,130 --> 00:42:38,750
+اليمين does not exist it suffices to
+
+314
+00:42:38,750 --> 00:42:42,710
+show يكفي
+
+315
+00:42:42,710 --> 00:42:52,650
+اثبات ان ال function g of x is not bounded on
+
+316
+00:42:56,170 --> 00:43:05,850
+on a right .. on a right neighborhood
+
+317
+00:43:05,850 --> 00:43:13,670
+.. on a right neighborhood اللي هو سفر دلتا of
+
+318
+00:43:13,670 --> 00:43:15,230
+zero
+
+319
+00:43:25,230 --> 00:43:28,670
+أخذنا قبل ذلك نظرية بتقول إيه ده عشان أثبت أنه ال
+
+320
+00:43:28,670 --> 00:43:35,710
+limit ل function عن نقطة معينة مش موجودة يكفي أثبت
+
+321
+00:43:35,710 --> 00:43:43,910
+أنه أنه الدالة unbounded عند أي unbounded
+
+322
+00:43:43,910 --> 00:43:48,650
+عند أي neighborhood
+
+323
+00:43:48,650 --> 00:43:56,210
+للنقطة الآن بالنسبة لل one-sided limitعشان أقول إن
+
+324
+00:43:56,210 --> 00:44:02,230
+ال limit ل function زي هذه g of x لما x تقول إلى
+
+325
+00:44:02,230 --> 00:44:09,430
+سفر من اليمين does not exist فهي
+
+326
+00:44:09,430 --> 00:44:16,390
+السفر و ال x تقول لسفر من اليمين فبدل ما أخد delta
+
+327
+00:44:16,390 --> 00:44:20,290
+neighborhood للسفر
+
+328
+00:44:20,290 --> 00:44:29,960
+فباخد right neighborhoodright neighborhood للسفر
+
+329
+00:44:29,960 --> 00:44:35,960
+فيكفي ان ال function هذه ماهياش bounded عن كل
+
+330
+00:44:35,960 --> 00:44:41,780
+right neighborhood يعني جوار من اليمين للسفر لان
+
+331
+00:44:41,780 --> 00:44:46,000
+انا بتعامل مع نهاية من اليمين لكن لما كنت اتعامل
+
+332
+00:44:46,000 --> 00:44:51,240
+مع نهاية من الطرفين فكنت ااخد delta neighborhood
+
+333
+00:44:51,240 --> 00:44:56,840
+كاملفلو أثبتت إن الـ function هذه ماهياش bounded
+
+334
+00:44:56,840 --> 00:45:01,280
+عند أي right neighborhood للصفر على الصورة هذه
+
+335
+00:45:01,280 --> 00:45:06,220
+فحسب نظرية سابقة الدالة مش ممكن يكون لها limit من
+
+336
+00:45:06,220 --> 00:45:09,960
+اليمين عند الصفر لأن لو كان لها limit عند الصفر من
+
+337
+00:45:09,960 --> 00:45:15,820
+اليمين فلازم تكون bounded على some neighborhood ..
+
+338
+00:45:15,820 --> 00:45:25,650
+right neighborhood للصفرOkay تمام و لإثبات ذلك to
+
+339
+00:45:25,650 --> 00:45:29,270
+see
+
+340
+00:45:29,270 --> 00:45:39,290
+this we use ال inequality التالية وهي T أكبر من
+
+341
+00:45:39,290 --> 00:45:47,010
+سفر دايما أصغر من E أس T for all T أكبر من سفر هذه
+
+342
+00:45:47,010 --> 00:45:54,200
+المتباينةهذه المتباينة موجودة
+
+343
+00:45:54,200 --> 00:46:01,780
+برهانة C Chapter 8 برهانة
+
+344
+00:46:01,780 --> 00:46:07,600
+موجودة في Chapter 8 اللي هتاخدوه لاحقا فهنستخدم
+
+345
+00:46:07,600 --> 00:46:11,460
+اللي هو المتباينة هذه في اثبات ان ال function
+
+346
+00:46:11,460 --> 00:46:17,420
+ماهياش bounded على neighborhood او right
+
+347
+00:46:17,420 --> 00:46:25,130
+neighborhood للصفرOkay عشان الوجد خلص بنوقف و
+
+348
+00:46:25,130 --> 00:46:29,590
+بناخد خمس دقايق break و بعدين بنكمل ان شاء الله
+
+349
+00:46:29,590 --> 00:46:35,550
+البرهانة فحنوقف و نكمل في الجزء التالي من المحاضرة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..7b74a92b25a4a3ccf094f695b0d7d8d392bca5c0
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/1Uemtyp4-IM_raw.json
@@ -0,0 +1 @@
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"text": "is convergent then is the series sigma للجدر التربيهي ولا لأ؟", "tokens": [271, 9652, 6930, 550, 307, 264, 2638, 12771, 24976, 7435, 3215, 2288, 16712, 2288, 21292, 3224, 1829, 49429, 5296, 10721, 22807], "avg_logprob": -0.2338423336094076, "compression_ratio": 0.9411764705882353, "no_speech_prob": 0.0, "words": [{"start": 114.21, "end": 114.61, "word": "is", "probability": 0.8974609375}, {"start": 114.61, "end": 115.63, "word": " convergent", "probability": 0.949951171875}, {"start": 115.63, "end": 121.23, "word": " then", "probability": 0.59619140625}, {"start": 121.23, "end": 123.15, "word": " is", "probability": 0.96337890625}, {"start": 123.15, "end": 125.23, "word": " the", "probability": 0.351318359375}, {"start": 125.23, "end": 125.59, "word": " series", "probability": 0.87744140625}, {"start": 125.59, "end": 126.31, "word": " sigma", "probability": 0.90869140625}, {"start": 126.31, "end": 128.07, "word": " للجدر", "probability": 0.863525390625}, {"start": 128.07, 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{"start": 411.18, "end": 411.44, "word": " is", "probability": 0.9140625}, {"start": 411.44, "end": 417.62, "word": " unbounded", "probability": 0.9300130208333334}, {"start": 417.62, "end": 419.0, "word": " في", "probability": 0.177734375}, {"start": 419.0, "end": 419.46, "word": " كان", "probability": 0.96337890625}, {"start": 419.46, "end": 419.92, "word": " مثال", "probability": 0.973876953125}, {"start": 419.92, "end": 420.72, "word": " سابق", "probability": 0.8507486979166666}, {"start": 420.72, "end": 422.84, "word": " بيقول", "probability": 0.9212239583333334}, {"start": 422.84, "end": 423.38, "word": " إنه", "probability": 0.7117919921875}], "temperature": 1.0}, {"id": 15, "seek": 45535, "start": 427.39, "end": 455.35, "text": "إن الـ sequence هذه من n بساوي واحد to infinity is unbounded is unbounded حسب مثال سألت إذا لما أضربها ال sequence هذه لما أضرب حدودها أو أضربها في ثابت موجة تبقى unbounded", "tokens": [28814, 1863, 2423, 39184, 8310, 29538, 9154, 297, 4724, 3794, 995, 45865, 36764, 24401, 281, 13202, 307, 517, 18767, 292, 307, 517, 18767, 292, 11331, 35457, 50113, 6027, 8608, 10721, 1211, 2655, 11933, 15730, 5296, 15042, 5551, 11242, 25513, 11296, 2423, 8310, 29538, 5296, 15042, 5551, 11242, 25513, 11331, 3215, 23328, 11296, 34051, 5551, 11242, 25513, 11296, 8978, 38637, 16758, 2655, 3714, 29245, 3660, 6055, 3555, 4587, 7578, 517, 18767, 292], "avg_logprob": -0.2897135338021649, "compression_ratio": 1.5776397515527951, "no_speech_prob": 0.0, "words": [{"start": 427.39, "end": 427.73, "word": "إن", "probability": 0.52947998046875}, {"start": 427.73, "end": 427.95, "word": " الـ", "probability": 0.437255859375}, {"start": 427.95, "end": 428.49, "word": " sequence", "probability": 0.71484375}, {"start": 428.49, "end": 429.07, "word": " هذه", "probability": 0.72607421875}, {"start": 429.07, "end": 431.17, "word": " من", "probability": 0.7919921875}, {"start": 431.17, "end": 431.47, "word": " n", "probability": 0.401123046875}, {"start": 431.47, "end": 432.17, "word": " بساوي", "probability": 0.711181640625}, {"start": 432.17, "end": 432.73, "word": " واحد", "probability": 0.937744140625}, {"start": 432.73, "end": 432.87, "word": " to", "probability": 0.235107421875}, {"start": 432.87, "end": 433.45, "word": " infinity", "probability": 0.849609375}, {"start": 433.45, "end": 434.97, "word": " is", "probability": 0.74755859375}, {"start": 434.97, "end": 438.59, "word": " unbounded", "probability": 0.9488932291666666}, {"start": 438.59, "end": 439.07, "word": " is", "probability": 0.141845703125}, {"start": 439.07, "end": 441.27, "word": " unbounded", "probability": 0.9627278645833334}, {"start": 441.27, "end": 445.01, "word": " حسب", "probability": 0.91064453125}, {"start": 445.01, "end": 445.43, "word": " مثال", "probability": 0.966796875}, {"start": 445.43, "end": 446.93, "word": " سألت", "probability": 0.539642333984375}, {"start": 446.93, "end": 447.25, "word": " إذا", "probability": 0.3809814453125}, {"start": 447.25, "end": 447.45, "word": " لما", "probability": 0.856689453125}, {"start": 447.45, "end": 448.01, "word": " أضربها", "probability": 0.9049072265625}, {"start": 448.01, "end": 448.75, "word": " ال", "probability": 0.79541015625}, {"start": 448.75, "end": 449.09, "word": " sequence", "probability": 0.74853515625}, {"start": 449.09, "end": 449.67, "word": " هذه", "probability": 0.8603515625}, {"start": 449.67, "end": 450.63, "word": " لما", "probability": 0.9072265625}, {"start": 450.63, "end": 451.03, "word": " أضرب", "probability": 0.9650065104166666}, {"start": 451.03, "end": 451.83, "word": " حدودها", "probability": 0.99462890625}, {"start": 451.83, "end": 452.79, "word": " أو", "probability": 0.477294921875}, {"start": 452.79, "end": 453.13, "word": " أضربها", "probability": 0.9593505859375}, {"start": 453.13, "end": 453.31, "word": " في", "probability": 0.90869140625}, {"start": 453.31, "end": 453.75, "word": " ثابت", "probability": 0.99169921875}, {"start": 453.75, "end": 454.19, "word": " موجة", "probability": 0.9412434895833334}, {"start": 454.19, "end": 454.55, "word": " تبقى", "probability": 0.729736328125}, {"start": 454.55, "end": 455.35, "word": " unbounded", "probability": 0.9482421875}], "temperature": 1.0}, {"id": 16, "seek": 48685, "start": 459.07, "end": 486.85, "text": "وبالتالي إذا SM هذا بيقدي ان ال sequence SM is unbounded therefore ال limit ل SM لما انتقل ل infinity does not exist", "tokens": [37746, 6027, 2655, 6027, 1829, 11933, 15730, 13115, 23758, 4724, 1829, 4587, 16254, 16472, 2423, 8310, 13115, 307, 517, 18767, 292, 4412, 2423, 4948, 5296, 13115, 5296, 15042, 16472, 2655, 4587, 1211, 5296, 13202, 775, 406, 2514], "avg_logprob": -0.46546053180569097, "compression_ratio": 1.188976377952756, "no_speech_prob": 0.0, "words": [{"start": 459.07, "end": 460.23, "word": "وبالتالي", "probability": 0.880517578125}, {"start": 460.23, "end": 460.51, "word": " إذا", "probability": 0.53179931640625}, {"start": 460.51, "end": 461.19, "word": " SM", "probability": 0.07830810546875}, {"start": 461.19, "end": 462.71, "word": " هذا", "probability": 0.56640625}, {"start": 462.71, "end": 463.31, "word": " بيقدي", "probability": 0.7235107421875}, {"start": 463.31, "end": 463.61, "word": " ان", "probability": 0.477294921875}, {"start": 463.61, "end": 463.93, "word": " ال", "probability": 0.4501953125}, {"start": 463.93, "end": 464.53, "word": " sequence", "probability": 0.83203125}, {"start": 464.53, "end": 465.31, "word": " SM", "probability": 0.8232421875}, {"start": 465.31, "end": 468.77, "word": " is", "probability": 0.89794921875}, {"start": 468.77, "end": 472.61, "word": " unbounded", "probability": 0.9422200520833334}, {"start": 472.61, "end": 474.83, "word": " therefore", "probability": 0.324951171875}, {"start": 474.83, "end": 479.87, "word": " ال", "probability": 0.5859375}, {"start": 479.87, "end": 480.29, "word": " limit", "probability": 0.9541015625}, {"start": 480.29, "end": 481.89, "word": " ل", "probability": 0.95556640625}, {"start": 481.89, "end": 482.51, "word": " SM", "probability": 0.5888671875}, {"start": 482.51, "end": 482.87, "word": " لما", "probability": 0.3265380859375}, {"start": 482.87, "end": 483.51, "word": " انتقل", "probability": 0.63372802734375}, {"start": 483.51, "end": 483.65, "word": " ل", "probability": 0.72705078125}, {"start": 483.65, "end": 484.19, "word": " infinity", "probability": 0.71484375}, {"start": 484.19, "end": 486.09, "word": " does", "probability": 0.50048828125}, {"start": 486.09, "end": 486.33, "word": " not", "probability": 0.9619140625}, {"start": 486.33, "end": 486.85, "word": " exist", "probability": 0.953125}], "temperature": 1.0}, {"id": 17, "seek": 51573, "start": 488.55, "end": 515.73, "text": "and therefore the series sigma dn diverges لان احنا قلنا قبلك ان اي infinite series بتكون convergent if and only if the sequence of partial sums is convergent لان هذا هو الحل okay تمام في أي أسئلة تانية في section تلاتة سبعة", "tokens": [474, 4412, 264, 2638, 12771, 274, 77, 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"end": 494.39, "word": " diverges", "probability": 0.788330078125}, {"start": 494.39, "end": 496.33, "word": " لان", "probability": 0.3184814453125}, {"start": 496.33, "end": 496.51, "word": " احنا", "probability": 0.83984375}, {"start": 496.51, "end": 496.77, "word": " قلنا", "probability": 0.9013671875}, {"start": 496.77, "end": 497.25, "word": " قبلك", "probability": 0.7548014322916666}, {"start": 497.25, "end": 497.43, "word": " ان", "probability": 0.7880859375}, {"start": 497.43, "end": 498.13, "word": " اي", "probability": 0.5721435546875}, {"start": 498.13, "end": 498.45, "word": " infinite", "probability": 0.468505859375}, {"start": 498.45, "end": 498.83, "word": " series", "probability": 0.8916015625}, {"start": 498.83, "end": 499.17, "word": " بتكون", "probability": 0.9521484375}, {"start": 499.17, "end": 499.97, "word": " convergent", "probability": 0.853759765625}, {"start": 499.97, "end": 501.01, "word": " if", "probability": 0.720703125}, {"start": 501.01, "end": 501.23, "word": " and", "probability": 0.83642578125}, {"start": 501.23, "end": 501.55, "word": " only", "probability": 0.92822265625}, {"start": 501.55, "end": 501.97, "word": " if", "probability": 0.96728515625}, {"start": 501.97, "end": 502.95, "word": " the", "probability": 0.77978515625}, {"start": 502.95, "end": 503.45, "word": " sequence", "probability": 0.9541015625}, {"start": 503.45, "end": 503.67, "word": " of", "probability": 0.97265625}, {"start": 503.67, "end": 504.01, "word": " partial", "probability": 0.93603515625}, {"start": 504.01, "end": 504.37, "word": " sums", "probability": 0.96630859375}, {"start": 504.37, "end": 504.57, "word": " is", "probability": 0.90673828125}, {"start": 504.57, "end": 505.25, "word": " convergent", "probability": 0.983154296875}, {"start": 505.25, "end": 507.09, "word": " لان", "probability": 0.800537109375}, {"start": 507.09, "end": 507.35, "word": " هذا", "probability": 0.720703125}, {"start": 507.35, "end": 507.65, "word": " هو", "probability": 0.98974609375}, {"start": 507.65, "end": 508.33, "word": " الحل", "probability": 0.96728515625}, {"start": 508.33, "end": 508.89, "word": " okay", "probability": 0.248779296875}, {"start": 508.89, "end": 509.69, "word": " تمام", "probability": 0.946044921875}, {"start": 509.69, "end": 512.87, "word": " في", "probability": 0.441162109375}, {"start": 512.87, "end": 513.07, "word": " أي", "probability": 0.38427734375}, {"start": 513.07, "end": 513.47, "word": " أسئلة", "probability": 0.885986328125}, {"start": 513.47, "end": 514.03, "word": " تانية", "probability": 0.9959309895833334}, {"start": 514.03, "end": 514.41, "word": " في", "probability": 0.8349609375}, {"start": 514.41, "end": 514.83, "word": " section", "probability": 0.79296875}, {"start": 514.83, "end": 515.33, "word": " تلاتة", "probability": 0.9027099609375}, {"start": 515.33, "end": 515.73, "word": " سبعة", "probability": 0.9195963541666666}], "temperature": 1.0}, {"id": 18, "seek": 55436, "start": 533.34, "end": 554.36, "text": "مفهوم الحل؟ في أسئلة تانية في ال section هذا أو أي section سابق؟ فسؤال سبعة هذا المماثل بيشبه مثال تلاتة سبعة ستة", "tokens": [2304, 5172, 3224, 20498, 21542, 1211, 22807, 8978, 5551, 3794, 19986, 37977, 6055, 7649, 10632, 8978, 2423, 3541, 23758, 34051, 36632, 3541, 8608, 16758, 4587, 22807, 6156, 3794, 33604, 6027, 8608, 3555, 27884, 23758, 9673, 2304, 5718, 104, 1211, 4724, 1829, 8592, 3555, 3224, 50113, 6027, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 8608, 2655, 3660], "avg_logprob": -0.19983553258996262, "compression_ratio": 1.446969696969697, "no_speech_prob": 0.0, "words": [{"start": 533.34, "end": 534.22, "word": "مفهوم", "probability": 0.9595947265625}, {"start": 534.22, "end": 535.1, "word": " الحل؟", "probability": 0.8924153645833334}, {"start": 535.1, "end": 538.32, "word": " في", "probability": 0.74169921875}, {"start": 538.32, "end": 538.74, "word": " أسئلة", "probability": 0.8824462890625}, {"start": 538.74, "end": 539.22, "word": " تانية", "probability": 0.99609375}, {"start": 539.22, "end": 539.42, "word": " في", "probability": 0.91015625}, {"start": 539.42, "end": 539.52, "word": " ال", "probability": 0.794921875}, {"start": 539.52, "end": 539.84, "word": " section", "probability": 0.927734375}, {"start": 539.84, "end": 540.22, "word": " هذا", "probability": 0.42626953125}, {"start": 540.22, "end": 540.58, "word": " أو", "probability": 0.767578125}, {"start": 540.58, "end": 540.78, "word": " أي", "probability": 0.58251953125}, {"start": 540.78, "end": 541.2, "word": " section", "probability": 0.8896484375}, {"start": 541.2, "end": 543.8, "word": " سابق؟", "probability": 0.8787841796875}, {"start": 543.8, "end": 544.22, "word": " فسؤال", "probability": 0.67156982421875}, {"start": 544.22, "end": 544.62, "word": " سبعة", "probability": 0.83447265625}, {"start": 544.62, "end": 551.18, "word": " هذا", "probability": 0.8623046875}, {"start": 551.18, "end": 552.06, "word": " المماثل", "probability": 0.8349609375}, {"start": 552.06, "end": 552.66, "word": " بيشبه", "probability": 0.8447265625}, {"start": 552.66, "end": 553.12, "word": " مثال", "probability": 0.67333984375}, {"start": 553.12, "end": 553.58, "word": " تلاتة", "probability": 0.841796875}, {"start": 553.58, "end": 553.88, "word": " سبعة", "probability": 0.857421875}, {"start": 553.88, "end": 554.36, "word": " ستة", "probability": 0.9544270833333334}], "temperature": 1.0}, {"id": 19, "seek": 57716, "start": 555.15, "end": 577.17, "text": "فاقرأي المثال حاولي تطبقي نفس الطريقة مشروحليك مثال فحاولي اتجلدي المثال في اي اسئلة تانية؟ مان لديها سؤال؟", "tokens": [5172, 995, 4587, 2288, 10721, 1829, 9673, 12984, 6027, 11331, 995, 12610, 1829, 6055, 9566, 3555, 38436, 8717, 36178, 41950, 16572, 28671, 37893, 32887, 5016, 20292, 4117, 50113, 6027, 6156, 5016, 995, 12610, 1829, 1975, 2655, 7435, 1211, 16254, 9673, 12984, 6027, 8978, 1975, 1829, 24525, 19986, 37977, 6055, 7649, 10632, 22807, 3714, 7649, 5296, 16254, 11296, 8608, 33604, 6027, 22807], "avg_logprob": -0.2384072597469053, "compression_ratio": 1.6147540983606556, "no_speech_prob": 0.0, "words": [{"start": 555.15, "end": 555.75, "word": "فاقرأي", "probability": 0.69140625}, {"start": 555.75, "end": 556.21, "word": " المثال", "probability": 0.9763997395833334}, {"start": 556.21, "end": 556.85, "word": " حاولي", "probability": 0.86328125}, {"start": 556.85, "end": 557.85, "word": " تطبقي", "probability": 0.65576171875}, {"start": 557.85, "end": 558.97, "word": " نفس", "probability": 0.9873046875}, {"start": 558.97, "end": 559.59, "word": " الطريقة", "probability": 0.9876302083333334}, {"start": 559.59, "end": 560.31, "word": " مشروحليك", "probability": 0.5947509765625}, {"start": 560.31, "end": 560.75, "word": " مثال", "probability": 0.984130859375}, {"start": 560.75, "end": 562.33, "word": " فحاولي", "probability": 0.94287109375}, {"start": 562.33, "end": 562.71, "word": " اتجلدي", "probability": 0.8369140625}, {"start": 562.71, "end": 563.47, "word": " المثال", "probability": 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992, 1500, 5296, 33456, 23758, 8978, 47356, 5551, 3794, 25720, 4032, 2304, 16095], "avg_logprob": -0.2944444550408257, "compression_ratio": 1.3333333333333333, "no_speech_prob": 0.0, "words": [{"start": 596.85, "end": 597.09, "word": "في", "probability": 0.53857421875}, {"start": 597.09, "end": 597.67, "word": " عندكم", "probability": 0.934814453125}, {"start": 597.67, "end": 598.25, "word": " أسرة", "probability": 0.705322265625}, {"start": 598.25, "end": 611.85, "word": " طيب", "probability": 0.9239908854166666}, {"start": 611.85, "end": 612.13, "word": " لما", "probability": 0.60211181640625}, {"start": 612.13, "end": 612.73, "word": " تفكروا", "probability": 0.9539794921875}, {"start": 612.73, "end": 612.81, "word": " في", "probability": 0.94970703125}, {"start": 612.81, "end": 613.23, "word": " أسرة", "probability": 0.982421875}, {"start": 613.23, "end": 613.53, "word": " بدي", "probability": 0.5499267578125}, {"start": 613.53, "end": 613.75, "word": " أنا", "probability": 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"سؤال اتماشي section تلاتة .. سابعة قوشيز condensation test", "tokens": [3794, 33604, 6027, 1975, 2655, 2304, 33599, 1829, 3541, 6055, 1211, 9307, 3660, 4386, 8608, 16758, 27884, 12174, 2407, 8592, 1829, 11622, 2224, 35292, 1500], "avg_logprob": -0.5012019322468684, "compression_ratio": 1.0121951219512195, "no_speech_prob": 0.0, "words": [{"start": 638.68, "end": 639.78, "word": "سؤال", "probability": 0.6831461588541666}, {"start": 639.78, "end": 641.3, "word": " اتماشي", "probability": 0.596484375}, {"start": 641.3, "end": 642.24, "word": " section", "probability": 0.319091796875}, {"start": 642.24, "end": 644.38, "word": " تلاتة", "probability": 0.8851318359375}, {"start": 644.38, "end": 644.52, "word": " ..", "probability": 0.347900390625}, {"start": 644.52, "end": 645.9, "word": " سابعة", "probability": 0.75927734375}, {"start": 645.9, "end": 656.66, "word": " قوشيز", "probability": 0.76435546875}, {"start": 656.66, "end": 660.76, "word": " condensation", "probability": 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{"start": 1093.91, "end": 1094.57, "word": " أكبر", "probability": 0.8595377604166666}, {"start": 1094.57, "end": 1094.81, "word": " من", "probability": 0.96337890625}, {"start": 1094.81, "end": 1095.05, "word": " أو", "probability": 0.8447265625}, {"start": 1095.05, "end": 1095.55, "word": " سوى", "probability": 0.7894694010416666}, {"start": 1095.55, "end": 1096.23, "word": " الواحد", "probability": 0.7906087239583334}, {"start": 1096.23, "end": 1097.93, "word": " choose", "probability": 0.350830078125}, {"start": 1097.93, "end": 1101.35, "word": " using", "probability": 0.89208984375}, {"start": 1101.35, "end": 1102.23, "word": " Archimedean", "probability": 0.9326171875}, {"start": 1102.23, "end": 1103.07, "word": " property", "probability": 0.93994140625}, {"start": 1103.07, "end": 1114.05, "word": " choose", "probability": 0.26416015625}, {"start": 1114.05, "end": 1115.39, "word": " k", "probability": 0.5078125}, {"start": 1115.39, "end": 1116.17, "word": " بحيث", "probability": 0.84765625}, {"start": 1116.17, "end": 1116.69, "word": " أنه", "probability": 0.776611328125}], "temperature": 1.0}, {"id": 37, "seek": 114249, "start": 1117.53, "end": 1142.49, "text": "two to K أكبر من M لأي عدد طبيعي ممكن ألاقي عدد طبيعي بحياتي two to K أكبر من M then ال summation from N equals zero to M", "tokens": [20534, 281, 591, 5551, 4117, 26890, 9154, 376, 5296, 10721, 1829, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 3714, 43020, 5551, 15040, 38436, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 9307, 1829, 732, 281, 591, 5551, 4117, 26890, 9154, 376, 550, 2423, 28811, 490, 426, 6915, 4018, 281, 376], "avg_logprob": -0.29540092990083516, "compression_ratio": 1.3852459016393444, "no_speech_prob": 0.0, "words": [{"start": 1117.53, "end": 1117.91, "word": "two", "probability": 0.04144287109375}, {"start": 1117.91, "end": 1118.27, "word": " to", "probability": 0.85107421875}, {"start": 1118.27, "end": 1118.75, "word": " K", "probability": 0.4306640625}, {"start": 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"word": " to", "probability": 0.966796875}, {"start": 1142.09, "end": 1142.49, "word": " M", "probability": 0.98193359375}], "temperature": 1.0}, {"id": 38, "seek": 117115, "start": 1143.87, "end": 1171.15, "text": "لان هذا بيطلع أصغر من a0 زائد a1 زائد a2 زائد a3 زائد a4 زائد a5 زائد a6 زائد a7 زائد a8", "tokens": [1211, 7649, 23758, 4724, 1829, 9566, 1211, 3615, 5551, 9381, 17082, 2288, 9154, 257, 15, 30767, 16373, 3215, 257, 16, 30767, 16373, 3215, 257, 17, 30767, 16373, 3215, 257, 18, 30767, 16373, 3215, 257, 19, 30767, 16373, 3215, 257, 20, 30767, 16373, 3215, 257, 21, 30767, 16373, 3215, 257, 22, 30767, 16373, 3215, 257, 23], "avg_logprob": -0.14620536406125342, "compression_ratio": 1.7125, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1143.87, "end": 1145.01, "word": "لان", "probability": 0.4478759765625}, {"start": 1145.01, "end": 1146.17, "word": " هذا", "probability": 0.80419921875}, {"start": 1146.17, "end": 1146.69, "word": " بيطلع", "probability": 0.710693359375}, {"start": 1146.69, "end": 1147.35, "word": " أصغر", "probability": 0.86944580078125}, {"start": 1147.35, "end": 1147.99, "word": " من", "probability": 0.98876953125}, {"start": 1147.99, "end": 1152.63, "word": " a0", "probability": 0.5294189453125}, {"start": 1152.63, "end": 1153.85, "word": " زائد", "probability": 0.7150065104166666}, {"start": 1153.85, "end": 1154.75, "word": " a1", "probability": 0.949462890625}, {"start": 1154.75, "end": 1155.89, "word": " زائد", "probability": 0.9544270833333334}, {"start": 1155.89, "end": 1159.71, "word": " a2", "probability": 0.9794921875}, {"start": 1159.71, "end": 1160.61, "word": " زائد", "probability": 0.9469401041666666}, {"start": 1160.61, "end": 1161.65, "word": " a3", "probability": 0.99169921875}, {"start": 1161.65, "end": 1162.59, "word": " زائد", "probability": 0.962890625}, {"start": 1162.59, "end": 1164.89, "word": " a4", "probability": 0.991455078125}, {"start": 1164.89, "end": 1165.53, "word": " زائد", "probability": 0.9597981770833334}, {"start": 1165.53, "end": 1166.33, "word": " a5", "probability": 0.992919921875}, {"start": 1166.33, "end": 1167.11, "word": " زائد", "probability": 0.9724934895833334}, {"start": 1167.11, "end": 1167.93, "word": " a6", "probability": 0.995849609375}, {"start": 1167.93, "end": 1168.73, "word": " زائد", "probability": 0.9772135416666666}, {"start": 1168.73, "end": 1169.65, "word": " a7", "probability": 0.9970703125}, {"start": 1169.65, "end": 1170.41, "word": " زائد", "probability": 0.9842122395833334}, {"start": 1170.41, "end": 1171.15, "word": " a8", "probability": 0.98046875}], "temperature": 1.0}, {"id": 39, "seek": 120395, "start": 1175.39, "end": 1203.95, "text": "مع بعض زائد و هكذا إلى اتنين أسكت زائد اتنين أسكت زائد واحد زائد و هكذا إلى اتنين أسكت زائد واحد سالب واحد", "tokens": [2304, 3615, 45030, 11242, 30767, 16373, 3215, 4032, 8032, 4117, 15730, 30731, 1975, 2655, 1863, 9957, 5551, 3794, 4117, 2655, 30767, 16373, 3215, 1975, 2655, 1863, 9957, 5551, 3794, 4117, 2655, 30767, 16373, 3215, 36764, 24401, 30767, 16373, 3215, 4032, 8032, 4117, 15730, 30731, 1975, 2655, 1863, 9957, 5551, 3794, 4117, 2655, 30767, 16373, 3215, 36764, 24401, 8608, 6027, 3555, 36764, 24401], "avg_logprob": -0.10869295351088994, "compression_ratio": 2.2093023255813953, "no_speech_prob": 0.0, "words": [{"start": 1175.39, "end": 1175.83, "word": "مع", "probability": 0.884033203125}, {"start": 1175.83, "end": 1176.37, "word": " بعض", "probability": 0.99365234375}, {"start": 1176.37, "end": 1179.05, "word": " زائد", "probability": 0.7090657552083334}, {"start": 1179.05, "end": 1179.23, "word": " و", "probability": 0.61328125}, {"start": 1179.23, "end": 1179.97, "word": " هكذا", "probability": 0.8435872395833334}, {"start": 1179.97, "end": 1180.93, "word": " إلى", "probability": 0.64990234375}, {"start": 1180.93, "end": 1187.67, "word": " اتنين", "probability": 0.791015625}, {"start": 1187.67, "end": 1188.47, "word": " أسكت", "probability": 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1494.87, "text": "by monotone convergence theorem السيريز هذي convergence ومجموعة بساول limit ل sequence of partial sums اللي هي أصغر من أو ساول عددها okay إذا نسمي المتباينة هذه اتنين إذا من المتباينة واحد واتنين", "tokens": [2322, 1108, 310, 546, 32181, 20904, 21136, 13546, 1829, 11622, 8032, 8848, 1829, 32181, 4032, 2304, 7435, 2304, 2407, 27884, 4724, 3794, 995, 12610, 4948, 5296, 8310, 295, 14641, 34499, 13672, 1829, 39896, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 12610, 6225, 3215, 3215, 11296, 1392, 11933, 15730, 8717, 38251, 1829, 9673, 2655, 3555, 995, 9957, 3660, 29538, 1975, 2655, 1863, 9957, 11933, 15730, 9154, 9673, 2655, 3555, 995, 9957, 3660, 36764, 24401, 4032, 9307, 1863, 9957], "avg_logprob": -0.25573576402060594, "compression_ratio": 1.5210526315789474, "no_speech_prob": 0.0, "words": [{"start": 1468.87, "end": 1469.19, "word": "by", "probability": 0.2020263671875}, {"start": 1469.19, "end": 1469.73, "word": " monotone", "probability": 0.79052734375}, 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"probability": 0.91845703125}, {"start": 1479.93, "end": 1480.33, "word": " sums", "probability": 0.97314453125}, {"start": 1480.33, "end": 1480.53, "word": " اللي", "probability": 0.779052734375}, {"start": 1480.53, "end": 1480.97, "word": " هي", "probability": 0.8046875}, {"start": 1480.97, "end": 1482.51, "word": " أصغر", "probability": 0.9478759765625}, {"start": 1482.51, "end": 1482.67, "word": " من", "probability": 0.90966796875}, {"start": 1482.67, "end": 1482.87, "word": " أو", "probability": 0.5126953125}, {"start": 1482.87, "end": 1483.27, "word": " ساول", "probability": 0.9055989583333334}, {"start": 1483.27, "end": 1483.87, "word": " عددها", "probability": 0.9385986328125}, {"start": 1483.87, "end": 1484.71, "word": " okay", "probability": 0.42626953125}, {"start": 1484.71, "end": 1485.77, "word": " إذا", "probability": 0.713623046875}, {"start": 1485.77, "end": 1486.83, "word": " نسمي", "probability": 0.8668619791666666}, {"start": 1486.83, "end": 1487.67, "word": " المتباينة", "probability": 0.9466145833333334}, {"start": 1487.67, "end": 1488.15, "word": " هذه", "probability": 0.4130859375}, {"start": 1488.15, "end": 1488.87, "word": " اتنين", "probability": 0.8319091796875}, {"start": 1488.87, "end": 1490.97, "word": " إذا", "probability": 0.630859375}, {"start": 1490.97, "end": 1491.71, "word": " من", "probability": 0.98828125}, {"start": 1491.71, "end": 1493.03, "word": " المتباينة", "probability": 0.98828125}, {"start": 1493.03, "end": 1494.17, "word": " واحد", "probability": 0.972900390625}, {"start": 1494.17, "end": 1494.87, "word": " واتنين", "probability": 0.837890625}], "temperature": 1.0}, {"id": 50, "seek": 152495, "start": 1511.87, "end": 1524.95, "text": "الان بمقارنة مباشرة الاختلافات المتباينات واحدة و اتنين بيقدوا", "tokens": [6027, 7649, 4724, 2304, 4587, 9640, 1863, 3660, 3714, 3555, 33599, 25720, 2423, 47283, 2655, 15040, 5172, 9307, 9673, 2655, 3555, 995, 9957, 9307, 36764, 24401, 3660, 4032, 1975, 2655, 1863, 9957, 4724, 1829, 28543, 14407], "avg_logprob": -0.506334485234441, "compression_ratio": 1.380952380952381, "no_speech_prob": 0.0, "words": [{"start": 1511.87, "end": 1513.27, "word": "الان", "probability": 0.656494140625}, {"start": 1513.27, "end": 1514.67, "word": " بمقارنة", "probability": 0.5627034505208334}, {"start": 1514.67, "end": 1514.67, "word": " مباشرة", "probability": 0.6322021484375}, {"start": 1514.67, "end": 1519.13, "word": " الاختلافات", "probability": 0.6164449055989584}, {"start": 1519.13, "end": 1521.33, "word": " المتباينات", "probability": 0.8240559895833334}, {"start": 1521.33, "end": 1522.03, "word": " واحدة", "probability": 0.6936848958333334}, {"start": 1522.03, "end": 1522.45, "word": " و", "probability": 0.74951171875}, {"start": 1522.45, "end": 1524.19, "word": " اتنين", "probability": 0.7679443359375}, {"start": 1524.19, "end": 1524.95, "word": " بيقدوا", "probability": 0.65521240234375}], "temperature": 1.0}, {"id": 51, "seek": 155578, "start": 1526.92, "end": 1555.78, "text": "السيريز sigma a n converges if and only if السيريز sigma اثنين اثنين a اثنين اثنين converges تعالى نشوف لو كانت السيريز هذه convergent فالسيريز هذه convergent", "tokens": [6027, 3794, 13546, 1829, 11622, 12771, 257, 297, 9652, 2880, 498, 293, 787, 498, 2423, 3794, 13546, 1829, 11622, 12771, 1975, 12984, 1863, 9957, 1975, 12984, 1863, 9957, 257, 1975, 12984, 1863, 9957, 1975, 12984, 1863, 9957, 9652, 2880, 37279, 6027, 7578, 8717, 8592, 38688, 45164, 25961, 2655, 21136, 13546, 1829, 11622, 29538, 9652, 6930, 6156, 6027, 3794, 13546, 1829, 11622, 29538, 9652, 6930], "avg_logprob": -0.28870192307692305, "compression_ratio": 1.8688524590163935, "no_speech_prob": 0.0, "words": [{"start": 1526.9199999999998, "end": 1528.32, "word": "السيريز", "probability": 0.645556640625}, {"start": 1528.32, "end": 1529.72, "word": " sigma", "probability": 0.260986328125}, {"start": 1529.72, "end": 1530.14, "word": " a", "probability": 0.350341796875}, {"start": 1530.14, "end": 1530.64, "word": " n", "probability": 0.5537109375}, {"start": 1530.64, "end": 1531.98, "word": " converges", "probability": 0.60595703125}, {"start": 1531.98, "end": 1533.48, "word": " if", "probability": 0.51904296875}, {"start": 1533.48, "end": 1533.78, "word": " and", "probability": 0.92333984375}, {"start": 1533.78, "end": 1534.1, "word": " only", "probability": 0.896484375}, {"start": 1534.1, "end": 1534.54, "word": " if", "probability": 0.9814453125}, {"start": 1534.54, "end": 1536.16, "word": " السيريز", "probability": 0.824609375}, {"start": 1536.16, "end": 1536.78, "word": " sigma", "probability": 0.8642578125}, {"start": 1536.78, "end": 1538.92, "word": " اثنين", "probability": 0.7100830078125}, {"start": 1538.92, "end": 1539.72, "word": " اثنين", "probability": 0.61279296875}, {"start": 1539.72, "end": 1540.06, "word": " a", "probability": 0.658203125}, {"start": 1540.06, "end": 1540.76, "word": " اثنين", "probability": 0.9112548828125}, {"start": 1540.76, "end": 1542.26, "word": " اثنين", "probability": 0.9473876953125}, {"start": 1542.26, "end": 1544.18, "word": " converges", "probability": 0.6865234375}, {"start": 1544.18, "end": 1547.82, "word": " تعالى", "probability": 0.8474934895833334}, {"start": 1547.82, "end": 1548.22, "word": " نشوف", "probability": 0.9943033854166666}, {"start": 1548.22, "end": 1548.42, "word": " لو", "probability": 0.8203125}, {"start": 1548.42, "end": 1548.86, "word": " كانت", "probability": 0.973388671875}, {"start": 1548.86, "end": 1549.48, "word": " السيريز", "probability": 0.961181640625}, {"start": 1549.48, "end": 1549.78, "word": " هذه", "probability": 0.31005859375}, {"start": 1549.78, "end": 1550.6, "word": " convergent", "probability": 0.88134765625}, {"start": 1550.6, "end": 1554.68, "word": " فالسيريز", "probability": 0.9308268229166666}, {"start": 1554.68, "end": 1554.96, "word": " هذه", "probability": 0.80859375}, {"start": 1554.96, "end": 1555.78, "word": " convergent", "probability": 0.904296875}], "temperature": 1.0}, {"id": 52, "seek": 158444, "start": 1558.08, "end": 1584.44, "text": "وبالتالي طبعا أن هذا صحيح لكل M بالمناسبة بقدر أن هذه أيضا sequence of partial sums هذه ال limit تبعتها exist وبالتالي ال infinite series هذه إذا أن ال ممكن نقول أن هذا الكلام صحيح", "tokens": [37746, 6027, 2655, 6027, 1829, 23032, 3555, 3615, 995, 14739, 23758, 20328, 5016, 1829, 5016, 5296, 28820, 376, 20666, 2304, 8315, 35457, 3660, 4724, 28543, 2288, 14739, 29538, 36632, 11242, 995, 8310, 295, 14641, 34499, 29538, 2423, 4948, 6055, 3555, 34268, 11296, 2514, 46599, 6027, 2655, 6027, 1829, 2423, 13785, 2638, 29538, 11933, 15730, 14739, 2423, 3714, 43020, 8717, 39648, 14739, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016], "avg_logprob": -0.27611606802259175, "compression_ratio": 1.5730337078651686, "no_speech_prob": 0.0, "words": [{"start": 1558.08, "end": 1558.84, "word": "وبالتالي", "probability": 0.836767578125}, {"start": 1558.84, "end": 1559.24, "word": " طبعا", "probability": 0.9171142578125}, {"start": 1559.24, "end": 1559.38, "word": " أن", "probability": 0.11846923828125}, {"start": 1559.38, "end": 1559.56, "word": " هذا", "probability": 0.90087890625}, {"start": 1559.56, "end": 1559.98, "word": " صحيح", "probability": 0.990478515625}, {"start": 1559.98, "end": 1560.52, "word": " لكل", "probability": 0.967041015625}, {"start": 1560.52, "end": 1560.78, "word": " M", "probability": 0.58349609375}, {"start": 1560.78, "end": 1562.6, "word": " بالمناسبة", "probability": 0.916015625}, {"start": 1562.6, "end": 1563.32, "word": " بقدر", "probability": 0.6993815104166666}, {"start": 1563.32, "end": 1563.5, "word": " أن", "probability": 0.383544921875}, {"start": 1563.5, "end": 1564.08, "word": " هذه", "probability": 0.64208984375}, {"start": 1564.08, "end": 1566.5, "word": " أيضا", "probability": 0.810546875}, {"start": 1566.5, "end": 1567.24, "word": " sequence", "probability": 0.30029296875}, {"start": 1567.24, "end": 1567.66, "word": " of", "probability": 0.9052734375}, {"start": 1567.66, "end": 1567.98, "word": " partial", "probability": 0.96826171875}, {"start": 1567.98, "end": 1568.36, "word": " sums", "probability": 0.935546875}, {"start": 1568.36, "end": 1568.58, "word": " هذه", "probability": 0.6533203125}, {"start": 1568.58, "end": 1568.68, "word": " ال", "probability": 0.33837890625}, {"start": 1568.68, "end": 1568.88, "word": " limit", "probability": 0.64013671875}, {"start": 1568.88, "end": 1569.58, "word": " تبعتها", "probability": 0.859619140625}, {"start": 1569.58, "end": 1570.24, "word": " exist", "probability": 0.69140625}, {"start": 1570.24, "end": 1571.36, "word": " وبالتالي", "probability": 0.8859375}, {"start": 1571.36, "end": 1571.64, "word": " ال", "probability": 0.8095703125}, {"start": 1571.64, "end": 1571.9, "word": " infinite", "probability": 0.399658203125}, {"start": 1571.9, "end": 1572.36, "word": " series", "probability": 0.9306640625}, {"start": 1572.36, "end": 1573.16, "word": " هذه", "probability": 0.8857421875}, {"start": 1573.16, "end": 1579.46, "word": " إذا", "probability": 0.5277099609375}, {"start": 1579.46, "end": 1579.76, "word": " أن", "probability": 0.5107421875}, {"start": 1579.76, "end": 1580.06, "word": " ال", "probability": 0.71533203125}, {"start": 1580.06, "end": 1582.78, "word": " ممكن", "probability": 0.7154541015625}, {"start": 1582.78, "end": 1583.08, "word": " نقول", "probability": 0.947509765625}, {"start": 1583.08, "end": 1583.24, "word": " أن", "probability": 0.57568359375}, {"start": 1583.24, "end": 1583.48, "word": " هذا", "probability": 0.96142578125}, {"start": 1583.48, "end": 1583.88, "word": " الكلام", "probability": 0.9075520833333334}, {"start": 1583.88, "end": 1584.44, "word": " صحيح", "probability": 0.9949951171875}], "temperature": 1.0}, {"id": 53, "seek": 160167, "start": 1585.69, "end": 1601.67, "text": "الان لو كانت ال series هادي convergent فنضربها في ثابت اتنين تطلع convergent وبالتالي ال series هادي convergent by direct comparison test العكس لو كانت ال series هادي convergent", "tokens": [6027, 7649, 45164, 25961, 2655, 2423, 2638, 8032, 995, 16254, 9652, 6930, 6156, 1863, 11242, 25513, 11296, 8978, 38637, 16758, 2655, 1975, 2655, 1863, 9957, 6055, 9566, 1211, 3615, 9652, 6930, 46599, 6027, 2655, 6027, 1829, 2423, 2638, 8032, 995, 16254, 9652, 6930, 538, 2047, 9660, 1500, 18863, 4117, 3794, 45164, 25961, 2655, 2423, 2638, 8032, 995, 16254, 9652, 6930], "avg_logprob": -0.1900614773640867, "compression_ratio": 1.7571428571428571, "no_speech_prob": 0.0, "words": [{"start": 1585.69, "end": 1586.53, "word": "الان", "probability": 0.6207275390625}, {"start": 1586.53, "end": 1586.73, "word": " لو", "probability": 0.8671875}, {"start": 1586.73, "end": 1587.11, "word": " كانت", "probability": 0.97314453125}, {"start": 1587.11, "end": 1587.25, "word": " ال", "probability": 0.78955078125}, {"start": 1587.25, "end": 1587.57, "word": " series", "probability": 0.83154296875}, {"start": 1587.57, "end": 1587.87, "word": " هادي", "probability": 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{"id": 54, "seek": 163380, "start": 1604.46, "end": 1633.8, "text": "فلما أضفلها حد عدد موجب بيبقى conversion وبالتالي by direct comparison test ال series الأصغر بتطلع conversion okay تمام؟ لأن هذا بثبت koshi condensation test هذا ال test قوي كتير ويله فوائد كتيرة فمن الفوائد تبعته يعني هذه مثال", "tokens": [5172, 1211, 15042, 5551, 11242, 5172, 1211, 11296, 11331, 3215, 6225, 3215, 3215, 3714, 29245, 3555, 4724, 1829, 3555, 4587, 7578, 14298, 46599, 6027, 2655, 6027, 1829, 538, 2047, 9660, 1500, 2423, 2638, 16247, 9381, 17082, 2288, 39894, 9566, 1211, 3615, 14298, 1392, 46811, 10943, 22807, 5296, 33456, 23758, 4724, 12984, 3555, 2655, 350, 17392, 2224, 35292, 1500, 23758, 2423, 1500, 12174, 45865, 9122, 2655, 13546, 4032, 26895, 3224, 6156, 2407, 16373, 3215, 9122, 2655, 48923, 6156, 27842, 27188, 2407, 16373, 3215, 6055, 3555, 34268, 3224, 37495, 22653, 29538, 50113, 6027], "avg_logprob": -0.27972147743339126, "compression_ratio": 1.527027027027027, "no_speech_prob": 0.0, "words": 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"start": 1673.04, "end": 1701.68, "text": "تعملين تلتاش سيكشن تلاتة سبعة ايش بيقول هذا if if P أكبر من السفر is a real number discuss the convergence", "tokens": [2655, 25957, 1211, 9957, 6055, 1211, 2655, 33599, 8608, 1829, 4117, 8592, 1863, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 1975, 1829, 8592, 4724, 1829, 39648, 23758, 498, 498, 430, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 307, 257, 957, 1230, 2248, 264, 32181], "avg_logprob": -0.310416669315762, "compression_ratio": 1.16793893129771, "no_speech_prob": 0.0, "words": [{"start": 1673.04, "end": 1673.76, "word": "تعملين", "probability": 0.672454833984375}, {"start": 1673.76, "end": 1674.5, "word": " تلتاش", "probability": 0.72479248046875}, {"start": 1674.5, "end": 1675.06, "word": " سيكشن", "probability": 0.67041015625}, {"start": 1675.06, "end": 1675.68, "word": " تلاتة", "probability": 0.8145751953125}, {"start": 1675.68, "end": 1676.22, "word": " سبعة", "probability": 0.94189453125}, {"start": 1676.22, "end": 1678.54, 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"word": " to", "probability": 0.921875}, {"start": 1803.81, "end": 1804.35, "word": " infinity", "probability": 0.8671875}, {"start": 1804.35, "end": 1806.93, "word": " لاتنين", "probability": 0.77392578125}, {"start": 1806.93, "end": 1807.83, "word": " اص", "probability": 0.964111328125}, {"start": 1807.83, "end": 1809.63, "word": " واحد", "probability": 0.97998046875}, {"start": 1809.63, "end": 1810.23, "word": " minus", "probability": 0.96240234375}, {"start": 1810.23, "end": 1810.93, "word": " P", "probability": 0.9072265625}], "temperature": 1.0}, {"id": 61, "seek": 184062, "start": 1812.42, "end": 1840.62, "text": "الكل أسئلة وهدي is a geometric series is a geometric series وبالتالي مظبوط هذا عبارة عن geometric series لو بدى أكتب حدود تبعتها", "tokens": [6027, 28820, 5551, 3794, 19986, 37977, 37037, 16254, 307, 257, 33246, 2638, 307, 257, 33246, 2638, 46599, 6027, 2655, 6027, 1829, 3714, 19913, 3555, 2407, 9566, 23758, 6225, 3555, 9640, 3660, 18871, 33246, 2638, 45164, 47525, 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0.8857421875}, {"start": 1824.82, "end": 1827.02, "word": " series", "probability": 0.93310546875}, {"start": 1827.02, "end": 1833.68, "word": " وبالتالي", "probability": 0.8724609375}, {"start": 1833.68, "end": 1834.88, "word": " مظبوط", "probability": 0.78896484375}, {"start": 1834.88, "end": 1835.96, "word": " هذا", "probability": 0.483154296875}, {"start": 1835.96, "end": 1836.24, "word": " عبارة", "probability": 0.85992431640625}, {"start": 1836.24, "end": 1836.42, "word": " عن", "probability": 0.990234375}, {"start": 1836.42, "end": 1836.8, "word": " geometric", "probability": 0.744140625}, {"start": 1836.8, "end": 1837.34, "word": " series", "probability": 0.92822265625}, {"start": 1837.34, "end": 1837.54, "word": " لو", "probability": 0.86328125}, {"start": 1837.54, "end": 1837.82, "word": " بدى", "probability": 0.56591796875}, {"start": 1837.82, "end": 1838.32, "word": " أكتب", "probability": 0.889892578125}, {"start": 1838.32, "end": 1839.92, "word": " حدود", "probability": 0.9420572916666666}, {"start": 1839.92, "end": 1840.62, "word": " تبعتها", "probability": 0.930419921875}], "temperature": 1.0}, {"id": 62, "seek": 186998, "start": 1843.1, "end": 1869.98, "text": "فاول حد عبارة عن اتنين اص واحد minus P الحد التاني اتنين اص واحد minus P الكل تربية و هكذا فالحد الاول اتنين اص واحد minus P الحد التاني اتنين اص واحد minus P و هكذا with ratio", "tokens": [5172, 995, 12610, 11331, 3215, 6225, 3555, 9640, 3660, 18871, 1975, 2655, 1863, 9957, 1975, 9381, 36764, 24401, 3175, 430, 21542, 3215, 16712, 7649, 1829, 1975, 2655, 1863, 9957, 1975, 9381, 36764, 24401, 3175, 430, 2423, 28820, 6055, 25513, 10632, 4032, 8032, 4117, 15730, 6156, 6027, 24401, 42963, 12610, 1975, 2655, 1863, 9957, 1975, 9381, 36764, 24401, 3175, 430, 21542, 3215, 16712, 7649, 1829, 1975, 2655, 1863, 9957, 1975, 9381, 36764, 24401, 3175, 430, 4032, 8032, 4117, 15730, 365, 8509], "avg_logprob": -0.17746913580246915, "compression_ratio": 2.256, "no_speech_prob": 0.0, "words": [{"start": 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"end": 1862.36, "word": " واحد", "probability": 0.9951171875}, {"start": 1862.36, "end": 1862.7, "word": " minus", "probability": 0.99267578125}, {"start": 1862.7, "end": 1863.18, "word": " P", "probability": 0.9853515625}, {"start": 1863.18, "end": 1864.48, "word": " الحد", "probability": 0.947509765625}, {"start": 1864.48, "end": 1864.88, "word": " التاني", "probability": 0.98876953125}, {"start": 1864.88, "end": 1865.22, "word": " اتنين", "probability": 0.9910888671875}, {"start": 1865.22, "end": 1865.48, "word": " اص", "probability": 0.976318359375}, {"start": 1865.48, "end": 1865.74, "word": " واحد", "probability": 0.99755859375}, {"start": 1865.74, "end": 1866.12, "word": " minus", "probability": 0.99365234375}, {"start": 1866.12, "end": 1866.46, "word": " P", "probability": 0.98876953125}, {"start": 1866.46, "end": 1866.62, "word": " و", "probability": 0.8779296875}, {"start": 1866.62, "end": 1867.1, "word": " هكذا", "probability": 0.9475911458333334}, {"start": 1867.1, "end": 1868.4, "word": " with", "probability": 0.48486328125}, {"start": 1868.4, "end": 1869.98, "word": " ratio", "probability": 0.96435546875}], "temperature": 1.0}, {"id": 63, "seek": 190179, "start": 1874.09, "end": 1901.79, "text": "with ratio with ratio R بساوي اتنين اص واحد minus P", "tokens": [11820, 8509, 365, 8509, 497, 4724, 3794, 995, 45865, 1975, 2655, 1863, 9957, 1975, 9381, 36764, 24401, 3175, 430], "avg_logprob": -0.2781249865889549, "compression_ratio": 1.0307692307692307, "no_speech_prob": 0.0, "words": [{"start": 1874.09, "end": 1875.49, "word": "with", "probability": 0.65625}, {"start": 1875.49, "end": 1876.89, "word": " ratio", "probability": 0.97265625}, {"start": 1876.89, "end": 1888.71, "word": " with", "probability": 0.58642578125}, {"start": 1888.71, "end": 1889.61, "word": " ratio", "probability": 0.98193359375}, {"start": 1889.61, "end": 1894.83, "word": " R", "probability": 0.296630859375}, {"start": 1894.83, "end": 1895.83, "word": " بساوي", "probability": 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1919.39, "end": 1919.75, "word": " if", "probability": 0.9619140625}, {"start": 1919.75, "end": 1920.47, "word": " absolute", "probability": 0.78564453125}, {"start": 1920.47, "end": 1921.35, "word": " R", "probability": 0.52001953125}, {"start": 1921.35, "end": 1922.45, "word": " بيساوي", "probability": 0.67493896484375}, {"start": 1922.45, "end": 1924.65, "word": " اتنين", "probability": 0.73956298828125}, {"start": 1924.65, "end": 1925.05, "word": " أس", "probability": 0.439453125}, {"start": 1925.05, "end": 1925.57, "word": " واحد", "probability": 0.956787109375}, {"start": 1925.57, "end": 1925.95, "word": " minus", "probability": 0.91845703125}, {"start": 1925.95, "end": 1926.45, "word": " P", "probability": 0.91845703125}, {"start": 1926.45, "end": 1928.55, "word": " أصغر", "probability": 0.9307861328125}, {"start": 1928.55, "end": 1928.75, "word": " من", "probability": 0.9951171875}, {"start": 1928.75, "end": 1929.23, "word": " واحد", "probability": 0.98388671875}, {"start": 1929.23, "end": 1932.11, "word": " وهذا", "probability": 0.7119140625}, {"start": 1932.11, "end": 1933.11, "word": " بتحقق", "probability": 0.81072998046875}, {"start": 1933.11, "end": 1934.57, "word": " اتنين", "probability": 0.9378662109375}, {"start": 1934.57, "end": 1934.85, "word": " أس", "probability": 0.911865234375}, {"start": 1934.85, "end": 1935.25, "word": " واحد", "probability": 0.985107421875}, {"start": 1935.25, "end": 1935.65, "word": " minus", "probability": 0.990234375}, {"start": 1935.65, "end": 1936.09, "word": " P", "probability": 0.984375}, {"start": 1936.09, "end": 1936.67, "word": " أصغر", "probability": 0.964111328125}, {"start": 1936.67, "end": 1936.83, "word": " من", "probability": 0.99658203125}, {"start": 1936.83, "end": 1937.29, "word": " واحد", "probability": 0.994873046875}], "temperature": 1.0}, {"id": 65, "seek": 196275, "start": 1938.67, "end": 1962.75, "text": "فنقول if واحد minus P اذا كان واحد minus P أصغر من السفر سالم لأن لو كان واحد minus P موجب فاتنين أس أي عدد موجب عمره ما بيكون أصغر من واحد", "tokens": [5172, 1863, 39648, 498, 36764, 24401, 3175, 430, 1975, 15730, 25961, 36764, 24401, 3175, 430, 5551, 9381, 17082, 2288, 9154, 21136, 5172, 2288, 8608, 45340, 5296, 33456, 45164, 25961, 36764, 24401, 3175, 430, 3714, 29245, 3555, 6156, 9307, 1863, 9957, 5551, 3794, 36632, 6225, 3215, 3215, 3714, 29245, 3555, 6225, 29973, 3224, 19446, 4724, 1829, 30544, 5551, 9381, 17082, 2288, 9154, 36764, 24401], "avg_logprob": -0.23449707054533064, "compression_ratio": 1.6330935251798562, "no_speech_prob": 0.0, "words": [{"start": 1938.67, "end": 1939.53, "word": "فنقول", "probability": 0.6533203125}, {"start": 1939.53, "end": 1939.87, "word": " if", "probability": 0.11798095703125}, {"start": 1939.87, "end": 1940.49, "word": " واحد", "probability": 0.580078125}, {"start": 1940.49, "end": 1940.93, "word": " minus", "probability": 0.85546875}, {"start": 1940.93, "end": 1941.49, "word": " P", "probability": 0.6201171875}, {"start": 1941.49, 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{"start": 2078.78, "end": 2079.18, "word": " شيء", "probability": 0.90966796875}, {"start": 2079.18, "end": 2079.28, "word": " في", "probability": 0.9462890625}, {"start": 2079.28, "end": 2079.64, "word": " الأسئلة", "probability": 0.9814453125}, {"start": 2079.64, "end": 2080.28, "word": " دي", "probability": 0.60009765625}, {"start": 2080.28, "end": 2081.34, "word": " أو", "probability": 0.5703125}, {"start": 2081.34, "end": 2081.78, "word": " أسئلة", "probability": 0.9649658203125}, {"start": 2081.78, "end": 2087.16, "word": " تانية؟", "probability": 0.8970947265625}, {"start": 2087.16, "end": 2096.5, "word": " في", "probability": 0.56787109375}, {"start": 2096.5, "end": 2096.88, "word": " عندكم", "probability": 0.9541015625}, {"start": 2096.88, "end": 2097.18, "word": " أي", "probability": 0.8623046875}, {"start": 2097.18, "end": 2098.18, "word": " أسئلة؟", "probability": 0.9826171875}], "temperature": 1.0}, {"id": 71, "seek": 213556, "start": 2113.0, "end": 2135.56, "text": "إذا سيكشن واحد تلاتة سبعة في أي سؤال تاني عندكم في الأسئلة هذه أو السيكاشن السابقة أو سيكشن أربعة واحد إذا بتحبه سيكشن أربعة واحد", "tokens": [28814, 15730, 8608, 1829, 4117, 8592, 1863, 36764, 24401, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 8978, 36632, 8608, 33604, 6027, 6055, 7649, 1829, 43242, 24793, 8978, 16247, 3794, 19986, 37977, 29538, 34051, 21136, 1829, 4117, 33599, 1863, 21136, 16758, 28671, 34051, 8608, 1829, 4117, 8592, 1863, 5551, 25513, 27884, 36764, 24401, 11933, 15730, 39894, 5016, 3555, 3224, 8608, 1829, 4117, 8592, 1863, 5551, 25513, 27884, 36764, 24401], "avg_logprob": -0.21229620169902194, "compression_ratio": 1.8425196850393701, "no_speech_prob": 0.0, "words": [{"start": 2113.0, "end": 2113.44, "word": "إذا", "probability": 0.51287841796875}, {"start": 2113.44, "end": 2114.58, "word": " سيكشن", "probability": 0.68984375}, {"start": 2114.58, "end": 2115.5, "word": " واحد", "probability": 0.6873779296875}, {"start": 2115.5, "end": 2115.98, "word": " تلاتة", "probability": 0.8577880859375}, {"start": 2115.98, "end": 2116.36, "word": " سبعة", "probability": 0.82666015625}, {"start": 2116.36, "end": 2116.56, "word": " في", "probability": 0.410400390625}, {"start": 2116.56, "end": 2116.8, "word": " أي", "probability": 0.7578125}, {"start": 2116.8, "end": 2117.34, "word": " سؤال", "probability": 0.9755859375}, {"start": 2117.34, "end": 2118.9, "word": " تاني", "probability": 0.853515625}, {"start": 2118.9, "end": 2119.4, "word": " عندكم", "probability": 0.94580078125}, {"start": 2119.4, "end": 2119.56, "word": " في", "probability": 0.95263671875}, {"start": 2119.56, "end": 2119.98, "word": " الأسئلة", "probability": 0.786865234375}, {"start": 2119.98, "end": 2120.38, "word": " هذه", "probability": 0.78662109375}, {"start": 2120.38, "end": 2125.54, "word": " أو", "probability": 0.77734375}, {"start": 2125.54, "end": 2126.2, "word": " السيكاشن", "probability": 0.742578125}, {"start": 2126.2, "end": 2126.94, "word": " السابقة", "probability": 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"end": 2172.95, "word": " نواصل", "probability": 0.8997802734375}, {"start": 2172.95, "end": 2173.07, "word": " ..", "probability": 0.59912109375}, {"start": 2173.07, "end": 2176.19, "word": " نكمل", "probability": 0.99462890625}, {"start": 2176.19, "end": 2176.75, "word": " المحاضرة", "probability": 0.8189697265625}, {"start": 2176.75, "end": 2176.95, "word": " في", "probability": 0.4658203125}, {"start": 2176.95, "end": 2177.49, "word": " السابقة", "probability": 0.98974609375}], "temperature": 1.0}, {"id": 73, "seek": 222767, "start": 2209.09, "end": 2227.67, "text": "المرة الأخرى اتحدثنا عن ال two-sided limits و عن ال one-sided limits و أخدنا بعض النظريات و قلنا إن جميع النظريات اللي برهنناها هو one-sided limit صحيحة لل two-sided limits", "tokens": [45340, 25720, 16247, 34740, 7578, 1975, 2655, 24401, 12984, 8315, 18871, 2423, 732, 12, 30941, 10406, 4032, 18871, 2423, 472, 12, 30941, 10406, 4032, 5551, 9778, 3215, 8315, 45030, 11242, 28239, 19913, 16572, 9307, 4032, 12174, 1211, 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"end": 2213.03, "word": " و", "probability": 0.97509765625}, {"start": 2213.03, "end": 2213.25, "word": " عن", "probability": 0.796875}, {"start": 2213.25, "end": 2213.47, "word": " ال", "probability": 0.99169921875}, {"start": 2213.47, "end": 2213.65, "word": " one", "probability": 0.96826171875}, {"start": 2213.65, "end": 2214.01, "word": "-sided", "probability": 0.9501953125}, {"start": 2214.01, "end": 2214.61, "word": " limits", "probability": 0.97802734375}, {"start": 2214.61, "end": 2215.93, "word": " و", "probability": 0.76025390625}, {"start": 2215.93, "end": 2216.65, "word": " أخدنا", "probability": 0.91455078125}, {"start": 2216.65, "end": 2217.01, "word": " بعض", "probability": 0.9873046875}, {"start": 2217.01, "end": 2217.85, "word": " النظريات", "probability": 0.97265625}, {"start": 2217.85, "end": 2219.81, "word": " و", "probability": 0.689453125}, {"start": 2219.81, "end": 2220.13, "word": " قلنا", "probability": 0.8375651041666666}, {"start": 2220.13, "end": 2220.35, 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واحد ال limit من اليسار يساوي سالب واحد مش متساوي اتين so by theorem حسب النظرية اللي أخدناها theorem اربعة تلاتة تلاتة بيطلع عندي ال limit او ال two sided limit", "tokens": [6027, 3794, 6027, 3555, 36764, 24401, 4724, 1829, 9566, 1211, 3615, 21136, 6027, 3555, 36764, 24401, 16472, 1975, 8315, 18871, 16254, 2423, 4948, 9154, 45595, 10943, 9957, 7251, 3794, 995, 45865, 36764, 24401, 2423, 4948, 9154, 45595, 3794, 9640, 7251, 3794, 995, 45865, 8608, 6027, 3555, 36764, 24401, 37893, 44650, 3794, 995, 45865, 1975, 2655, 9957, 370, 538, 20904, 11331, 35457, 28239, 19913, 2288, 10632, 13672, 1829, 5551, 9778, 3215, 8315, 11296, 20904, 1975, 25513, 27884, 6055, 1211, 9307, 3660, 6055, 1211, 9307, 3660, 4724, 1829, 9566, 1211, 3615, 18871, 16254, 2423, 4948, 1975, 2407, 2423, 732, 41651, 4948], "avg_logprob": -0.2410937462747097, "compression_ratio": 2.0614525139664805, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2370.29, "end": 2370.89, "word": "السالب", "probability": 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so", "probability": 0.2437744140625}, {"start": 2383.65, "end": 2384.23, "word": " by", "probability": 0.86572265625}, {"start": 2384.23, "end": 2384.83, "word": " theorem", "probability": 0.845703125}, {"start": 2384.83, "end": 2386.59, "word": " حسب", "probability": 0.95068359375}, {"start": 2386.59, "end": 2387.11, "word": " النظرية", "probability": 0.851806640625}, {"start": 2387.11, "end": 2387.25, "word": " اللي", "probability": 0.86865234375}, {"start": 2387.25, "end": 2387.93, "word": " أخدناها", "probability": 0.8865234375}, {"start": 2387.93, "end": 2388.41, "word": " theorem", "probability": 0.78857421875}, {"start": 2388.41, "end": 2390.15, "word": " اربعة", "probability": 0.7596028645833334}, {"start": 2390.15, "end": 2390.77, "word": " تلاتة", "probability": 0.924072265625}, {"start": 2390.77, "end": 2392.13, "word": " تلاتة", "probability": 0.9864501953125}, {"start": 2392.13, "end": 2395.63, "word": " بيطلع", "probability": 0.9380859375}, {"start": 2395.63, "end": 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1211, 11703, 9957, 1829, 1975, 8315, 1975, 9778, 3215, 855, 300, 2423, 4948, 24976, 2445, 308, 16070, 14407, 24401, 15844, 2031], "avg_logprob": -0.44270832273695204, "compression_ratio": 1.2112676056338028, "no_speech_prob": 0.0, "words": [{"start": 2400.4, "end": 2400.78, "word": "للـ", "probability": 0.5323079427083334}, {"start": 2400.78, "end": 2401.08, "word": " signal", "probability": 0.5146484375}, {"start": 2401.08, "end": 2401.72, "word": " function", "probability": 0.82373046875}, {"start": 2401.72, "end": 2402.08, "word": " لما", "probability": 0.72265625}, {"start": 2402.08, "end": 2402.32, "word": " x", "probability": 0.52685546875}, {"start": 2402.32, "end": 2402.64, "word": " تقول", "probability": 0.708984375}, {"start": 2402.64, "end": 2403.28, "word": " السفر", "probability": 0.7350260416666666}, {"start": 2403.28, "end": 2404.1, "word": " does", "probability": 0.168701171875}, {"start": 2404.1, "end": 2404.46, "word": " not", "probability": 0.96435546875}, {"start": 2404.46, "end": 2405.04, "word": " exist", "probability": 0.974609375}, {"start": 2405.04, "end": 2409.56, "word": " تمام؟", "probability": 0.577392578125}, {"start": 2409.56, "end": 2411.7, "word": " طيب", "probability": 0.88232421875}, {"start": 2411.7, "end": 2412.1, "word": " خلّيني", "probability": 0.7275390625}, {"start": 2412.1, "end": 2412.26, "word": " انا", "probability": 0.5521240234375}, {"start": 2412.26, "end": 2414.06, "word": " اخد", "probability": 0.8170572916666666}, {"start": 2414.06, "end": 2422.28, "word": " show", "probability": 0.87109375}, {"start": 2422.28, "end": 2423.0, "word": " that", "probability": 0.8955078125}, {"start": 2423.0, "end": 2427.38, "word": " ال", "probability": 0.64208984375}, {"start": 2427.38, "end": 2427.64, "word": " limit", "probability": 0.865234375}, {"start": 2427.64, "end": 2427.92, "word": " لل", "probability": 0.68017578125}, {"start": 2427.92, "end": 2428.38, "word": " function", "probability": 0.80419921875}, {"start": 2428.38, "end": 2428.78, "word": " e", "probability": 0.3173828125}, {"start": 2428.78, "end": 2429.3, "word": " والواحد", "probability": 0.508697509765625}, {"start": 2429.3, "end": 2429.52, "word": " على", "probability": 0.70361328125}, {"start": 2429.52, "end": 2429.92, "word": " x", "probability": 0.93359375}], "temperature": 1.0}, {"id": 81, "seek": 245871, "start": 2430.91, "end": 2458.71, "text": "لما x تقول إلى سفر من اليمين does not exist and من ال limit لنفس ال function e to واحد على x لما x تقول إلى سفر من اليسار تطلع موجودة و بساوي سفر", "tokens": [1211, 15042, 2031, 6055, 39648, 30731, 8608, 5172, 2288, 9154, 2423, 32640, 9957, 775, 406, 2514, 293, 9154, 2423, 4948, 5296, 1863, 36178, 2423, 2445, 308, 281, 36764, 24401, 15844, 2031, 5296, 15042, 2031, 6055, 39648, 30731, 8608, 5172, 2288, 9154, 45595, 3794, 9640, 6055, 9566, 1211, 3615, 3714, 29245, 23328, 3660, 4032, 4724, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.27049180914144044, "compression_ratio": 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"word": " من", "probability": 0.11248779296875}, {"start": 2443.91, "end": 2444.27, "word": " ال", "probability": 0.69921875}, {"start": 2444.27, "end": 2446.95, "word": " limit", "probability": 0.78857421875}, {"start": 2446.95, "end": 2447.75, "word": " لنفس", "probability": 0.7306315104166666}, {"start": 2447.75, "end": 2447.91, "word": " ال", "probability": 0.9619140625}, {"start": 2447.91, "end": 2448.33, "word": " function", "probability": 0.90869140625}, {"start": 2448.33, "end": 2448.63, "word": " e", "probability": 0.5126953125}, {"start": 2448.63, "end": 2448.83, "word": " to", "probability": 0.473388671875}, {"start": 2448.83, "end": 2449.23, "word": " واحد", "probability": 0.773681640625}, {"start": 2449.23, "end": 2449.41, "word": " على", "probability": 0.802734375}, {"start": 2449.41, "end": 2449.79, "word": " x", "probability": 0.953125}, {"start": 2449.79, "end": 2450.89, "word": " لما", "probability": 0.9609375}, {"start": 2450.89, "end": 2451.17, "word": " x", "probability": 0.970703125}, {"start": 2451.17, "end": 2451.45, "word": " تقول", "probability": 0.985107421875}, {"start": 2451.45, "end": 2451.69, "word": " إلى", "probability": 0.5078125}, {"start": 2451.69, "end": 2452.09, "word": " سفر", "probability": 0.9803059895833334}, {"start": 2452.09, "end": 2452.33, "word": " من", "probability": 0.98974609375}, {"start": 2452.33, "end": 2453.05, "word": " اليسار", "probability": 0.9923502604166666}, {"start": 2453.05, "end": 2454.37, "word": " تطلع", "probability": 0.9468994140625}, {"start": 2454.37, "end": 2456.91, "word": " موجودة", "probability": 0.9896240234375}, {"start": 2456.91, "end": 2457.43, "word": " و", "probability": 0.65283203125}, {"start": 2457.43, "end": 2458.19, "word": " بساوي", "probability": 0.727783203125}, {"start": 2458.19, "end": 2458.71, "word": " سفر", "probability": 0.96826171875}], "temperature": 1.0}, {"id": 82, "seek": 249405, "start": 2483.83, "end": 2494.05, "text": "طيب ال .. نحاول نبرهن الجزء الأول", "tokens": [9566, 1829, 3555, 2423, 4386, 8717, 5016, 995, 12610, 8717, 26890, 3224, 1863, 25724, 11622, 38207, 16247, 12610], "avg_logprob": -0.10444078633659765, "compression_ratio": 1.0545454545454545, "no_speech_prob": 0.0, "words": [{"start": 2483.83, "end": 2484.41, "word": "طيب", "probability": 0.9599609375}, {"start": 2484.41, "end": 2484.77, "word": " ال", "probability": 0.70361328125}, {"start": 2484.77, "end": 2491.01, "word": " ..", "probability": 0.81396484375}, {"start": 2491.01, "end": 2492.33, "word": " نحاول", "probability": 0.8768310546875}, {"start": 2492.33, "end": 2493.19, "word": " نبرهن", "probability": 0.8931884765625}, {"start": 2493.19, "end": 2493.61, "word": " الجزء", "probability": 0.9847005208333334}, {"start": 2493.61, "end": 2494.05, "word": " الأول", "probability": 0.970947265625}], "temperature": 1.0}, {"id": 83, "seek": 254326, "start": 2516.42, "end": 2543.26, "text": "بناخد الجزء الأول let z of x بساوي e to 1 على x حفة x لا تساوي 0 وبدنا نثبت to show ان ال limit", "tokens": [3555, 1863, 47283, 3215, 25724, 11622, 38207, 16247, 12610, 718, 710, 295, 2031, 4724, 3794, 995, 45865, 308, 281, 502, 15844, 2031, 11331, 5172, 3660, 2031, 20193, 6055, 3794, 995, 45865, 1958, 4032, 44510, 8315, 8717, 12984, 3555, 2655, 281, 855, 16472, 2423, 4948], "avg_logprob": -0.38958333333333334, "compression_ratio": 1.165289256198347, "no_speech_prob": 0.0, "words": [{"start": 2516.42, "end": 2517.06, "word": "بناخد", "probability": 0.74871826171875}, {"start": 2517.06, "end": 2517.48, "word": " الجزء", "probability": 0.9285481770833334}, {"start": 2517.48, "end": 2518.08, "word": " الأول", "probability": 0.940185546875}, {"start": 2518.08, "end": 2523.38, "word": " let", "probability": 0.275634765625}, {"start": 2523.38, "end": 2525.16, "word": " z", "probability": 0.3125}, {"start": 2525.16, "end": 2525.4, "word": " of", "probability": 0.576171875}, {"start": 2525.4, "end": 2525.78, "word": " x", "probability": 0.951171875}, {"start": 2525.78, "end": 2526.52, "word": " بساوي", "probability": 0.7236328125}, {"start": 2526.52, "end": 2526.9, "word": " e", "probability": 0.388427734375}, {"start": 2526.9, "end": 2527.1, "word": " to", "probability": 0.254150390625}, {"start": 2527.1, "end": 2527.46, "word": " 1", "probability": 0.37841796875}, {"start": 2527.46, "end": 2527.72, "word": " على", "probability": 0.646484375}, {"start": 2527.72, "end": 2528.16, "word": " x", "probability": 0.91015625}, {"start": 2528.16, "end": 2529.44, "word": " حفة", "probability": 0.3765869140625}, {"start": 2529.44, "end": 2529.7, "word": " x", "probability": 0.84912109375}, {"start": 2529.7, "end": 2529.94, "word": " لا", "probability": 0.55224609375}, {"start": 2529.94, "end": 2530.52, "word": " تساوي", "probability": 0.9566650390625}, {"start": 2530.52, "end": 2530.88, "word": " 0", "probability": 0.52685546875}, {"start": 2530.88, "end": 2533.54, "word": " وبدنا", "probability": 0.5719401041666666}, {"start": 2533.54, "end": 2535.56, "word": " نثبت", "probability": 0.992919921875}, {"start": 2535.56, "end": 2539.26, "word": " to", "probability": 0.7880859375}, {"start": 2539.26, "end": 2540.02, "word": " show", "probability": 0.974609375}, {"start": 2540.02, "end": 2542.7, "word": " ان", "probability": 0.46044921875}, {"start": 2542.7, "end": 2542.88, "word": " ال", "probability": 0.8017578125}, {"start": 2542.88, "end": 2543.26, "word": " limit", "probability": 0.919921875}], "temperature": 1.0}, {"id": 84, "seek": 257265, "start": 2544.65, "end": 2572.65, "text": "لـ g of x لما x تقول لصفر من اليمين does not exist it suffices to show يكفي اثبات ان ال function g of x is not bounded on", "tokens": [1211, 39184, 290, 295, 2031, 5296, 15042, 2031, 6055, 39648, 5296, 9381, 5172, 2288, 9154, 2423, 32640, 9957, 775, 406, 2514, 309, 3889, 1473, 281, 855, 7251, 4117, 41185, 1975, 12984, 3555, 9307, 16472, 2423, 2445, 290, 295, 2031, 307, 406, 37498, 322], "avg_logprob": -0.2356179045005278, "compression_ratio": 1.2301587301587302, 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"probability": 0.96923828125}, {"start": 2551.43, "end": 2552.13, "word": " exist", "probability": 0.962890625}, {"start": 2552.13, "end": 2554.33, "word": " it", "probability": 0.890625}, {"start": 2554.33, "end": 2555.47, "word": " suffices", "probability": 0.9169921875}, {"start": 2555.47, "end": 2558.75, "word": " to", "probability": 0.92822265625}, {"start": 2558.75, "end": 2559.37, "word": " show", "probability": 0.95263671875}, {"start": 2559.37, "end": 2562.71, "word": " يكفي", "probability": 0.8819986979166666}, {"start": 2562.71, "end": 2563.27, "word": " اثبات", "probability": 0.89013671875}, {"start": 2563.27, "end": 2563.49, "word": " ان", "probability": 0.60205078125}, {"start": 2563.49, "end": 2563.67, "word": " ال", "probability": 0.86865234375}, {"start": 2563.67, "end": 2564.07, "word": " function", "probability": 0.7236328125}, {"start": 2564.07, "end": 2564.55, "word": " g", "probability": 0.88671875}, {"start": 2564.55, "end": 2564.77, "word": " of", "probability": 0.85302734375}, {"start": 2564.77, "end": 2565.17, "word": " x", "probability": 0.99072265625}, {"start": 2565.17, "end": 2566.41, "word": " is", "probability": 0.95458984375}, {"start": 2566.41, "end": 2566.93, "word": " not", "probability": 0.96630859375}, {"start": 2566.93, "end": 2569.01, "word": " bounded", "probability": 0.93115234375}, {"start": 2569.01, "end": 2572.65, "word": " on", "probability": 0.95263671875}], "temperature": 1.0}, {"id": 85, "seek": 259523, "start": 2576.17, "end": 2595.23, "text": "on a right .. on a right neighborhood .. on a right neighborhood اللي هو سفر دلتا of zero", "tokens": [266, 257, 558, 4386, 322, 257, 558, 7630, 4386, 322, 257, 558, 7630, 13672, 1829, 31439, 8608, 5172, 2288, 11778, 1211, 2655, 995, 295, 4018], "avg_logprob": -0.14227764824262032, "compression_ratio": 1.36, "no_speech_prob": 0.0, "words": [{"start": 2576.17, "end": 2576.67, "word": "on", "probability": 0.8388671875}, {"start": 2576.67, "end": 2577.33, "word": " a", "probability": 0.97998046875}, {"start": 2577.33, "end": 2578.37, "word": " right", "probability": 0.916015625}, {"start": 2578.37, "end": 2578.53, "word": " ..", "probability": 0.64501953125}, {"start": 2578.53, "end": 2581.03, "word": " on", "probability": 0.90234375}, {"start": 2581.03, "end": 2581.35, "word": " a", "probability": 0.99462890625}, {"start": 2581.35, "end": 2581.85, "word": " right", "probability": 0.93701171875}, {"start": 2581.85, "end": 2585.85, "word": " neighborhood", "probability": 0.49560546875}, {"start": 2585.85, "end": 2587.81, "word": " ..", "probability": 0.578125}, {"start": 2587.81, "end": 2588.07, "word": " on", "probability": 0.9130859375}, {"start": 2588.07, "end": 2588.25, "word": " a", "probability": 0.9873046875}, {"start": 2588.25, "end": 2588.53, "word": " right", "probability": 0.9296875}, {"start": 2588.53, "end": 2589.29, "word": " neighborhood", "probability": 0.8740234375}, {"start": 2589.29, "end": 2590.29, "word": " اللي", "probability": 0.970703125}, {"start": 2590.29, "end": 2590.45, "word": " هو", "probability": 0.98876953125}, {"start": 2590.45, "end": 2591.07, "word": " سفر", "probability": 0.9112955729166666}, {"start": 2591.07, "end": 2592.29, "word": " دلتا", "probability": 0.862548828125}, {"start": 2592.29, "end": 2593.67, "word": " of", "probability": 0.943359375}, {"start": 2593.67, "end": 2595.23, "word": " zero", "probability": 0.82666015625}], "temperature": 1.0}, {"id": 86, "seek": 263421, "start": 2605.23, "end": 2634.21, "text": "أخذنا قبل ذلك نظرية بتقول إيه ده عشان أثبت أنه ال limit ل function عن نقطة معينة مش موجودة يكفي أثبت أنه أنه الدالة unbounded عند أي unbounded عند أي neighborhood للنقطة الآن بالنسبة لل one-sided limit", "tokens": [10721, 9778, 8848, 8315, 12174, 36150, 29910, 23275, 8717, 19913, 2288, 10632, 39894, 39648, 11933, 1829, 3224, 11778, 3224, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 14739, 3224, 2423, 4948, 5296, 2445, 18871, 8717, 47432, 3660, 20449, 9957, 3660, 37893, 3714, 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"word": " عشان", "probability": 0.82470703125}, {"start": 2608.01, "end": 2608.33, "word": " أثبت", "probability": 0.87054443359375}, {"start": 2608.33, "end": 2608.61, "word": " أنه", "probability": 0.39794921875}, {"start": 2608.61, "end": 2608.67, "word": " ال", "probability": 0.5625}, {"start": 2608.67, "end": 2609.01, "word": " limit", "probability": 0.78466796875}, {"start": 2609.01, "end": 2610.09, "word": " ل", "probability": 0.80810546875}, {"start": 2610.09, "end": 2610.69, "word": " function", "probability": 0.62646484375}, {"start": 2610.69, "end": 2610.95, "word": " عن", "probability": 0.486083984375}, {"start": 2610.95, "end": 2611.41, "word": " نقطة", "probability": 0.9762369791666666}, {"start": 2611.41, "end": 2612.23, "word": " معينة", "probability": 0.9928385416666666}, {"start": 2612.23, "end": 2613.71, "word": " مش", "probability": 0.93310546875}, {"start": 2613.71, "end": 2614.47, "word": " موجودة", "probability": 0.989501953125}, {"start": 2614.47, "end": 2615.07, "word": " يكفي", "probability": 0.9207356770833334}, {"start": 2615.07, "end": 2615.71, "word": " أثبت", "probability": 0.9781494140625}, {"start": 2615.71, "end": 2616.27, "word": " أنه", "probability": 0.842041015625}, {"start": 2616.27, "end": 2617.61, "word": " أنه", "probability": 0.67822265625}, {"start": 2617.61, "end": 2618.21, "word": " الدالة", "probability": 0.8370768229166666}, {"start": 2618.21, "end": 2619.43, "word": " unbounded", "probability": 0.8883463541666666}, {"start": 2619.43, "end": 2619.75, "word": " عند", "probability": 0.87744140625}, {"start": 2619.75, "end": 2620.29, "word": " أي", "probability": 0.92236328125}, {"start": 2620.29, "end": 2623.91, "word": " unbounded", "probability": 0.6698811848958334}, {"start": 2623.91, "end": 2624.15, "word": " عند", "probability": 0.95263671875}, {"start": 2624.15, "end": 2624.65, "word": " أي", "probability": 0.923828125}, {"start": 2624.65, "end": 2628.65, "word": " neighborhood", "probability": 0.732421875}, {"start": 2628.65, "end": 2629.93, "word": " للنقطة", "probability": 0.85406494140625}, {"start": 2629.93, "end": 2631.69, "word": " الآن", "probability": 0.6390380859375}, {"start": 2631.69, "end": 2632.33, "word": " بالنسبة", "probability": 0.9945068359375}, {"start": 2632.33, "end": 2632.71, "word": " لل", "probability": 0.88916015625}, {"start": 2632.71, "end": 2633.33, "word": " one", "probability": 0.609375}, {"start": 2633.33, "end": 2633.75, "word": "-sided", "probability": 0.725830078125}, {"start": 2633.75, "end": 2634.21, "word": " limit", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 87, "seek": 266387, "start": 2635.05, "end": 2663.87, "text": "عشان أقول إن ال limit ل function زي هذه g of x لما x تقول إلى سفر من اليمين does not exist فهي السفر و ال x تقول لسفر من اليمين فبدل ما أخد delta neighborhood للسفر فباخد right neighborhood", "tokens": [3615, 8592, 7649, 5551, 39648, 36145, 2423, 4948, 5296, 2445, 30767, 1829, 29538, 290, 295, 2031, 5296, 15042, 2031, 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انا بتعامل مع نهاية من اليمين لكن لما كنت اتعامل مع نهاية من الطرفين فكنت ااخد delta neighborhood كامل", "tokens": [1938, 7630, 24976, 3794, 5172, 2288, 6156, 1829, 4117, 41185, 16472, 2423, 2445, 29538, 19446, 3224, 1829, 33599, 37498, 18871, 28242, 558, 7630, 37495, 22653, 10874, 2407, 9640, 9154, 2423, 32640, 9957, 24976, 3794, 5172, 2288, 5296, 7649, 1975, 8315, 39894, 3615, 10943, 1211, 20449, 8717, 11296, 10632, 9154, 2423, 32640, 9957, 44381, 5296, 15042, 9122, 29399, 1975, 2655, 3615, 10943, 1211, 20449, 8717, 11296, 10632, 9154, 41950, 28480, 9957, 6156, 19452, 2655, 1975, 47283, 3215, 8289, 7630, 9122, 10943, 1211], "avg_logprob": -0.25342989258649873, "compression_ratio": 1.8579234972677596, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2666.1, "end": 2666.6, "word": "right", "probability": 0.4599609375}, {"start": 2666.6, "end": 2667.42, "word": " neighborhood", "probability": 0.64208984375}, {"start": 2667.42, "end": 2669.96, "word": " للسفر", "probability": 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"probability": 0.97314453125}, {"start": 2709.24, "end": 2709.76, "word": " الصفر", "probability": 0.9765625}, {"start": 2709.76, "end": 2709.96, "word": " من", "probability": 0.98974609375}, {"start": 2709.96, "end": 2710.44, "word": " اليمين", "probability": 0.8741861979166666}, {"start": 2710.44, "end": 2711.28, "word": " فلازم", "probability": 0.93896484375}, {"start": 2711.28, "end": 2711.66, "word": " تكون", "probability": 0.99072265625}, {"start": 2711.66, "end": 2712.14, "word": " bounded", "probability": 0.92626953125}, {"start": 2712.14, "end": 2713.04, "word": " على", "probability": 0.91015625}, {"start": 2713.04, "end": 2713.58, "word": " some", "probability": 0.8330078125}, {"start": 2713.58, "end": 2715.14, "word": " neighborhood", "probability": 0.7900390625}, {"start": 2715.14, "end": 2715.82, "word": " ..", "probability": 0.381103515625}, {"start": 2715.82, "end": 2716.76, "word": " right", "probability": 0.4580078125}, {"start": 2716.76, "end": 2717.22, "word": " neighborhood", "probability": 0.83544921875}, {"start": 2717.22, "end": 2717.8, "word": " للصفر", "probability": 0.867724609375}], "temperature": 1.0}, {"id": 90, "seek": 274797, "start": 2720.13, "end": 2747.97, "text": "Okay تمام و لإثبات ذلك to see this we use ال inequality التالية وهي T أكبر من سفر دايما أصغر من E أس T for all T أكبر من سفر هذه المتباينة", "tokens": [8297, 46811, 10943, 4032, 5296, 28814, 12984, 3555, 9307, 29910, 23275, 281, 536, 341, 321, 764, 2423, 16970, 16712, 6027, 10632, 37037, 1829, 314, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 11778, 47302, 15042, 5551, 9381, 17082, 2288, 9154, 462, 5551, 3794, 314, 337, 439, 314, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 29538, 9673, 2655, 3555, 995, 9957, 3660], "avg_logprob": -0.21606045179679745, "compression_ratio": 1.335483870967742, "no_speech_prob": 0.0, "words": [{"start": 2720.13, "end": 2720.65, "word": "Okay", "probability": 0.578125}, {"start": 2720.65, "end": 2721.23, "word": " تمام", "probability": 0.89306640625}, {"start": 2721.23, "end": 2722.17, "word": " و", "probability": 0.471923828125}, {"start": 2722.17, "end": 2722.95, "word": " لإثبات", "probability": 0.79404296875}, {"start": 2722.95, "end": 2723.59, "word": " ذلك", "probability": 0.9912109375}, {"start": 2723.59, "end": 2725.65, "word": " to", "probability": 0.231201171875}, {"start": 2725.65, "end": 2729.27, "word": " see", "probability": 0.923828125}, {"start": 2729.27, "end": 2729.81, "word": " this", "probability": 0.96484375}, {"start": 2729.81, "end": 2731.89, "word": " we", "probability": 0.7353515625}, {"start": 2731.89, "end": 2732.55, "word": " use", "probability": 0.87744140625}, {"start": 2732.55, "end": 2734.03, "word": " ال", "probability": 0.408447265625}, {"start": 2734.03, "end": 2734.71, "word": " inequality", "probability": 0.92626953125}, {"start": 2734.71, "end": 2737.29, "word": " التالية", "probability": 0.97314453125}, {"start": 2737.29, "end": 2738.01, "word": " وهي", "probability": 0.7105712890625}, {"start": 2738.01, "end": 2738.37, "word": " T", "probability": 0.63916015625}, {"start": 2738.37, "end": 2738.99, "word": " أكبر", "probability": 0.912109375}, {"start": 2738.99, "end": 2739.29, "word": " من", "probability": 0.99267578125}, {"start": 2739.29, "end": 2739.63, "word": " سفر", "probability": 0.7749837239583334}, {"start": 2739.63, "end": 2740.11, "word": " دايما", "probability": 0.83837890625}, {"start": 2740.11, "end": 2741.41, "word": " أصغر", "probability": 0.9305419921875}, {"start": 2741.41, "end": 2741.71, "word": " من", "probability": 0.98974609375}, {"start": 2741.71, "end": 2742.15, "word": " E", "probability": 0.943359375}, {"start": 2742.15, "end": 2742.51, "word": " أس", "probability": 0.3536376953125}, {"start": 2742.51, "end": 2742.87, "word": " T", "probability": 0.94482421875}, {"start": 2742.87, "end": 2743.97, "word": " for", "probability": 0.9140625}, {"start": 2743.97, "end": 2744.51, "word": " all", "probability": 0.9423828125}, {"start": 2744.51, "end": 2744.93, "word": " T", "probability": 0.91552734375}, {"start": 2744.93, "end": 2745.51, "word": " أكبر", "probability": 0.9733072916666666}, {"start": 2745.51, "end": 2745.77, "word": " من", "probability": 0.99609375}, {"start": 2745.77, "end": 2746.19, "word": " سفر", "probability": 0.97900390625}, {"start": 2746.19, "end": 2747.01, "word": " هذه", "probability": 0.488037109375}, {"start": 2747.01, "end": 2747.97, "word": " المتباينة", "probability": 0.9327799479166666}], "temperature": 1.0}, {"id": 91, "seek": 277920, "start": 2749.44, "end": 2779.2, "text": "هذه المتباينة موجودة برهانة C Chapter 8 برهانة موجودة في Chapter 8 اللي هتاخدوه لاحقا فهنستخدم اللي هو المتباينة هذه في اثبات ان ال function ماهياش bounded على neighborhood او right neighborhood للصفر", "tokens": [3224, 24192, 9673, 2655, 3555, 995, 9957, 3660, 3714, 29245, 23328, 3660, 4724, 2288, 3224, 7649, 3660, 383, 18874, 1649, 4724, 2288, 3224, 7649, 3660, 3714, 29245, 23328, 3660, 8978, 18874, 1649, 13672, 1829, 8032, 2655, 47283, 3215, 2407, 3224, 20193, 5016, 4587, 995, 6156, 3224, 1863, 14851, 9778, 40448, 13672, 1829, 31439, 9673, 2655, 3555, 995, 9957, 3660, 29538, 8978, 1975, 12984, 3555, 9307, 16472, 2423, 2445, 19446, 3224, 1829, 33599, 37498, 15844, 7630, 1975, 2407, 558, 7630, 24976, 9381, 5172, 2288], "avg_logprob": -0.1934523866290138, "compression_ratio": 1.6868131868131868, "no_speech_prob": 0.0, "words": [{"start": 2749.44, "end": 2749.78, "word": "هذه", "probability": 0.869384765625}, {"start": 2749.78, "end": 2750.72, "word": " المتباينة", "probability": 0.9444173177083334}, {"start": 2750.72, "end": 2754.2, "word": " موجودة", "probability": 0.9471435546875}, {"start": 2754.2, "end": 2755.08, "word": " برهانة", "probability": 0.9388671875}, {"start": 2755.08, "end": 2756.9, "word": " C", "probability": 0.173828125}, {"start": 2756.9, "end": 2757.62, "word": " Chapter", "probability": 0.3271484375}, {"start": 2757.62, "end": 2758.14, "word": " 8", "probability": 0.78955078125}, {"start": 2758.14, "end": 2761.78, "word": " برهانة", "probability": 0.876416015625}, {"start": 2761.78, "end": 2762.2, "word": " موجودة", "probability": 0.9473876953125}, {"start": 2762.2, "end": 2762.26, "word": " في", "probability": 0.8994140625}, {"start": 2762.26, "end": 2762.54, "word": " Chapter", "probability": 0.5498046875}, {"start": 2762.54, "end": 2762.94, "word": " 8", "probability": 0.958984375}, {"start": 2762.94, "end": 2763.36, "word": " اللي", "probability": 0.7548828125}, {"start": 2763.36, "end": 2765.08, "word": " هتاخدوه", "probability": 0.85498046875}, {"start": 2765.08, "end": 2765.74, "word": " لاحقا", "probability": 0.93115234375}, {"start": 2765.74, "end": 2767.6, "word": " فهنستخدم", "probability": 0.8881022135416666}, {"start": 2767.6, "end": 2767.84, "word": " اللي", "probability": 0.6072998046875}, {"start": 2767.84, "end": 2768.04, "word": " هو", "probability": 0.84716796875}, {"start": 2768.04, "end": 2768.88, "word": " المتباينة", "probability": 0.98974609375}, {"start": 2768.88, "end": 2769.36, "word": " هذه", "probability": 0.80615234375}, {"start": 2769.36, "end": 2770.16, "word": " في", "probability": 0.486572265625}, {"start": 2770.16, "end": 2770.68, "word": " اثبات", "probability": 0.82501220703125}, {"start": 2770.68, "end": 2770.92, "word": " ان", "probability": 0.5595703125}, {"start": 2770.92, "end": 2771.08, "word": " ال", "probability": 0.91259765625}, {"start": 2771.08, "end": 2771.46, "word": " function", "probability": 0.84228515625}, {"start": 2771.46, "end": 2772.08, "word": " ماهياش", "probability": 0.91748046875}, {"start": 2772.08, "end": 2772.56, "word": " bounded", "probability": 0.798828125}, {"start": 2772.56, "end": 2773.68, "word": " على", "probability": 0.8466796875}, {"start": 2773.68, "end": 2775.94, "word": " neighborhood", "probability": 0.66943359375}, {"start": 2775.94, "end": 2777.12, "word": " او", "probability": 0.7047119140625}, {"start": 2777.12, "end": 2777.42, "word": " right", "probability": 0.9404296875}, {"start": 2777.42, "end": 2777.94, "word": " neighborhood", "probability": 0.82470703125}, {"start": 2777.94, "end": 2779.2, "word": " للصفر", "probability": 0.8348388671875}], "temperature": 1.0}, {"id": 92, "seek": 279555, "start": 2780.29, "end": 2795.55, "text": "Okay عشان الوجد خلص بنوقف و بناخد خمس دقايق break و بعدين بنكمل ان شاء الله البرهانة فحنوقف و نكمل في الجزء التالي من المحاضرة", "tokens": [8297, 6225, 8592, 7649, 2423, 29245, 3215, 16490, 1211, 9381, 44945, 30543, 5172, 4032, 44945, 47283, 3215, 16490, 2304, 3794, 11778, 4587, 47302, 4587, 1821, 4032, 39182, 9957, 44945, 24793, 1211, 16472, 13412, 16606, 21984, 2423, 26890, 3224, 7649, 3660, 6156, 5016, 1863, 30543, 5172, 4032, 8717, 24793, 1211, 8978, 25724, 11622, 38207, 16712, 6027, 1829, 9154, 9673, 5016, 46958, 25720], "avg_logprob": -0.13961693764694275, "compression_ratio": 1.5103448275862068, "no_speech_prob": 0.0, "words": [{"start": 2780.29, "end": 2780.87, "word": "Okay", "probability": 0.900390625}, {"start": 2780.87, "end": 2781.57, "word": " عشان", "probability": 0.8740234375}, {"start": 2781.57, "end": 2782.03, "word": " الوجد", "probability": 0.603759765625}, {"start": 2782.03, "end": 2783.03, "word": " خلص", "probability": 0.9734700520833334}, {"start": 2783.03, "end": 2784.25, "word": " بنوقف", "probability": 0.8885091145833334}, {"start": 2784.25, "end": 2785.13, "word": " و", "probability": 0.8994140625}, {"start": 2785.13, "end": 2785.53, "word": " بناخد", "probability": 0.92626953125}, {"start": 2785.53, "end": 2785.87, "word": " خمس", "probability": 0.97021484375}, {"start": 2785.87, "end": 2786.29, "word": " دقايق", "probability": 0.8475341796875}, {"start": 2786.29, "end": 2786.81, "word": " break", "probability": 0.99267578125}, {"start": 2786.81, "end": 2787.81, "word": " و", "probability": 0.833984375}, {"start": 2787.81, "end": 2788.31, "word": " بعدين", "probability": 0.92529296875}, {"start": 2788.31, "end": 2788.99, "word": " بنكمل", "probability": 0.9807942708333334}, {"start": 2788.99, "end": 2789.17, "word": " ان", "probability": 0.720703125}, {"start": 2789.17, "end": 2789.43, "word": " شاء", "probability": 0.9794921875}, {"start": 2789.43, "end": 2789.59, "word": " الله", "probability": 0.9609375}, {"start": 2789.59, "end": 2790.59, "word": " البرهانة", "probability": 0.83046875}, {"start": 2790.59, "end": 2791.57, "word": " فحنوقف", "probability": 0.82861328125}, {"start": 2791.57, "end": 2792.27, "word": " و", "probability": 0.98779296875}, {"start": 2792.27, "end": 2792.91, "word": " نكمل", "probability": 0.8585611979166666}, {"start": 2792.91, "end": 2793.83, "word": " في", "probability": 0.9052734375}, {"start": 2793.83, "end": 2794.37, "word": " الجزء", "probability": 0.9900716145833334}, {"start": 2794.37, "end": 2794.83, "word": " التالي", "probability": 0.91650390625}, {"start": 2794.83, "end": 2794.97, "word": " من", "probability": 0.986328125}, {"start": 2794.97, "end": 2795.55, "word": " المحاضرة", "probability": 0.86376953125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2795.82175, "duration_after_vad": 2359.755624999989}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ.srt
new file mode 100644
index 0000000000000000000000000000000000000000..9afae4a531f990afa63271e9ac09a7de57c645e4
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/2qaKB7theEQ.srt
@@ -0,0 +1,1607 @@
+1
+00:00:19,940 --> 00:00:25,840
+السلام عليكم هنكمل
+
+2
+00:00:25,840 --> 00:00:33,420
+اليوم section أربعة اثنين في ال section هذا كان
+
+3
+00:00:33,420 --> 00:00:38,780
+اتبقى بس إن أحنا نثبت النظرية اللي كتبتها على
+
+4
+00:00:38,780 --> 00:00:44,700
+اللوح النظرية هذه بتنص على إن لو كان في هندية
+
+5
+00:00:44,700 --> 00:00:53,020
+function من A إلى R و c cluster point للset A وإذا كان
+
+6
+00:00:53,020 --> 00:00:59,120
+limit ال function عن c exist وموجبة أو 
+
+7
+00:00:59,120 --> 00:01:04,880
+على التوالي إذا كانت limit f of x عن c موجودة
+
+8
+00:01:04,880 --> 00:01:09,080
+وسالبة يوجد
+
+9
+00:01:09,080 --> 00:01:14,990
+نقدر نلاقي delta neighborhood دي delta لل c بحيث إن
+
+10
+00:01:14,990 --> 00:01:19,510
+الدالة هتكون إذا كانت ال limit موجبة فالدالة هتكون
+
+11
+00:01:19,510 --> 00:01:26,670
+موجبة على ال delta neighborhood ل C وإذا 
+
+12
+00:01:26,670 --> 00:01:31,950
+كانت ال limit سالبة فالدالة هتكون سالبة على جوار
+
+13
+00:01:31,950 --> 00:01:38,890
+delta ل C هذه
+
+14
+00:01:38,890 --> 00:01:43,210
+نظرية تشبه نظرية سابقة بخصوص limits of sequences
+
+15
+00:01:44,920 --> 00:01:48,400
+النظرية اللي فاتت بتاعة ال sequences المشابهة 
+
+16
+00:01:48,400 --> 00:01:52,640
+بتقول لو كانت ال sequence النهاية بتاعتها limit x
+
+17
+00:01:52,640 --> 00:01:58,200
+exist وموجبة فلازم ال sequence تكون حدودها من
+
+18
+00:01:58,200 --> 00:02:02,420
+capital N وأنت طالع كلها موجبة ولو كانت ال limit
+
+19
+00:02:02,420 --> 00:02:06,640
+لل sequence exist وسالبة فلازم حدود ال sequence
+
+20
+00:02:06,640 --> 00:02:10,900
+من capital N وأنت طالع كلها تكون سالبة فهذه شبيهة
+
+21
+00:02:10,900 --> 00:02:18,100
+فيها والبرهان سهل وشبيه بالبرهان تبع النظرية
+
+22
+00:02:18,100 --> 00:02:24,320
+المشابهة في حالة ال sequences فناخد الحالة assume
+
+23
+00:02:24,320 --> 00:02:27,360
+ناخد 
+
+24
+00:02:27,360 --> 00:02:32,400
+الحالة اللي فيها ال limit ل
+
+25
+00:02:32,400 --> 00:02:40,060
+f of x at c exists and equals عدد l موجب
+
+26
+00:02:53,880 --> 00:03:00,200
+فإذا كانت ال limit موجبة بنا أثبت إن يوجد delta
+
+27
+00:03:00,200 --> 00:03:07,240
+neighborhood إلى آخر A فخلّينا
+
+28
+00:03:07,240 --> 00:03:17,740
+ناخد let epsilon في الحالة دي let epsilon بيساوي
+
+29
+00:03:17,740 --> 00:03:26,600
+L على 2 فهذا عدد موجب الآن by definition of limit
+
+30
+00:03:26,600 --> 00:03:32,640
+of function by epsilon delta definition لأي
+
+31
+00:03:32,640 --> 00:03:38,140
+epsilon موجبة زي هذه يوجد delta تعتمد على L على 2
+
+32
+00:03:38,140 --> 00:03:43,280
+اللي هي ال epsilon عدد موجب بحيث إنه لو كان X
+
+33
+00:03:45,890 --> 00:03:51,090
+ينتمي إلى A و absolute x minus c أصغر من delta
+
+34
+00:03:51,090 --> 00:03:59,790
+أكبر من 0 فهذا بتضمن إن absolute f of x minus L
+
+35
+00:03:59,790 --> 00:04:04,510
+أصغر من epsilon اللي هي عبارة عن L ع 2
+
+36
+00:04:08,170 --> 00:04:15,990
+فحل المتباينة هذه في f of x فتصير f of x minus L
+
+37
+00:04:15,990 --> 00:04:24,930
+أصغر من L على 2 أكبر من سالب L على 2 وهذا
+
+38
+00:04:24,930 --> 00:04:29,210
+بيؤدي إلى إن
+
+39
+00:04:29,210 --> 00:04:30,350
+f of x
+
+40
+00:04:34,740 --> 00:04:45,980
+من هنا F of X تطلع أكبر من L على 2 لأنه لما أخد
+
+41
+00:04:45,980 --> 00:04:50,240
+سالب L أنقلها عن ناحية التانية فتصير F of X أكبر
+
+42
+00:04:50,240 --> 00:04:55,700
+من L سالب L على 2 تطلع L على 2 وال L موجبة إذا L
+
+43
+00:04:55,700 --> 00:05:06,860
+على 2 موجبة إذا هيك بنكون أثبتنا إن ال F of X طلعت
+
+44
+00:05:06,860 --> 00:05:18,580
+أكبر من صفر لمين لكل X تنتمي إلى A ومن 
+
+45
+00:05:18,580 --> 00:05:28,080
+المتباينة هذه هذا معناه X لا تساوي C إن ال X ينتمي
+
+46
+00:05:28,080 --> 00:05:34,480
+إلى A ولا تساوي C يعني موجودة في A ومش موجودة في
+
+47
+00:05:34,480 --> 00:05:44,280
+singleton set C والمتباينة هذه هذه معناها إن X
+
+48
+00:05:44,280 --> 00:05:46,460
+ينتمي إلى V Delta
+
+49
+00:05:56,010 --> 00:06:00,790
+x-c أصغر من دلتا بكافئ
+
+50
+00:06:10,030 --> 00:06:17,770
+إن X أصغر من C زائد Delta أكبر من C سالب Delta
+
+51
+00:06:17,770 --> 00:06:21,550
+فهذا
+
+52
+00:06:21,550 --> 00:06:27,890
+معناه X تنتمي لفترة مفتوحة Delta-neighborhood ل-C
+
+53
+00:06:29,570 --> 00:06:34,830
+Okay تمام إذا f of x اللي أعطاها موجبة لكل x في a
+
+54
+00:06:34,830 --> 00:06:43,250
+ومختلفة عن c وأيضا من هنا ال x أيضا تنتمي ل delta
+
+55
+00:06:43,250 --> 00:06:48,850
+neighborhood ل c وبالتالي تنتمي لتقاطع المجموعتين
+
+56
+00:06:48,850 --> 00:06:55,270
+إذا هذا بيثبت النظرية في حالة لما يكون ال limit
+
+57
+00:06:55,270 --> 00:06:56,490
+تبعتها موجبة
+
+58
+00:07:00,240 --> 00:07:08,880
+لو كانت ال limit سالبة فالبرهان مشابه لأن ال
+
+59
+00:07:08,880 --> 00:07:18,680
+proof of the case لما تكون ال limit ل f of x لما x
+
+60
+00:07:18,680 --> 00:07:30,600
+تؤول ل c بيساوي العدد سالب l is similar to
+
+61
+00:07:30,600 --> 00:07:38,200
+above case مشابه
+
+62
+00:07:38,200 --> 00:07:50,120
+للبرهان السابق  في الحالة هذه take start with
+
+63
+00:07:50,120 --> 00:07:55,920
+epsilon بيساوي سالب ال ع اتنين وهذا بيطلع عدد موجب
+
+64
+00:07:55,920 --> 00:08:01,060
+يعني ابدأوا البرهان بدل ما نبدأ ب epsilon بيساوي ال ع
+
+65
+00:08:01,060 --> 00:08:04,340
+اتنين ابدأوا epsilon ... epsilon بيساوي
+
+66
+00:08:14,820 --> 00:08:20,320
+البرهان البرهان البرهان البرهان البرهان البرهان 
+
+67
+00:08:20,320 --> 00:08:20,800
+البرهان البرهان البرهان البرهان البرهان البرهان 
+
+68
+00:08:20,800 --> 00:08:20,860
+البرهان البرهان البرهان البرهان البرهان البرهان 
+
+69
+00:08:20,860 --> 00:08:24,600
+البرهان البرهان البرهان البرهان البرهان البرهان 
+
+70
+00:08:24,600 --> 00:08:24,640
+البرهان البرهان البرهان البرهان البرهان البرهان
+
+71
+00:08:24,640 --> 00:08:25,080
+البرهان البرهان البرهان البرهان البرهان البرهان 
+
+72
+00:08:25,080 --> 00:08:33,380
+البرهان البرهان
+
+73
+00:08:34,550 --> 00:08:40,910
+لأن ال L سالبة وهذا صحيح لكل X في جوار Delta ل C
+
+74
+00:08:40,910 --> 00:08:46,530
+وفي A minus single to C okay؟ لأن حاسبكم أنتم 
+
+75
+00:08:46,530 --> 00:08:50,990
+تكتبوا البرهان تبع الحالة التانية تمام؟ واضح؟ في
+
+76
+00:08:50,990 --> 00:08:54,850
+أي سؤال أو سفسار؟ تمام؟ 
+
+77
+00:09:00,180 --> 00:09:08,120
+Okay إذا نبدأ section جديد
+
+78
+00:09:08,120 --> 00:09:26,200
+section
+
+79
+00:09:26,200 --> 00:09:29,460
+أربعة ثلاثة
+
+80
+00:09:33,950 --> 00:09:47,190
+بعض التطبيقات .. بعض التطبيقات ل
+
+81
+00:09:47,190 --> 00:09:53,590
+limit concept
+
+82
+00:10:04,410 --> 00:10:12,090
+بعض التعاملات أو توصية بعض مفاهيم النهايات احنا
+
+83
+00:10:12,090 --> 00:10:16,470
+قبل هيك درسنا في section 4.1 و 4.2 ال limit of
+
+84
+00:10:16,470 --> 00:10:20,450
+function أو ال two sided limit لل function عن نقطة
+
+85
+00:10:20,450 --> 00:10:24,810
+معينة اليوم هندرس ال one sided limit ل function عن
+
+86
+00:10:24,810 --> 00:10:30,960
+نقطة and cluster point للمجال تبعها مقصود بال one
+
+87
+00:10:30,960 --> 00:10:34,040
+sided limit اللي هو limit من اليمين أو limit من
+
+88
+00:10:34,040 --> 00:10:38,740
+اليسار ونشوف ما هي علاقة ال one sided limit بال
+
+89
+00:10:38,740 --> 00:10:43,320
+two sided limit فنعرف 
+
+90
+00:10:43,320 --> 00:10:47,880
+الأول definition نعرف ال one sided limit
+
+91
+00:10:47,880 --> 00:10:53,000
+definition let
+
+92
+00:10:55,520 --> 00:11:03,780
+f be a function from a to r and c be a cluster point
+
+93
+00:11:03,780 --> 00:11:06,840
+cluster
+
+94
+00:11:06,840 --> 00:11:22,340
+point of a واحد أو
+
+95
+00:11:22,340 --> 00:11:34,220
+خلّيالـ cluster point of المجموعة A
+
+96
+00:11:34,220 --> 00:11:40,880
+تقاطع الفترة المفتوحة من C إلى infinity اللي هي كل
+
+97
+00:11:40,880 --> 00:11:47,240
+ال X مجموعة كل العناصر X تنتمي إلى A حيث X أكبر من
+
+98
+00:11:47,240 --> 00:11:53,520
+C نقول
+
+99
+00:11:57,140 --> 00:12:03,280
+إن العدد l ينتمي إلى R is
+
+100
+00:12:03,280 --> 00:12:14,600
+a right .. is a right hand limit .. right hand
+
+101
+00:12:14,600 --> 00:12:29,490
+limit of the function F at ..x بيساوي c if الشرط 
+
+102
+00:12:29,490 --> 00:12:35,570
+التالي بيتحقق لكل
+
+103
+00:12:35,570 --> 00:12:40,630
+epsilon given
+
+104
+00:12:40,630 --> 00:12:43,970
+epsilon
+
+105
+00:12:43,970 --> 00:12:51,040
+أكبر من الصفر يوجد delta تعتمد على epsilon عدد موجبة 
+
+106
+00:12:51,040 --> 00:13:01,180
+بحيث إنه لو كان x ينتمي ل a و x minus c أكبر من
+
+107
+00:13:01,180 --> 00:13:07,760
+صفر أصغر من delta فهذا بتضمن إن absolute f of x
+
+108
+00:13:07,760 --> 00:13:13,540
+minus l أصغر من epsilon in
+
+109
+00:13:13,540 --> 00:13:17,720
+this case in
+
+110
+00:13:17,720 --> 00:13:28,480
+this case we write نكتب إن ال limit لل function f
+
+111
+00:13:28,480 --> 00:13:37,920
+عندما x تؤول إلى c من اليمين بيساوي العدد l .. 
+
+112
+00:13:37,920 --> 00:13:44,920
+تمام؟ لأن هذا تعريف ال limit from the right أو ال
+
+113
+00:13:44,920 --> 00:13:49,460
+right hand limit لل function f عند النقطة c
+
+114
+00:14:05,180 --> 00:14:13,440
+إذا أنا عندي هذه خط الأعداد وهي النقطة C وأنا عندي
+
+115
+00:14:13,440 --> 00:14:21,900
+ال C هي cluster point ل
+
+116
+00:14:21,900 --> 00:14:26,180
+A .. لكل ال X موجود في A وأكبر من C
+
+117
+00:14:32,710 --> 00:14:37,650
+فبنقول إن ال limit عند x بيساوي c أو ال function 
+
+118
+00:14:37,650 --> 00:14:42,150
+في إلها right-hand limit وال right-hand limit هي
+
+119
+00:14:42,150 --> 00:14:48,410
+العدد L إذا كان لأي epsilon أكبر من الصفر بتقدر
+
+120
+00:14:48,410 --> 00:14:53,390
+نلاقي delta عدد موجبة بيعتمد على epsilon بحيث لكل x
+
+121
+00:14:53,390 --> 00:15:01,510
+في المجموعة A إذا كانت ال X هذه على يمين ال C
+
+122
+00:15:05,140 --> 00:15:12,460
+والمسافة بينها وبين ال C أصغر من Delta فبتطلع
+
+123
+00:15:12,460 --> 00:15:19,860
+المسافة بين F و X L أصغر من Y بالمثل
+
+124
+00:15:19,860 --> 00:15:24,100
+ممكن نعرف ال limit from the right يعني أنا بدي
+
+125
+00:15:24,100 --> 00:15:27,800
+أعرف ال limit from the right أو ال right hand
+
+126
+00:15:27,800 --> 00:15:34,750
+limit هي نفس let f be a function from A to R و C
+
+127
+00:15:34,750 --> 00:15:41,010
+cluster point للمجموعة A تقاطع الفترة المفتوحة من
+
+128
+00:15:41,010 --> 00:15:50,850
+سالب مالانهاية إلى C اللي هي كل ال X في A حيث X
+
+129
+00:15:50,850 --> 00:15:58,490
+هتكون أصغر من مرة هذه أصغر من C فنقول إن ال real
+
+130
+00:15:58,490 --> 00:16:06,070
+number L هو بدل right hand limit هيكون left hand
+
+131
+00:16:06,070 --> 00:16:10,550
+limit of f at c if given epsilon there exists
+
+132
+00:16:10,550 --> 00:16:15,310
+delta depends on epsilon بحيث إنه لكل x ينتمي إلى
+
+133
+00:16:15,310 --> 00:16:21,900
+A لكل X تنتمي إلى A وال X طبعا موجودة في الفترة هذه
+
+134
+00:16:21,900 --> 00:16:28,600
+يعني ال X المرة هذه على يسار المرة هذه ال X هتكون
+
+135
+00:16:31,420 --> 00:16:35,260
+موجودة في A وفي الفترة المفتوحة من سالب مالانهاية
+
+136
+00:16:35,260 --> 00:16:44,720
+إلى C يعني ال X هتكون على يسار ال C وبالتالي هنا 
+
+137
+00:16:44,720 --> 00:16:51,240
+ال C minus المسافة بين X و C absolute X minus C
+
+138
+00:16:51,240 --> 00:16:57,640
+هتطلع بيساوي C minus X فلو كانت المسافة هذه أصغر من 
+
+139
+00:16:57,640 --> 00:16:59,660
+Delta وطبعا أكبر من صفر
+
+140
+00:17:09,550 --> 00:17:18,370
+هذا الشرط سيصبح c-x أصغر من دلتا أكبر من صفر فهذا
+
+141
+00:17:18,370 --> 00:17:22,910
+لازم يضمن أن absolute of f of x minus l أصغر من
+
+142
+00:17:22,910 --> 00:17:29,610
+إبسيلون في الحالة هذه بيقول إن ال limit لf of x لما
+
+143
+00:17:29,610 --> 00:17:31,850
+x تقول لc من اليسار
+
+144
+00:17:34,230 --> 00:17:39,550
+بس n بساوي l okay إنّها تعريف ال left hand limit
+
+145
+00:17:39,550 --> 00:17:44,890
+أو ال limit from the left okay تعديل
+
+146
+00:17:44,890 --> 00:17:50,770
+بسيط بس طيب
+
+147
+00:17:50,770 --> 00:17:58,170
+ال limits هذه هنشوف يعني بعد شوية إنّ ال one sided
+
+148
+00:17:58,170 --> 00:18:03,210
+limits ده functional نقطة ممكن يعني التنتين يكونوا
+
+149
+00:18:03,210 --> 00:18:09,750
+موجودين عند النقطة و ليهم نفس القيمة أو ممكن
+
+150
+00:18:09,750 --> 00:18:15,430
+التنتين يكونوا موجودين عند نقطة لكن قيمهم مختلفة
+
+151
+00:18:15,430 --> 00:18:20,270
+زي ال Signum function عند الصفر شوفنا إنّ ال limit
+
+152
+00:18:20,270 --> 00:18:23,350
+تبعتها من اليمين واحد و ال limit تبعتها من اليسار
+
+153
+00:18:23,350 --> 00:18:27,850
+سالب واحد إذا ممكن ال two sided limits يكونوا
+
+154
+00:18:27,850 --> 00:18:33,630
+موجودات لكن they are different مختلفات، ممكن برضه
+
+155
+00:18:33,630 --> 00:18:37,790
+one sided limit تكون موجودة and the other may not
+
+156
+00:18:37,790 --> 00:18:42,090
+exist، ممكن ما تكونش موجودة من أساسه
+
+157
+00:18:44,810 --> 00:18:54,930
+ممكن ال one sided limits ولا واحدة فيهم تكون
+
+158
+00:18:54,930 --> 00:19:03,250
+موجودة فكل الحالات هذه هنشوفها في أمثلة لاحقة لكن
+
+159
+00:19:03,250 --> 00:19:08,030
+الأول خلّينا نبرهن النظرية التالية
+
+160
+00:19:13,670 --> 00:19:18,750
+طبعا هنا بنحب ال ...
+
+161
+00:19:18,750 --> 00:19:24,870
+النوّه إنّ كل نظريات اللي أثبتناها في section 4.1
+
+162
+00:19:24,870 --> 00:19:31,790
+أو 4.2 بخصوص ال two sided limit هتكون صحيحة بخصوص
+
+163
+00:19:31,790 --> 00:19:38,030
+ال right limit و كذلك صحيحة بخصوص ال left hand
+
+164
+00:19:38,030 --> 00:19:38,430
+limit
+
+165
+00:19:41,100 --> 00:19:46,240
+فعلى سبيل المثال وليس الحصر إحنا أخدنا sequential
+
+166
+00:19:46,240 --> 00:19:51,240
+criterion sequential criterion for two sided limit
+
+167
+00:19:51,240 --> 00:19:56,580
+الآن هنكتب برضه sequential criterion for right
+
+168
+00:19:56,580 --> 00:20:09,420
+limit sequential criterion for right 
+
+169
+00:20:09,420 --> 00:20:10,100
+hand
+
+170
+00:20:25,970 --> 00:20:35,670
+limits let f from a to r be a function and c be a
+
+171
+00:20:35,670 --> 00:20:37,330
+cluster
+
+172
+00:20:39,230 --> 00:20:47,910
+point of A then the following statements are
+
+173
+00:20:47,910 --> 00:20:54,190
+equivalent العبارات التالية متكافئة واحد ال limit
+
+174
+00:20:54,190 --> 00:21:01,810
+ل F of X as X tends to C from the right exists
+
+175
+00:21:01,810 --> 00:21:06,030
+و بساوي عدد L اتنين
+
+176
+00:21:13,830 --> 00:21:20,370
+for every sequence
+
+177
+00:21:20,370 --> 00:21:36,530
+x n contained in a تقاطع c إلى infinity such that
+
+178
+00:21:38,210 --> 00:21:47,290
+limit x n as n tends to infinity بيساوي c we have
+
+179
+00:21:47,290 --> 00:21:51,090
+limit
+
+180
+00:21:51,090 --> 00:21:59,170
+لل image of the sequence x n بيساوي العدد L
+
+181
+00:22:10,340 --> 00:22:17,060
+البرهان شبيه بالبرهان الخاص بالـ two-sided limit
+
+182
+00:22:17,060 --> 00:22:24,100
+فمثلا لو بدنا نبرهن proof لو بدنا نبرهن العبارة
+
+183
+00:22:24,100 --> 00:22:30,720
+الأولى بتأدي للتانية فبنقول assume .. نبدأ ب
+
+184
+00:22:30,720 --> 00:22:38,240
+assume إنّ ال limit ال right limitالـ F عند الـ C
+
+185
+00:22:38,240 --> 00:22:44,240
+exists بساوي L و بدنا
+
+186
+00:22:44,240 --> 00:22:48,680
+نثبت إنّ الـ two بيطلع العبارة اتنين بتطلع صحيحة
+
+187
+00:22:48,680 --> 00:22:55,400
+لبرهان العبارة to prove two
+
+188
+00:22:55,400 --> 00:22:56,260
+holds
+
+189
+00:22:59,500 --> 00:23:08,320
+لتبدأ لت xn contained in a تقاطع c إلى infinity
+
+190
+00:23:08,320 --> 00:23:12,360
+ب sequence
+
+191
+00:23:12,360 --> 00:23:19,040
+such that ال limit تبعتها as n tends to infinity
+
+192
+00:23:19,040 --> 00:23:24,440
+بيساوى c إذا أنا باخد sequence في المجموعة a
+
+193
+00:23:24,440 --> 00:23:30,410
+و حدودها كلها أكبر من c و بفرض إنّ ال limit لل
+
+194
+00:23:30,410 --> 00:23:38,870
+sequence بيساوي العدد c نحتاج إنّنا نظهر عشان
+
+195
+00:23:38,870 --> 00:23:46,250
+نثبت اتنين باقي نثبت إنّ ال limit نحتاج إنّنا نظهر إنّ
+
+196
+00:23:46,250 --> 00:23:53,530
+ال limit لل image of the sequence xn as n tends to
+
+197
+00:23:53,530 --> 00:24:01,630
+infinity بساوي L هيك بنكون أثبتنا إنّ العبارة 2
+
+198
+00:24:01,630 --> 00:24:10,150
+صحيحة، مصبوط، صح؟ طيب لبرهان ذلك to
+
+199
+00:24:10,150 --> 00:24:11,130
+see this
+
+200
+00:24:16,090 --> 00:24:19,390
+نبدأ نثبت إنّ ال limit لل sequence هذه بساوي عدد L
+
+201
+00:24:19,390 --> 00:24:23,890
+فبستخدم تعريف epsilon capital N لل limit فلازم
+
+202
+00:24:23,890 --> 00:24:31,510
+نبدأ with epsilon أكبر من الصفر ب given طيب مش
+
+203
+00:24:31,510 --> 00:24:41,970
+إحنا فرضنا Since الـ right limit ل F and C موجود أو
+
+204
+00:24:41,970 --> 00:24:46,770
+بيساوي L من تعريف ال right limit there exists
+
+205
+00:24:46,770 --> 00:24:52,230
+delta depends on epsilon positive number بحيث إنّه
+
+206
+00:24:52,230 --> 00:25:02,200
+لو كانت ال X تنتمي إلى A و X minus C أكبر من 0 أصغر
+
+207
+00:25:02,200 --> 00:25:09,760
+من دلتا هذا معناه بيؤدي إنّ absolute f of x minus
+
+208
+00:25:09,760 --> 00:25:22,600
+L أصغر من إبسيليون نسمي ال implication هذي star now
+
+209
+00:25:22,600 --> 00:25:30,880
+for the above الدلتا أكبر من الصفر لدلتا هذه
+
+210
+00:25:30,880 --> 00:25:34,720
+العدد الموجبة لدلتا هذه العدد الموجبة لدلتا
+
+211
+00:25:34,720 --> 00:25:36,800
+هذه العدد الموجبة لدلتا هذه العدد الموجبة
+
+212
+00:25:36,800 --> 00:25:38,440
+لدلتا هذه العدد الموجبة لدلتا هذه العدد
+
+213
+00:25:38,440 --> 00:25:41,400
+الموجبة لدلتا هذه العدد الموجبة لدلتا هذه
+
+214
+00:25:41,400 --> 00:25:41,660
+العدد الموجبة لدلتا هذه العدد الموجبة لدلتا
+
+215
+00:25:41,660 --> 00:25:42,040
+هذه العدد الموجبة لدلتا هذه العدد الموجبة
+
+216
+00:25:42,040 --> 00:25:43,160
+لدلتا هذه العدد الموجبة لدلتا هذه العدد
+
+217
+00:25:43,160 --> 00:25:51,140
+الموجبة لدلتا هذه العدد الموجبة لدلتا هذه
+
+218
+00:25:51,140 --> 00:25:57,180
+العدد الموجبة لدلتا هذه العدد الموجبة لدلتا
+
+219
+00:25:57,830 --> 00:26:05,110
+natural number عدد طبيعي بحيث إنّه لو كان ال N أكبر
+
+220
+00:26:05,110 --> 00:26:12,510
+من أو يساوي capital N فهذا بيضمن إنّ absolute xn
+
+221
+00:26:12,510 --> 00:26:20,090
+minus c أصغر من delta نسمي ال implication هذه
+
+222
+00:26:20,090 --> 00:26:21,050
+double star
+
+223
+00:26:30,480 --> 00:26:44,680
+hence و بالتالي star and double star imply بيؤدّيان
+
+224
+00:26:44,680 --> 00:26:51,860
+إلى ما يلي إنّه لو كانت ال N أكبر من أو يساوي
+
+225
+00:26:51,860 --> 00:26:56,360
+capital N فمن
+
+226
+00:26:56,360 --> 00:26:57,380
+double star
+
+227
+00:26:59,860 --> 00:27:04,940
+لو كانت n أكبر من أو يساوي capital N فمن double
+
+228
+00:27:04,940 --> 00:27:21,340
+star بيطلع absolute xn minus c أصغر من delta هذا
+
+229
+00:27:21,340 --> 00:27:24,800
+بيؤدي إنّ xn
+
+230
+00:27:26,360 --> 00:27:35,400
+minus C أكبر من صفر أصغر من Delta ليه؟ لأنّ ال xn
+
+231
+00:27:35,400 --> 00:27:44,420
+موجودة تنتمي لإيه؟ هو أكبر من C، لذلك هذا لأنّ xn
+
+232
+00:27:44,420 --> 00:27:48,400
+أكبر
+
+233
+00:27:48,400 --> 00:27:58,050
+من C فبالتالي absolute xn-c أكبر من 0 و بالتالي
+
+234
+00:27:58,050 --> 00:28:06,790
+absolute xn-c absolute عدد موجب بيساوي نفسه لأنّ ال
+
+235
+00:28:06,790 --> 00:28:15,750
+absolute value هنا ل xn-c بساوي xn-c لأنّ xn أكبر
+
+236
+00:28:15,750 --> 00:28:18,110
+من c و طبعا
+
+237
+00:28:21,830 --> 00:28:32,590
+هذا أكبر من الصفر لأنّ xn لا تساوي c أكبر من c الآن
+
+238
+00:28:32,590 --> 00:28:39,310
+من ال star هذا بيؤدي by star ال star بتقول إذا
+
+239
+00:28:39,310 --> 00:28:45,330
+كانت ال X أو هنا في الحالة تبعتنا xn ال xn هذه
+
+240
+00:28:45,330 --> 00:28:49,990
+تنتمي لإيه؟ ال xn هي تنتمي لإيه؟ و بعدين هي عندي
+
+241
+00:28:49,990 --> 00:28:56,470
+xn سالب C أكبر من صفر أصغر من Delta إذا by star
+
+242
+00:28:56,470 --> 00:29:06,950
+بيطلع absolute F of xn minus L أصغر من epsilon تمام؟
+
+243
+00:29:10,290 --> 00:29:18,790
+الآن نلاحظ إنّ epsilon was arbitrary إبسيليون
+
+244
+00:29:18,790 --> 00:29:28,230
+was arbitrary since
+
+245
+00:29:28,230 --> 00:29:36,890
+إبسيليون أكبر من الصفر was arbitrary إذاً هيك بنكون
+
+246
+00:29:36,890 --> 00:29:43,470
+إحنا أثبتنا إنّه لأي إبسيليون أو لكل إبسيليون يوجد Delta
+
+247
+00:29:43,470 --> 00:29:50,890
+لكل إبسيليون يوجد capital N يعتمد على ال Delta
+
+248
+00:29:50,890 --> 00:29:55,070
+و بالتالي تعتمد على إبسيليون لأنّ ال Delta تعتمد على
+
+249
+00:29:55,070 --> 00:30:01,410
+إبسيليون بحيث إنّه لكل N أكبر من أو يساوي capital N طلع
+
+250
+00:30:01,410 --> 00:30:06,190
+عندي absolute f of xn minus L أصغر من إبسيليون إذاً
+
+251
+00:30:06,190 --> 00:30:12,750
+by epsilon capital N definition of limit بيطلع هيك
+
+252
+00:30:12,750 --> 00:30:18,710
+بيكون أثبتنا إنّ limit ال sequence f of x n as n
+
+253
+00:30:18,710 --> 00:30:23,710
+tends to infinity بساوي L و هذا اللي بدنا يعني هذا
+
+254
+00:30:23,710 --> 00:30:26,570
+اللي إحنا إيه اللي عايزين نثبته
+
+255
+00:30:29,870 --> 00:30:35,490
+إذاً هيك بنكون أثبتنا إنّه إيه اتنين holds و بالتالي
+
+256
+00:30:35,490 --> 00:30:41,670
+هيك هذا بيكمل برهان واحد implies two okay تمام؟
+
+257
+00:30:41,670 --> 00:30:46,490
+بالمثل ممكن إنّنا نبرهن اتنين implies one
+
+258
+00:30:55,620 --> 00:31:03,360
+the proof of اتنين implies العبارة
+
+259
+00:31:03,360 --> 00:31:11,200
+التانية implies الأولى is similar is
+
+260
+00:31:11,200 --> 00:31:18,600
+similar to is
+
+261
+00:31:18,600 --> 00:31:22,000
+similar to the proof of
+
+262
+00:31:24,130 --> 00:31:34,570
+the sequential criterion for two-sided limit
+
+263
+00:31:34,570 --> 00:31:45,850
+exercises
+
+264
+00:31:45,850 --> 00:31:50,980
+يعني اتمرّنوا عليها أنا ارجع لبرهان ال sequential
+
+265
+00:31:50,980 --> 00:31:55,520
+criterion for two-sided limit و شوفوا اقرأوا
+
+266
+00:31:55,520 --> 00:31:59,600
+البرهان و اعملوا التعديلات البسيطة على البرهان لأنّ
+
+267
+00:31:59,600 --> 00:32:03,920
+هنا إحنا بنتعامل مع right hand limit أو limit from
+
+268
+00:32:03,920 --> 00:32:07,500
+the right rather than two-sided limit زي ما عملنا
+
+269
+00:32:07,500 --> 00:32:12,920
+في البرهان تبع واحد implies اتنين okay فحاسيبكم
+
+270
+00:32:12,920 --> 00:32:17,740
+انتوا تكتبوا البرهان تبع اتنين بيؤدي لواحد بنفس
+
+271
+00:32:17,740 --> 00:32:21,700
+الطريقة اللي برهناها في حالة ال two sided limit
+
+272
+00:32:21,700 --> 00:32:30,080
+okay تمام في أي سؤال طبعا ممكن برضه أيضا يوجد
+
+273
+00:32:30,080 --> 00:32:35,500
+ممكننا نثبت sequential criterion for left hand
+
+274
+00:32:35,500 --> 00:32:42,620
+limit أو limit from the left بنفس الطريقة okay يعني
+
+275
+00:32:42,620 --> 00:32:47,080
+إحنا مش هنكتب طبعا نظرية دي هنعتبرها نظرية قائمة و
+
+276
+00:32:47,080 --> 00:32:53,010
+صحيحة و مش بدون برهان okay تمام؟ إذن هذه واحدة من
+
+277
+00:32:53,010 --> 00:32:58,650
+النظريات اللي برهناها في section 4.1 و 4.2 و
+
+278
+00:32:58,650 --> 00:33:04,470
+بالمثل كل نظريات اللي برهناهم لـ two sided limit
+
+279
+00:33:04,470 --> 00:33:10,590
+في section 4.1 و 4.2 هنعتبرهم قائمين أو نعتبر
+
+280
+00:33:10,590 --> 00:33:15,330
+نظريات هذه صحيحة لـ left limit و right limit 
+
+281
+00:33:22,080 --> 00:33:37,560
+في نظرية أخرى مهمة وهي التعطيل
+
+282
+00:33:37,560 --> 00:33:43,000
+العلاقة بين الـ two sided limits و الـ one sided 
+
+283
+00:33:43,000 --> 00:33:49,100
+limits ف
+
+284
+00:33:51,870 --> 00:34:01,250
+if f is a function from a to r and let c be a cluster
+
+285
+00:34:01,250 --> 00:34:05,450
+point
+
+286
+00:34:05,450 --> 00:34:08,690
+of 
+
+287
+00:34:08,690 --> 00:34:15,310
+المجموعة a تقاطع الفترة المفتوحة from c to
+
+288
+00:34:15,310 --> 00:34:24,070
+infinity and of a تقاطع الـ open interval from
+
+289
+00:34:24,070 --> 00:34:32,250
+negative infinity to c then
+
+290
+00:34:32,250 --> 00:34:42,450
+الـ two-sided limit للـ function f and c بتكون 
+
+291
+00:34:42,450 --> 00:34:47,730
+موجودة وبتساوي
+
+292
+00:34:47,730 --> 00:34:54,760
+عدد L if and only if الـ one-sided limit أو الـ
+
+293
+00:34:54,760 --> 00:35:02,120
+limit from the right the limit at C from the right
+
+294
+00:35:02,120 --> 00:35:12,860
+exist و بتساوي L and the limit of f at C from the
+
+295
+00:35:12,860 --> 00:35:19,360
+left exist و بتساوي نفس العدد L وهذه نظرية أخذناها 
+
+296
+00:35:19,360 --> 00:35:21,520
+في تفاضل ألف إذا بتذكروا
+
+297
+00:35:24,420 --> 00:35:29,460
+متى ال limit عند نقطة في مجالها أو cluster point
+
+298
+00:35:29,460 --> 00:35:34,940
+لمجالها بتكون exist بالساوية عدد إذا كانت ال limit
+
+299
+00:35:34,940 --> 00:35:37,980
+من اليمين موجودة و ال limit من اليسار موجودة و
+
+300
+00:35:37,980 --> 00:35:47,600
+الاثنتين متساويتين و بتساوي نفس العدد هناك
+
+301
+00:35:47,600 --> 00:35:50,500
+بس ماكنش البرهان المطلوب منكم المرة دي احنا
+
+302
+00:35:50,500 --> 00:35:58,420
+مطالبين بالبرهان البرهان يعني كتير سهل ينتج من 
+
+303
+00:35:58,420 --> 00:36:06,780
+التعريفات proof ف .. هحاول أبرهن لكم الـ f part هذا
+
+304
+00:36:06,780 --> 00:36:16,520
+مسمى الـ f part يعني هفرض أنه assume أنه
+
+305
+00:36:16,520 --> 00:36:17,780
+الـ one sided limits
+
+306
+00:36:24,700 --> 00:36:29,160
+the limit from the right exist و بتساوي L وكذلك
+
+307
+00:36:29,160 --> 00:36:36,000
+limit from the left موجودة
+
+308
+00:36:36,000 --> 00:36:41,840
+و بتساوي العدد L وعايز اثبت ان ال limit from the two
+
+309
+00:36:41,840 --> 00:36:48,940
+sides exist إذا هنا هذا الفرض المطلوب
+
+310
+00:37:02,770 --> 00:37:09,030
+أكلم الـ two-sided limit لـ الـ function f at x
+
+311
+00:37:09,030 --> 00:37:14,530
+بتساوي c exist و بتساوي نفس القيمة أو نفس الأعداد L
+
+312
+00:37:14,530 --> 00:37:27,010
+لبرهان ذلك to see this لبرهان ذلك بنحاول نطبق
+
+313
+00:37:27,010 --> 00:37:33,000
+تعريف epsilon delta للـ limit of function فبنبدأ
+
+314
+00:37:33,000 --> 00:37:41,140
+بنقول let epsilon أكبر من الصفر be given طيب 
+
+315
+00:37:41,140 --> 00:37:47,380
+أنا من الفرض أنا فارض تعالى نستفيد من الفرض للوصول 
+
+316
+00:37:47,380 --> 00:37:51,720
+إلى المطلوب هذا برهان مباشر البرهان المباشر ده
+
+317
+00:37:51,720 --> 00:37:57,420
+ناخد الفرض بنشتغل عليه بنحط عليه شوية برات و بعدين 
+
+318
+00:37:57,420 --> 00:38:05,060
+بنطلع منه المطلوب فمن الفرض فرضين احنا ان ال limit
+
+319
+00:38:05,060 --> 00:38:14,060
+لـ f of x as x tends to c positive لما انه ال limit
+
+320
+00:38:14,060 --> 00:38:20,020
+من اليمين عن c بتساوي L y أكبر من الصفر given by 
+
+321
+00:38:20,020 --> 00:38:25,190
+definition there exists delta واحد بالساوي delta 
+
+322
+00:38:25,190 --> 00:38:32,830
+واحد تعتمد على epsilon عدد موجب بحيث أنه لو كان x
+
+323
+00:38:32,830 --> 00:38:40,650
+ينتمي إلى a و x minus c أكبر من الصفر أصغر من delta
+
+324
+00:38:40,650 --> 00:38:48,350
+واحد فهذا بتضمن أن absolute f of x minus l أصغر من
+
+325
+00:38:48,350 --> 00:38:48,810
+epsilon
+
+326
+00:38:52,780 --> 00:39:00,720
+نسمي الـ implication head star also كذلك بما أن
+
+327
+00:39:00,720 --> 00:39:08,420
+احنا فرضين ان ال limit لـ f of x as x tends to c
+
+328
+00:39:08,420 --> 00:39:13,900
+from the left exist و equal نفس العدد L، إذا by
+
+329
+00:39:13,900 --> 00:39:18,980
+definition of left hand limit there exists delta
+
+330
+00:39:18,980 --> 00:39:21,880
+ثانية مش صارت الـ delta هذه تكون نفس الـ delta
+
+331
+00:39:21,880 --> 00:39:27,040
+اللي فوق ماحد بيقدر يجزم بذلك فنسميها delta ثانية
+
+332
+00:39:27,040 --> 00:39:32,980
+there exists delta two depends طبعا بالتأكيد تعتمد
+
+333
+00:39:32,980 --> 00:39:38,560
+على إبسلون وعدد موجب بحيث أنه حسب التعريف لكل x
+
+334
+00:39:39,250 --> 00:39:46,270
+تنتمي إلى a و c minus x أكبر من الصفر أصغر من delta
+
+335
+00:39:46,270 --> 00:39:54,290
+و 2 طبعا هذا بتضمن أن absolute f of x minus n less
+
+336
+00:39:54,290 --> 00:40:00,710
+than epsilon نسمي الـ implication هذه double star
+
+337
+00:40:00,710 --> 00:40:05,390
+خلينا
+
+338
+00:40:05,390 --> 00:40:11,530
+ناخد كالعادة delta نعرف delta على إنها minimum ال
+
+339
+00:40:11,530 --> 00:40:17,530
+minimum الأصغر بين delta واحد و delta اثنين طبعا
+
+340
+00:40:17,530 --> 00:40:21,890
+هذه بالتأكيد هيطلع الصغيرة بين الاتنين هتكون واحدة
+
+341
+00:40:21,890 --> 00:40:27,770
+منهم وبالتالي تطلع عدد موجب وتعتمد على epsilon إذن
+
+342
+00:40:27,770 --> 00:40:30,930
+هيثبت أن يوجد delta تعتمد على epsilon و ال delta
+
+343
+00:40:30,930 --> 00:40:36,110
+هي عدد موجب الان for this delta تعالى نشوف
+
+344
+00:40:40,450 --> 00:40:49,310
+لو كان x ينتمي ل a و absolute x minus c أكبر من
+
+345
+00:40:49,310 --> 00:40:54,510
+الصفر أصغر من دلتا الان 
+
+346
+00:40:54,510 --> 00:40:57,850
+بناخد delta بتساوي ال minimum ل delta واحد و delta
+
+347
+00:40:57,850 --> 00:41:02,060
+اثنين طبعا بما ان دلتا واحد ودلتا اثنين اعداد موجبة
+
+348
+00:41:02,060 --> 00:41:06,080
+اذا دلتا عدد موجب وكذلك تعتمد على epsilon لان
+
+349
+00:41:06,080 --> 00:41:10,380
+دلتا واحد ودلتا اثنين تعتمد على epsilon الان لو
+
+350
+00:41:10,380 --> 00:41:16,720
+أخدت x تنتمي لمجموعة a و ال x صارت مختلفة عن ال c
+
+351
+00:41:16,720 --> 00:41:23,320
+و المسافة بينها و بين ال c أصغر من دلتا هذا معناه 
+
+352
+00:41:23,320 --> 00:41:34,510
+هذا معناه أنه ال x لا تساوي c وبالتالي 
+
+353
+00:41:34,510 --> 00:41:48,230
+ال x ممكن تكون أصغر من c أو ال x أكبر من c فهذا
+
+354
+00:41:48,230 --> 00:41:55,630
+بيقدي أن ال .. ال 
+
+355
+00:41:55,630 --> 00:42:04,400
+.. ال .. إذا كانت ال x إذا كانت الـ x أكبر من c لو
+
+356
+00:42:04,400 --> 00:42:08,980
+كانت الـ x أكبر من c فهذا بقدي أن absolute x
+
+357
+00:42:08,980 --> 00:42:15,200
+minus c بتساوي x minus c بصير الـ absolute value
+
+358
+00:42:15,200 --> 00:42:20,560
+هذه عبارة عن x minus c هو أكبر من 0 أصغر من delta
+
+359
+00:42:20,560 --> 00:42:29,800
+ولو كانت ال x أصغر من c فال absolute value هذه
+
+360
+00:42:29,800 --> 00:42:37,360
+بيصير c minus x أكبر من الصفر أصغر من delta في
+
+361
+00:42:37,360 --> 00:42:41,460
+الحالة الأولى ال delta تبعتي هذه أصغر من أو ساوي 
+
+362
+00:42:41,460 --> 00:42:47,120
+delta واحد صح؟ ال delta هذه هي ال minimum ل delta
+
+363
+00:42:47,120 --> 00:42:50,760
+واحد و delta اثنين وبالتالي أصغر من أو ساوي delta
+
+364
+00:42:50,760 --> 00:42:58,090
+واحد وبالتالي من ال star إذا كانت x تنتمي إلى a و x
+
+365
+00:42:58,090 --> 00:43:03,990
+minus c أكبر من الصفر أصغر من دلتا واحد من ال star
+
+366
+00:43:03,990 --> 00:43:11,770
+بيطلع عندي absolute f of x minus l أصغر من يو إذا 
+
+367
+00:43:11,770 --> 00:43:17,510
+كانت ال x أصغر من ال c فبيطلع absolute x سالب c
+
+368
+00:43:17,510 --> 00:43:22,870
+بيطلع بيساوي c سالب x أصغر من delta وطبعا x مستويش 
+
+369
+00:43:22,870 --> 00:43:29,190
+c أكبر من 0 وال delta هذه من تعريفها أصغر من أو 
+
+370
+00:43:29,190 --> 00:43:35,300
+يساوي delta 2 باستخدام double star ال implication
+
+371
+00:43:35,300 --> 00:43:41,420
+double star لما يكون ال x تنتمي ل a و c minus x 
+
+372
+00:43:41,420 --> 00:43:46,640
+أكبر من 0 أصغر من delta 2 هذا بيقدر أن absolute f
+
+373
+00:43:46,640 --> 00:43:53,680
+of x minus l أصغر من إبسن إذن في كل الأحوال هذه
+
+374
+00:43:53,680 --> 00:43:58,180
+بتقدر أن absolute f of x minus l أصغر من إبسن
+
+375
+00:43:58,180 --> 00:43:59,400
+تمام؟
+
+376
+00:44:02,170 --> 00:44:06,090
+طب ما هذا هو تعريف epsilon delta للـ limit of
+
+377
+00:44:06,090 --> 00:44:12,270
+function صح؟ إذا نيجي بنقول هنا since epsilon أكبر
+
+378
+00:44:12,270 --> 00:44:15,870
+من الصفر was arbitrary
+
+379
+00:44:17,410 --> 00:44:22,850
+إذا احنا بنكون أثبتنا لكل إبسلون أكبر من الصفر يوجد
+
+380
+00:44:22,850 --> 00:44:27,950
+delta تعتمد على إبسلون عدد موجب بحيث لكل x تنتمي ل
+
+381
+00:44:27,950 --> 00:44:32,210
+a و absolute x minus c أكبر من الصفر أصغر من delta
+
+382
+00:44:32,210 --> 00:44:37,810
+طلع عندي absolute f of x في الحالتين minus l أصغر
+
+383
+00:44:37,810 --> 00:44:41,630
+من إبسلون وبالتالي إذا هذا صحيح لكل إبسلون 
+
+384
+00:44:41,630 --> 00:44:45,620
+وبالتالي by epsilon delta definition of limit أو 
+
+385
+00:44:45,620 --> 00:44:54,600
+function we have أثبتنا أن ال limit ل f of x as x
+
+386
+00:44:54,600 --> 00:45:01,380
+tends to c بتساوي العدد l okay تمام، إذا هذا بيثبت 
+
+387
+00:45:01,380 --> 00:45:04,840
+اللي هو لو كان ال two sided limits موجودين
+
+388
+00:45:04,840 --> 00:45:10,730
+متساويتين، لأ لو كان ال one sided limits كلا هما 
+
+389
+00:45:10,730 --> 00:45:15,530
+موجودة و بتساوي قيمة مشتركة l ف ال two sided limit
+
+390
+00:45:15,530 --> 00:45:20,130
+بتطلع exist و قيمتها بتساوي القيمة المشتركة الان 
+
+391
+00:45:20,130 --> 00:45:28,210
+برهان العكس أسهل لذلك هكتب هنا ال proof of 
+
+392
+00:45:28,210 --> 00:45:36,650
+the converse is easier أسهل
+
+393
+00:45:38,760 --> 00:45:44,180
+So exercise it يعني
+
+394
+00:45:44,180 --> 00:45:49,780
+تمرن عليها لو كانت ال two-sided limit exist فمن
+
+395
+00:45:49,780 --> 00:45:55,840
+السهل أن نثبت أن ال right hand limit exist و ال
+
+396
+00:45:55,840 --> 00:46:00,600
+left hand limit exist و كلهم لهم نفس القيمة okay
+
+397
+00:46:00,600 --> 00:46:05,170
+تمام؟ إذا هنوقف هنا و في المحاضرة الجاية إن شاء
+
+398
+00:46:05,170 --> 00:46:09,710
+الله هناخد أمثلة على one-sided limits إما في اثنين
+
+399
+00:46:09,710 --> 00:46:13,350
+موجودين و متساوياتين أو اثنين موجودين و مختلفتين
+
+400
+00:46:13,350 --> 00:46:18,190
+أو واحدة موجودة و اثنين مش موجودة و هكذا، هنشوف كل
+
+401
+00:46:18,190 --> 00:46:24,670
+الأنواع و كل ال situations، تمام؟ okay شكرا لكم و
+
+402
+00:46:24,670 --> 00:46:26,550
+نشوفكم إن شاء الله المرة القادمة
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7K-d4aAzbLs_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7K-d4aAzbLs_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce5c5624a863444f44e103dab95c543ccf2175d9
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7K-d4aAzbLs_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4749, "start": 21.41, "end": 47.49, "text": "السلام عليكم اليوم في اللقاء الأول هناخد مناقشة و اعتقد ان احنا في المناقشة السابقة وصلنا ل section تلاتة خمسة، أصبع؟ فممكن اليوم", "tokens": [6027, 3794, 37440, 25894, 24793, 45595, 20498, 8978, 13672, 4587, 16606, 16247, 12610, 8032, 1863, 47283, 3215, 3714, 8315, 4587, 8592, 3660, 4032, 1975, 34268, 28543, 16472, 1975, 5016, 8315, 8978, 9673, 8315, 4587, 8592, 3660, 21136, 16758, 28671, 4032, 36520, 8315, 5296, 3541, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 12399, 5551, 9381, 3555, 3615, 22807, 6156, 2304, 43020, 45595, 20498], "avg_logprob": -0.16979980235919356, "compression_ratio": 1.5165562913907285, "no_speech_prob": 0.0, "words": [{"start": 21.41, "end": 22.01, "word": "السلام", "probability": 0.8857421875}, {"start": 22.01, "end": 22.69, "word": " عليكم", "probability": 0.947265625}, {"start": 22.69, "end": 23.63, "word": " اليوم", "probability": 0.83642578125}, {"start": 23.63, "end": 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{"start": 129.2, "end": 131.9, "word": " y", "probability": 0.93359375}, {"start": 131.9, "end": 132.42, "word": " in", "probability": 0.962890625}], "temperature": 1.0}, {"id": 5, "seek": 16180, "start": 134.78, "end": 161.8, "text": "ب such that limit x in ضرب y in limit حصل ضرب لما n تقول لinfinity الساوي L ينتمي إلى R يعني exists in R شو مطلوب", "tokens": [3555, 1270, 300, 4948, 2031, 294, 48812, 25513, 288, 294, 4948, 11331, 36520, 48812, 25513, 5296, 15042, 297, 6055, 4587, 2407, 1211, 5296, 259, 5194, 507, 21136, 995, 45865, 441, 7251, 29399, 2304, 1829, 30731, 497, 37495, 22653, 8198, 294, 497, 13412, 2407, 3714, 9566, 1211, 37746], "avg_logprob": -0.5292968812088171, "compression_ratio": 1.1818181818181819, "no_speech_prob": 0.0, "words": [{"start": 134.78, "end": 135.78, "word": "ب", "probability": 0.1190185546875}, {"start": 135.78, "end": 136.78, "word": " such", "probability": 0.1473388671875}, {"start": 136.78, "end": 137.78, "word": " that", "probability": 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" لinfinity", "probability": 0.5810546875}, {"start": 147.7, "end": 151.98, "word": " الساوي", "probability": 0.684326171875}, {"start": 151.98, "end": 152.7, "word": " L", "probability": 0.206298828125}, {"start": 152.7, "end": 154.88, "word": " ينتمي", "probability": 0.85693359375}, {"start": 154.88, "end": 155.2, "word": " إلى", "probability": 0.2056884765625}, {"start": 155.2, "end": 155.46, "word": " R", "probability": 0.64892578125}, {"start": 155.46, "end": 155.68, "word": " يعني", "probability": 0.775146484375}, {"start": 155.68, "end": 156.2, "word": " exists", "probability": 0.5595703125}, {"start": 156.2, "end": 156.48, "word": " in", "probability": 0.6513671875}, {"start": 156.48, "end": 156.9, "word": " R", "probability": 0.96923828125}, {"start": 156.9, "end": 160.62, "word": " شو", "probability": 0.616455078125}, {"start": 160.62, "end": 161.8, "word": " مطلوب", "probability": 0.93798828125}], "temperature": 1.0}, {"id": 6, "seek": 19187, "start": 162.93, "end": 191.87, "text": "ثم اثبت اظهر ان سيكوينس ين يتعامل بالزيرو حل السؤال هذا بعتمد", "tokens": [12984, 2304, 1975, 12984, 3555, 2655, 1975, 19913, 3224, 2288, 16472, 8608, 1829, 4117, 2407, 9957, 3794, 7251, 1863, 7251, 2655, 3615, 10943, 1211, 20666, 11622, 1829, 32887, 11331, 1211, 21136, 33604, 6027, 23758, 4724, 34268, 2304, 3215], "avg_logprob": -0.6722756318556957, "compression_ratio": 1.2758620689655173, "no_speech_prob": 0.0, "words": [{"start": 162.93, "end": 163.23, "word": "ثم", "probability": 0.376953125}, {"start": 163.23, "end": 163.63, "word": " اثبت", "probability": 0.633636474609375}, {"start": 163.63, "end": 165.29, "word": " اظهر", "probability": 0.612091064453125}, {"start": 165.29, "end": 166.51, "word": " ان", "probability": 0.69677734375}, {"start": 166.51, "end": 168.29, "word": " سيكوينس", "probability": 0.6178690592447916}, {"start": 168.29, "end": 170.03, "word": " ين", "probability": 0.31781005859375}, {"start": 170.03, "end": 174.41, "word": " يتعامل", "probability": 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"probability": 0.269775390625}, {"start": 212.43, "end": 214.09, "word": " x", "probability": 0.6142578125}, {"start": 214.09, "end": 214.57, "word": " n", "probability": 0.55859375}, {"start": 214.57, "end": 215.23, "word": " أكبر", "probability": 0.8523763020833334}, {"start": 215.23, "end": 215.47, "word": " من", "probability": 0.978515625}, {"start": 215.47, "end": 216.09, "word": " سفر", "probability": 0.7980143229166666}, {"start": 216.09, "end": 217.87, "word": " لكل", "probability": 0.97021484375}, {"start": 217.87, "end": 218.35, "word": " n", "probability": 0.736328125}, {"start": 218.35, "end": 220.19, "word": " عدد", "probability": 0.9072265625}, {"start": 220.19, "end": 220.95, "word": " طبيعي", "probability": 0.9246826171875}], "temperature": 1.0}, {"id": 8, "seek": 25035, "start": 222.63, "end": 250.35, "text": "و ال then limit xn بساوي zero if and only if limit واحد على xn as n tends to infinity بساوي plus infinity", "tokens": [2407, 2423, 550, 4948, 2031, 77, 4724, 3794, 995, 45865, 4018, 498, 293, 787, 498, 4948, 36764, 24401, 15844, 2031, 77, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 1804, 13202], "avg_logprob": -0.3169981060606061, "compression_ratio": 1.2019230769230769, "no_speech_prob": 0.0, "words": [{"start": 222.63, "end": 223.21, "word": "و", "probability": 0.4990234375}, {"start": 223.21, "end": 224.15, "word": " ال", "probability": 0.1884765625}, {"start": 224.15, "end": 230.31, "word": " then", "probability": 0.28662109375}, {"start": 230.31, "end": 231.91, "word": " limit", "probability": 0.9658203125}, {"start": 231.91, "end": 234.23, "word": " xn", "probability": 0.5853271484375}, {"start": 234.23, "end": 236.15, "word": " بساوي", "probability": 0.68658447265625}, {"start": 236.15, "end": 236.83, "word": " zero", "probability": 0.295654296875}, {"start": 236.83, "end": 237.45, "word": " if", "probability": 0.82080078125}, {"start": 237.45, "end": 238.83, "word": " and", "probability": 0.93505859375}, {"start": 238.83, "end": 239.17, "word": " only", "probability": 0.84765625}, {"start": 239.17, "end": 239.65, "word": " if", "probability": 0.97998046875}, {"start": 239.65, "end": 241.99, "word": " limit", "probability": 0.97802734375}, {"start": 241.99, "end": 243.77, "word": " واحد", "probability": 0.875244140625}, {"start": 243.77, "end": 243.99, "word": " على", "probability": 0.5302734375}, {"start": 243.99, "end": 245.03, "word": " xn", "probability": 0.84814453125}, {"start": 245.03, "end": 245.75, "word": " as", "probability": 0.78173828125}, {"start": 245.75, "end": 246.19, "word": " n", "probability": 0.5478515625}, {"start": 246.19, "end": 246.51, "word": " tends", "probability": 0.509765625}, {"start": 246.51, "end": 246.67, "word": " to", "probability": 0.88330078125}, {"start": 246.67, "end": 247.27, "word": " infinity", "probability": 0.89013671875}, {"start": 247.27, "end": 249.17, "word": " بساوي", "probability": 0.968017578125}, {"start": 249.17, "end": 249.75, "word": " plus", "probability": 0.9384765625}, {"start": 249.75, "end": 250.35, "word": " infinity", "probability": 0.88671875}], "temperature": 1.0}, {"id": 9, "seek": 28219, "start": 260.79, "end": 282.19, "text": "Okay لأن في سؤال طلعتها إذا كانت xn حدود sequence حدودها موجة بقى و ف limit ال sequence xn بساوي سفر if and only if limit مقلوب ال sequence xn بساوي plus infinity", "tokens": [8297, 5296, 33456, 8978, 8608, 33604, 6027, 23032, 1211, 34268, 11296, 11933, 15730, 25961, 2655, 2031, 77, 11331, 3215, 23328, 8310, 11331, 3215, 23328, 11296, 3714, 29245, 3660, 4724, 4587, 7578, 4032, 6156, 4948, 2423, 8310, 2031, 77, 4724, 3794, 995, 45865, 8608, 5172, 2288, 498, 293, 787, 498, 4948, 3714, 4587, 1211, 37746, 2423, 8310, 2031, 77, 4724, 3794, 995, 45865, 1804, 13202], "avg_logprob": -0.20264423076923077, "compression_ratio": 1.4516129032258065, "no_speech_prob": 0.0, "words": [{"start": 260.78999999999996, "end": 261.83, "word": "Okay", "probability": 0.08160400390625}, {"start": 261.83, "end": 262.87, "word": " لأن", "probability": 0.523193359375}, {"start": 262.87, "end": 264.45, "word": " في", "probability": 0.8837890625}, {"start": 264.45, "end": 264.81, "word": " سؤال", "probability": 0.9925130208333334}, {"start": 264.81, "end": 265.47, "word": " طلعتها", "probability": 0.77435302734375}, {"start": 265.47, "end": 265.69, "word": " إذا", "probability": 0.875}, {"start": 265.69, "end": 266.05, "word": " كانت", "probability": 0.92529296875}, {"start": 266.05, "end": 266.55, "word": " xn", "probability": 0.3482666015625}, {"start": 266.55, "end": 266.99, "word": " حدود", "probability": 0.9791666666666666}, {"start": 266.99, "end": 267.67, "word": " sequence", "probability": 0.38232421875}, {"start": 267.67, "end": 269.21, "word": " حدودها", "probability": 0.970947265625}, {"start": 269.21, "end": 269.63, "word": " موجة", "probability": 0.9737955729166666}, {"start": 269.63, "end": 270.19, "word": " بقى", "probability": 0.9866536458333334}, {"start": 270.19, "end": 271.77, "word": " و", "probability": 0.6015625}, {"start": 271.77, "end": 272.83, "word": " ف", "probability": 0.826171875}, {"start": 272.83, "end": 273.35, "word": " limit", "probability": 0.9296875}, {"start": 273.35, "end": 273.65, "word": " ال", "probability": 0.4072265625}, {"start": 273.65, "end": 274.09, "word": " sequence", "probability": 0.990234375}, {"start": 274.09, "end": 274.83, "word": " xn", "probability": 0.94140625}, {"start": 274.83, "end": 275.65, "word": " بساوي", "probability": 0.81396484375}, {"start": 275.65, "end": 276.29, "word": " سفر", "probability": 0.8836263020833334}, {"start": 276.29, "end": 277.15, "word": " if", "probability": 0.875}, {"start": 277.15, "end": 277.43, "word": " and", "probability": 0.9306640625}, {"start": 277.43, "end": 277.73, "word": " only", "probability": 0.91552734375}, {"start": 277.73, "end": 278.09, "word": " if", "probability": 0.9794921875}, {"start": 278.09, "end": 278.95, "word": " limit", "probability": 0.9580078125}, {"start": 278.95, "end": 279.55, "word": " مقلوب", "probability": 0.9544677734375}, {"start": 279.55, "end": 279.71, "word": " ال", "probability": 0.8505859375}, {"start": 279.71, "end": 280.11, "word": " sequence", "probability": 0.990234375}, {"start": 280.11, "end": 280.75, "word": " xn", "probability": 0.970703125}, {"start": 280.75, "end": 281.35, "word": " بساوي", "probability": 0.9539794921875}, {"start": 281.35, "end": 281.73, "word": " plus", "probability": 0.9345703125}, {"start": 281.73, "end": 282.19, "word": " infinity", "probability": 0.7666015625}], "temperature": 1.0}, {"id": 10, "seek": 30397, "start": 286.23, "end": 303.97, "text": "و طبعا في كمان ممكن نثبت ان لو كانت ال Xn حدودها سالبة ف limit Xn بساوي صفر F and only F limit واحد على Xn بساوي negative infinity", "tokens": [2407, 23032, 3555, 3615, 995, 8978, 9122, 2304, 7649, 3714, 43020, 8717, 12984, 3555, 2655, 16472, 45164, 25961, 2655, 2423, 1783, 77, 11331, 3215, 23328, 11296, 8608, 6027, 49401, 6156, 4948, 1783, 77, 4724, 3794, 995, 45865, 20328, 5172, 2288, 479, 293, 787, 479, 4948, 36764, 24401, 15844, 1783, 77, 4724, 3794, 995, 45865, 3671, 13202], "avg_logprob": -0.2554824477747867, "compression_ratio": 1.3450704225352113, "no_speech_prob": 0.0, "words": [{"start": 286.23, "end": 286.51, "word": "و", "probability": 0.53564453125}, {"start": 286.51, "end": 286.93, "word": " طبعا", "probability": 0.857666015625}, {"start": 286.93, "end": 287.17, "word": " في", "probability": 0.638671875}, {"start": 287.17, "end": 288.43, "word": " كمان", "probability": 0.8810221354166666}, {"start": 288.43, "end": 288.79, "word": " ممكن", "probability": 0.95263671875}, {"start": 288.79, "end": 289.41, "word": " نثبت", "probability": 0.99169921875}, {"start": 289.41, "end": 289.59, "word": " ان", "probability": 0.54638671875}, {"start": 289.59, "end": 289.81, "word": " لو", "probability": 0.9248046875}, {"start": 289.81, "end": 290.95, "word": " كانت", "probability": 0.96484375}, {"start": 290.95, "end": 291.23, "word": " ال", "probability": 0.8505859375}, {"start": 291.23, "end": 291.89, "word": " Xn", "probability": 0.34814453125}, {"start": 291.89, "end": 292.69, "word": " حدودها", "probability": 0.9915771484375}, {"start": 292.69, "end": 293.33, "word": " سالبة", "probability": 0.89599609375}, {"start": 293.33, "end": 294.31, "word": " ف", "probability": 0.779296875}, {"start": 294.31, "end": 297.23, "word": " limit", "probability": 0.83203125}, {"start": 297.23, "end": 297.77, "word": " Xn", "probability": 0.845458984375}, {"start": 297.77, "end": 298.27, "word": " بساوي", "probability": 0.69769287109375}, {"start": 298.27, "end": 298.71, "word": " صفر", "probability": 0.7724609375}, {"start": 298.71, "end": 298.93, "word": " F", "probability": 0.53466796875}, {"start": 298.93, "end": 299.17, "word": " and", "probability": 0.69873046875}, {"start": 299.17, "end": 299.41, "word": " only", "probability": 0.8193359375}, {"start": 299.41, "end": 299.75, "word": " F", "probability": 0.712890625}, {"start": 299.75, "end": 300.47, "word": " limit", "probability": 0.763671875}, {"start": 300.47, "end": 300.95, "word": " واحد", "probability": 0.7227783203125}, {"start": 300.95, "end": 301.11, "word": " على", "probability": 0.70654296875}, {"start": 301.11, "end": 301.57, "word": " Xn", "probability": 0.963623046875}, {"start": 301.57, "end": 302.21, "word": " بساوي", "probability": 0.9517822265625}, {"start": 302.21, "end": 302.65, "word": " negative", "probability": 0.90087890625}, {"start": 302.65, "end": 303.97, "word": " infinity", "probability": 0.299560546875}], "temperature": 1.0}, {"id": 11, "seek": 34018, "start": 315.22, "end": 340.18, "text": "بما أن xn هو بشكل صحيح ديبيرزينت ثم قيمة xn بساوي إفينتي أو قيمة xn بساوي نيجاتيف إفينتي", "tokens": [3555, 15042, 14739, 2031, 77, 31439, 4724, 8592, 28820, 20328, 5016, 1829, 5016, 11778, 1829, 3555, 13546, 11622, 9957, 2655, 38637, 2304, 12174, 32640, 3660, 2031, 77, 4724, 3794, 995, 45865, 11933, 5172, 9957, 31371, 34051, 12174, 32640, 3660, 2031, 77, 4724, 3794, 995, 45865, 8717, 1829, 7435, 9307, 33911, 11933, 5172, 9957, 31371], "avg_logprob": -0.566761337627064, "compression_ratio": 1.4854368932038835, "no_speech_prob": 0.0, "words": [{"start": 315.22, "end": 315.96, "word": "بما", "probability": 0.411376953125}, {"start": 315.96, "end": 316.0, "word": " أن", "probability": 0.67822265625}, {"start": 316.0, "end": 318.36, "word": " xn", "probability": 0.435791015625}, {"start": 318.36, "end": 319.8, "word": " هو", "probability": 0.304443359375}, {"start": 319.8, "end": 320.48, "word": " بشكل", "probability": 0.5055135091145834}, {"start": 320.48, "end": 321.46, "word": " صحيح", "probability": 0.8748779296875}, {"start": 321.46, "end": 324.48, "word": " ديبيرزينت", "probability": 0.4816981724330357}, {"start": 324.48, "end": 327.1, "word": " ثم", "probability": 0.56439208984375}, {"start": 327.1, "end": 329.34, "word": " قيمة", "probability": 0.4974772135416667}, {"start": 329.34, "end": 331.5, "word": " xn", "probability": 0.9013671875}, {"start": 331.5, "end": 332.4, "word": " بساوي", "probability": 0.5351409912109375}, {"start": 332.4, "end": 333.48, "word": " إفينتي", "probability": 0.276153564453125}, {"start": 333.48, "end": 334.8, "word": " أو", "probability": 0.7724609375}, {"start": 334.8, "end": 336.4, "word": " قيمة", "probability": 0.97705078125}, {"start": 336.4, "end": 337.98, "word": " xn", "probability": 0.928955078125}, {"start": 337.98, "end": 338.96, "word": " بساوي", "probability": 0.955078125}, {"start": 338.96, "end": 339.58, "word": " نيجاتيف", "probability": 0.580126953125}, {"start": 339.58, "end": 340.18, "word": " إفينتي", "probability": 0.91845703125}], "temperature": 1.0}, {"id": 12, "seek": 37154, "start": 345.46, "end": 371.54, "text": "case one ناخد الحالة الأولى اللى فيها limit xm بساوي infinity by exercise رقم تلاتة section تلاتة ستة", "tokens": [9765, 472, 8717, 47283, 3215, 21542, 6027, 3660, 16247, 12610, 7578, 13672, 7578, 8978, 11296, 4948, 2031, 76, 4724, 3794, 995, 45865, 13202, 538, 5380, 12602, 4587, 2304, 6055, 1211, 9307, 3660, 3541, 6055, 1211, 9307, 3660, 8608, 2655, 3660], "avg_logprob": -0.25114329849801414, "compression_ratio": 1.2066115702479339, "no_speech_prob": 0.0, "words": [{"start": 345.46, "end": 346.0, "word": "case", "probability": 0.0467529296875}, {"start": 346.0, "end": 346.46, "word": " one", "probability": 0.49365234375}, {"start": 346.46, "end": 346.92, "word": " ناخد", "probability": 0.75390625}, {"start": 346.92, "end": 347.4, "word": " الحالة", "probability": 0.94384765625}, {"start": 347.4, "end": 348.82, "word": " الأولى", "probability": 0.96728515625}, {"start": 348.82, "end": 349.4, "word": " اللى", "probability": 0.833740234375}, {"start": 349.4, "end": 349.76, "word": " فيها", "probability": 0.826171875}, {"start": 349.76, "end": 350.24, "word": " limit", "probability": 0.95556640625}, {"start": 350.24, "end": 352.22, "word": " xm", "probability": 0.508544921875}, {"start": 352.22, "end": 353.66, "word": " بساوي", "probability": 0.684783935546875}, {"start": 353.66, "end": 354.36, "word": " infinity", "probability": 0.67578125}, {"start": 354.36, "end": 359.86, "word": " by", "probability": 0.82763671875}, {"start": 359.86, "end": 363.56, "word": " exercise", "probability": 0.96484375}, {"start": 363.56, "end": 366.06, "word": " رقم", "probability": 0.9259440104166666}, {"start": 366.06, "end": 367.02, "word": " تلاتة", "probability": 0.9378662109375}, {"start": 367.02, "end": 369.94, "word": " section", "probability": 0.7626953125}, {"start": 369.94, "end": 370.78, "word": " تلاتة", "probability": 0.94580078125}, {"start": 370.78, "end": 371.54, "word": " ستة", "probability": 0.9484049479166666}], "temperature": 1.0}, {"id": 13, "seek": 39984, "start": 373.22, "end": 399.84, "text": "والـ exercise اللى فوق هذا معناه انه we have هيطلع انه limit مطلوب ال sequence xn as n tends to infinity بفلع صفر يعني اعتبرى هذه هي xn", "tokens": [2407, 6027, 39184, 5380, 13672, 7578, 6156, 30543, 23758, 20449, 8315, 3224, 16472, 3224, 321, 362, 8032, 1829, 9566, 1211, 3615, 16472, 3224, 4948, 3714, 9566, 1211, 37746, 2423, 8310, 2031, 77, 382, 297, 12258, 281, 13202, 4724, 5172, 1211, 3615, 20328, 5172, 2288, 37495, 22653, 1975, 34268, 26890, 7578, 29538, 39896, 2031, 77], "avg_logprob": -0.4562499859116294, "compression_ratio": 1.2847682119205297, "no_speech_prob": 0.0, "words": [{"start": 373.22, "end": 373.76, "word": "والـ", "probability": 0.3745524088541667}, {"start": 373.76, "end": 374.3, "word": " exercise", "probability": 0.77294921875}, {"start": 374.3, "end": 374.56, "word": " اللى", "probability": 0.7039794921875}, {"start": 374.56, "end": 375.02, "word": " فوق", "probability": 0.963134765625}, {"start": 375.02, "end": 378.1, "word": " هذا", "probability": 0.708984375}, {"start": 378.1, "end": 378.84, "word": " معناه", "probability": 0.9510091145833334}, {"start": 378.84, "end": 380.32, "word": " انه", "probability": 0.701416015625}, {"start": 380.32, "end": 381.0, "word": " we", "probability": 0.35986328125}, {"start": 381.0, "end": 382.16, "word": " have", "probability": 0.95556640625}, {"start": 382.16, "end": 384.3, "word": " هيطلع", "probability": 0.753271484375}, {"start": 384.3, "end": 384.54, "word": " انه", "probability": 0.33642578125}, {"start": 384.54, "end": 384.98, "word": " limit", "probability": 0.368896484375}, {"start": 384.98, "end": 386.96, "word": " مطلوب", "probability": 0.8062744140625}, {"start": 386.96, "end": 387.16, "word": " ال", "probability": 0.4267578125}, {"start": 387.16, "end": 387.72, "word": " sequence", "probability": 0.859375}, {"start": 387.72, "end": 388.6, "word": " xn", "probability": 0.4425048828125}, {"start": 388.6, "end": 390.28, "word": " as", "probability": 0.791015625}, {"start": 390.28, "end": 390.68, "word": " n", "probability": 0.67724609375}, {"start": 390.68, "end": 390.94, "word": " tends", "probability": 0.720703125}, {"start": 390.94, "end": 391.1, "word": " to", "probability": 0.89111328125}, {"start": 391.1, "end": 391.58, "word": " infinity", "probability": 0.92431640625}, {"start": 391.58, "end": 392.06, "word": " بفلع", "probability": 0.438873291015625}, {"start": 392.06, "end": 392.42, "word": " صفر", "probability": 0.7787272135416666}, {"start": 392.42, "end": 398.0, "word": " يعني", "probability": 0.869873046875}, {"start": 398.0, "end": 398.54, "word": " اعتبرى", "probability": 0.77520751953125}, {"start": 398.54, "end": 398.84, "word": " هذه", "probability": 0.6953125}, {"start": 398.84, "end": 399.14, "word": " هي", "probability": 0.80419921875}, {"start": 399.14, "end": 399.84, "word": " xn", "probability": 0.888427734375}], "temperature": 1.0}, {"id": 14, "seek": 42869, "start": 401.05, "end": 428.69, "text": "تعتبر ال 1 على xn هي xn فإذا كان limit xn بساوي infinity فlimit مقلوب ال xn اللي هنا مقلوب اللي هو ايه بتطلع سفر ولا عكس يعني هنا نفس ال exercise بس badly xn بواحد على xn فهذه نتيجة صحية تمام hence", "tokens": [2655, 34268, 26890, 2423, 502, 15844, 2031, 77, 39896, 2031, 77, 6156, 28814, 15730, 25961, 4948, 2031, 77, 4724, 3794, 995, 45865, 13202, 6156, 4197, 270, 3714, 4587, 1211, 37746, 2423, 2031, 77, 13672, 1829, 34105, 3714, 4587, 1211, 37746, 13672, 1829, 31439, 1975, 1829, 3224, 39894, 9566, 1211, 3615, 8608, 5172, 2288, 49429, 6225, 4117, 3794, 37495, 22653, 34105, 8717, 36178, 2423, 5380, 4724, 3794, 13425, 2031, 77, 4724, 14407, 24401, 15844, 2031, 77, 6156, 3224, 24192, 8717, 31371, 7435, 3660, 20328, 5016, 10632, 46811, 10943, 16678], "avg_logprob": -0.34919240501489535, "compression_ratio": 1.5380710659898478, "no_speech_prob": 0.0, "words": [{"start": 401.05, "end": 401.63, "word": "تعتبر", "probability": 0.6489054361979166}, {"start": 401.63, "end": 401.93, "word": " ال", "probability": 0.482666015625}, {"start": 401.93, "end": 402.15, "word": " 1", "probability": 0.1944580078125}, {"start": 402.15, "end": 402.37, "word": " على", "probability": 0.69091796875}, {"start": 402.37, "end": 402.71, "word": " xn", "probability": 0.624267578125}, {"start": 402.71, "end": 402.93, "word": " هي", "probability": 0.8525390625}, {"start": 402.93, "end": 403.45, "word": " xn", "probability": 0.91552734375}, {"start": 403.45, "end": 404.43, "word": " فإذا", "probability": 0.6809895833333334}, {"start": 404.43, "end": 404.71, "word": " كان", "probability": 0.984375}, {"start": 404.71, "end": 405.01, "word": " limit", "probability": 0.78076171875}, {"start": 405.01, "end": 406.03, "word": " xn", "probability": 0.947265625}, {"start": 406.03, "end": 406.53, "word": " بساوي", "probability": 0.560302734375}, {"start": 406.53, "end": 407.07, "word": " infinity", "probability": 0.732421875}, {"start": 407.07, "end": 408.17, "word": " فlimit", "probability": 0.83251953125}, {"start": 408.17, "end": 408.67, "word": " مقلوب", "probability": 0.9063720703125}, {"start": 408.67, "end": 408.83, "word": " ال", "probability": 0.415771484375}, {"start": 408.83, "end": 409.35, "word": " xn", 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{"start": 424.77, "end": 426.15, "word": " تمام", "probability": 0.75537109375}, {"start": 426.15, "end": 428.69, "word": " hence", "probability": 0.775390625}], "temperature": 1.0}, {"id": 15, "seek": 45931, "start": 433.03, "end": 459.31, "text": "الـ limit ل YN as intense infinity بساوي ال limit ال YN ممكن كتبتها على صورة على صورة", "tokens": [6027, 39184, 4948, 5296, 398, 45, 382, 9447, 13202, 4724, 3794, 995, 45865, 2423, 4948, 2423, 398, 45, 3714, 43020, 9122, 2655, 3555, 2655, 11296, 15844, 20328, 13063, 3660, 15844, 20328, 13063, 3660], "avg_logprob": -0.3667279385468539, "compression_ratio": 1.1730769230769231, "no_speech_prob": 0.0, "words": [{"start": 433.03, "end": 433.41, "word": "الـ", "probability": 0.514404296875}, {"start": 433.41, "end": 433.75, "word": " limit", "probability": 0.64697265625}, {"start": 433.75, "end": 436.81, "word": " ل", "probability": 0.46923828125}, {"start": 436.81, "end": 438.59, "word": " YN", "probability": 0.28717041015625}, {"start": 438.59, "end": 440.73, "word": " as", "probability": 0.61767578125}, {"start": 440.73, "end": 441.47, "word": " intense", "probability": 0.2119140625}, {"start": 441.47, "end": 442.11, "word": " infinity", "probability": 0.50146484375}, {"start": 442.11, "end": 442.99, "word": " بساوي", "probability": 0.6693115234375}, {"start": 442.99, "end": 443.15, "word": " ال", "probability": 0.52685546875}, {"start": 443.15, "end": 444.01, "word": " limit", "probability": 0.5947265625}, {"start": 444.01, "end": 449.29, "word": " ال", "probability": 0.86279296875}, {"start": 449.29, "end": 449.97, "word": " YN", "probability": 0.81640625}, {"start": 449.97, "end": 450.39, "word": " ممكن", "probability": 0.985595703125}, {"start": 450.39, "end": 451.29, "word": " كتبتها", "probability": 0.95546875}, {"start": 451.29, "end": 451.65, "word": " على", "probability": 0.90234375}, {"start": 451.65, "end": 452.67, "word": " صورة", "probability": 0.9388020833333334}, {"start": 452.67, "end": 458.11, "word": " على", "probability": 0.5703125}, {"start": 458.11, "end": 459.31, "word": " صورة", "probability": 0.9923502604166666}], "temperature": 1.0}, {"id": 16, "seek": 48985, "start": 466.21, "end": 489.85, "text": "xn في yn ضرب 1 على xn صح نظبط هيك ال yn هي عبارة عن xn في yn في 1 على xn", "tokens": [87, 77, 8978, 17861, 48812, 25513, 502, 15844, 2031, 77, 20328, 5016, 8717, 19913, 3555, 9566, 39896, 4117, 2423, 17861, 39896, 6225, 3555, 9640, 3660, 18871, 2031, 77, 8978, 17861, 8978, 502, 15844, 2031, 77], "avg_logprob": -0.3118489682674408, "compression_ratio": 1.244186046511628, "no_speech_prob": 0.0, "words": [{"start": 466.21, "end": 467.33, "word": "xn", "probability": 0.197113037109375}, {"start": 467.33, "end": 467.99, "word": " في", "probability": 0.63671875}, {"start": 467.99, "end": 468.69, "word": " yn", "probability": 0.8056640625}, {"start": 468.69, "end": 471.61, "word": " ضرب", "probability": 0.83984375}, {"start": 471.61, "end": 475.77, "word": " 1", "probability": 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488.47, "word": " yn", "probability": 0.97021484375}, {"start": 488.47, "end": 488.73, "word": " في", "probability": 0.9794921875}, {"start": 488.73, "end": 489.03, "word": " 1", "probability": 0.92578125}, {"start": 489.03, "end": 489.41, "word": " على", "probability": 0.88671875}, {"start": 489.41, "end": 489.85, "word": " xn", "probability": 0.990478515625}], "temperature": 1.0}, {"id": 17, "seek": 51588, "start": 492.88, "end": 515.88, "text": "الان ال limit هذه لحد الأول exist و limit ل واحد على xn برضه exist اذا ال limit حاصل ضرب بساوي حاصل ضرب ال limits بقدر استخدم القانون هذا هطبق انه limit حاصل ضرب two sequences بساوي limit الأولى اللي هي حاصل ضرب xn yn", "tokens": [6027, 7649, 2423, 4948, 29538, 5296, 24401, 16247, 12610, 2514, 4032, 4948, 5296, 36764, 24401, 15844, 2031, 77, 4724, 43042, 3224, 2514, 1975, 15730, 2423, 4948, 11331, 33546, 1211, 48812, 25513, 4724, 3794, 995, 45865, 11331, 33546, 1211, 48812, 25513, 2423, 10406, 4724, 28543, 2288, 44713, 9778, 40448, 25062, 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0.232666015625}], "temperature": 1.0}, {"id": 18, "seek": 54591, "start": 517.96, "end": 545.92, "text": "درب limit الـ sequence التانية هي واحد على X end as n tends to infinity و ال limit الأولى مش سامناها عدد L لما exist ضرب ال limit التانية سفر فبطلع عندي سفر و هو المطلوب فهنا أثبتنا في الحالة التانية case two", "tokens": [3215, 25513, 4948, 2423, 39184, 8310, 16712, 7649, 10632, 39896, 36764, 24401, 15844, 1783, 917, 382, 297, 12258, 281, 13202, 4032, 2423, 4948, 16247, 12610, 7578, 37893, 8608, 10943, 8315, 11296, 6225, 3215, 3215, 441, 5296, 15042, 2514, 48812, 25513, 2423, 4948, 16712, 7649, 10632, 8608, 5172, 2288, 6156, 3555, 9566, 1211, 3615, 18871, 16254, 8608, 5172, 2288, 4032, 31439, 9673, 9566, 1211, 37746, 6156, 3224, 8315, 5551, 12984, 3555, 2655, 8315, 8978, 21542, 6027, 3660, 16712, 7649, 10632, 1389, 732], "avg_logprob": -0.37423780996625017, "compression_ratio": 1.5879396984924623, "no_speech_prob": 0.0, "words": [{"start": 517.96, "end": 518.48, "word": "درب", 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{"start": 526.04, "end": 526.2, "word": " to", "probability": 0.828125}, {"start": 526.2, "end": 526.68, "word": " infinity", "probability": 0.8486328125}, {"start": 526.68, "end": 528.18, "word": " و", "probability": 0.3330078125}, {"start": 528.18, "end": 528.44, "word": " ال", "probability": 0.50341796875}, {"start": 528.44, "end": 528.72, "word": " limit", "probability": 0.8046875}, {"start": 528.72, "end": 529.42, "word": " الأولى", "probability": 0.9314778645833334}, {"start": 529.42, "end": 529.74, "word": " مش", "probability": 0.172119140625}, {"start": 529.74, "end": 530.74, "word": " سامناها", "probability": 0.7557373046875}, {"start": 530.74, "end": 531.18, "word": " عدد", "probability": 0.9747721354166666}, {"start": 531.18, "end": 531.46, "word": " L", "probability": 0.460205078125}, {"start": 531.46, "end": 531.78, "word": " لما", "probability": 0.85498046875}, {"start": 531.78, "end": 532.28, "word": " exist", "probability": 0.80517578125}, {"start": 532.28, "end": 532.94, "word": " ضرب", "probability": 0.818115234375}, {"start": 532.94, "end": 533.94, "word": " ال", "probability": 0.794921875}, {"start": 533.94, "end": 534.18, "word": " limit", "probability": 0.734375}, {"start": 534.18, "end": 534.76, "word": " التانية", "probability": 0.916015625}, {"start": 534.76, "end": 535.36, "word": " سفر", "probability": 0.8424479166666666}, {"start": 535.36, "end": 536.04, "word": " فبطلع", "probability": 0.86396484375}, {"start": 536.04, "end": 536.38, "word": " عندي", "probability": 0.637939453125}, {"start": 536.38, "end": 536.98, "word": " سفر", "probability": 0.9059244791666666}, {"start": 536.98, "end": 537.58, "word": " و", "probability": 0.464599609375}, {"start": 537.58, "end": 537.78, "word": " هو", "probability": 0.8681640625}, {"start": 537.78, "end": 538.62, "word": " المطلوب", "probability": 0.9791259765625}, {"start": 538.62, "end": 541.6, "word": " فهنا", "probability": 0.5892333984375}, {"start": 541.6, "end": 542.1, "word": " أثبتنا", "probability": 0.96259765625}, {"start": 542.1, "end": 542.24, "word": " في", "probability": 0.876953125}, {"start": 542.24, "end": 542.72, "word": " الحالة", "probability": 0.9798177083333334}, {"start": 542.72, "end": 544.86, "word": " التانية", "probability": 0.78515625}, {"start": 544.86, "end": 545.46, "word": " case", "probability": 0.8447265625}, {"start": 545.46, "end": 545.92, "word": " two", "probability": 0.50927734375}], "temperature": 1.0}, {"id": 19, "seek": 57638, "start": 550.14, "end": 576.38, "text": "لو كانت ال limit لـ xn بساوي negative infinity ففي الحالة هذه بيطلع عندي برضه by exercise تلاتة section تلاتة ستة بس هنا مع التعديل هيطلع ان ال limit", "tokens": [1211, 2407, 25961, 2655, 2423, 4948, 5296, 39184, 2031, 77, 4724, 3794, 995, 45865, 3671, 13202, 6156, 41185, 21542, 6027, 3660, 29538, 4724, 1829, 9566, 1211, 3615, 18871, 16254, 4724, 43042, 3224, 538, 5380, 6055, 1211, 9307, 3660, 3541, 6055, 1211, 9307, 3660, 8608, 2655, 3660, 4724, 3794, 34105, 20449, 16712, 3615, 16254, 1211, 39896, 9566, 1211, 3615, 16472, 2423, 4948], "avg_logprob": -0.285534274674231, "compression_ratio": 1.40625, "no_speech_prob": 0.0, "words": [{"start": 550.14, "end": 550.54, "word": "لو", "probability": 0.6339111328125}, {"start": 550.54, "end": 550.98, "word": " كانت", "probability": 0.927490234375}, {"start": 550.98, "end": 551.08, "word": " ال", "probability": 0.338134765625}, {"start": 551.08, "end": 551.4, "word": " limit", "probability": 0.642578125}, {"start": 551.4, "end": 551.92, "word": " لـ", "probability": 0.3499755859375}, {"start": 551.92, "end": 552.68, "word": " xn", "probability": 0.442138671875}, {"start": 552.68, "end": 554.4, "word": " بساوي", "probability": 0.6470947265625}, {"start": 554.4, "end": 554.86, "word": " negative", "probability": 0.70849609375}, {"start": 554.86, "end": 555.66, "word": " infinity", "probability": 0.89208984375}, {"start": 555.66, "end": 564.2, "word": " ففي", "probability": 0.605712890625}, {"start": 564.2, 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589.23, "end": 590.35, "word": " rest", "probability": 0.94384765625}, {"start": 590.35, "end": 591.67, "word": " of", "probability": 0.96435546875}, {"start": 591.67, "end": 593.33, "word": " the", "probability": 0.80615234375}, {"start": 593.33, "end": 593.87, "word": " proof", "probability": 0.9580078125}, {"start": 593.87, "end": 594.83, "word": " is", "probability": 0.892578125}, {"start": 594.83, "end": 595.45, "word": " similar", "probability": 0.94970703125}, {"start": 595.45, "end": 598.73, "word": " to", "probability": 0.6015625}, {"start": 598.73, "end": 599.15, "word": " case", "probability": 0.90478515625}, {"start": 599.15, "end": 599.45, "word": " one", "probability": 0.7763671875}], "temperature": 1.0}, {"id": 21, "seek": 62480, "start": 603.65, "end": 624.81, "text": "Okay تمام اذا هذا اللي هو البرهام ان الادكارة تعتمد على exercise ثلاثة المهم وهو ان limit ل sequence بيساوي infinity if and only if limit مقلوب ال sequence بيساوي سفر او لعكس تمام و هذا", "tokens": [8297, 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0.81005859375}, {"start": 615.87, "end": 616.49, "word": " sequence", "probability": 0.75048828125}, {"start": 616.49, "end": 617.03, "word": " بيساوي", "probability": 0.736669921875}, {"start": 617.03, "end": 617.55, "word": " infinity", "probability": 0.81005859375}, {"start": 617.55, "end": 618.31, "word": " if", "probability": 0.93798828125}, {"start": 618.31, "end": 618.51, "word": " and", "probability": 0.93408203125}, {"start": 618.51, "end": 618.69, "word": " only", "probability": 0.91748046875}, {"start": 618.69, "end": 619.03, "word": " if", "probability": 0.97900390625}, {"start": 619.03, "end": 619.75, "word": " limit", "probability": 0.97412109375}, {"start": 619.75, "end": 620.25, "word": " مقلوب", "probability": 0.8194580078125}, {"start": 620.25, "end": 620.39, "word": " ال", "probability": 0.73974609375}, {"start": 620.39, "end": 620.73, "word": " sequence", "probability": 0.9755859375}, {"start": 620.73, "end": 621.15, "word": " بيساوي", "probability": 0.96650390625}, {"start": 621.15, "end": 621.47, "word": " سفر", "probability": 0.8333333333333334}, {"start": 621.47, "end": 621.63, "word": " او", "probability": 0.888916015625}, {"start": 621.63, "end": 622.39, "word": " لعكس", "probability": 0.8690185546875}, {"start": 622.39, "end": 624.77, "word": " تمام", "probability": 0.888427734375}, {"start": 624.77, "end": 624.81, "word": " و", "probability": 0.52685546875}, {"start": 624.81, "end": 624.81, "word": " هذا", "probability": 0.64453125}], "temperature": 1.0}, {"id": 22, "seek": 64967, "start": 634.46, "end": 649.68, "text": "في عنكم أسئلة تانية؟ في أسئلة تانية section تلاتة ستة الفرق بيه من سؤال تسعة", "tokens": [41185, 18871, 24793, 5551, 3794, 19986, 37977, 6055, 7649, 10632, 22807, 8978, 5551, 3794, 19986, 37977, 6055, 7649, 10632, 3541, 6055, 1211, 9307, 3660, 8608, 2655, 3660, 27188, 2288, 4587, 4724, 1829, 3224, 9154, 8608, 33604, 6027, 6055, 3794, 27884], "avg_logprob": -0.22465701510266559, "compression_ratio": 1.3232323232323233, 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"probability": 0.469970703125}, {"start": 694.78, "end": 696.4, "word": " بعدين", "probability": 0.901611328125}, {"start": 696.4, "end": 701.38, "word": " السؤال", "probability": 0.87353515625}, {"start": 701.38, "end": 701.92, "word": " تسعة", "probability": 0.8566080729166666}, {"start": 701.92, "end": 702.42, "word": " section", "probability": 0.5927734375}, {"start": 702.42, "end": 702.92, "word": " تلاتة", "probability": 0.882568359375}, {"start": 702.92, "end": 703.06, "word": " ع", "probability": 0.84619140625}, {"start": 703.06, "end": 703.6, "word": " ستة", "probability": 0.9754231770833334}], "temperature": 1.0}, {"id": 24, "seek": 74244, "start": 713.32, "end": 742.44, "text": "لت XIN و YIN بيكونوا عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين من عاملين", "tokens": [1211, 2655, 1783, 1464, 4032, 398, 1464, 4724, 1829, 30544, 14407, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957, 9154, 6225, 10943, 1211, 9957], "avg_logprob": -0.3168604609578155, "compression_ratio": 5.175438596491228, "no_speech_prob": 0.0, "words": [{"start": 713.32, "end": 713.84, "word": "لت", "probability": 0.384063720703125}, {"start": 713.84, "end": 714.68, "word": " XIN", "probability": 0.3646240234375}, {"start": 714.68, "end": 716.4, "word": " و", "probability": 0.90771484375}, {"start": 716.4, "end": 717.44, "word": " YIN", "probability": 0.7509765625}, {"start": 717.44, "end": 718.5, "word": " بيكونوا", "probability": 0.609375}, {"start": 718.5, "end": 718.9, "word": " عاملين", "probability": 0.40338134765625}, {"start": 718.9, "end": 720.64, "word": " من", "probability": 0.373291015625}, {"start": 720.64, "end": 723.06, "word": " عاملين", "probability": 0.700225830078125}, {"start": 723.06, "end": 723.06, "word": " من", "probability": 0.06561279296875}, {"start": 723.06, "end": 724.38, "word": " عاملين", "probability": 0.941650390625}, {"start": 724.38, "end": 724.4, "word": " من", "probability": 0.477294921875}, {"start": 724.4, "end": 724.44, "word": " عاملين", "probability": 0.96923828125}, {"start": 724.44, "end": 724.46, "word": " من", "probability": 0.6396484375}, {"start": 724.46, "end": 726.72, "word": " عاملين", "probability": 0.978759765625}, {"start": 726.72, "end": 726.72, "word": " من", "probability": 0.71435546875}, {"start": 726.72, "end": 726.8, "word": " عاملين", "probability": 0.9840087890625}, {"start": 726.8, "end": 726.8, "word": " من", "probability": 0.75732421875}, {"start": 726.8, "end": 726.8, "word": " عاملين", "probability": 0.9871826171875}, {"start": 726.8, "end": 726.8, "word": " من", "probability": 0.79345703125}, {"start": 726.8, "end": 726.86, "word": " عاملين", "probability": 0.98876953125}, {"start": 726.86, "end": 726.86, "word": " من", "probability": 0.7958984375}, {"start": 726.86, "end": 726.94, "word": " عاملين", "probability": 0.9906005859375}, {"start": 726.94, "end": 727.02, "word": " من", "probability": 0.8037109375}, {"start": 727.02, "end": 727.12, "word": " عاملين", "probability": 0.9918212890625}, {"start": 727.12, "end": 727.12, "word": " من", "probability": 0.78955078125}, {"start": 727.12, "end": 727.24, "word": " عاملين", "probability": 0.99267578125}, {"start": 727.24, "end": 727.24, "word": " من", "probability": 0.79150390625}, {"start": 727.24, "end": 727.34, "word": " عاملين", "probability": 0.9935302734375}, {"start": 727.34, "end": 727.34, "word": " من", "probability": 0.8037109375}, {"start": 727.34, "end": 727.78, "word": " عاملين", "probability": 0.9940185546875}, {"start": 727.78, "end": 729.1, "word": " من", "probability": 0.81591796875}, {"start": 729.1, "end": 730.8, "word": " عاملين", "probability": 0.99462890625}, {"start": 730.8, "end": 732.86, "word": " من", "probability": 0.845703125}, {"start": 732.86, "end": 742.44, "word": " عاملين", "probability": 0.994873046875}], "temperature": 1.0}, {"id": 25, "seek": 77359, "start": 751.89, "end": 773.59, "text": "مطلوب الأول هو show if limit yn بساوي infinity then limit xn بساوي infinity", "tokens": [2304, 9566, 1211, 37746, 16247, 12610, 31439, 855, 498, 4948, 17861, 4724, 3794, 995, 45865, 13202, 550, 4948, 2031, 77, 4724, 3794, 995, 45865, 13202], "avg_logprob": -0.4477163346914145, "compression_ratio": 1.1975308641975309, "no_speech_prob": 0.0, "words": [{"start": 751.89, "end": 753.29, "word": "مطلوب", "probability": 0.677276611328125}, {"start": 753.29, "end": 753.79, "word": " الأول", "probability": 0.6378173828125}, {"start": 753.79, "end": 754.43, "word": " هو", "probability": 0.1396484375}, {"start": 754.43, "end": 756.73, "word": " show", "probability": 0.43408203125}, {"start": 756.73, "end": 757.29, "word": " if", "probability": 0.8740234375}, {"start": 757.29, "end": 760.03, "word": " limit", "probability": 0.8134765625}, {"start": 760.03, "end": 762.65, "word": " yn", "probability": 0.2452392578125}, {"start": 762.65, "end": 763.55, "word": " بساوي", "probability": 0.697021484375}, {"start": 763.55, "end": 764.27, "word": " infinity", "probability": 0.76318359375}, {"start": 764.27, "end": 767.21, "word": " then", "probability": 0.263916015625}, {"start": 767.21, "end": 771.37, "word": " limit", "probability": 0.93212890625}, {"start": 771.37, "end": 772.27, "word": " xn", "probability": 0.95654296875}, {"start": 772.27, "end": 772.97, "word": " بساوي", "probability": 0.9593505859375}, {"start": 772.97, "end": 773.59, "word": " infinity", "probability": 0.83935546875}], "temperature": 1.0}, {"id": 26, "seek": 80560, "start": 776.82, "end": 805.6, "text": "والجزء التاني show if x in is bounded then limit y in is serviceable طبعا", "tokens": [2407, 6027, 7435, 11622, 38207, 16712, 7649, 1829, 855, 498, 2031, 294, 307, 37498, 550, 4948, 288, 294, 307, 2643, 712, 23032, 3555, 3615, 995], "avg_logprob": -0.28290264308452606, "compression_ratio": 1.0, "no_speech_prob": 0.0, "words": [{"start": 776.82, "end": 777.6, "word": "والجزء", "probability": 0.7919921875}, {"start": 777.6, "end": 778.22, "word": " التاني", "probability": 0.8585611979166666}, {"start": 778.22, "end": 778.72, "word": " show", "probability": 0.72314453125}, {"start": 778.72, "end": 779.46, "word": " if", "probability": 0.89404296875}, {"start": 779.46, "end": 781.58, "word": " x", "probability": 0.71826171875}, {"start": 781.58, "end": 782.08, "word": " in", "probability": 0.6845703125}, {"start": 782.08, "end": 784.24, "word": " is", "probability": 0.6142578125}, {"start": 784.24, "end": 784.78, "word": " bounded", "probability": 0.98046875}, {"start": 784.78, "end": 790.66, "word": " then", "probability": 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tends to infinity بساوي plus infinity then by exercise تلاتة section تلاتة ستة if limit sequence بساوي infinity", "tokens": [87, 77, 15844, 17861, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 1804, 13202, 550, 538, 5380, 6055, 1211, 9307, 3660, 3541, 6055, 1211, 9307, 3660, 8608, 2655, 3660, 498, 4948, 8310, 4724, 3794, 995, 45865, 13202], "avg_logprob": -0.31330127746630937, "compression_ratio": 1.3217391304347825, "no_speech_prob": 0.0, "words": [{"start": 840.29, "end": 841.31, "word": "xn", "probability": 0.29779052734375}, {"start": 841.31, "end": 841.61, "word": " على", "probability": 0.367431640625}, {"start": 841.61, "end": 842.25, "word": " yn", "probability": 0.91064453125}, {"start": 842.25, "end": 844.03, "word": " as", "probability": 0.311767578125}, {"start": 844.03, "end": 844.39, "word": " n", "probability": 0.595703125}, {"start": 844.39, "end": 844.69, "word": " tends", "probability": 0.357177734375}, {"start": 844.69, "end": 844.79, "word": " to", 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"probability": 0.953125}, {"start": 931.96, "end": 933.88, "word": " ففي", "probability": 0.794921875}, {"start": 933.88, "end": 934.32, "word": " الحالة", "probability": 0.97119140625}, {"start": 934.32, "end": 934.8, "word": " هذه", "probability": 0.89990234375}, {"start": 934.8, "end": 936.32, "word": " إذا", "probability": 0.80908203125}, {"start": 936.32, "end": 936.72, "word": " كانت", "probability": 0.96435546875}, {"start": 936.72, "end": 936.98, "word": " limit", "probability": 0.7392578125}, {"start": 936.98, "end": 937.2, "word": " ال", "probability": 0.845703125}, {"start": 937.2, "end": 937.74, "word": " sequence", "probability": 0.92138671875}, {"start": 937.74, "end": 938.04, "word": " اللي", "probability": 0.905029296875}, {"start": 938.04, "end": 938.48, "word": " تحت", "probability": 0.9423828125}, {"start": 938.48, "end": 939.4, "word": " convergent", "probability": 0.621337890625}, {"start": 939.4, "end": 947.2, "word": " إذا", "probability": 0.60516357421875}, {"start": 947.2, "end": 947.6, "word": " كانت", "probability": 0.989501953125}, {"start": 947.6, "end": 947.86, "word": " limit", "probability": 0.94287109375}, {"start": 947.86, "end": 948.08, "word": " ال", "probability": 0.93212890625}, {"start": 948.08, "end": 948.68, "word": " sequence", "probability": 0.99169921875}, {"start": 948.68, "end": 949.68, "word": " اللي", "probability": 0.98583984375}, {"start": 949.68, "end": 950.24, "word": " تحت", "probability": 0.991943359375}], "temperature": 1.0}, {"id": 32, "seek": 96634, "start": 956.1, "end": 966.34, "text": "لأ limit ال sequence اللي فوق اللي هي yn هنا infinity فبطلع limit xn بال 7 infinity اللي هو جزء 11", "tokens": [1211, 10721, 4948, 2423, 8310, 13672, 1829, 6156, 30543, 13672, 1829, 39896, 17861, 34105, 13202, 6156, 3555, 9566, 1211, 3615, 4948, 2031, 77, 20666, 1614, 13202, 13672, 1829, 31439, 10874, 11622, 38207, 2975], "avg_logprob": -0.3421415432411082, "compression_ratio": 1.3366336633663367, "no_speech_prob": 0.0, "words": [{"start": 956.1, "end": 956.56, "word": "لأ", "probability": 0.605712890625}, {"start": 956.56, "end": 956.96, "word": " limit", "probability": 0.240478515625}, {"start": 956.96, "end": 957.18, "word": " ال", "probability": 0.6708984375}, {"start": 957.18, "end": 957.68, "word": " sequence", "probability": 0.8076171875}, {"start": 957.68, "end": 957.98, "word": " اللي", "probability": 0.730712890625}, {"start": 957.98, "end": 959.16, "word": " فوق", "probability": 0.969970703125}, {"start": 959.16, "end": 959.84, "word": " اللي", "probability": 0.765625}, {"start": 959.84, "end": 960.1, "word": " هي", "probability": 0.64501953125}, {"start": 960.1, "end": 961.58, "word": " yn", "probability": 0.1826171875}, {"start": 961.58, "end": 961.96, "word": " هنا", "probability": 0.74755859375}, {"start": 961.96, "end": 962.66, "word": " infinity", "probability": 0.693359375}, {"start": 962.66, "end": 963.74, "word": " فبطلع", "probability": 0.89140625}, {"start": 963.74, "end": 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نبرهنه", "tokens": [2407, 24793, 7649, 1975, 15730, 25961, 2655, 2423, 8310, 13672, 7578, 6156, 7578, 9673, 4587, 10943, 37498, 13672, 7578, 8032, 7578, 2031, 294, 34105, 23032, 3555, 3615, 995, 6156, 7578, 9673, 4587, 10943, 37498, 6156, 2288, 4587, 3660, 2423, 8310, 13672, 7578, 6156, 7578, 2423, 19432, 6055, 9566, 1211, 3615, 7251, 3794, 995, 45865, 1958, 37037, 15730, 31439, 25724, 11622, 38207, 16712, 7649, 7578, 23758, 11331, 35457, 23758, 45164, 47525, 8315, 8717, 14851, 9778, 40448, 5380, 12602, 4587, 2304, 1614, 4032, 9566, 3555, 3615, 995, 5296, 31377, 2304, 8717, 26890, 3224, 1863, 3224], "avg_logprob": -0.15516955468882906, "compression_ratio": 1.7628865979381443, "no_speech_prob": 0.0, "words": [{"start": 968.22, "end": 969.0, "word": "وكمان", "probability": 0.750244140625}, {"start": 969.0, "end": 969.54, "word": " اذا", "probability": 0.6488037109375}, {"start": 969.54, "end": 969.9, "word": " كانت", "probability": 0.989990234375}, {"start": 969.9, "end": 970.06, "word": " ال", "probability": 0.916015625}, {"start": 970.06, "end": 970.64, "word": " sequence", "probability": 0.84619140625}, {"start": 970.64, "end": 971.94, "word": " اللى", "probability": 0.760986328125}, {"start": 971.94, "end": 972.24, "word": " فى", "probability": 0.886962890625}, {"start": 972.24, "end": 972.98, "word": " المقام", "probability": 0.9890950520833334}, {"start": 972.98, "end": 973.62, "word": " bounded", "probability": 0.8955078125}, {"start": 973.62, "end": 973.98, "word": " اللى", "probability": 0.928466796875}, {"start": 973.98, "end": 974.26, "word": " هى", "probability": 0.94287109375}, {"start": 974.26, "end": 974.56, "word": " x", "probability": 0.58984375}, {"start": 974.56, "end": 974.84, "word": " in", "probability": 0.281005859375}, {"start": 974.84, "end": 975.04, "word": " هنا", "probability": 0.755859375}, {"start": 975.04, "end": 975.32, "word": " طبعا", "probability": 0.90771484375}, {"start": 975.32, "end": 975.46, "word": " فى", "probability": 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"end": 981.5, "word": " وهذا", "probability": 0.641357421875}, {"start": 981.5, "end": 981.66, "word": " هو", "probability": 0.9501953125}, {"start": 981.66, "end": 982.2, "word": " الجزء", "probability": 0.9886067708333334}, {"start": 982.2, "end": 983.42, "word": " التانى", "probability": 0.9490559895833334}, {"start": 983.42, "end": 983.62, "word": " هذا", "probability": 0.77490234375}, {"start": 983.62, "end": 984.02, "word": " حسب", "probability": 0.876220703125}, {"start": 984.02, "end": 984.84, "word": " هذا", "probability": 0.6328125}, {"start": 984.84, "end": 984.98, "word": " لو", "probability": 0.54052734375}, {"start": 984.98, "end": 985.18, "word": " بدنا", "probability": 0.95556640625}, {"start": 985.18, "end": 985.74, "word": " نستخدم", "probability": 0.9937744140625}, {"start": 985.74, "end": 986.74, "word": " exercise", "probability": 0.8916015625}, {"start": 986.74, "end": 987.82, "word": " رقم", "probability": 0.9783528645833334}, {"start": 987.82, "end": 988.2, "word": " 7", "probability": 0.6806640625}, {"start": 988.2, "end": 989.7, "word": " وطبعا", "probability": 0.9623046875}, {"start": 989.7, "end": 990.06, "word": " لازم", "probability": 0.9573567708333334}, {"start": 990.06, "end": 990.7, "word": " نبرهنه", "probability": 0.9595703125}], "temperature": 1.0}, {"id": 34, "seek": 101481, "start": 992.33, "end": 1014.81, "text": "لكن ممكن نعطي برهان مباشر بدون ما يستخدم exercise السابعة وبالتالي إذا في حال تاني أو برهان تاني باستخدام التعريفات وال comparison tests", "tokens": [1211, 19452, 3714, 43020, 8717, 3615, 9566, 1829, 4724, 2288, 3224, 7649, 3714, 3555, 33599, 2288, 47525, 11536, 19446, 7251, 14851, 9778, 40448, 5380, 21136, 16758, 27884, 46599, 6027, 2655, 6027, 1829, 11933, 15730, 8978, 11331, 6027, 6055, 7649, 1829, 34051, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 4724, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 9307, 16070, 9660, 6921], "avg_logprob": -0.12096973851559654, "compression_ratio": 1.474025974025974, 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"seek": 104209, "start": 1016.26, "end": 1042.1, "text": "باستخدام التعريفات زايد ال comparison tests اختبارات المقارنة ال proof رقم اتنين since اننا ننسى هذا القرآن انا عند هذه الفرض since limit ل xn over yn هذا عبارة عن sequence", "tokens": [3555, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 9307, 30767, 995, 25708, 2423, 9660, 6921, 1975, 46456, 3555, 9640, 9307, 9673, 4587, 9640, 1863, 3660, 2423, 8177, 12602, 4587, 2304, 1975, 2655, 1863, 9957, 1670, 16472, 8315, 8717, 1863, 3794, 7578, 23758, 25062, 2288, 148, 48506, 1975, 8315, 43242, 29538, 27188, 43042, 1670, 4948, 5296, 2031, 77, 670, 288, 77, 23758, 6225, 3555, 9640, 3660, 18871, 8310], "avg_logprob": -0.3656249936137881, "compression_ratio": 1.4775280898876404, "no_speech_prob": 0.0, "words": [{"start": 1016.26, "end": 1017.2, "word": "باستخدام", "probability": 0.7483113606770834}, {"start": 1017.2, "end": 1017.9, "word": " التعريفات", "probability": 0.89140625}, {"start": 1017.9, "end": 1018.32, "word": " زايد", "probability": 0.5538736979166666}, {"start": 1018.32, "end": 1018.48, "word": " ال", "probability": 0.517578125}, {"start": 1018.48, "end": 1018.98, "word": " comparison", "probability": 0.5087890625}, {"start": 1018.98, "end": 1019.66, "word": " tests", "probability": 0.5126953125}, {"start": 1019.66, "end": 1020.6, "word": " اختبارات", "probability": 0.806298828125}, {"start": 1020.6, "end": 1021.38, "word": " المقارنة", "probability": 0.98544921875}, {"start": 1021.38, "end": 1021.82, "word": " ال", "probability": 0.272705078125}, {"start": 1021.82, "end": 1022.38, "word": " proof", "probability": 0.9404296875}, {"start": 1022.38, "end": 1023.06, "word": " رقم", "probability": 0.9231770833333334}, {"start": 1023.06, "end": 1025.0, "word": " اتنين", "probability": 0.8155517578125}, {"start": 1025.0, "end": 1029.24, "word": " since", "probability": 0.38623046875}, {"start": 1029.24, "end": 1030.12, "word": " اننا", "probability": 0.2208251953125}, {"start": 1030.12, "end": 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"word": " yn", "probability": 0.664306640625}, {"start": 1040.88, "end": 1041.2, "word": " هذا", "probability": 0.4033203125}, {"start": 1041.2, "end": 1041.52, "word": " عبارة", "probability": 0.8572998046875}, {"start": 1041.52, "end": 1041.72, "word": " عن", "probability": 0.99658203125}, {"start": 1041.72, "end": 1042.1, "word": " sequence", "probability": 0.90625}], "temperature": 1.0}, {"id": 36, "seek": 107119, "start": 1043.65, "end": 1071.19, "text": "لأن الـ limit إلا بالساقر plus infinity then given Alpha أي real number Alpha من تعريف الـ improper convergence لأي Alpha there exists capital N يعتمد على Alpha عدد قضية", "tokens": [1211, 33456, 2423, 39184, 4948, 11933, 15040, 20666, 3794, 995, 4587, 2288, 1804, 13202, 550, 2212, 20588, 36632, 957, 1230, 20588, 9154, 37279, 16572, 5172, 2423, 39184, 40651, 32181, 5296, 10721, 1829, 20588, 456, 8198, 4238, 426, 7251, 34268, 2304, 3215, 15844, 20588, 6225, 3215, 3215, 12174, 11242, 10632], "avg_logprob": -0.48031251192092894, 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y", "probability": 0.1658935546875}, {"start": 1221.76, "end": 1222.18, "word": " in", "probability": 0.916015625}, {"start": 1222.18, "end": 1223.58, "word": " بسرعه", "probability": 0.481103515625}, {"start": 1223.58, "end": 1223.92, "word": " نصف", "probability": 0.4425862630208333}, {"start": 1223.92, "end": 1227.4, "word": " طيب", "probability": 0.7618001302083334}, {"start": 1227.4, "end": 1232.42, "word": " الجزء", "probability": 0.8525390625}, {"start": 1232.42, "end": 1233.66, "word": " دي", "probability": 0.945556640625}, {"start": 1233.66, "end": 1235.4, "word": " since", "probability": 0.67578125}, {"start": 1235.4, "end": 1237.96, "word": " x", "probability": 0.79638671875}, {"start": 1237.96, "end": 1238.62, "word": " in", "probability": 0.9111328125}, {"start": 1238.62, "end": 1239.22, "word": " is", "probability": 0.94580078125}, {"start": 1239.22, "end": 1239.86, "word": " bounded", "probability": 0.94482421875}, {"start": 1239.86, "end": 1243.34, "word": " إذن", 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"probability": 0.8133138020833334}, {"start": 1269.14, "end": 1271.54, "word": " to", "probability": 0.72705078125}, {"start": 1271.54, "end": 1272.02, "word": " show", "probability": 0.904296875}], "temperature": 1.0}, {"id": 44, "seek": 129909, "start": 1273.7, "end": 1299.1, "text": "limit yn بساوي zero let epsilon بنستخدم تعريف epsilon capital M لإن هي let epsilon أكبر من الصفر be given من epsilon على M بيطلع عدد موجب من", "tokens": [4197, 270, 17861, 4724, 3794, 995, 45865, 4018, 718, 17889, 4724, 1863, 14851, 9778, 40448, 37279, 16572, 5172, 17889, 4238, 376, 5296, 28814, 1863, 39896, 718, 17889, 5551, 4117, 26890, 9154, 31767, 5172, 2288, 312, 2212, 9154, 17889, 15844, 376, 4724, 1829, 9566, 1211, 3615, 6225, 3215, 3215, 3714, 29245, 3555, 9154], "avg_logprob": -0.29274763251250646, "compression_ratio": 1.3150684931506849, "no_speech_prob": 0.0, "words": [{"start": 1273.6999999999998, "end": 1274.6399999999999, "word": "limit", "probability": 0.77294921875}, {"start": 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0.5205078125}, {"start": 1283.76, "end": 1284.92, "word": " let", "probability": 0.951171875}, {"start": 1284.92, "end": 1285.32, "word": " epsilon", "probability": 0.9443359375}, {"start": 1285.32, "end": 1285.76, "word": " أكبر", "probability": 0.93994140625}, {"start": 1285.76, "end": 1285.94, "word": " من", "probability": 0.9921875}, {"start": 1285.94, "end": 1286.48, "word": " الصفر", "probability": 0.801513671875}, {"start": 1286.48, "end": 1286.7, "word": " be", "probability": 0.603515625}, {"start": 1286.7, "end": 1287.18, "word": " given", "probability": 0.88037109375}, {"start": 1287.18, "end": 1292.06, "word": " من", "probability": 0.28466796875}, {"start": 1292.06, "end": 1294.76, "word": " epsilon", "probability": 0.77001953125}, {"start": 1294.76, "end": 1295.08, "word": " على", "probability": 0.6845703125}, {"start": 1295.08, "end": 1295.34, "word": " M", "probability": 0.84033203125}, {"start": 1295.34, "end": 1295.78, "word": " بيطلع", "probability": 0.784033203125}, {"start": 1295.78, "end": 1296.12, "word": " عدد", "probability": 0.97998046875}, {"start": 1296.12, "end": 1296.64, "word": " موجب", "probability": 0.8655598958333334}, {"start": 1296.64, "end": 1299.1, "word": " من", "probability": 0.77197265625}], "temperature": 1.0}, {"id": 45, "seek": 132159, "start": 1301.39, "end": 1321.59, "text": "العدد الموجب يعتمد على إبسلون على م يعتبر إبسلون على م", "tokens": [6027, 3615, 3215, 3215, 9673, 29245, 3555, 7251, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536, 15844, 3714, 7251, 34268, 26890, 11933, 3555, 3794, 1211, 11536, 15844, 3714], "avg_logprob": -0.4895833114782969, "compression_ratio": 1.4202898550724639, "no_speech_prob": 0.0, "words": [{"start": 1301.3899999999999, "end": 1302.79, "word": "العدد", "probability": 0.5425262451171875}, {"start": 1302.79, "end": 1303.75, "word": " الموجب", "probability": 0.9768880208333334}, {"start": 1303.75, "end": 1312.91, "word": " يعتمد", "probability": 0.79144287109375}, {"start": 1312.91, "end": 1313.27, "word": " على", "probability": 0.86767578125}, {"start": 1313.27, "end": 1314.99, "word": " إبسلون", "probability": 0.7670654296875}, {"start": 1314.99, "end": 1315.19, "word": " على", "probability": 0.79736328125}, {"start": 1315.19, "end": 1315.57, "word": " م", "probability": 0.251708984375}, {"start": 1315.57, "end": 1318.83, "word": " يعتبر", "probability": 0.6276652018229166}, {"start": 1318.83, "end": 1321.07, "word": " إبسلون", "probability": 0.90888671875}, {"start": 1321.07, "end": 1321.23, "word": " على", "probability": 0.9052734375}, {"start": 1321.23, "end": 1321.59, "word": " م", "probability": 0.68994140625}], "temperature": 1.0}, {"id": 46, "seek": 135117, "start": 1336.45, "end": 1351.17, "text": "أنا عندي ايش عندي بدي أثبت ان limit yn بالساوي سفر فخلينا نشوف", "tokens": [10721, 8315, 18871, 16254, 1975, 1829, 8592, 18871, 16254, 4724, 16254, 5551, 12984, 3555, 2655, 16472, 4948, 17861, 20666, 3794, 995, 45865, 8608, 5172, 2288, 6156, 9778, 1211, 9957, 995, 8717, 8592, 38688], "avg_logprob": -0.38878676996511574, "compression_ratio": 1.2209302325581395, "no_speech_prob": 0.0, "words": [{"start": 1336.45, "end": 1336.85, "word": "أنا", "probability": 0.482452392578125}, {"start": 1336.85, "end": 1337.21, "word": " عندي", "probability": 0.838134765625}, {"start": 1337.21, "end": 1337.41, "word": " ايش", "probability": 0.671142578125}, {"start": 1337.41, "end": 1337.89, "word": " عندي", "probability": 0.779296875}, {"start": 1337.89, "end": 1344.15, "word": " بدي", "probability": 0.41656494140625}, {"start": 1344.15, "end": 1344.59, "word": " أثبت", "probability": 0.8873291015625}, {"start": 1344.59, "end": 1344.73, "word": " ان", "probability": 0.71337890625}, {"start": 1344.73, "end": 1345.11, "word": " limit", "probability": 0.6396484375}, {"start": 1345.11, "end": 1345.91, "word": " yn", "probability": 0.408447265625}, {"start": 1345.91, "end": 1347.39, "word": " بالساوي", "probability": 0.62982177734375}, {"start": 1347.39, "end": 1347.97, "word": " سفر", "probability": 0.86376953125}, {"start": 1347.97, "end": 1350.65, "word": " فخلينا", "probability": 0.75849609375}, {"start": 1350.65, "end": 1351.17, "word": " نشوف", "probability": 0.99560546875}], "temperature": 1.0}, {"id": 47, "seek": 138682, "start": 1363.72, "end": 1386.82, "text": "طيب أنا لازم أستخدم .. آه لازم أستخدم .. طيب since .. طيب بس هنا يعني خليني أقول since", "tokens": [9566, 1829, 3555, 41850, 5296, 31377, 2304, 5551, 14851, 9778, 40448, 4386, 19753, 3224, 5296, 31377, 2304, 5551, 14851, 9778, 40448, 4386, 23032, 1829, 3555, 1670, 4386, 23032, 1829, 3555, 4724, 3794, 34105, 37495, 22653, 16490, 1211, 9957, 1829, 5551, 39648, 1670], "avg_logprob": -0.17087572258572245, "compression_ratio": 1.4526315789473685, "no_speech_prob": 0.0, "words": [{"start": 1363.7199999999998, "end": 1365.12, "word": "طيب", "probability": 0.9103190104166666}, {"start": 1365.12, "end": 1366.02, "word": " أنا", "probability": 0.453857421875}, {"start": 1366.02, "end": 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بيطلع عندي absolute yn على xn minus ال zero أصغر من epsilon على n تمام؟", "tokens": [3615, 3215, 3215, 23032, 21292, 3615, 1829, 5296, 5016, 1829, 12984, 16472, 3224, 5296, 28820, 297, 5551, 4117, 26890, 9154, 34051, 3794, 3615, 2423, 4238, 426, 4724, 1829, 9566, 1211, 3615, 18871, 16254, 8236, 17861, 15844, 2031, 77, 3175, 2423, 4018, 5551, 9381, 17082, 2288, 9154, 17889, 15844, 297, 46811, 10943, 22807], "avg_logprob": -0.32547171160859883, "compression_ratio": 1.2328767123287672, "no_speech_prob": 0.0, "words": [{"start": 1418.73, "end": 1419.29, "word": "عدد", "probability": 0.65008544921875}, {"start": 1419.29, "end": 1420.09, "word": " طبيعي", "probability": 0.92578125}, {"start": 1420.09, "end": 1421.43, "word": " لحيث", "probability": 0.82421875}, {"start": 1421.43, "end": 1423.21, "word": " انه", "probability": 0.7841796875}, {"start": 1423.21, "end": 1424.81, "word": " لكل", "probability": 0.951171875}, {"start": 1424.81, "end": 1425.37, "word": " n", "probability": 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"text": "لو كان n أكبر من أو ساوي capital N بتثبت أن ال absolute value ل y in minus 0 أصغر من epsilon عشان أثبت أن ال limit ل y in بساوي سفر بتثبت أن ال absolute value ل y in minus 0 أصغر من ال given epsilon طيب هذا بساوي absolute", "tokens": [1211, 2407, 25961, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 39894, 12984, 3555, 2655, 14739, 2423, 8236, 2158, 5296, 288, 294, 3175, 1958, 5551, 9381, 17082, 2288, 9154, 17889, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 14739, 2423, 4948, 5296, 288, 294, 4724, 3794, 995, 45865, 8608, 5172, 2288, 39894, 12984, 3555, 2655, 14739, 2423, 8236, 2158, 5296, 288, 294, 3175, 1958, 5551, 9381, 17082, 2288, 9154, 2423, 2212, 17889, 23032, 1829, 3555, 23758, 4724, 3794, 995, 45865, 8236], "avg_logprob": -0.2639508992433548, "compression_ratio": 1.8313253012048192, "no_speech_prob": 0.0, "words": [{"start": 1475.6399999999999, "end": 1476.36, "word": "لو", "probability": 0.3331298828125}, {"start": 1476.36, "end": 1476.82, "word": " 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ثلاثة طبعا هكذا بنثبت تسعة و سبعة بالمناسبة زيه بنفس الطريقة بافكار مشابه ممكن اثباته بنفس السلوب بنفس النمط", "tokens": [3224, 15730, 15844, 1975, 34268, 3555, 9640, 16472, 1975, 5016, 8315, 5380, 8608, 3555, 27884, 19446, 44945, 3615, 28480, 2407, 8592, 4724, 3794, 8978, 28242, 16247, 5016, 2407, 6027, 1975, 5016, 8315, 44713, 9778, 40448, 8315, 5380, 38637, 1211, 5718, 104, 3660, 23032, 3555, 3615, 995, 8032, 4117, 15730, 44945, 12984, 3555, 2655, 6055, 3794, 27884, 4032, 8608, 3555, 27884, 20666, 2304, 8315, 35457, 3660, 30767, 1829, 3224, 44945, 36178, 41950, 16572, 28671, 4724, 31845, 4117, 9640, 37893, 16758, 3224, 3714, 43020, 1975, 12984, 3555, 9307, 3224, 44945, 36178, 21136, 1211, 37746, 44945, 36178, 28239, 2304, 9566], "avg_logprob": -0.1627072685835313, "compression_ratio": 1.78125, "no_speech_prob": 0.0, "words": [{"start": 1594.06, "end": 1594.4, "word": "هذا", "probability": 0.937744140625}, {"start": 1594.4, "end": 1594.5, "word": " على", "probability": 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بنفس", "probability": 0.927001953125}, {"start": 1613.74, "end": 1614.4, "word": " الطريقة", "probability": 0.98876953125}, {"start": 1614.4, "end": 1615.62, "word": " بافكار", "probability": 0.87109375}, {"start": 1615.62, "end": 1616.34, "word": " مشابه", "probability": 0.9430338541666666}, {"start": 1616.34, "end": 1618.66, "word": " ممكن", "probability": 0.856689453125}, {"start": 1618.66, "end": 1619.22, "word": " اثباته", "probability": 0.86240234375}, {"start": 1619.22, "end": 1619.6, "word": " بنفس", "probability": 0.978515625}, {"start": 1619.6, "end": 1620.1, "word": " السلوب", "probability": 0.7294921875}, {"start": 1620.1, "end": 1620.48, "word": " بنفس", "probability": 0.92236328125}, {"start": 1620.48, "end": 1621.02, "word": " النمط", "probability": 0.9532877604166666}], "temperature": 1.0}, {"id": 56, "seek": 164575, "start": 1623.51, "end": 1645.75, "text": "كمان في أي أسئلة تانية في section تلاتة سبعة؟ إذا مافيش خلينا ننتقل ل section تلاتة سبعة تبع ال series هذا في عندكم أي أسئلة في section تلاتة سبعة؟ تلاتة خمسة؟ تلاتة سبعة؟", "tokens": [24793, 7649, 8978, 36632, 5551, 3794, 19986, 37977, 6055, 7649, 10632, 8978, 3541, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 22807, 11933, 15730, 19446, 41185, 8592, 16490, 20292, 8315, 8717, 29399, 4587, 1211, 5296, 3541, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 6055, 3555, 3615, 2423, 2638, 23758, 8978, 43242, 24793, 36632, 5551, 3794, 19986, 37977, 8978, 3541, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 22807, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 22807, 6055, 1211, 9307, 3660, 8608, 3555, 27884, 22807], "avg_logprob": -0.1783854239398525, "compression_ratio": 1.9652777777777777, "no_speech_prob": 0.0, "words": [{"start": 1623.51, "end": 1623.93, "word": "كمان", "probability": 0.59600830078125}, {"start": 1623.93, "end": 1624.15, "word": " في", "probability": 0.77685546875}, {"start": 1624.15, "end": 1624.39, "word": " أي", "probability": 0.55517578125}, {"start": 1624.39, "end": 1624.79, "word": 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برز واحد على ستة سيطلع سالب واحد على ستة لإن بيساوي خمسة سيطلع ثاندي سالب واحد على ستة فمين اللي بيضل على المحدود يعني بتاعي هذا ستس هيبقى هذا هيروح", "tokens": [2407, 3714, 43020, 9122, 2304, 7649, 4724, 2288, 11622, 36764, 24401, 15844, 8608, 2655, 3660, 1975, 2407, 8978, 4724, 2288, 11622, 36764, 24401, 15844, 8608, 2655, 3660, 8608, 1829, 9566, 1211, 3615, 8608, 6027, 3555, 36764, 24401, 15844, 8608, 2655, 3660, 5296, 28814, 1863, 4724, 1829, 3794, 995, 45865, 16490, 2304, 3794, 3660, 8608, 1829, 9566, 1211, 3615, 38637, 7649, 16254, 8608, 6027, 3555, 36764, 24401, 15844, 8608, 2655, 3660, 6156, 2304, 9957, 13672, 1829, 4724, 1829, 11242, 1211, 15844, 9673, 24401, 23328, 37495, 22653, 39894, 45761, 1829, 23758, 8608, 2655, 3794, 39896, 3555, 4587, 7578, 23758, 39896, 32887, 5016], "avg_logprob": -0.29130569188901695, "compression_ratio": 2.1298701298701297, "no_speech_prob": 0.0, "words": [{"start": 2156.98, "end": 2157.28, "word": "و", "probability": 0.7041015625}, {"start": 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2544.05, "end": 2544.71, "word": " أخدنا", "probability": 0.9825439453125}, {"start": 2544.71, "end": 2544.71, "word": " هذا", "probability": 0.83203125}, {"start": 2544.71, "end": 2545.97, "word": " و", "probability": 0.92822265625}, {"start": 2545.97, "end": 2547.91, "word": " أخدنا", "probability": 0.982177734375}, {"start": 2547.91, "end": 2548.33, "word": " هذا", "probability": 0.7998046875}, {"start": 2548.33, "end": 2548.59, "word": " و", "probability": 0.919921875}, {"start": 2548.59, "end": 2553.83, "word": " أخدنا", "probability": 0.982177734375}, {"start": 2553.83, "end": 2553.83, "word": " هذا", "probability": 0.79150390625}, {"start": 2553.83, "end": 2554.29, "word": " و", "probability": 0.916015625}, {"start": 2554.29, "end": 2555.99, "word": " أ", "probability": 0.98388671875}], "temperature": 1.0}, {"id": 88, "seek": 258438, "start": 2556.8, "end": 2584.38, "text": "وحد على X لما X تقول لسفر does not exist in R فلبرحان ذلك let F of X تساوي صين واحد على X و X لا تساوي سفر", "tokens": [2407, 24401, 15844, 1783, 5296, 15042, 1783, 6055, 39648, 5296, 3794, 5172, 2288, 775, 406, 2514, 294, 497, 6156, 46152, 2288, 5016, 7649, 29910, 23275, 718, 479, 295, 1783, 6055, 3794, 995, 45865, 20328, 9957, 36764, 24401, 15844, 1783, 4032, 1783, 20193, 6055, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.31124999761581423, "compression_ratio": 1.272, "no_speech_prob": 0.0, "words": [{"start": 2556.8, "end": 2557.52, "word": "وحد", "probability": 0.2965087890625}, {"start": 2557.52, "end": 2557.84, "word": " على", "probability": 0.421142578125}, {"start": 2557.84, "end": 2558.34, "word": " X", "probability": 0.32470703125}, {"start": 2558.34, "end": 2559.94, "word": " لما", "probability": 0.757568359375}, {"start": 2559.94, "end": 2560.54, "word": " X", "probability": 0.92138671875}, {"start": 2560.54, "end": 2561.06, "word": " تقول", "probability": 0.94775390625}, {"start": 2561.06, "end": 2561.76, "word": " لسفر", "probability": 0.739501953125}, {"start": 2561.76, "end": 2563.88, "word": " does", "probability": 0.2315673828125}, {"start": 2563.88, "end": 2564.26, "word": " not", "probability": 0.966796875}, {"start": 2564.26, "end": 2564.9, "word": " exist", "probability": 0.97021484375}, {"start": 2564.9, "end": 2566.64, "word": " in", "probability": 0.55615234375}, {"start": 2566.64, "end": 2567.08, "word": " R", "probability": 0.98974609375}, {"start": 2567.08, "end": 2570.4, "word": " فلبرحان", "probability": 0.78544921875}, {"start": 2570.4, "end": 2573.0, "word": " ذلك", "probability": 0.992919921875}, {"start": 2573.0, "end": 2576.7, "word": " let", "probability": 0.513671875}, {"start": 2576.7, "end": 2577.04, "word": " F", "probability": 0.625}, {"start": 2577.04, "end": 2577.32, "word": " of", "probability": 0.65673828125}, {"start": 2577.32, "end": 2577.78, "word": " X", "probability": 0.99072265625}, {"start": 2577.78, "end": 2579.48, "word": " تساوي", "probability": 0.709716796875}, {"start": 2579.48, "end": 2580.2, "word": " صين", "probability": 0.6494140625}, {"start": 2580.2, "end": 2580.8, "word": " واحد", "probability": 0.931884765625}, {"start": 2580.8, "end": 2581.0, "word": " على", "probability": 0.7900390625}, {"start": 2581.0, "end": 2581.46, "word": " X", "probability": 0.9912109375}, {"start": 2581.46, "end": 2582.76, "word": " و", "probability": 0.7822265625}, {"start": 2582.76, "end": 2583.08, "word": " X", "probability": 0.74072265625}, {"start": 2583.08, "end": 2583.38, "word": " لا", "probability": 0.64013671875}, {"start": 2583.38, "end": 2583.92, "word": " تساوي", "probability": 0.9666748046875}, {"start": 2583.92, "end": 2584.38, "word": " سفر", "probability": 0.96923828125}], "temperature": 1.0}, {"id": 89, "seek": 261803, "start": 2590.73, "end": 2618.03, "text": "و بعدين we consider two sequences واحدة xn الحد لعام تبعها أدارة عن واحد على واحد على n πاي و n ينتمي ل z", "tokens": [2407, 39182, 9957, 321, 1949, 732, 22978, 36764, 24401, 3660, 2031, 77, 21542, 3215, 5296, 3615, 10943, 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2641.3, "word": " numbers", "probability": 0.88525390625}], "temperature": 1.0}, {"id": 91, "seek": 267369, "start": 2644.95, "end": 2673.69, "text": "واضح ان ال limit ل xn as n tends to infinity بساوي 0 وكذلك ال limit ل yn لما n تقول infinity برضه بساوي 0 لان المقان لما n تقول infinity المقان بيروح ل infinity طيب الآن ال limit", "tokens": [2407, 46958, 5016, 16472, 2423, 4948, 5296, 2031, 77, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 1958, 4032, 4117, 8848, 23275, 2423, 4948, 5296, 17861, 5296, 15042, 297, 6055, 39648, 13202, 4724, 43042, 3224, 4724, 3794, 995, 45865, 1958, 5296, 7649, 9673, 4587, 7649, 5296, 15042, 297, 6055, 39648, 13202, 9673, 4587, 7649, 4724, 13546, 2407, 5016, 5296, 13202, 23032, 1829, 3555, 6024, 48506, 2423, 4948], "avg_logprob": -0.3405797118725984, "compression_ratio": 1.7816901408450705, "no_speech_prob": 0.0, "words": [{"start": 2644.95, "end": 2645.59, "word": "واضح", "probability": 0.9402669270833334}, {"start": 2645.59, "end": 2645.87, "word": " ان", "probability": 0.517578125}, {"start": 2645.87, "end": 2647.03, "word": " ال", "probability": 0.29638671875}, {"start": 2647.03, "end": 2647.39, "word": " limit", "probability": 0.5751953125}, {"start": 2647.39, "end": 2648.69, "word": " ل", "probability": 0.54931640625}, {"start": 2648.69, "end": 2649.55, "word": " xn", "probability": 0.18341064453125}, {"start": 2649.55, "end": 2650.15, "word": " as", "probability": 0.7900390625}, {"start": 2650.15, "end": 2650.51, "word": " n", "probability": 0.83642578125}, {"start": 2650.51, "end": 2650.81, "word": " tends", "probability": 0.59130859375}, {"start": 2650.81, "end": 2650.93, "word": " to", "probability": 0.94580078125}, {"start": 2650.93, "end": 2651.41, "word": " infinity", "probability": 0.84716796875}, {"start": 2651.41, "end": 2652.13, "word": " بساوي", "probability": 0.7550048828125}, {"start": 2652.13, "end": 2652.49, "word": " 0", "probability": 0.470458984375}, {"start": 2652.49, "end": 2654.85, "word": " وكذلك", "probability": 0.8287353515625}, {"start": 2654.85, "end": 2655.11, "word": " ال", "probability": 0.36865234375}, {"start": 2655.11, "end": 2655.43, "word": " limit", "probability": 0.97900390625}, {"start": 2655.43, "end": 2657.29, "word": " ل", "probability": 0.95947265625}, {"start": 2657.29, "end": 2658.05, "word": " yn", "probability": 0.5576171875}, {"start": 2658.05, "end": 2658.59, "word": " لما", "probability": 0.806640625}, {"start": 2658.59, "end": 2658.87, "word": " n", "probability": 0.27587890625}, {"start": 2658.87, "end": 2659.27, "word": " تقول", "probability": 0.852294921875}, {"start": 2659.27, "end": 2659.53, "word": " infinity", "probability": 0.3798828125}, {"start": 2659.53, "end": 2660.29, "word": " برضه", "probability": 0.8165690104166666}, {"start": 2660.29, "end": 2660.77, "word": " بساوي", "probability": 0.966064453125}, {"start": 2660.77, "end": 2661.15, "word": " 0", "probability": 0.9208984375}, {"start": 2661.15, "end": 2662.27, "word": " لان", "probability": 0.6527099609375}, {"start": 2662.27, "end": 2662.91, "word": " المقان", "probability": 0.6102701822916666}, {"start": 2662.91, "end": 2663.41, "word": " لما", "probability": 0.93603515625}, {"start": 2663.41, "end": 2663.61, "word": " n", "probability": 0.55712890625}, {"start": 2663.61, "end": 2663.81, "word": " تقول", "probability": 0.98681640625}, {"start": 2663.81, "end": 2664.27, "word": " infinity", "probability": 0.873046875}, {"start": 2664.27, "end": 2664.73, "word": " المقان", "probability": 0.8548177083333334}, {"start": 2664.73, "end": 2665.03, "word": " بيروح", "probability": 0.864013671875}, {"start": 2665.03, "end": 2665.13, "word": " ل", "probability": 0.75390625}, {"start": 2665.13, "end": 2665.59, "word": " infinity", "probability": 0.611328125}, {"start": 2665.59, "end": 2672.09, "word": " طيب", "probability": 0.9269205729166666}, {"start": 2672.09, "end": 2672.59, "word": " الآن", "probability": 0.6302490234375}, {"start": 2672.59, "end": 2673.31, "word": " ال", "probability": 0.411376953125}, {"start": 2673.31, "end": 2673.69, "word": " limit", "probability": 0.98486328125}], "temperature": 1.0}, {"id": 92, "seek": 270562, "start": 2677.86, "end": 2705.62, "text": "الـ image لـ sequence xn لما n تقول الانفلتين بساوي ال limit ل sign xn لما n تقول الانفلتين وهذا بساوي ال limit ل sign n في pi لما n تقول الانفلتين", "tokens": [6027, 39184, 3256, 5296, 39184, 8310, 2031, 77, 5296, 15042, 297, 6055, 39648, 2423, 7649, 5172, 1211, 2655, 9957, 4724, 3794, 995, 45865, 2423, 4948, 5296, 1465, 2031, 77, 5296, 15042, 297, 6055, 39648, 2423, 7649, 5172, 1211, 2655, 9957, 37037, 15730, 4724, 3794, 995, 45865, 2423, 4948, 5296, 1465, 297, 8978, 3895, 5296, 15042, 297, 6055, 39648, 2423, 7649, 5172, 1211, 2655, 9957], "avg_logprob": -0.2639423076923077, "compression_ratio": 2.0, "no_speech_prob": 0.0, "words": [{"start": 2677.86, "end": 2678.46, "word": "الـ", "probability": 0.5634765625}, {"start": 2678.46, "end": 2678.78, "word": " image", "probability": 0.52880859375}, {"start": 2678.78, "end": 2679.08, "word": " لـ", "probability": 0.583740234375}, {"start": 2679.08, "end": 2679.62, "word": " sequence", "probability": 0.9052734375}, {"start": 2679.62, "end": 2680.46, "word": " xn", "probability": 0.48663330078125}, {"start": 2680.46, "end": 2680.82, "word": " لما", "probability": 0.824462890625}, {"start": 2680.82, "end": 2681.16, "word": " n", "probability": 0.50830078125}, {"start": 2681.16, "end": 2681.58, "word": " تقول", "probability": 0.91259765625}, {"start": 2681.58, "end": 2682.42, "word": " الانفلتين", "probability": 0.58660888671875}, {"start": 2682.42, "end": 2684.06, "word": " بساوي", "probability": 0.815673828125}, {"start": 2684.06, "end": 2685.32, "word": " ال", "probability": 0.476806640625}, {"start": 2685.32, "end": 2685.8, "word": " limit", "probability": 0.88818359375}, {"start": 2685.8, "end": 2687.7, "word": " ل", "probability": 0.9482421875}, {"start": 2687.7, "end": 2688.36, "word": " sign", "probability": 0.194580078125}, {"start": 2688.36, "end": 2690.78, "word": " xn", "probability": 0.945556640625}, {"start": 2690.78, "end": 2691.48, "word": " لما", "probability": 0.958984375}, {"start": 2691.48, "end": 2691.74, "word": " n", "probability": 0.849609375}, {"start": 2691.74, "end": 2692.14, "word": " تقول", "probability": 0.9853515625}, {"start": 2692.14, "end": 2693.0, "word": " الانفلتين", "probability": 0.97265625}, {"start": 2693.0, "end": 2695.5, "word": " وهذا", "probability": 0.7181396484375}, {"start": 2695.5, "end": 2696.06, "word": " بساوي", "probability": 0.9241943359375}, {"start": 2696.06, "end": 2696.22, "word": " ال", "probability": 0.482421875}, {"start": 2696.22, "end": 2696.58, "word": " limit", "probability": 0.96728515625}, {"start": 2696.58, "end": 2699.06, "word": " ل", "probability": 0.96533203125}, {"start": 2699.06, "end": 2699.88, "word": " sign", "probability": 0.8701171875}, {"start": 2699.88, "end": 2701.58, "word": " n", "probability": 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13202, 4724, 3794, 995, 45865, 8608, 5172, 2288, 293, 4948], "avg_logprob": -0.2511574212047789, "compression_ratio": 1.416, "no_speech_prob": 0.0, "words": [{"start": 2706.34, "end": 2707.2, "word": "Sin", "probability": 0.47308349609375}, {"start": 2707.2, "end": 2707.64, "word": " N", "probability": 0.2371826171875}, {"start": 2707.64, "end": 2707.98, "word": " في", "probability": 0.7041015625}, {"start": 2707.98, "end": 2708.38, "word": " Pi", "probability": 0.22900390625}, {"start": 2708.38, "end": 2709.74, "word": " بساوي", "probability": 0.7806396484375}, {"start": 2709.74, "end": 2710.6, "word": " واحد", "probability": 0.859619140625}, {"start": 2710.6, "end": 2712.0, "word": " بساوي", "probability": 0.875732421875}, {"start": 2712.0, "end": 2712.36, "word": " سفر", "probability": 0.8759765625}, {"start": 2712.36, "end": 2713.48, "word": " لكل", "probability": 0.96435546875}, {"start": 2713.48, "end": 2713.8, "word": " N", "probability": 0.90869140625}, {"start": 2713.8, "end": 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"end": 2725.24, "word": " سفر", "probability": 0.9772135416666666}, {"start": 2725.24, "end": 2726.68, "word": " and", "probability": 0.65234375}, {"start": 2726.68, "end": 2732.6, "word": " limit", "probability": 0.9248046875}], "temperature": 1.0}, {"id": 94, "seek": 274136, "start": 2733.9, "end": 2741.36, "text": "الإمج للسيكوينس YM لما N تقول انفينيتي بساوي limit", "tokens": [6027, 28814, 2304, 7435, 24976, 3794, 1829, 4117, 2407, 9957, 3794, 398, 44, 5296, 15042, 426, 6055, 39648, 16472, 5172, 9957, 1829, 31371, 4724, 3794, 995, 45865, 4948], "avg_logprob": -0.6099137684394573, "compression_ratio": 1.024390243902439, "no_speech_prob": 0.0, "words": [{"start": 2733.9, "end": 2734.68, "word": "الإمج", "probability": 0.508209228515625}, {"start": 2734.68, "end": 2735.64, "word": " للسيكوينس", "probability": 0.6970738002232143}, {"start": 2735.64, "end": 2736.58, "word": " YM", "probability": 0.2542724609375}, {"start": 2736.58, "end": 2738.04, "word": " لما", "probability": 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2769.85, "word": " المفروض", "probability": 0.6524658203125}, {"start": 2769.85, "end": 2770.21, "word": " يكون", "probability": 0.90283203125}, {"start": 2770.21, "end": 2776.37, "word": " sign", "probability": 0.1986083984375}, {"start": 2776.37, "end": 2776.93, "word": " 1", "probability": 0.402587890625}, {"start": 2776.93, "end": 2777.45, "word": " على", "probability": 0.501953125}, {"start": 2777.45, "end": 2778.27, "word": " xn", "probability": 0.494873046875}, {"start": 2778.27, "end": 2779.83, "word": " وهذا", "probability": 0.6228841145833334}, {"start": 2779.83, "end": 2780.33, "word": " المفروض", "probability": 0.9747314453125}, {"start": 2780.33, "end": 2780.71, "word": " يكون", "probability": 0.938720703125}, {"start": 2780.71, "end": 2781.29, "word": " sign", "probability": 0.8583984375}, {"start": 2781.29, "end": 2781.77, "word": " 1", "probability": 0.91015625}, {"start": 2781.77, "end": 2782.27, "word": " على", "probability": 0.89208984375}, {"start": 2782.27, "end": 2782.83, "word": " yn", "probability": 0.60205078125}], "temperature": 1.0}, {"id": 96, "seek": 280890, "start": 2783.48, "end": 2808.9, "text": "مقلوب y in بيطلع بساوي πاية اتنين ازايد اتنين in πاية وهذا المقدار دايما بساوي واحد لكل in اذا انا في عندي limit لل sequence بالحد العام تبعها واحد", "tokens": [2304, 4587, 1211, 37746, 288, 294, 4724, 1829, 9566, 1211, 3615, 4724, 3794, 995, 45865, 4654, 995, 10632, 1975, 2655, 1863, 9957, 1975, 11622, 995, 25708, 1975, 2655, 1863, 9957, 294, 4654, 995, 10632, 37037, 15730, 9673, 28543, 9640, 11778, 47302, 15042, 4724, 3794, 995, 45865, 36764, 24401, 5296, 28820, 294, 1975, 15730, 1975, 8315, 8978, 18871, 16254, 4948, 24976, 8310, 20666, 24401, 18863, 10943, 6055, 3555, 3615, 11296, 36764, 24401], "avg_logprob": -0.3439670205116272, "compression_ratio": 1.5569620253164558, "no_speech_prob": 0.0, "words": [{"start": 2783.48, "end": 2784.14, "word": "مقلوب", "probability": 0.7713623046875}, {"start": 2784.14, "end": 2784.44, "word": " y", "probability": 0.2183837890625}, {"start": 2784.44, "end": 2784.84, "word": " in", "probability": 0.422119140625}, {"start": 2784.84, "end": 2785.88, "word": " بيطلع", "probability": 0.652197265625}, {"start": 2785.88, "end": 2786.68, "word": " بساوي", "probability": 0.8564453125}, {"start": 2786.68, "end": 2788.1, "word": " πاية", "probability": 0.4497884114583333}, {"start": 2788.1, "end": 2788.76, "word": " اتنين", "probability": 0.901611328125}, {"start": 2788.76, "end": 2789.86, "word": " ازايد", "probability": 0.63751220703125}, {"start": 2789.86, "end": 2790.58, "word": " اتنين", "probability": 0.9337158203125}, {"start": 2790.58, "end": 2790.94, "word": " in", "probability": 0.51416015625}, {"start": 2790.94, "end": 2791.5, "word": " πاية", "probability": 0.6826985677083334}, {"start": 2791.5, "end": 2794.54, "word": " وهذا", "probability": 0.744140625}, {"start": 2794.54, "end": 2795.56, "word": " المقدار", "probability": 0.95703125}, {"start": 2795.56, "end": 2796.16, "word": " دايما", "probability": 0.8343098958333334}, {"start": 2796.16, "end": 2798.94, "word": " بساوي", "probability": 0.71673583984375}, {"start": 2798.94, "end": 2800.28, "word": " واحد", "probability": 0.982421875}, {"start": 2800.28, "end": 2800.98, "word": " لكل", "probability": 0.96826171875}, {"start": 2800.98, "end": 2801.3, "word": " in", "probability": 0.7724609375}, {"start": 2801.3, "end": 2803.36, "word": " اذا", "probability": 0.687255859375}, {"start": 2803.36, "end": 2803.6, "word": " انا", "probability": 0.849853515625}, {"start": 2803.6, "end": 2803.74, "word": " في", "probability": 0.3037109375}, {"start": 2803.74, "end": 2804.04, "word": " عندي", "probability": 0.59149169921875}, {"start": 2804.04, "end": 2804.62, "word": " limit", "probability": 0.77880859375}, {"start": 2804.62, "end": 2806.14, "word": " لل", "probability": 0.6923828125}, {"start": 2806.14, "end": 2806.66, "word": " sequence", "probability": 0.7734375}, {"start": 2806.66, "end": 2807.26, "word": " بالحد", "probability": 0.639404296875}, {"start": 2807.26, "end": 2807.7, "word": " العام", "probability": 0.701416015625}, {"start": 2807.7, "end": 2808.28, "word": " تبعها", "probability": 0.9071044921875}, {"start": 2808.28, "end": 2808.9, "word": " واحد", "probability": 0.985595703125}], "temperature": 1.0}, {"id": 97, "seek": 283731, "start": 2810.77, "end": 2837.31, "text": "السيكوانس تابعة واحد وهذا بالساوية واحد ان ان انا في عندي two sequences Xm تقولها سفر و limit صورتها بالساوية سفر و في عندي سيكوانس تانية Ym ال limit تبعتها ايضا بالساوية سفر لكن limit صورتها بالساوية واحد وبالتالي", "tokens": [6027, 3794, 1829, 4117, 2407, 7649, 3794, 6055, 16758, 27884, 36764, 24401, 37037, 15730, 20666, 3794, 995, 2407, 10632, 36764, 24401, 16472, 16472, 1975, 8315, 8978, 18871, 16254, 732, 22978, 1783, 76, 6055, 39648, 11296, 8608, 5172, 2288, 4032, 4948, 20328, 13063, 2655, 11296, 20666, 3794, 995, 2407, 10632, 8608, 5172, 2288, 4032, 8978, 18871, 16254, 8608, 1829, 4117, 2407, 7649, 3794, 6055, 7649, 10632, 398, 76, 2423, 4948, 6055, 3555, 34268, 11296, 1975, 1829, 11242, 995, 20666, 3794, 995, 2407, 10632, 8608, 5172, 2288, 44381, 4948, 20328, 13063, 2655, 11296, 20666, 3794, 995, 2407, 10632, 36764, 24401, 46599, 6027, 2655, 6027, 1829], "avg_logprob": -0.2916165805206849, "compression_ratio": 2.111764705882353, "no_speech_prob": 0.0, "words": [{"start": 2810.77, "end": 2811.77, "word": "السيكوانس", "probability": 0.5807931082589286}, {"start": 2811.77, "end": 2812.27, "word": " تابعة", "probability": 0.5104166666666666}, {"start": 2812.27, "end": 2812.79, "word": " واحد", "probability": 0.912109375}, {"start": 2812.79, "end": 2813.67, "word": " وهذا", "probability": 0.6815185546875}, {"start": 2813.67, "end": 2814.09, "word": " بالساوية", "probability": 0.738037109375}, {"start": 2814.09, "end": 2814.47, "word": " واحد", "probability": 0.9912109375}, {"start": 2814.47, "end": 2816.97, "word": " ان", "probability": 0.1966552734375}, {"start": 2816.97, "end": 2817.11, "word": " ان", "probability": 0.291015625}, {"start": 2817.11, "end": 2817.25, "word": " انا", "probability": 0.897705078125}, {"start": 2817.25, "end": 2817.39, "word": " في", "probability": 0.36279296875}, {"start": 2817.39, "end": 2817.67, "word": " عندي", "probability": 0.6673583984375}, {"start": 2817.67, "end": 2817.87, "word": " two", "probability": 0.67626953125}, {"start": 2817.87, "end": 2818.63, "word": " sequences", "probability": 0.90380859375}, {"start": 2818.63, "end": 2819.47, "word": " Xm", "probability": 0.4698486328125}, {"start": 2819.47, "end": 2820.61, "word": " تقولها", "probability": 0.5611165364583334}, {"start": 2820.61, "end": 2821.03, "word": " سفر", "probability": 0.7837727864583334}, {"start": 2821.03, "end": 2822.43, "word": " و", "probability": 0.62451171875}, {"start": 2822.43, "end": 2822.91, "word": " limit", "probability": 0.8740234375}, {"start": 2822.91, "end": 2823.71, "word": " صورتها", "probability": 0.9556884765625}, {"start": 2823.71, "end": 2824.55, "word": " بالساوية", "probability": 0.96357421875}, {"start": 2824.55, "end": 2824.99, "word": " سفر", "probability": 0.95263671875}, {"start": 2824.99, "end": 2826.01, "word": " و", "probability": 0.93212890625}, {"start": 2826.01, "end": 2826.13, "word": " في", "probability": 0.5625}, {"start": 2826.13, "end": 2826.39, "word": " عندي", "probability": 0.959228515625}, {"start": 2826.39, "end": 2826.91, "word": " سيكوانس", "probability": 0.8582356770833334}, {"start": 2826.91, "end": 2827.25, "word": " تانية", "probability": 0.9622395833333334}, {"start": 2827.25, "end": 2827.99, "word": " Ym", "probability": 0.5626220703125}, {"start": 2827.99, "end": 2828.83, "word": " ال", "probability": 0.297119140625}, {"start": 2828.83, "end": 2829.05, "word": " limit", "probability": 0.91162109375}, {"start": 2829.05, "end": 2829.51, "word": " تبعتها", "probability": 0.7576904296875}, {"start": 2829.51, "end": 2829.79, "word": " ايضا", "probability": 0.82843017578125}, {"start": 2829.79, "end": 2830.31, "word": " بالساوية", "probability": 0.973828125}, {"start": 2830.31, "end": 2830.79, "word": " سفر", "probability": 0.9928385416666666}, {"start": 2830.79, "end": 2832.29, "word": " لكن", "probability": 0.86474609375}, {"start": 2832.29, "end": 2832.63, "word": " limit", "probability": 0.7080078125}, {"start": 2832.63, "end": 2833.15, "word": " صورتها", "probability": 0.978759765625}, {"start": 2833.15, "end": 2833.65, "word": " بالساوية", "probability": 0.96494140625}, {"start": 2833.65, "end": 2834.21, "word": " واحد", "probability": 0.994384765625}, {"start": 2834.21, "end": 2837.31, "word": " وبالتالي", "probability": 0.90029296875}], "temperature": 1.0}, {"id": 98, "seek": 286834, "start": 2840.36, "end": 2868.34, "text": "by sequential criterion ال limit لل function f of x لما x تقول ل0 does not exist in R مش ممكن تكون موجودة في R لأن لو كانت ال limit هذه موجودة", "tokens": [2322, 42881, 46691, 2423, 4948, 24976, 2445, 283, 295, 2031, 5296, 15042, 2031, 6055, 39648, 5296, 15, 775, 406, 2514, 294, 497, 37893, 3714, 43020, 6055, 30544, 3714, 29245, 23328, 3660, 8978, 497, 5296, 33456, 45164, 25961, 2655, 2423, 4948, 29538, 3714, 29245, 23328, 3660], "avg_logprob": -0.2778532679962075, "compression_ratio": 1.2972972972972974, "no_speech_prob": 0.0, "words": [{"start": 2840.36, "end": 2840.86, "word": "by", "probability": 0.0987548828125}, {"start": 2840.86, "end": 2841.9, "word": " sequential", "probability": 0.7568359375}, {"start": 2841.9, "end": 2842.78, "word": " criterion", "probability": 0.96142578125}, {"start": 2842.78, "end": 2848.34, "word": " ال", "probability": 0.30517578125}, {"start": 2848.34, "end": 2848.76, "word": " limit", "probability": 0.52490234375}, {"start": 2848.76, "end": 2850.08, "word": " لل", "probability": 0.7666015625}, {"start": 2850.08, "end": 2850.58, "word": " function", "probability": 0.59765625}, {"start": 2850.58, "end": 2851.06, "word": " f", "probability": 0.84521484375}, {"start": 2851.06, "end": 2851.32, "word": " of", "probability": 0.3955078125}, {"start": 2851.32, "end": 2851.72, "word": " x", "probability": 0.9248046875}, {"start": 2851.72, "end": 2852.76, "word": " لما", "probability": 0.689453125}, {"start": 2852.76, "end": 2853.1, "word": " x", "probability": 0.84033203125}, {"start": 2853.1, "end": 2853.5, "word": " تقول", "probability": 0.72119140625}, {"start": 2853.5, "end": 2854.12, "word": " ل0", "probability": 0.37615966796875}, {"start": 2854.12, "end": 2855.14, "word": " does", "probability": 0.69970703125}, {"start": 2855.14, "end": 2855.48, "word": " not", "probability": 0.96484375}, {"start": 2855.48, "end": 2856.08, "word": " exist", "probability": 0.96435546875}, {"start": 2856.08, "end": 2856.56, "word": " in", "probability": 0.92431640625}, {"start": 2856.56, "end": 2857.04, "word": " R", "probability": 0.92822265625}, {"start": 2857.04, "end": 2858.2, "word": " مش", "probability": 0.2486572265625}, {"start": 2858.2, "end": 2858.44, "word": " ممكن", "probability": 0.989013671875}, {"start": 2858.44, "end": 2858.88, "word": " تكون", "probability": 0.984619140625}, {"start": 2858.88, "end": 2859.68, "word": " موجودة", "probability": 0.984130859375}, {"start": 2859.68, "end": 2859.9, "word": " في", "probability": 0.89794921875}, {"start": 2859.9, "end": 2860.32, "word": " R", "probability": 0.87939453125}, {"start": 2860.32, "end": 2866.44, "word": " لأن", "probability": 0.6954345703125}, {"start": 2866.44, "end": 2866.62, "word": " لو", "probability": 0.9853515625}, {"start": 2866.62, "end": 2866.98, "word": " كانت", "probability": 0.980224609375}, {"start": 2866.98, "end": 2867.12, "word": " ال", "probability": 0.921875}, {"start": 2867.12, "end": 2867.32, "word": " limit", "probability": 0.96923828125}, {"start": 2867.32, "end": 2867.62, "word": " هذه", "probability": 0.7607421875}, {"start": 2867.62, "end": 2868.34, "word": " موجودة", "probability": 0.9906005859375}], "temperature": 1.0}, {"id": 99, "seek": 289478, "start": 2869.88, "end": 2894.78, "text": "فالمفروض limit صورة xn بما أن xn تقول السفر نكتب since otherwise لأن لو كلاك ذلك لو كانت هذه موجودة if limit", "tokens": [5172, 45340, 5172, 32887, 11242, 4948, 20328, 13063, 3660, 2031, 77, 4724, 15042, 14739, 2031, 77, 6055, 39648, 21136, 5172, 2288, 8717, 4117, 2655, 3555, 1670, 5911, 5296, 33456, 45164, 9122, 15040, 4117, 29910, 23275, 45164, 25961, 2655, 29538, 3714, 29245, 23328, 3660, 498, 4948], "avg_logprob": -0.278192946444387, "compression_ratio": 1.2595419847328244, "no_speech_prob": 0.0, "words": [{"start": 2869.88, "end": 2870.82, "word": "فالمفروض", "probability": 0.83017578125}, {"start": 2870.82, "end": 2872.28, "word": " limit", "probability": 0.430908203125}, {"start": 2872.28, "end": 2873.14, "word": " صورة", "probability": 0.9371744791666666}, {"start": 2873.14, "end": 2873.9, "word": " xn", "probability": 0.4776611328125}, {"start": 2873.9, "end": 2875.82, "word": " بما", "probability": 0.5125732421875}, {"start": 2875.82, "end": 2875.96, "word": " أن", "probability": 0.56103515625}, {"start": 2875.96, "end": 2876.38, "word": " xn", "probability": 0.810302734375}, {"start": 2876.38, "end": 2876.64, "word": " تقول", "probability": 0.697998046875}, {"start": 2876.64, "end": 2878.2, "word": " السفر", "probability": 0.8518880208333334}, {"start": 2878.2, "end": 2882.06, "word": " نكتب", "probability": 0.8585205078125}, {"start": 2882.06, "end": 2882.5, "word": " since", "probability": 0.73046875}, {"start": 2882.5, "end": 2883.32, "word": " otherwise", "probability": 0.78076171875}, {"start": 2883.32, "end": 2887.18, "word": " لأن", "probability": 0.827392578125}, {"start": 2887.18, "end": 2887.4, "word": " لو", "probability": 0.75146484375}, {"start": 2887.4, "end": 2887.92, "word": " كلاك", "probability": 0.5961100260416666}, {"start": 2887.92, "end": 2888.44, "word": " ذلك", "probability": 0.98388671875}, {"start": 2888.44, "end": 2889.26, "word": " لو", "probability": 0.82177734375}, {"start": 2889.26, "end": 2889.74, "word": " كانت", "probability": 0.958251953125}, {"start": 2889.74, "end": 2890.0, "word": " هذه", "probability": 0.72216796875}, {"start": 2890.0, "end": 2891.3, "word": " موجودة", "probability": 0.9892578125}, {"start": 2891.3, "end": 2894.22, "word": " if", "probability": 0.7666015625}, {"start": 2894.22, "end": 2894.78, "word": " limit", "probability": 0.9287109375}], "temperature": 1.0}, {"id": 100, "seek": 290521, "start": 2900.17, "end": 2905.21, "text": "فى limit ل F of X لما X تقول لسة exist", "tokens": [5172, 7578, 4948, 5296, 479, 295, 1783, 5296, 15042, 1783, 6055, 39648, 5296, 3794, 3660, 2514], "avg_logprob": -0.5643382352941176, "compression_ratio": 0.8947368421052632, "no_speech_prob": 0.0, "words": [{"start": 2900.17, "end": 2900.63, "word": "فى", "probability": 0.51336669921875}, {"start": 2900.63, "end": 2900.93, "word": " limit", "probability": 0.300537109375}, {"start": 2900.93, "end": 2901.11, "word": " ل", "probability": 0.75830078125}, {"start": 2901.11, "end": 2901.35, "word": " F", "probability": 0.33642578125}, {"start": 2901.35, "end": 2901.59, "word": " of", "probability": 0.56201171875}, {"start": 2901.59, "end": 2901.95, "word": " X", "probability": 0.94677734375}, {"start": 2901.95, "end": 2902.85, "word": " لما", "probability": 0.6075439453125}, {"start": 2902.85, "end": 2903.19, "word": " X", "probability": 0.9033203125}, {"start": 2903.19, "end": 2903.53, "word": " تقول", "probability": 0.8662109375}, {"start": 2903.53, "end": 2904.01, "word": " لسة", "probability": 0.5765380859375}, {"start": 2904.01, "end": 2905.21, "word": " exist", "probability": 0.37060546875}], "temperature": 1.0}, {"id": 101, "seek": 293979, "start": 2912.71, "end": 2939.79, "text": "then المفروض ال limit ل f of x n لما n تقول infinity بتساوي ال limit ل f of y n as n tends to infinity وهذا مستحيل which is impossible وهذا زي ما شوفنا مستحيل impossible", "tokens": [19096, 9673, 5172, 32887, 11242, 2423, 4948, 5296, 283, 295, 2031, 297, 5296, 15042, 297, 6055, 39648, 13202, 39894, 3794, 995, 45865, 2423, 4948, 5296, 283, 295, 288, 297, 382, 297, 12258, 281, 13202, 37037, 15730, 3714, 14851, 5016, 26895, 597, 307, 6243, 37037, 15730, 30767, 1829, 19446, 13412, 38688, 8315, 3714, 14851, 5016, 26895, 6243], "avg_logprob": -0.23930921261770682, "compression_ratio": 1.5238095238095237, "no_speech_prob": 0.0, "words": [{"start": 2912.71, "end": 2913.21, "word": "then", "probability": 0.12335205078125}, {"start": 2913.21, "end": 2914.35, "word": " المفروض", "probability": 0.978515625}, {"start": 2914.35, "end": 2914.51, "word": " ال", "probability": 0.415283203125}, {"start": 2914.51, "end": 2914.89, "word": " limit", "probability": 0.8671875}, {"start": 2914.89, "end": 2916.43, "word": " ل", "probability": 0.9169921875}, {"start": 2916.43, "end": 2916.73, "word": " f", "probability": 0.39501953125}, {"start": 2916.73, "end": 2916.95, "word": " of", "probability": 0.5732421875}, {"start": 2916.95, "end": 2917.33, "word": " x", "probability": 0.9443359375}, {"start": 2917.33, "end": 2917.79, "word": " n", "probability": 0.28125}, {"start": 2917.79, "end": 2919.47, "word": " لما", "probability": 0.831787109375}, {"start": 2919.47, "end": 2919.73, "word": " n", "probability": 0.52880859375}, {"start": 2919.73, "end": 2919.99, "word": " تقول", "probability": 0.6473388671875}, {"start": 2919.99, "end": 2920.65, "word": " infinity", "probability": 0.62109375}, {"start": 2920.65, "end": 2922.21, "word": " بتساوي", "probability": 0.91455078125}, {"start": 2922.21, "end": 2922.51, "word": " ال", "probability": 0.79052734375}, {"start": 2922.51, "end": 2922.91, "word": " limit", "probability": 0.98291015625}, {"start": 2922.91, "end": 2924.37, "word": " ل", "probability": 0.9619140625}, {"start": 2924.37, "end": 2924.67, "word": " f", "probability": 0.7509765625}, {"start": 2924.67, "end": 2924.91, "word": " of", "probability": 0.94677734375}, {"start": 2924.91, "end": 2925.27, "word": " y", "probability": 0.8583984375}, {"start": 2925.27, "end": 2925.81, "word": " n", "probability": 0.677734375}, {"start": 2925.81, "end": 2926.39, "word": " as", "probability": 0.9150390625}, {"start": 2926.39, "end": 2926.81, "word": " n", "probability": 0.8583984375}, {"start": 2926.81, "end": 2927.11, "word": " tends", "probability": 0.378662109375}, {"start": 2927.11, "end": 2927.27, "word": " to", "probability": 0.96337890625}, {"start": 2927.27, "end": 2927.69, "word": " infinity", "probability": 0.84326171875}, {"start": 2927.69, "end": 2929.71, "word": " وهذا", "probability": 0.847412109375}, {"start": 2929.71, "end": 2930.41, "word": " مستحيل", "probability": 0.9862060546875}, {"start": 2930.41, "end": 2933.23, "word": " which", "probability": 0.791015625}, {"start": 2933.23, "end": 2933.95, "word": " is", "probability": 0.970703125}, {"start": 2933.95, "end": 2934.75, "word": " impossible", "probability": 0.8505859375}, {"start": 2934.75, "end": 2935.39, "word": " وهذا", "probability": 0.823974609375}, {"start": 2935.39, "end": 2937.07, "word": " زي", "probability": 0.837890625}, {"start": 2937.07, "end": 2937.19, "word": " ما", "probability": 0.98583984375}, {"start": 2937.19, "end": 2937.83, "word": " شوفنا", "probability": 0.90380859375}, {"start": 2937.83, "end": 2939.33, "word": " مستحيل", "probability": 0.981201171875}, {"start": 2939.33, "end": 2939.79, "word": " impossible", "probability": 0.57568359375}], "temperature": 1.0}, {"id": 102, "seek": 296803, "start": 2941.45, "end": 2968.03, "text": "لأن طولها limit f of x in بالساوي سفر و limit f of y in بالساوي واحد إذن هنا استخدمنا sequential criterion في إثبات إن ال limit لل function f of x بالساوي صين واحد على x غير موجودة عند السفر طيب هناخد break خمس دقايق و بعدين نواصل", "tokens": [1211, 33456, 23032, 12610, 11296, 4948, 283, 295, 2031, 294, 20666, 3794, 995, 45865, 8608, 5172, 2288, 4032, 4948, 283, 295, 288, 294, 20666, 3794, 995, 45865, 36764, 24401, 11933, 8848, 1863, 34105, 44713, 9778, 40448, 8315, 42881, 46691, 8978, 11933, 12984, 3555, 9307, 36145, 2423, 4948, 24976, 2445, 283, 295, 2031, 20666, 3794, 995, 45865, 20328, 9957, 36764, 24401, 15844, 2031, 32771, 13546, 3714, 29245, 23328, 3660, 43242, 21136, 5172, 2288, 23032, 1829, 3555, 8032, 1863, 47283, 3215, 1821, 16490, 2304, 3794, 11778, 4587, 47302, 4587, 4032, 39182, 9957, 8717, 2407, 33546, 1211], "avg_logprob": -0.2054276356571599, "compression_ratio": 1.5963302752293578, "no_speech_prob": 0.0, "words": [{"start": 2941.45, "end": 2941.81, "word": "لأن", "probability": 0.645263671875}, {"start": 2941.81, "end": 2942.11, "word": " طولها", "probability": 0.669189453125}, {"start": 2942.11, "end": 2942.35, "word": " limit", "probability": 0.798828125}, {"start": 2942.35, "end": 2942.65, "word": " f", "probability": 0.65625}, {"start": 2942.65, "end": 2942.87, "word": " of", "probability": 0.45947265625}, {"start": 2942.87, "end": 2943.15, "word": " x", "probability": 0.92333984375}, {"start": 2943.15, "end": 2943.37, "word": " in", "probability": 0.1611328125}, {"start": 2943.37, "end": 2943.77, "word": " بالساوي", "probability": 0.6429443359375}, {"start": 2943.77, "end": 2944.21, "word": " سفر", "probability": 0.6842447916666666}, {"start": 2944.21, "end": 2944.73, "word": " و", "probability": 0.5029296875}, {"start": 2944.73, "end": 2944.97, "word": " limit", "probability": 0.87353515625}, {"start": 2944.97, "end": 2945.51, "word": " f", "probability": 0.810546875}, {"start": 2945.51, "end": 2945.71, "word": " of", "probability": 0.9765625}, {"start": 2945.71, "end": 2945.93, "word": " y", "probability": 0.98681640625}, {"start": 2945.93, "end": 2946.13, "word": " in", "probability": 0.9521484375}, {"start": 2946.13, "end": 2946.81, "word": " بالساوي", "probability": 0.968505859375}, {"start": 2946.81, "end": 2947.27, "word": " واحد", "probability": 0.968505859375}, {"start": 2947.27, "end": 2949.39, "word": " إذن", "probability": 0.478515625}, {"start": 2949.39, "end": 2950.15, "word": " هنا", "probability": 0.81689453125}, {"start": 2950.15, "end": 2950.77, "word": " استخدمنا", "probability": 0.8994140625}, {"start": 2950.77, "end": 2951.21, "word": " sequential", "probability": 0.75927734375}, {"start": 2951.21, "end": 2951.99, "word": " criterion", "probability": 0.95947265625}, {"start": 2951.99, "end": 2952.63, "word": " في", "probability": 0.83154296875}, {"start": 2952.63, "end": 2953.17, "word": " إثبات", "probability": 0.84857177734375}, {"start": 2953.17, "end": 2953.33, "word": " إن", "probability": 0.47705078125}, {"start": 2953.33, "end": 2953.47, "word": " ال", "probability": 0.78857421875}, {"start": 2953.47, "end": 2953.81, "word": " limit", "probability": 0.876953125}, {"start": 2953.81, "end": 2954.55, "word": " لل", "probability": 0.50634765625}, {"start": 2954.55, "end": 2954.97, "word": " function", "probability": 0.8154296875}, {"start": 2954.97, "end": 2955.25, "word": " f", "probability": 0.931640625}, {"start": 2955.25, "end": 2955.43, "word": " of", "probability": 0.88330078125}, {"start": 2955.43, "end": 2955.65, "word": " x", "probability": 0.98583984375}, {"start": 2955.65, "end": 2956.05, "word": " بالساوي", "probability": 0.963134765625}, {"start": 2956.05, "end": 2956.43, "word": " صين", "probability": 0.6827392578125}, {"start": 2956.43, "end": 2956.79, "word": " واحد", "probability": 0.989990234375}, {"start": 2956.79, "end": 2956.95, "word": " على", "probability": 0.6953125}, {"start": 2956.95, "end": 2957.33, "word": " x", "probability": 0.8525390625}, {"start": 2957.33, "end": 2958.29, "word": " غير", "probability": 0.982421875}, {"start": 2958.29, "end": 2959.07, "word": " موجودة", "probability": 0.9901123046875}, {"start": 2959.07, "end": 2959.77, "word": " عند", "probability": 0.98828125}, {"start": 2959.77, "end": 2960.39, "word": " السفر", "probability": 0.92578125}, {"start": 2960.39, "end": 2964.49, "word": " طيب", "probability": 0.9169921875}, {"start": 2964.49, "end": 2965.15, "word": " هناخد", "probability": 0.8431396484375}, {"start": 2965.15, "end": 2965.81, "word": " break", "probability": 0.98974609375}, {"start": 2965.81, "end": 2966.15, "word": " خمس", "probability": 0.9619140625}, {"start": 2966.15, "end": 2966.73, "word": " دقايق", "probability": 0.8887939453125}, {"start": 2966.73, "end": 2966.85, "word": " و", "probability": 0.671875}, {"start": 2966.85, "end": 2967.29, "word": " بعدين", "probability": 0.968017578125}, {"start": 2967.29, "end": 2968.03, "word": " نواصل", "probability": 0.95166015625}], "temperature": 1.0}, {"id": 103, "seek": 297184, "start": 2968.72, "end": 2971.84, "text": "المحاضرة التانية", "tokens": [45340, 5016, 46958, 25720, 16712, 7649, 10632], "avg_logprob": -0.029571533668786287, "compression_ratio": 0.7948717948717948, "no_speech_prob": 0.0, "words": [{"start": 2968.72, "end": 2970.12, "word": "المحاضرة", "probability": 0.985107421875}, {"start": 2970.12, "end": 2971.84, "word": " التانية", "probability": 0.9527994791666666}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2975.49775, "duration_after_vad": 2592.6056249999897}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7K-d4aAzbLs_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7K-d4aAzbLs_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..c2bb9fe844a9d76168be48dbb206ace887881d22
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7K-d4aAzbLs_raw.srt
@@ -0,0 +1,1448 @@
+1
+00:00:21,410 --> 00:00:29,070
+السلام عليكم اليوم في اللقاء الأول هناخد مناقشة و
+
+2
+00:00:29,070 --> 00:00:36,710
+اعتقد ان احنا في المناقشة السابقة وصلنا ل section
+
+3
+00:00:36,710 --> 00:00:46,810
+تلاتة خمسة، أصبع؟ فممكن
+
+4
+00:00:46,810 --> 00:01:07,930
+اليومبنناقش section تلاتة ستة أو تلاتة سبعة في
+
+5
+00:01:07,930 --> 00:01:14,710
+أي أسل عندكم في section تلاتة خمسة أو section
+
+6
+00:01:14,710 --> 00:01:25,230
+تلاتة ستةثلاثة ستة السؤال ستة أي سؤال سؤال ستة ستة
+
+7
+00:01:25,230 --> 00:01:35,470
+سؤال
+
+8
+00:01:35,470 --> 00:01:37,650
+ستة section تلاتة ستة
+
+9
+00:01:46,160 --> 00:01:58,020
+let x in let the sequence x in be properly die
+
+10
+00:01:58,020 --> 00:02:05,920
+there and let
+
+11
+00:02:05,920 --> 00:02:24,780
+and let y inب such that limit x in ضرب y in limit
+
+12
+00:02:24,780 --> 00:02:31,980
+حصل ضرب لما n تقول لinfinity الساوي
+
+13
+00:02:31,980 --> 00:02:40,620
+L ينتمي إلى R يعني exists in R شو
+
+14
+00:02:40,620 --> 00:02:54,410
+مطلوبثم اثبت اظهر ان سيكوينس ين يتعامل
+
+15
+00:02:54,410 --> 00:03:10,230
+بالزيرو حل
+
+16
+00:03:10,230 --> 00:03:17,950
+السؤال هذا بعتمدعلى سؤال سابق اللي هو سؤال تلاتة
+
+17
+00:03:17,950 --> 00:03:27,190
+فالسؤال
+
+18
+00:03:27,190 --> 00:03:32,430
+هذا بيقول ان f
+
+19
+00:03:32,430 --> 00:03:50,310
+x n أكبر من سفر لكل n عدد طبيعيو ال then
+
+20
+00:03:50,310 --> 00:04:03,990
+limit xn بساوي zero if and only if limit واحد على
+
+21
+00:04:03,990 --> 00:04:10,350
+xn as n tends to infinity بساوي plus infinity
+
+22
+00:04:20,790 --> 00:04:27,670
+Okay لأن في سؤال طلعتها إذا كانت xn حدود sequence
+
+23
+00:04:27,670 --> 00:04:36,290
+حدودها موجة بقى و ف limit ال sequence xn بساوي سفر
+
+24
+00:04:36,290 --> 00:04:41,350
+if and only if limit مقلوب ال sequence xn بساوي
+
+25
+00:04:41,350 --> 00:04:42,190
+plus infinity
+
+26
+00:04:46,230 --> 00:04:52,690
+و طبعا في كمان ممكن نثبت ان لو كانت ال Xn حدودها
+
+27
+00:04:52,690 --> 00:05:00,950
+سالبة ف limit Xn بساوي صفر F and only F limit واحد
+
+28
+00:05:00,950 --> 00:05:03,970
+على Xn بساوي negative infinity
+
+29
+00:05:15,220 --> 00:05:24,480
+بما أن xn هو بشكل صحيح ديبيرزينت
+
+30
+00:05:24,480 --> 00:05:39,580
+ثم قيمة xn بساوي إفينتي أو قيمة xn بساوي نيجاتيف
+
+31
+00:05:39,580 --> 00:05:40,180
+إفينتي
+
+32
+00:05:45,460 --> 00:05:52,220
+case one ناخد الحالة الأولى اللى فيها limit xm
+
+33
+00:05:52,220 --> 00:05:59,860
+بساوي infinity by
+
+34
+00:05:59,860 --> 00:06:03,560
+exercise
+
+35
+00:06:03,560 --> 00:06:15,020
+رقم تلاتة section تلاتة ستةوالـ exercise اللى فوق
+
+36
+00:06:15,020 --> 00:06:18,100
+هذا
+
+37
+00:06:18,100 --> 00:06:27,160
+معناه انه we have هيطلع انه limit مطلوب ال
+
+38
+00:06:27,160 --> 00:06:38,000
+sequence xn as n tends to infinity بفلع صفر يعني
+
+39
+00:06:38,000 --> 00:06:44,710
+اعتبرى هذه هي xnتعتبر ال 1 على xn هي xn فإذا كان
+
+40
+00:06:44,710 --> 00:06:49,510
+limit xn بساوي infinity فlimit مقلوب ال xn اللي
+
+41
+00:06:49,510 --> 00:06:57,530
+هنا مقلوب اللي هو ايه بتطلع سفر ولا عكس يعني هنا
+
+42
+00:06:57,530 --> 00:07:03,850
+نفس ال exercise بس badly xn بواحد على xn فهذه
+
+43
+00:07:03,850 --> 00:07:08,690
+نتيجة صحية تمام hence
+
+44
+00:07:13,030 --> 00:07:16,810
+الـ limit ل
+
+45
+00:07:16,810 --> 00:07:29,290
+YN as intense infinity بساوي ال limit ال
+
+46
+00:07:29,290 --> 00:07:38,110
+YN ممكن كتبتها على صورة على
+
+47
+00:07:38,110 --> 00:07:39,310
+صورة
+
+48
+00:07:46,210 --> 00:07:55,770
+xn في yn ضرب 1
+
+49
+00:07:55,770 --> 00:08:01,290
+على xn صح
+
+50
+00:08:01,290 --> 00:08:09,850
+نظبط هيك ال yn هي عبارة عن xn في yn في 1 على xn
+
+51
+00:08:12,880 --> 00:08:18,360
+الان ال limit هذه لحد الأول exist و limit ل واحد
+
+52
+00:08:18,360 --> 00:08:22,660
+على xn برضه exist اذا ال limit حاصل ضرب بساوي حاصل
+
+53
+00:08:22,660 --> 00:08:27,540
+ضرب ال limits بقدر استخدم القانون هذا هطبق انه
+
+54
+00:08:27,540 --> 00:08:32,360
+limit حاصل ضرب two sequences بساوي limit الأولى
+
+55
+00:08:32,360 --> 00:08:41,100
+اللي هي حاصل ضرب xn ynدرب limit الـ sequence
+
+56
+00:08:41,100 --> 00:08:48,180
+التانية هي واحد على X end as n tends to infinity و
+
+57
+00:08:48,180 --> 00:08:53,940
+ال limit الأولى مش سامناها عدد L لما exist ضرب ال
+
+58
+00:08:53,940 --> 00:09:01,600
+limit التانية سفر فبطلع عندي سفر و هو المطلوب فهنا
+
+59
+00:09:01,600 --> 00:09:05,920
+أثبتنا في الحالة التانية case two
+
+60
+00:09:10,140 --> 00:09:24,200
+لو كانت ال limit لـ xn بساوي negative infinity ففي
+
+61
+00:09:24,200 --> 00:09:29,580
+الحالة هذه بيطلع عندي برضه by exercise تلاتة
+
+62
+00:09:29,580 --> 00:09:36,020
+section تلاتة ستة بس هنا مع التعديل هيطلع ان ال
+
+63
+00:09:36,020 --> 00:09:44,910
+limitلا واحد على اكس ان مثلا سفر و باقي البرهان
+
+64
+00:09:44,910 --> 00:09:58,730
+and the rest of the proof is similar to
+
+65
+00:09:58,730 --> 00:09:59,450
+case one
+
+66
+00:10:03,650 --> 00:10:09,850
+Okay تمام اذا هذا اللي هو البرهام ان الادكارة
+
+67
+00:10:09,850 --> 00:10:15,870
+تعتمد على exercise ثلاثة المهم وهو ان limit ل
+
+68
+00:10:15,870 --> 00:10:19,750
+sequence بيساوي infinity if and only if limit
+
+69
+00:10:19,750 --> 00:10:24,810
+مقلوب ال sequence بيساوي سفر او لعكس تمام و هذا
+
+70
+00:10:34,460 --> 00:10:39,400
+في عنكم أسئلة تانية؟
+
+71
+00:10:39,400 --> 00:10:45,260
+في
+
+72
+00:10:45,260 --> 00:10:49,220
+أسئلة تانية section تلاتة ستة الفرق بيه من سؤال
+
+73
+00:10:49,220 --> 00:10:49,680
+تسعة
+
+74
+00:11:33,220 --> 00:11:41,380
+حاول نكتب السؤال و بعدين السؤال
+
+75
+00:11:41,380 --> 00:11:43,600
+تسعة section تلاتة ع ستة
+
+76
+00:11:53,320 --> 00:12:04,400
+لت XIN و YIN بيكونوا عاملين من عاملين من عاملين من
+
+77
+00:12:04,400 --> 00:12:06,860
+عاملين من عاملين من عاملين من عاملين من عاملين من
+
+78
+00:12:06,860 --> 00:12:09,100
+عاملين من عاملين من عاملين من عاملين من عاملين من
+
+79
+00:12:09,100 --> 00:12:22,440
+عاملين من عاملين
+
+80
+00:12:31,890 --> 00:12:44,270
+مطلوب الأول هو show if limit yn بساوي infinity
+
+81
+00:12:44,270 --> 00:12:51,370
+then limit
+
+82
+00:12:51,370 --> 00:12:53,590
+xn بساوي infinity
+
+83
+00:12:56,820 --> 00:13:10,660
+والجزء التاني show if x in is bounded then
+
+84
+00:13:10,660 --> 00:13:15,680
+limit
+
+85
+00:13:15,680 --> 00:13:25,600
+y in is serviceable طبعا
+
+86
+00:13:30,880 --> 00:13:39,520
+في برهانين لل ..
+
+87
+00:13:39,520 --> 00:13:47,540
+لل exercise هذا البرهان الأول باستخدام
+
+88
+00:13:47,540 --> 00:13:55,580
+exercise 7 اللي جابله يعني هنا since
+
+89
+00:13:55,580 --> 00:14:04,790
+من الفرض لما انه limitxn على yn as n tends to
+
+90
+00:14:04,790 --> 00:14:15,150
+infinity بساوي plus infinity then by exercise
+
+91
+00:14:15,150 --> 00:14:24,790
+تلاتة section تلاتة ستة if limit sequence بساوي
+
+92
+00:14:24,790 --> 00:14:30,780
+infinityبطلع limit مقلوب الـ sequence اللي هو y in
+
+93
+00:14:30,780 --> 00:14:40,920
+على x and as n tends to infinity بساوي سبعة now
+
+94
+00:14:40,920 --> 00:14:44,480
+apply
+
+95
+00:14:44,480 --> 00:14:47,900
+exercise
+
+96
+00:14:47,900 --> 00:14:55,940
+رقم سبعة section تلاتة ستة to
+
+97
+00:14:55,940 --> 00:14:56,340
+get
+
+98
+00:14:59,870 --> 00:15:13,950
+the results in a and b وهذا بيعطيني مرغب لو بصيت و
+
+99
+00:15:13,950 --> 00:15:18,950
+لا ال exercise
+
+100
+00:15:18,950 --> 00:15:25,070
+سبعة في ال exercise سبعة بيقول ده كانت ال limit لل
+
+101
+00:15:25,070 --> 00:15:30,880
+quotientلـ quotient زي هذا بساوي صفر و x in و y in
+
+102
+00:15:30,880 --> 00:15:37,200
+حدودهم موجبة ففي الحالة هذه إذا كانت limit ال
+
+103
+00:15:37,200 --> 00:15:47,200
+sequence اللي تحت convergent إذا
+
+104
+00:15:47,200 --> 00:15:50,240
+كانت limit ال sequence اللي تحت
+
+105
+00:15:56,100 --> 00:16:01,960
+لأ limit ال sequence اللي فوق اللي هي yn هنا
+
+106
+00:16:01,960 --> 00:16:06,040
+infinity فبطلع limit xn بال 7 infinity اللي هو جزء
+
+107
+00:16:06,040 --> 00:16:12,980
+11وكمان اذا كانت ال sequence اللى فى المقام
+
+108
+00:16:12,980 --> 00:16:16,520
+bounded اللى هى x in هنا طبعا فى المقام bounded
+
+109
+00:16:16,520 --> 00:16:21,500
+فرقة ال sequence اللى فى ال bust تطلع يساوي 0 وهذا
+
+110
+00:16:21,500 --> 00:16:25,740
+هو الجزء التانى هذا حسب هذا لو بدنا نستخدم
+
+111
+00:16:25,740 --> 00:16:33,610
+exercise رقم 7 وطبعا لازم نبرهنهلكن ممكن نعطي
+
+112
+00:16:33,610 --> 00:16:39,710
+برهان مباشر بدون ما يستخدم exercise السابعة
+
+113
+00:16:39,710 --> 00:16:49,670
+وبالتالي إذا في حال تاني أو برهان تاني باستخدام
+
+114
+00:16:49,670 --> 00:16:57,900
+التعريفات وال comparison testsباستخدام التعريفات
+
+115
+00:16:57,900 --> 00:17:01,820
+زايد ال comparison tests اختبارات المقارنة ال
+
+116
+00:17:01,820 --> 00:17:09,240
+proof رقم اتنين since
+
+117
+00:17:09,240 --> 00:17:16,820
+اننا ننسى هذا القرآن انا عند هذه الفرض since limit
+
+118
+00:17:16,820 --> 00:17:24,630
+ل xn over yn هذا عبارة عن sequenceلأن الـ limit
+
+119
+00:17:24,630 --> 00:17:33,410
+إلا بالساقر plus infinity then given Alpha أي real
+
+120
+00:17:33,410 --> 00:17:41,610
+number Alpha من تعريف الـ improper convergence
+
+121
+00:17:41,610 --> 00:17:50,030
+لأي Alpha there exists capital N يعتمد على Alpha
+
+122
+00:17:50,030 --> 00:17:56,450
+عدد قضيةبحيث انه يكون M أكبر من أوسع ال capital M
+
+123
+00:17:56,450 --> 00:18:12,950
+بطلع عندي XM على YM أكبر من Alpha طبعا
+
+124
+00:18:12,950 --> 00:18:20,480
+وهذا بيقدي ان XM أكبر من Alpha في YMلما عندي yn
+
+125
+00:18:20,480 --> 00:18:26,120
+هنا موجبة لما أضرب الطرفين في yn التباينة إشارتها
+
+126
+00:18:26,120 --> 00:18:32,340
+تبقى كما هي إذا
+
+127
+00:18:32,340 --> 00:18:40,880
+أنا عندي الان الكلام هذا صحيح لكل n أكبر من أوسع
+
+128
+00:18:40,880 --> 00:18:46,540
+كابتن الان الان
+
+129
+00:18:46,540 --> 00:18:47,360
+by
+
+130
+00:18:49,930 --> 00:18:56,930
+بمعنى الـ Direct Comparison Test بما
+
+131
+00:18:56,930 --> 00:19:02,970
+انه limit yn
+
+132
+00:19:02,970 --> 00:19:04,130
+بالساوي infinity
+
+133
+00:19:25,130 --> 00:19:33,010
+ناخد alpha في واحد ممكن
+
+134
+00:19:33,010 --> 00:19:37,750
+اه ناخد alpha في واحد صح دي من ال alpha دي ثاني
+
+135
+00:19:37,750 --> 00:19:49,320
+واحد يعني ثاني ال R مظبوط فده واحد وبتاني واحدبما
+
+136
+00:19:49,320 --> 00:19:54,600
+ان ال limit ل yn
+
+137
+00:19:54,600 --> 00:20:02,180
+بساوي infinity نحن نحصل على limit ل xn بساوي
+
+138
+00:20:02,180 --> 00:20:06,640
+infinity لان هذا بثبت الجزء الأول انت بدك الجزء
+
+139
+00:20:06,640 --> 00:20:08,160
+التاني صح؟ طيب
+
+140
+00:20:15,760 --> 00:20:19,840
+بنشوف الجزء التاني إذا كانت ال sequence x in
+
+141
+00:20:19,840 --> 00:20:27,400
+bounded فبنلمط y in بسرعه نصف طيب
+
+142
+00:20:27,400 --> 00:20:32,420
+الجزء
+
+143
+00:20:32,420 --> 00:20:43,340
+دي since x in is bounded إذن
+
+144
+00:20:43,340 --> 00:20:48,760
+في عدد موجبThere exists m positive number بحيث انه
+
+145
+00:20:48,760 --> 00:20:57,360
+absolute xm أصغر من أو ساوي m لكل m في n هذا من
+
+146
+00:20:57,360 --> 00:21:05,280
+تعريف الboundary نفسي طيب بالمنفذ بتاعنا ايه؟ ان
+
+147
+00:21:05,280 --> 00:21:16,480
+ال limit ل ym بساعة صفر طيب to showlimit yn بساوي
+
+148
+00:21:16,480 --> 00:21:23,260
+zero let epsilon بنستخدم تعريف epsilon capital M
+
+149
+00:21:23,260 --> 00:21:32,060
+لإن هي let epsilon أكبر من الصفر be given من
+
+150
+00:21:32,060 --> 00:21:39,100
+epsilon على M بيطلع عدد موجب من
+
+151
+00:21:41,390 --> 00:21:52,910
+العدد الموجب يعتمد
+
+152
+00:21:52,910 --> 00:21:58,830
+على إبسلون على م يعتبر
+
+153
+00:21:58,830 --> 00:22:01,590
+إبسلون على م
+
+154
+00:22:16,450 --> 00:22:24,150
+أنا عندي ايش عندي بدي
+
+155
+00:22:24,150 --> 00:22:31,170
+أثبت ان limit yn بالساوي سفر فخلينا نشوف
+
+156
+00:22:43,720 --> 00:22:54,900
+طيب أنا لازم أستخدم .. آه لازم أستخدم ..
+
+157
+00:22:54,900 --> 00:23:01,840
+طيب since ..
+
+158
+00:23:01,840 --> 00:23:06,820
+طيب بس هنا يعني خليني أقول since
+
+159
+00:23:11,850 --> 00:23:20,390
+بما أن ال limit ل yn على xn as n tends to infinity
+
+160
+00:23:20,390 --> 00:23:24,410
+أنا عندي المقلوب هذا ال limit تبعته infinity، هذا
+
+161
+00:23:24,410 --> 00:23:29,890
+ال limit تبعته سفر وهي عندي epsilon على m عدد موجة
+
+162
+00:23:29,890 --> 00:23:36,150
+given، there exists capital M يعتمد على epsilon
+
+163
+00:23:36,150 --> 00:23:47,110
+على mعدد طبيعي لحيث انه لكل n أكبر من أوسع ال
+
+164
+00:23:47,110 --> 00:23:56,850
+capital N بيطلع عندي absolute yn على xn minus ال
+
+165
+00:23:56,850 --> 00:24:05,850
+zero أصغر من epsilon على n تمام؟
+
+166
+00:24:07,690 --> 00:24:27,750
+طب ما هذا بيقدي فانا
+
+167
+00:24:27,750 --> 00:24:32,910
+بدي اثبت انه limit yn بالساو ستر يعني بدي اثبت انه
+
+168
+00:24:32,910 --> 00:24:38,980
+ال absolute valueلو كان n أكبر من أو ساوي capital
+
+169
+00:24:38,980 --> 00:24:44,920
+N بتثبت أن ال absolute value ل y in minus 0 أصغر
+
+170
+00:24:44,920 --> 00:24:50,020
+من epsilon عشان أثبت أن ال limit ل y in بساوي سفر
+
+171
+00:24:50,020 --> 00:24:55,520
+بتثبت أن ال absolute value ل y in minus 0 أصغر من
+
+172
+00:24:55,520 --> 00:25:00,500
+ال given epsilon طيب
+
+173
+00:25:00,500 --> 00:25:12,970
+هذا بساوي absoluteYn بيساوي أبقى عن Xn ضرب Yn
+
+174
+00:25:12,970 --> 00:25:26,270
+على Xn minus zero و هذا بيساوي absolute Xn في
+
+175
+00:25:26,270 --> 00:25:30,170
+absolute Yn على Xn
+
+176
+00:25:37,650 --> 00:25:44,410
+بتكون موضوع ممكن نحط سارة zero هنا طيب
+
+177
+00:25:44,410 --> 00:25:51,510
+هذا لكل N هذا أصغر من أو يساوي M وال absolute
+
+178
+00:25:51,510 --> 00:25:56,570
+value هذه لكل N أكبر من أو يساوي capital N هذا
+
+179
+00:25:56,570 --> 00:26:05,270
+أصغر من إبسلون على M إبسلون على M مش هبقى M مع N
+
+180
+00:26:05,270 --> 00:26:14,790
+بقى اللي عندي إبسلونطبعا؟ طيب since أكبر من السفر
+
+181
+00:26:14,790 --> 00:26:25,310
+was arbitrarily we get انه limit ل y in as in tens
+
+182
+00:26:25,310 --> 00:26:29,650
+of infinity بساوي zero و هو المفروض يعني هذا بيكمل
+
+183
+00:26:29,650 --> 00:26:35,380
+برعان الجزء بيه okay طبعا؟هذا على اعتبار ان احنا
+
+184
+00:26:35,380 --> 00:26:41,300
+exercise سبعة ما بنعرفوش بس في كل الأحوال احنا
+
+185
+00:26:41,300 --> 00:26:47,300
+استخدمنا exercise ثلاثة طبعا
+
+186
+00:26:47,300 --> 00:26:54,400
+هكذا بنثبت تسعة و سبعة بالمناسبة زيه بنفس الطريقة
+
+187
+00:26:54,400 --> 00:27:01,020
+بافكار مشابه ممكن اثباته بنفس السلوب بنفس النمط
+
+188
+00:27:03,510 --> 00:27:09,270
+كمان في أي أسئلة تانية في section تلاتة سبعة؟ إذا
+
+189
+00:27:09,270 --> 00:27:15,430
+مافيش خلينا ننتقل ل section تلاتة سبعة تبع
+
+190
+00:27:15,430 --> 00:27:20,110
+ال series هذا في
+
+191
+00:27:20,110 --> 00:27:24,550
+عندكم أي أسئلة في section تلاتة سبعة؟ تلاتة خمسة؟
+
+192
+00:27:24,550 --> 00:27:25,750
+تلاتة سبعة؟
+
+193
+00:27:44,990 --> 00:27:54,110
+في أي أسلة في section تلاتة سبعة أو تلاتة ستة
+
+194
+00:27:54,110 --> 00:28:07,910
+مافيش؟
+
+195
+00:28:07,910 --> 00:28:13,700
+السؤال تلاتة فرصة تلاتة سبعةالسؤال التالت الفارقة
+
+196
+00:28:13,700 --> 00:28:14,320
+السيه؟
+
+197
+00:28:28,470 --> 00:28:33,570
+استخدمت ال partial fractions؟ اه بس مش .. مش كله
+
+198
+00:28:33,570 --> 00:28:37,990
+بالغاية طلعت قيم A و C بيطلعوا نص و نص و B بيطلعوا
+
+199
+00:28:37,990 --> 00:28:43,770
+سالم واحد و بعد ما جيت اكمل مش كل الحدود بيطلعوا
+
+200
+00:28:43,770 --> 00:28:48,790
+بالطب معايا زي قمتي لو سؤاليني للجامعة اه عشان
+
+201
+00:28:48,790 --> 00:28:52,310
+هيكون تلات قصور يعني اه
+
+202
+00:28:54,820 --> 00:29:09,900
+بس لازم يكون فيه يعني تلاشي و فيه
+
+203
+00:29:09,900 --> 00:29:18,020
+.. نشوف
+
+204
+00:29:18,020 --> 00:29:22,040
+يعني مافيش تلاشي جيبت الارت برشا الصمت .. فيه
+
+205
+00:29:22,040 --> 00:29:26,940
+تلاشي بس فيه بيضغط اه بخلينا نشوفخلّيني أجرب
+
+206
+00:29:26,940 --> 00:29:44,520
+السؤال
+
+207
+00:29:44,520 --> 00:29:47,580
+تلاتة الفرق C سكتشن تلاتة سبعة
+
+208
+00:29:54,860 --> 00:29:59,300
+استخدم الـ partial fractions
+
+209
+00:29:59,300 --> 00:30:03,020
+لإظهار
+
+210
+00:30:03,020 --> 00:30:11,100
+أن عدد الـ infinite series sigma من ن يعني واحد
+
+211
+00:30:11,100 --> 00:30:21,320
+لإنفينيتي الواحد عشان ن في ن اضافة واحد لان اضافة
+
+212
+00:30:21,320 --> 00:30:23,600
+اثنين بساوي واحد اربعة
+
+213
+00:30:32,200 --> 00:30:37,060
+فبدنا نكتب هذا بتحلل و باستخدام ال partial
+
+214
+00:30:37,060 --> 00:30:43,740
+fractions إلى تلت قصور فجدتش
+
+215
+00:30:43,740 --> 00:30:48,080
+فرعة التواردة كانت دي؟ كان الأولى a بتسوي نص يعني
+
+216
+00:30:48,080 --> 00:30:57,340
+نص على n تانية سالب واحد سالب او زاد سالب واحد على
+
+217
+00:30:57,340 --> 00:31:09,270
+n plus one والاخيرة نصنص على n plus two تعالى
+
+218
+00:31:09,270 --> 00:31:18,590
+نحسب ال inf partial sum sn بسعر sigma من k بسعر
+
+219
+00:31:18,590 --> 00:31:31,370
+واحد الى n ل xk اللى هو واحد علىك في ك زائد واحد
+
+220
+00:31:31,370 --> 00:31:40,190
+في ك زائد اتنين بنبدل ن بالك وبعدين
+
+221
+00:31:40,190 --> 00:31:54,030
+هذا عبارة عن سيجما من ك بيسار واحد إلى ن و بنكتب
+
+222
+00:31:54,030 --> 00:31:57,350
+هذا واحد على
+
+223
+00:32:00,560 --> 00:32:07,980
+2k سالب واحد
+
+224
+00:32:07,980 --> 00:32:16,740
+على ك زائد واحد موجب خلينا
+
+225
+00:32:16,740 --> 00:32:21,800
+نحط الحاجات الموجبة مع بعض يعني زائد واحد على
+
+226
+00:32:21,800 --> 00:32:26,360
+اتنين ك زائد اربعة
+
+227
+00:32:28,860 --> 00:32:36,700
+-1 على K-1 و
+
+228
+00:32:36,700 --> 00:32:42,140
+بعدين نكتب أول شوية حدوث مهم جدا اللي كل ثوابت هذه
+
+229
+00:32:42,140 --> 00:32:48,080
+صح يعني في حد تاني جابهم متأكد من صحتهم لإن لو
+
+230
+00:32:48,080 --> 00:32:50,680
+فيهم خطأ مش هنقبلهم و نطلع الجواب
+
+231
+00:33:03,670 --> 00:33:08,910
+فنكتب أول حد هي .. أول حد هيكون لما كيب الساعة
+
+232
+00:33:08,910 --> 00:33:17,090
+واحد .. نص .. هيطلع نص زائد واحد على .. تمانية ..
+
+233
+00:33:17,090 --> 00:33:24,090
+اتنين .. لأ واحد على ستة .. واحد على ستة صح ناقص
+
+234
+00:33:24,090 --> 00:33:26,110
+نص .. سالب نص
+
+235
+00:33:31,480 --> 00:33:38,960
+زاد لحد التاني واحد على تلاتة زاد
+
+236
+00:33:38,960 --> 00:33:41,520
+.. واحد على اربع .. واحد على اربع .. اربع .. الاول
+
+237
+00:33:41,520 --> 00:33:48,800
+واحد على اربع او واحد على اربع الاول و بعدين واحد
+
+238
+00:33:48,800 --> 00:33:53,040
+على .. تمانية .. واحد على تمانية .. تمانية ناقص
+
+239
+00:33:53,040 --> 00:34:03,060
+تلت مايناس تلت طيب قولي بعدهواحد على ستة واحد على
+
+240
+00:34:03,060 --> 00:34:09,380
+ستة واحد على ايه؟ على ستة واحد على عشرة اتنين في
+
+241
+00:34:09,380 --> 00:34:14,500
+تلاتة بستة اه واحد على ستة زائد واحد على عشرة زائد
+
+242
+00:34:14,500 --> 00:34:25,360
+واحد على عشرة minus ربع minus ربع زائد
+
+243
+00:34:25,360 --> 00:34:39,020
+و هكذا الاخر حد هيكون1 على 2n زائد 1 على 2n زائد 4
+
+244
+00:34:39,020 --> 00:34:55,940
+مع بعض و بعدين الثاني 1 على n زائد 1 فنشوف
+
+245
+00:34:55,940 --> 00:35:01,570
+أيش اللي بتلاعش و أيش اللي بيطلععين نص هنا راح عين
+
+246
+00:35:01,570 --> 00:35:07,190
+نص و
+
+247
+00:35:07,190 --> 00:35:20,090
+ربع هنا راح مع الربع هنا قلت
+
+248
+00:35:20,090 --> 00:35:26,030
+لك هذا مش هيروح مع حد صح؟ هذا يبقى
+
+249
+00:35:30,600 --> 00:35:36,960
+لكن الطمن هيروح والصدرس هيروح لإن الصدرس في مجموعة
+
+250
+00:35:36,960 --> 00:35:42,560
+ليه صدرس و الطمن هيجمع ليه الطمن بس برضه هييجي
+
+251
+00:35:42,560 --> 00:35:46,620
+ناقص واحد على تمانية و هيظل واحد على تمانية فيه؟
+
+252
+00:35:46,620 --> 00:35:51,220
+اه لما نقعد بالقمة سوى سبعة هيطلع اننا ناقص واحد
+
+253
+00:35:51,220 --> 00:35:54,040
+على تمانية اه اشي ناقص واحد على تمانية
+
+254
+00:35:56,980 --> 00:36:01,600
+و ممكن كمان برز واحد على ستة او في برز واحد على
+
+255
+00:36:01,600 --> 00:36:08,020
+ستة سيطلع سالب واحد على ستة لإن بيساوي خمسة سيطلع
+
+256
+00:36:08,020 --> 00:36:14,700
+ثاندي سالب واحد على ستة فمين اللي بيضل على المحدود
+
+257
+00:36:14,700 --> 00:36:21,140
+يعني
+
+258
+00:36:21,140 --> 00:36:33,420
+بتاعي هذا ستس هيبقى هذا هيروحو هذا هيروحك يعني
+
+259
+00:36:33,420 --> 00:36:39,600
+شو اللي بضلف الآخر يعني
+
+260
+00:36:39,600 --> 00:36:46,060
+انا بتاعي اللي هيضلف الآخر اللي هو يمكن السدر
+
+261
+00:36:46,060 --> 00:36:53,860
+السادى ناقص تلت ناقص تلت و هنا
+
+262
+00:36:58,270 --> 00:37:05,290
+كل حد بيروح مع ادم فوضوح يروح مع اللي بعده فمش
+
+263
+00:37:05,290 --> 00:37:15,870
+هيروح مع حد فهيبقى واحد على اتنين يعني و
+
+264
+00:37:15,870 --> 00:37:19,030
+.. ايش هبقى كمان؟
+
+265
+00:37:30,970 --> 00:37:36,990
+هذا هيروح هيبقى له اتنين هدول اتالي ايه مظلوم زاد
+
+266
+00:37:36,990 --> 00:37:51,870
+واحد على اتنين ام زاد اربع سدس
+
+267
+00:37:51,870 --> 00:37:59,410
+minus تلت تطلع minus سدس وهذا مروح من صفر مش مظلوم
+
+268
+00:38:08,750 --> 00:38:15,650
+المفروض ال limit تطلع ربعها بالتالي
+
+269
+00:38:15,650 --> 00:38:19,710
+لازم احنا نكتب مزيد من الحدود عشان نشوف كيف النمط
+
+270
+00:38:19,710 --> 00:38:22,150
+.. كيف النمط هيصير
+
+271
+00:38:27,470 --> 00:38:34,430
+فبدأ عملية grouping للحدود تجميع ويعني حصول علامات
+
+272
+00:38:34,430 --> 00:38:41,310
+معينة مش عارف انا مش متأكد ان هذا هتكون صح يمكن
+
+273
+00:38:41,310 --> 00:38:55,770
+في شغلات تانية بتبقى واحنا ماذكرناش فال
+
+274
+00:38:55,770 --> 00:38:56,090
+..
+
+275
+00:38:59,080 --> 00:39:04,640
+ذا بده فحص اه فخلينا نقول try it again try it
+
+276
+00:39:04,640 --> 00:39:07,780
+again
+
+277
+00:39:07,780 --> 00:39:17,200
+خلينا نحاول فيه مرة تانية و نحاول يعني نقدر نخلي
+
+278
+00:39:17,200 --> 00:39:22,740
+يعني هذا يساوي ربع او يساوي حاجة ال limit بقتها في
+
+279
+00:39:22,740 --> 00:39:26,480
+النهاية هتطلع ربعوبالتالي ال limit لل sequence of
+
+280
+00:39:26,480 --> 00:39:29,180
+partial sums هيطلع ربعه وبالتالي ال series
+
+281
+00:39:29,180 --> 00:39:33,000
+conversion مجموعة بساوي ال limit لل partial sums
+
+282
+00:39:33,000 --> 00:39:38,700
+فهذا يعني مشكوك فيه شكله مش صح نحاول مرة تانية فيه
+
+283
+00:39:38,700 --> 00:39:43,100
+و بعدين نشوف يعني كيف مين اللي بصل للجواب الصح
+
+284
+00:39:43,100 --> 00:39:48,720
+نحاول نكتبه مرة تانية okay تمام لكن يعني ماهواش
+
+285
+00:39:48,720 --> 00:39:53,570
+مستحيل أو ماهواش يعني صعبممكن اي واحد يتواصل اليه
+
+286
+00:39:53,570 --> 00:39:58,990
+بس بده ايه مزيد من الحدود والاستنتاج نمط معين
+
+287
+00:39:58,990 --> 00:40:04,370
+فخلينا نسيبكم تفكروا فيه كمان مرة في اسئلة تانية
+
+288
+00:40:04,370 --> 00:40:08,990
+في ال section هذا فحاولوا
+
+289
+00:40:08,990 --> 00:40:11,170
+تفكروا فيه في اسئلة تانية
+
+290
+00:40:17,700 --> 00:40:21,980
+في أسئلة تانية في section تلاتة سبعة أو السكاشن
+
+291
+00:40:21,980 --> 00:40:32,680
+السابقة اللى تسبقه تلاتة ستة تلاتة خمسة في
+
+292
+00:40:32,680 --> 00:40:38,140
+كتير أسئلة يعني مطلوبة منكم واضح أن انتم مش محضرين
+
+293
+00:40:38,140 --> 00:40:42,420
+ولا دارسين الموضوع وبالتالي ماعندكم مش أسئلة
+
+294
+00:40:46,220 --> 00:40:54,880
+فإلى أن يكون عندكم أسئلة بنكمل المناقشة يوم السبت
+
+295
+00:40:54,880 --> 00:40:59,860
+الجاي أو تحضروا المناقشة مع الشعبة التانية يوم
+
+296
+00:40:59,860 --> 00:41:05,140
+الأربع خلينا
+
+297
+00:41:05,140 --> 00:41:14,690
+نرجع لل limits of functions وناخد المثال الأخيرفي
+
+298
+00:41:14,690 --> 00:41:16,930
+ال section هداك
+
+299
+00:41:46,290 --> 00:41:50,970
+المرة الجاية دخلنا اثبتنا
+
+300
+00:41:50,970 --> 00:42:02,310
+ان ال limit اثبتنا
+
+301
+00:42:02,310 --> 00:42:05,450
+ان ال candy انتي مثال رقم 2
+
+302
+00:42:08,630 --> 00:42:15,350
+لسفر function ل X لما X تقول لسفر does not exist
+
+303
+00:42:15,350 --> 00:42:19,270
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+304
+00:42:19,270 --> 00:42:21,290
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+305
+00:42:21,290 --> 00:42:21,570
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+306
+00:42:21,570 --> 00:42:21,890
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+307
+00:42:21,890 --> 00:42:22,210
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+308
+00:42:22,210 --> 00:42:22,610
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+309
+00:42:22,610 --> 00:42:22,610
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+310
+00:42:22,610 --> 00:42:25,970
+أخدنا هذا و أخدنا هذا و أخدنا هذا و أخدنا هذا و
+
+311
+00:42:25,970 --> 00:42:33,830
+أخدنا هذا و أخدنا
+
+312
+00:42:33,830 --> 00:42:46,640
+هذا و أوحد على X لما X تقول لسفر does not exist in
+
+313
+00:42:46,640 --> 00:42:50,400
+R فلبرحان
+
+314
+00:42:50,400 --> 00:42:56,700
+ذلك let
+
+315
+00:42:56,700 --> 00:43:04,380
+F of X تساوي صين واحد على X و X لا تساوي سفر
+
+316
+00:43:10,730 --> 00:43:16,210
+و بعدين we consider two
+
+317
+00:43:16,210 --> 00:43:20,870
+sequences واحدة
+
+318
+00:43:20,870 --> 00:43:33,750
+xn الحد لعام تبعها أدارة عن واحد على واحد
+
+319
+00:43:33,750 --> 00:43:38,030
+على n πاي و n ينتمي ل z
+
+320
+00:43:41,720 --> 00:43:47,620
+و Yn لحد الآن تبعها واحد على πاي على تمين زاد
+
+321
+00:43:47,620 --> 00:43:56,020
+اتنين N πاي و N ينتمي الى Z هذا عبارة عن Sequences
+
+322
+00:43:56,020 --> 00:44:01,300
+of positive numbers
+
+323
+00:44:04,950 --> 00:44:12,130
+واضح ان ال limit ل xn as n tends to infinity بساوي
+
+324
+00:44:12,130 --> 00:44:20,290
+0 وكذلك ال limit ل yn لما n تقول infinity برضه
+
+325
+00:44:20,290 --> 00:44:24,730
+بساوي 0 لان المقان لما n تقول infinity المقان
+
+326
+00:44:24,730 --> 00:44:32,090
+بيروح ل infinity طيب
+
+327
+00:44:32,090 --> 00:44:33,690
+الآن ال limit
+
+328
+00:44:37,860 --> 00:44:42,420
+الـ image لـ sequence xn لما n تقول الانفلتين
+
+329
+00:44:42,420 --> 00:44:55,500
+بساوي ال limit ل sign xn لما n تقول الانفلتين وهذا
+
+330
+00:44:55,500 --> 00:45:05,620
+بساوي ال limit ل sign n في pi لما n تقول الانفلتين
+
+331
+00:45:06,340 --> 00:45:16,300
+Sin N في Pi بساوي واحد بساوي سفر لكل N وبالتالي
+
+332
+00:45:16,300 --> 00:45:21,640
+هذا بساوي limit ال sequence سفر لما N طولة
+
+333
+00:45:21,640 --> 00:45:32,600
+infinity بساوي سفر and limit
+
+334
+00:45:33,900 --> 00:45:41,360
+الإمج للسيكوينس YM لما N تقول انفينيتي بساوي limit
+
+335
+00:46:08,450 --> 00:46:16,370
+وهذا المفروض يكون sign
+
+336
+00:46:16,370 --> 00:46:24,840
+1 على xn وهذا المفروض يكون sign 1 على ynمقلوب y in
+
+337
+00:46:24,840 --> 00:46:34,540
+بيطلع بساوي πاية اتنين ازايد اتنين in πاية وهذا
+
+338
+00:46:34,540 --> 00:46:44,040
+المقدار دايما بساوي واحد لكل in اذا انا في عندي
+
+339
+00:46:44,040 --> 00:46:51,770
+limit لل sequence بالحد العام تبعها واحدالسيكوانس
+
+340
+00:46:51,770 --> 00:46:57,670
+تابعة واحد وهذا بالساوية واحد ان ان انا في عندي
+
+341
+00:46:57,670 --> 00:47:03,710
+two sequences Xm تقولها سفر و limit صورتها
+
+342
+00:47:03,710 --> 00:47:09,050
+بالساوية سفر و في عندي سيكوانس تانية Ym ال limit
+
+343
+00:47:09,050 --> 00:47:13,650
+تبعتها ايضا بالساوية سفر لكن limit صورتها بالساوية
+
+344
+00:47:13,650 --> 00:47:17,310
+واحد وبالتالي
+
+345
+00:47:20,360 --> 00:47:28,340
+by sequential criterion ال
+
+346
+00:47:28,340 --> 00:47:35,480
+limit لل function f of x لما x تقول ل0 does not
+
+347
+00:47:35,480 --> 00:47:46,440
+exist in R مش ممكن تكون موجودة في R لأن
+
+348
+00:47:46,440 --> 00:47:53,900
+لو كانت ال limit هذه موجودةفالمفروض limit صورة xn
+
+349
+00:47:53,900 --> 00:48:02,060
+بما أن xn تقول السفر نكتب
+
+350
+00:48:02,060 --> 00:48:07,180
+since otherwise لأن
+
+351
+00:48:07,180 --> 00:48:14,780
+لو كلاك ذلك لو كانت هذه موجودة if limit
+
+352
+00:48:20,170 --> 00:48:25,210
+فى limit ل F of X لما X تقول لسة exist
+
+353
+00:48:32,710 --> 00:48:39,990
+then المفروض ال limit ل f of x n لما n تقول
+
+354
+00:48:39,990 --> 00:48:47,270
+infinity بتساوي ال limit ل f of y n as n tends to
+
+355
+00:48:47,270 --> 00:48:57,070
+infinity وهذا مستحيل which is impossible وهذا زي
+
+356
+00:48:57,070 --> 00:49:03,150
+ما شوفنا مستحيل impossibleلأن طولها limit f of x
+
+357
+00:49:03,150 --> 00:49:09,390
+in بالساوي سفر و limit f of y in بالساوي واحد إذن
+
+358
+00:49:09,390 --> 00:49:13,470
+هنا استخدمنا sequential criterion في إثبات إن ال
+
+359
+00:49:13,470 --> 00:49:17,330
+limit لل function f of x بالساوي صين واحد على x
+
+360
+00:49:17,330 --> 00:49:24,490
+غير موجودة عند السفر طيب
+
+361
+00:49:24,490 --> 00:49:30,120
+هناخد break خمس دقايق و بعدين نواصلالمحاضرة
+
+362
+00:49:30,120 --> 00:49:31,840
+التانية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA.srt
new file mode 100644
index 0000000000000000000000000000000000000000..cd05b10e398466c6907b18c58dbb2001a644ca72
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/7KfEZYA9kIA.srt
@@ -0,0 +1,1591 @@
+1
+00:00:21,840 --> 00:00:28,120
+المحاضرة اللي فاتت بدينا في عرض بعض ال
+
+2
+00:00:28,120 --> 00:00:32,760
+applications of the supremum property وبعتقد أن
+
+3
+00:00:32,760 --> 00:00:37,680
+احنا أخذنا أول مثال اللي هو المثال هذا مظبوط 
+
+4
+00:00:37,680 --> 00:00:40,980
+فقولنا
+
+5
+00:00:40,980 --> 00:00:46,160
+إن المثال هذا لو أخدت أي bounded set bounded
+
+6
+00:00:46,160 --> 00:00:56,150
+above وعرفت المجموعة a زائد s بالطريقة هذه فأثبتنا
+
+7
+00:00:56,150 --> 00:01:00,770
+وممكن بسهولة إثبات أن ال supremum للمجموعة الجديدة
+
+8
+00:01:00,770 --> 00:01:09,870
+A plus S بتساوي A plus ال supremum لـ S وشوفنا
+
+9
+00:01:09,870 --> 00:01:15,630
+البرهان بالتفصيل المرة اللي فاتت وكان هنا البرهان
+
+10
+00:01:15,630 --> 00:01:18,390
+بعتمد على أن الـ set اللي bounded above
+
+11
+00:01:32,250 --> 00:01:36,230
+الـ set S هي bounded above لأن ال supremum تبعها
+
+12
+00:01:36,230 --> 00:01:42,940
+exists by the supremum property وشوفنا بعد هيك أنه 
+
+13
+00:01:42,940 --> 00:01:49,660
+الـ .. العدد a زائد u بيطلع upper bound للـ set هذه و
+
+14
+00:01:49,660 --> 00:01:53,500
+بعدين أثبتنا أن هذا العدد هو أصغر upper bound أو
+
+15
+00:01:53,500 --> 00:01:59,320
+supremum للـ set هذه وبالتالي هيك بنكون أثبتنا أن 
+
+16
+00:01:59,320 --> 00:02:03,760
+supremum للـ set هذه موجود و بيساوي العدد a زائد u
+
+17
+00:02:03,760 --> 00:02:09,420
+اللي هو a زائد supremum S المثال الثاني
+
+18
+00:02:16,520 --> 00:02:20,320
+لو أخدت two functions المجال الـ domain تبعهم 
+
+19
+00:02:20,320 --> 00:02:25,300
+مجموعة D subset من R وكتبت 
+
+20
+00:02:25,300 --> 00:02:29,280
+F of D على أنها مجموعة كل العناصر F of X حيث و X
+
+21
+00:02:29,280 --> 00:02:34,400
+ينتمي لـ D فالـ set F of D هذه هي الـ range تبع الـ
+
+22
+00:02:34,400 --> 00:02:39,120
+function F صح؟ هي المدى تبع الـ function F  و كذلك
+
+23
+00:02:39,120 --> 00:02:46,000
+الـ set G of D هي الـ range تبع الـ function G 
+
+24
+00:02:48,510 --> 00:02:53,250
+فلو فرضنا أن الـ set f of d و الـ set g of d bounded
+
+25
+00:02:53,250 --> 00:03:01,530
+set R فطبعا حسب ال supremum property المجموعات دول 
+
+26
+00:03:01,530 --> 00:03:06,430
+كل واحدة لها supremum كذلك حسب ال infimum property
+
+27
+00:03:07,290 --> 00:03:11,050
+المجموعتين هذول كل واحدة فيهم إلها infimum، الـ
+
+28
+00:03:11,050 --> 00:03:15,350
+infimum تبعهم exists إذا نفرض إن المجمعتين هذول
+
+29
+00:03:15,350 --> 00:03:18,570
+bounded عشان إيه نضمن وجود ال supremum والinfimum 
+
+30
+00:03:18,570 --> 00:03:26,450
+لكل واحدة منهم الآن في عندي بدي أبرهن حاجة ثانية لو
+
+31
+00:03:26,450 --> 00:03:31,930
+كان الفرض f of x أصغر من أو يساوي g of x بتحقق لكل 
+
+32
+00:03:31,930 --> 00:03:38,040
+x ينتمي لـ D بيطلع ال supremum للمجموعة F of D بيطلع أصغر من
+
+33
+00:03:38,040 --> 00:03:44,660
+أو يساوي ال supremum للمجموعة G of D وبرهان هذا
+
+34
+00:03:44,660 --> 00:03:54,220
+البرهان يعني سهل أنا كاتب إنه easy exercise لكن 
+
+35
+00:03:54,220 --> 00:04:02,780
+ممكن تبرهنه ممكن تبرهنه بكل سهولة فهي نكتب الـ proof
+
+36
+00:04:06,320 --> 00:04:14,320
+of part one للجزء الأول فخلّينا 
+
+37
+00:04:14,320 --> 00:04:19,400
+نثبت fix x
+
+38
+00:04:19,400 --> 00:04:29,400
+ينتمي إلى d ناخد عنصر x ينتمي إلى d عشوائي by 
+
+39
+00:04:29,400 --> 00:04:31,240
+hypothesis من الفرض 
+
+40
+00:04:33,710 --> 00:04:40,970
+من الفرض أنا عندي f of x أصغر من أو يساوي g of x
+
+41
+00:04:40,970 --> 00:04:52,470
+للـ x هذه و لأي x دي صح هذا من الفرض و g of x g of
+
+42
+00:04:52,470 --> 00:05:00,550
+x أصغر من أو يساوي ال supremum للـ set g of d
+
+43
+00:05:04,610 --> 00:05:14,410
+طبعا هذا زي ما قلنا exists by supremum property
+
+44
+00:05:14,410 --> 00:05:20,970
+باستخدام خاصية الـ
+
+45
+00:05:20,970 --> 00:05:26,910
+supremum .. هذا .. هذا عنصر في الـ set هذا g of x عنصر
+
+46
+00:05:26,910 --> 00:05:32,550
+في الـ set g of d صح؟وهذا upper bound ال supremum لـ g of 
+
+47
+00:05:32,550 --> 00:05:38,690
+d و هذا عنصر في الـ set g of d فهذا أكيد أكبر من أو يساوي
+
+48
+00:05:38,690 --> 00:05:43,610
+ال upper bound للـ set اللي بينتمي إليها فهذا صحيح 
+
+49
+00:05:43,610 --> 00:05:56,610
+الآن هذا صحيح لكل x since x belonged to D was 
+
+50
+00:05:56,610 --> 00:05:57,610
+arbitrarily
+
+51
+00:06:03,450 --> 00:06:10,110
+arbitrary  إن إن بيطلع عندي F of X أصغر من أو يساوي 
+
+52
+00:06:10,110 --> 00:06:20,490
+ال supremum لـ G of D وهذا صحيح لكل X في D هذا 
+
+53
+00:06:20,490 --> 00:06:29,900
+معناه إنه العدد هذا هذا العدد أكبر من أو يساوي كل 
+
+54
+00:06:29,900 --> 00:06:36,960
+عناصر الـ set F of D صح؟ هي هذا معناه أن الـ 
+
+55
+00:06:36,960 --> 00:06:47,600
+supremum لـ set G of D is an upper bound an upper
+
+56
+00:06:47,600 --> 00:06:50,860
+bound
+
+57
+00:06:50,860 --> 00:06:53,780
+لمين؟ 
+
+58
+00:06:54,920 --> 00:07:01,100
+of set f of d بصح؟ 
+
+59
+00:07:01,100 --> 00:07:07,040
+لأن هيك كل عنصر f of x في f of d أصغر من أو يساوي
+
+60
+00:07:07,040 --> 00:07:18,980
+العدد هذا، صح؟ طيب since ال supremum لـ set f of d 
+
+61
+00:07:18,980 --> 00:07:25,890
+exists in R طبعا برضه by supremum property لأن احنا 
+
+62
+00:07:25,890 --> 00:07:31,890
+فرضين أن الـ set هذه bounded صح فال supremum تبعها
+
+63
+00:07:31,890 --> 00:07:37,110
+موجود الآن الـ set هذه ال supremum تبعها موجود
+
+64
+00:07:37,110 --> 00:07:42,750
+والعدد هذا هذا العدد عبارة عن upper bound لـ set
+
+65
+00:07:42,750 --> 00:07:46,850
+إذا ما العلاقة بين ال upper bound هذا للـ set وال
+
+66
+00:07:46,850 --> 00:07:53,650
+supremum للـ set؟ في واحد أكبر من أو يساوي الثاني لأن 
+
+67
+00:07:53,650 --> 00:07:59,770
+بما أن هذا الكلام صحيح نحن نحن نحن نحن نحن نحن نحن
+
+68
+00:07:59,770 --> 00:08:01,050
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+69
+00:08:01,050 --> 00:08:01,350
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+70
+00:08:01,350 --> 00:08:04,050
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن
+
+71
+00:08:04,050 --> 00:08:05,880
+نحن نحن نحن نحن نحن نحن نحن نحن نحن نحن هذا
+
+72
+00:08:05,880 --> 00:08:10,820
+أصغر upper bound للـ set f of d وهذا upper bound للـ set
+
+73
+00:08:10,820 --> 00:08:15,440
+f of d إذا ال supremum بيطلع أصغر من أو يساوي ال
+
+74
+00:08:15,440 --> 00:08:22,480
+upper bound اللي هو supremum g of d وهو المطلوب 
+
+75
+00:08:22,480 --> 00:08:29,800
+وهذا بيثبت الجزء الأول okay تمام إذا الجزء الأول مش 
+
+76
+00:08:29,800 --> 00:08:33,680
+صعب وهنا أثبتنا واضح
+
+77
+00:08:37,050 --> 00:08:42,310
+برهان الجزء الثاني برضه شبيه فيه الجزء الثاني، إيش
+
+78
+00:08:42,310 --> 00:08:47,510
+بيقول ليه؟ الفرض، لاحظوا الفرق بين الفرض تبع الجزء
+
+79
+00:08:47,510 --> 00:08:54,910
+الثاني والجزء الأول الفرض
+
+80
+00:08:54,910 --> 00:09:00,210
+هنا إن f of x أصغر من أو يساوي g of y لكل x و y في 
+
+81
+00:09:00,210 --> 00:09:00,450
+D
+
+82
+00:09:04,010 --> 00:09:09,170
+هذا أشمَل وهذا أعمل من هذا وهذا أقوى من هذا لاحظوا
+
+83
+00:09:09,170 --> 00:09:14,690
+إنه لو هذا صح فهذا بيطلع صح اللي فوق لكن الـ x مش 
+
+84
+00:09:14,690 --> 00:09:18,430
+صحيح طيب 
+
+85
+00:09:18,430 --> 00:09:22,130
+إذا .. إذا هذا الكلام صحيح فهذا بيقدي إن الـ
+
+86
+00:09:22,130 --> 00:09:26,410
+supremum لـ F of D بيطلع أصغر من أو يساوي ال infimum
+
+87
+00:09:26,410 --> 00:09:31,110
+لـ set G of D نشوف 
+
+88
+00:09:31,110 --> 00:09:32,710
+الـ .. نبرهن الكلام هذا
+
+89
+00:09:50,270 --> 00:10:02,090
+البرهان الجزء الثاني البرهان 
+
+90
+00:10:02,090 --> 00:10:05,030
+الجزء الثاني هذا conditional statement هي الفرض
+
+91
+00:10:05,030 --> 00:10:11,370
+وهي النتيجة الـ conclusion فبنفرض أن الفرض هذا صحيح 
+
+92
+00:10:11,370 --> 00:10:23,770
+و بنثبت يثبت يثبت عنصر Y في D من الفرض بيطلع عندي f
+
+93
+00:10:23,770 --> 00:10:29,530
+of x أصغر من أو يساوي g of y وهذا صحيح لكل x في دي
+
+94
+00:10:29,530 --> 00:10:38,280
+و الـ y ثابت يعني هذا من الفرض صحيح لكل x في دي طيب، 
+
+95
+00:10:38,280 --> 00:10:45,040
+الآن هذا معناه أن العدد هذا g of y هي في y أنصه 
+
+96
+00:10:45,040 --> 00:10:49,600
+ثابت هي أكبر .. هذا العدد أكبر من أو يساوي كل الـ F
+
+97
+00:10:49,600 --> 00:10:54,100
+of X لكل X دي معناه هذا upper bound للـ set F of D
+
+98
+00:10:54,100 --> 00:10:59,020
+الآن g of y عبارة عن upper bound للـ set F of D من
+
+99
+00:10:59,020 --> 00:11:01,860
+هنا، مظبوط؟ تمام؟
+
+100
+00:11:04,040 --> 00:11:07,840
+وبالتالي الـ least upper bound لـ F of D بيطلع أصغر 
+
+101
+00:11:07,840 --> 00:11:12,080
+من أو يساوي الـ upper bound لـ F of D اللي هو G of Y لأن 
+
+102
+00:11:12,080 --> 00:11:13,620
+هذه المتباينة صحيحة
+
+103
+00:11:18,050 --> 00:11:22,770
+اخترناها was arbitrary fixed احنا اخترناها عشوائي
+
+104
+00:11:22,770 --> 00:11:27,470
+arbitrary وثبتناها أن الكلام المتباينة هذه الآن صحيح
+
+105
+00:11:27,470 --> 00:11:33,110
+لكل y أن المتباينة هذه صحيحة true for every y في D
+
+106
+00:11:33,110 --> 00:11:39,510
+هذا معناه من المتباينة هذه percentage إنه العدد
+
+107
+00:11:39,510 --> 00:11:45,350
+ال supremum لـ F of D هذا عبارة عن lower bound
+
+108
+00:11:45,350 --> 00:11:51,030
+لمجموعة العناصر g of y حيث y ينتمي لـ d يعني العدد 
+
+109
+00:11:51,030 --> 00:11:58,210
+هذا عبارة عن lower bound للـ set g of d عظيم صح؟ طيب 
+
+110
+00:11:58,210 --> 00:12:04,230
+ال infimum لـ g of d exists وهذا العدد lower bound
+
+111
+00:12:04,230 --> 00:12:08,950
+للـ set هذه و ال infimum هذا عبارة عن الـ greatest 
+
+112
+00:12:08,950 --> 00:12:12,970
+lower bound لـ G و D إذا الـ greatest lower bound
+
+113
+00:12:12,970 --> 00:12:18,810
+دايما بيكون أكبر من أو يساوي أي lower bound إذا الـ 
+
+114
+00:12:18,810 --> 00:12:23,090
+lower bound هذا أصغر من أو يساوي الـ greatest lower 
+
+115
+00:12:23,090 --> 00:12:28,610
+bound لـ G و D و هذا اللي هو هذا النتيجة اللي احنا
+
+116
+00:12:28,610 --> 00:12:34,800
+عايزين نصل لها okay تمام واضح؟ إذن هذا برهاني جزء
+
+117
+00:12:34,800 --> 00:12:48,220
+الثاني الآن في ملاحظة الملاحظة هذه بتقول إنه يعني
+
+118
+00:12:48,220 --> 00:12:56,120
+ممكن طالبة طلعت تسأل أو تستفسر أو تتساءل طب ما هذا 
+
+119
+00:12:56,120 --> 00:13:01,400
+الشرط تبعين زي هذا ما فيش فرق بينهم فاحنا بنقول لأ
+
+120
+00:13:01,400 --> 00:13:05,480
+هذا الشرط التحت أقوى من اللي فوق اللي تحت لو كان 
+
+121
+00:13:05,480 --> 00:13:09,160
+التحت صحيح بيقدي للي فوق لكن لو كان اللي فوق صحيح
+
+122
+00:13:09,160 --> 00:13:14,300
+هذا ما بيقدي للي تحت هذا الشرط أقوى من اللي فوق 
+
+123
+00:13:14,300 --> 00:13:20,240
+فممكن واحدة فيكم تسأل تقول طب لو احنا أخذنا الفرض
+
+124
+00:13:20,240 --> 00:13:24,840
+هذا لو فرضنا أن هذا الكلام صح هل ممكن نحصل على 
+
+125
+00:13:24,840 --> 00:13:30,580
+نتيجة اللي تحته؟ الإجابة لأ، الإجابة لأ، هذا مش 
+
+126
+00:13:30,580 --> 00:13:36,900
+ممكن، إذا الـ .. لو شيلنا الفرض هذا و بدلناه بالفرض
+
+127
+00:13:36,900 --> 00:13:41,820
+اللي فوق فالنتيجة هذه لا يمكن نحصل عليها، مش شرط
+
+128
+00:13:41,820 --> 00:13:53,110
+تكون صحيحة أو مثال يوضح إنه لا يمكن استبدال الفرض
+
+129
+00:13:53,110 --> 00:13:58,610
+تبع الجزء الثاني بالفرض تبع الجزء الأول ونحصل نحصل
+
+130
+00:13:58,610 --> 00:14:00,630
+على نتيجة الجزء الثاني
+
+131
+00:14:12,790 --> 00:14:16,530
+فناخد على سبيل المثال أو counterexample بيسميه في 
+
+132
+00:14:16,530 --> 00:14:22,910
+رياضيات لو أخدت f of x بيساوي x تربيع دالة تربيع
+
+133
+00:14:22,910 --> 00:14:26,830
+و g of x الـ identity function و أخدت الـ domain
+
+134
+00:14:26,830 --> 00:14:30,950
+المشترك لـ f و g الـ closed unit interval 
+
+135
+00:14:34,300 --> 00:14:40,040
+فطبعا بنلاحظ أن f of x اللي هي x تربيع لكل x في الـ
+
+136
+00:14:40,040 --> 00:14:45,220
+closed unit interval x تربيع أصغر من أو يساوي x، 
+
+137
+00:14:45,220 --> 00:14:51,180
+مظبوط؟ و X بيساوي G of X فهي في عندي الـ two
+
+138
+00:14:51,180 --> 00:14:54,460
+functions هدول بالمناسبة الـ two functions هدول 
+
+139
+00:14:54,460 --> 00:14:59,220
+كلاهم كلاهم bounded bounded below by zero bounded
+
+140
+00:14:59,220 --> 00:15:08,940
+above by الـ range تبعهم الـ range تبعهم F of D و G of D
+
+141
+00:15:08,940 --> 00:15:14,900
+of D ك sets كمجموعات بطلوا subset من المجموعة من
+
+142
+00:15:14,900 --> 00:15:20,520
+السفر لواحد، وبالتالي كلا هما bounded above by واحد
+
+143
+00:15:20,520 --> 00:15:27,000
+و bounded below by صفر، إذن
+
+144
+00:15:27,000 --> 00:15:32,420
+هذه المجموعات هي bounded وهي عند ال function f of
+
+145
+00:15:32,420 --> 00:15:36,860
+x أصغر من أو يساوي g of x لكل x دي، هذا الفرض تبع
+
+146
+00:15:36,860 --> 00:15:41,600
+الجزء واحد اللي شوفناه قبل شوية، لكن النتيجة تبع
+
+147
+00:15:41,600 --> 00:15:45,560
+الجزء التالي لا تتحقق، تعالى نشوف هي ال supremum ل
+
+148
+00:15:45,560 --> 00:15:52,190
+f of d هي مجموعة f of d الواحد
+
+149
+00:15:52,190 --> 00:15:56,430
+أكبر 
+
+150
+00:15:56,430 --> 00:16:00,090
+من الصفر الصفر
+
+151
+00:16:00,090 --> 00:16:05,650
+برضه عبارة عن greatest lower bound أو الانفم من
+
+152
+00:16:05,650 --> 00:16:09,950
+المجموعة هذه، واضح أن الصفر lower bound للسفر هذه
+
+153
+00:16:09,950 --> 00:16:15,580
+وهو greatest lower bound، إذاً هي عند الـ supremum
+
+154
+00:16:15,580 --> 00:16:20,220
+لـ F of D أكبر من الـ infimum لـ G of D، وهذا نفي 
+
+155
+00:16:20,220 --> 00:16:23,700
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+156
+00:16:23,700 --> 00:16:24,240
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+157
+00:16:24,240 --> 00:16:26,120
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+158
+00:16:26,120 --> 00:16:26,240
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+159
+00:16:26,240 --> 00:16:26,440
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+160
+00:16:26,440 --> 00:16:32,560
+نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة نتيجة
+
+161
+00:16:49,380 --> 00:16:56,900
+كنتيجة على الـ completeness property في عندي نتيجة
+
+162
+00:16:56,900 --> 00:17:05,420
+كتير مهمة، وهنستخدمها كتير، معناها اللي هو ال
+
+163
+00:17:05,420 --> 00:17:10,120
+material اللي هناخدها لاحقا، اللي هو ال Archimedean
+
+164
+00:17:10,120 --> 00:17:16,220
+property أو خاصية Archimedes، إيه الخاصية هذه بتقول
+
+165
+00:17:17,950 --> 00:17:23,890
+لأي عدد حقيقي x في عدد طبيعي أكبر منه، أعطيني أي 
+
+166
+00:17:23,890 --> 00:17:29,650
+عدد حقيقي x سواء كان صفر أو موجب أو سالب، بقدر
+
+167
+00:17:29,650 --> 00:17:36,970
+أعطيكي عدد طبيعي أكبر منه أو بقدر أوجدلك عدد طبيعي
+
+168
+00:17:36,970 --> 00:17:42,760
+يكون أكبر منه، البرهان تبع النظرية هذه بيعتمد على
+
+169
+00:17:42,760 --> 00:17:47,040
+الـ completeness property، فلبرهان ذلك نبدأ بالـ
+
+170
+00:17:47,040 --> 00:17:54,320
+Fix X في R ونثبتها ونعمل برهان بالتناقض، نحن عايزين
+
+171
+00:17:54,320 --> 00:17:58,840
+نثبت أنه للـ Fix X اللي احنا ثبتناها يوجد
+
+172
+00:18:01,850 --> 00:18:07,810
+عايزين نثبت العبارة، أن العبارة هذه تكون صحيحة، يوجد
+
+173
+00:18:07,810 --> 00:18:12,430
+عدد طبيعي أكبر من X، فبدا أعمل برهان بالتناقض، بدا
+
+174
+00:18:12,430 --> 00:18:17,610
+أفرض أن نفي العبارة هذه هو الصح، إذا ن assume ال
+
+175
+00:18:17,610 --> 00:18:21,030
+contrary أن نفي العبارة هذه الصح، طب نفي العبارة
+
+176
+00:18:21,030 --> 00:18:27,750
+هذه الصح، there exist ما بصير لكل N في N عكس
+
+177
+00:18:27,750 --> 00:18:32,730
+المتباينة هذه اللي هو n أصغر من أو يساوي x، إذن هنا
+
+178
+00:18:32,730 --> 00:18:37,550
+ال contrary أو النفي، نفي النتيجة هذه، معناها أن كل 
+
+179
+00:18:37,550 --> 00:18:44,610
+الأعداد الطبيعية أصغر من أو يساوي x، هذا معناه أن ال
+
+180
+00:18:44,610 --> 00:18:51,230
+x هذا upper bound لـ set N وبالتالي الـ set N إلها
+
+181
+00:18:51,230 --> 00:18:54,850
+upper bound أو bounded above، إذا by the supremum
+
+182
+00:18:54,850 --> 00:19:00,590
+أو completeness of property، الـ set N بطلع يوجد
+
+183
+00:19:00,590 --> 00:19:04,970
+إلها supremum، الـ supremum تبعها exist and are،
+
+184
+00:19:04,970 --> 00:19:12,410
+سميه، فلنسميه u، فلنسميه u، تمام؟ في
+
+185
+00:19:12,410 --> 00:19:19,340
+لمة واحد اثنين عشر، لمة واحدة اثناء عشر كده بتقول لو كان
+
+186
+00:19:19,340 --> 00:19:28,300
+U أو u بساوي ال supremum لست S if and only if لكل
+
+187
+00:19:28,300 --> 00:19:35,920
+epsilon أكبر من الصفر نقدر نلاقي S epsilon في الست 
+
+188
+00:19:35,920 --> 00:19:42,460
+S بحيث انه U سالب epsilon أصغر من S epsilon
+
+189
+00:19:45,010 --> 00:19:50,110
+طب أقل، أنا عندي فيه U بساوي Supremum ل N، S بساوي 
+
+190
+00:19:50,110 --> 00:19:55,450
+6 N كل الأعداد الطبيعية، هي عندي Supremum ل N اللي
+
+191
+00:19:55,450 --> 00:20:01,890
+هو U exist، إذا حسب لمة واحد اثنين عشر لو أخدت epsilon 
+
+192
+00:20:01,890 --> 00:20:06,670
+لو أخدت epsilon بالساوية واحد، هذا عدد موجب، إذا لهذا
+
+193
+00:20:06,670 --> 00:20:11,690
+ال epsilon بقدر ألاقي عدد S epsilon هسمي M هنا بدل S
+
+194
+00:20:11,690 --> 00:20:16,930
+epsilon في اللمة، عدد طبيعي بحيث أنه لما أخد U minus
+
+195
+00:20:16,930 --> 00:20:20,670
+epsilon اللي هو الواحد، هذا بيطلع أصغر من S epsilon
+
+196
+00:20:20,670 --> 00:20:25,050
+اللي هو M، إذاً هذا نحصل عليه من لمة واحدة واثنين
+
+197
+00:20:25,050 --> 00:20:30,870
+عشر، طيب المتباين هذه، ودي واحد، نجري واحد على مين
+
+198
+00:20:30,870 --> 00:20:35,010
+فبيطلع U أصغر من M زائد واحد، طيب ال M عدد طبيعي
+
+199
+00:20:35,010 --> 00:20:40,130
+إذاً M زائد واحد عدد طبيعي صح؟ إذاً هذا M زائد
+
+200
+00:20:40,130 --> 00:20:47,360
+واحد عدد طبيعي وأكبر من U، و U قلنا ال U هو ال
+
+201
+00:20:47,360 --> 00:20:50,520
+supremum ل N يعني upper bound بيطلع upper bound ل
+
+202
+00:20:50,520 --> 00:20:55,860
+N، فكيف U upper bound ل set N للعداد الطبيعية، وفي
+
+203
+00:20:55,860 --> 00:20:59,620
+عنصر في العداد الطبيعية أكبر منه، لأن هذا بيديني
+
+204
+00:20:59,620 --> 00:21:06,380
+تناقض لكون U هو upper bound ل set للعداد الطبيعية
+
+205
+00:21:06,380 --> 00:21:13,060
+إذا وصلنا إلى تناقض، وبالتالي هذا بكمل البرهانة، إذا
+
+206
+00:21:13,060 --> 00:21:16,980
+الفرض تبعنا التناقض هذا، تقول إن ال assumption
+
+207
+00:21:16,980 --> 00:21:24,720
+تبعنا هذا، إن الكلام هذا صح كان خطر، إذا الصح نفيه
+
+208
+00:21:24,720 --> 00:21:29,480
+اللي هو المطلوب، okay، تمام، إذا هذه ال Archimedean
+
+209
+00:21:29,480 --> 00:21:35,460
+property هذه، ال Archimedean property، الآن ال
+
+210
+00:21:35,460 --> 00:21:39,580
+Archimedean property هذه أو خاصية Archimedes إلها
+
+211
+00:21:39,580 --> 00:21:45,520
+صور أخرى متعددة، وهذه الصور هي موجودة في كوريلري
+
+212
+00:21:45,520 --> 00:21:50,700
+واحد ستة عشر، إذا
+
+213
+00:21:50,700 --> 00:21:58,060
+النتيجة هذه في أن صور أخرى لـ ال Archimedean
+
+214
+00:21:58,060 --> 00:22:06,500
+property ف
+
+215
+00:22:07,840 --> 00:22:11,520
+Alternative forms يعني صور أخرى لـ Archimedean
+
+216
+00:22:11,520 --> 00:22:16,520
+property، let YUZ be positive real numbers، إذن 
+
+217
+00:22:16,520 --> 00:22:19,760
+YUZ تنتمي لمجموعة الأعداد الحقيقية الموجبة
+
+218
+00:22:22,550 --> 00:22:28,990
+أول نتيجة، يوجد n عدد طبيعي بحيث أن الـ z أصغر من n
+
+219
+00:22:28,990 --> 00:22:35,410
+مضروب في y، إذا لو عندي عددين حقيقين موجبين z وy
+
+220
+00:22:35,410 --> 00:22:39,790
+بقدر ألاقي عدد طبيعي بحيث أن ال z أصغر من n مضروب
+
+221
+00:22:39,790 --> 00:22:49,740
+في y، كذلك لأي عدد حقيقي موجب y بقدر ألاقي عدد طبيعي
+
+222
+00:22:49,740 --> 00:22:54,740
+مقلوبه أصغر من العدد الموجب Y، طبعا مقلوب العدد
+
+223
+00:22:54,740 --> 00:22:59,220
+الطبيعي دائما موجب، كذلك
+
+224
+00:22:59,220 --> 00:23:04,820
+لأي عدد حقيقي موجب Z بقدر ألاقي عدد طبيعي بحيث أن
+
+225
+00:23:04,820 --> 00:23:09,920
+العدد الموجب Z أكبر من أو يساوي N سالب واحد وأصغر
+
+226
+00:23:09,920 --> 00:23:16,770
+من N، إذن التلات خواص هدولة كل واحدة منهم بنسميها
+
+227
+00:23:16,770 --> 00:23:20,730
+Archimedean property أو صورة أخرى من ال
+
+228
+00:23:20,730 --> 00:23:25,590
+Archimedean property، الجزء
+
+229
+00:23:25,590 --> 00:23:30,250
+الأخير هذا هو عبارة عن مثال وليس ال Archimedean
+
+230
+00:23:30,250 --> 00:23:37,810
+يعني هذا استثناء، يعني مجرد set بالساوي ال sequence
+
+231
+00:23:37,810 --> 00:23:44,140
+واحد على n، متتالية العداد الحقيقية 1 على N حيث N
+
+232
+00:23:44,140 --> 00:23:49,540
+عدد طبيعي، فال set هذه هنثبت أن ال infimum إلها هو
+
+233
+00:23:49,540 --> 00:23:59,860
+الصفر، طيب إذا نشوف ونثبت العزاء الأولى، الجزء 
+
+234
+00:23:59,860 --> 00:24:00,780
+الأول
+
+235
+00:24:06,710 --> 00:24:15,270
+الجزء A لإثبات الجزء A خلّينا نعرف X بساوي Z على Y
+
+236
+00:24:15,270 --> 00:24:19,930
+طبعا Z وY أعداد حقيقية موجبة، إذن خارج قسمتهم أعداد
+
+237
+00:24:19,930 --> 00:24:26,090
+موجب، إذن هذا عبارة عن عدد حقيقي موجب، يعني ال X هذا
+
+238
+00:24:26,090 --> 00:24:33,170
+عبارة عن real number وموجب، فحسب ال Archimedean
+
+239
+00:24:33,170 --> 00:24:42,860
+property، لأي x عدد حقيقي يوجد عدد طبيعي أكبر من الـ
+
+240
+00:24:42,860 --> 00:24:48,000
+x، إذا الـ x اللي أنا أخده Z على y بقدر ألاقي عدد
+
+241
+00:24:48,000 --> 00:24:53,440
+طبيعي n أكبر منه، يعني Z على y أصغر من n، لو ضربت
+
+242
+00:24:53,440 --> 00:25:01,550
+المتباينة هذه في y، y عدد موجب، فهيصير عندي Z أصغر من
+
+243
+00:25:01,550 --> 00:25:08,110
+n في y، وهذه هي النتيجة تبع الجزء الأول، okay، إذا
+
+244
+00:25:08,110 --> 00:25:13,270
+هيك يكون أثبتنا الجزء الأول، واضح؟ لإثبات الجزء
+
+245
+00:25:13,270 --> 00:25:19,410
+الثاني، لو أخدنا في الجزء الأول لو أخدت Z بساوي
+
+246
+00:25:19,410 --> 00:25:30,500
+واحد، فهيصير عندي 1 أصغر من n في y، ال Z هذا عدد
+
+247
+00:25:30,500 --> 00:25:35,780
+موجب، فلو أخد ال Z بالساوية واحد، هذا عدد موجب، فحسب
+
+248
+00:25:35,780 --> 00:25:41,420
+النتيجة a بيطلع عندي Z أصغر من n، يوجد عدد طبيعي n
+
+249
+00:25:41,420 --> 00:25:48,080
+بحيث أن Z أصغر من ny، يعني 1 أصغر من ny، الآن نضرب
+
+250
+00:25:48,080 --> 00:25:53,910
+في 1 على n، 1 على n عدد موجب، لو ضربنا الطرفين بالعدد
+
+251
+00:25:53,910 --> 00:25:57,850
+الموجب بواحد علينا بيطلع 1 علينا أصغر من Y، وهذا
+
+252
+00:25:57,850 --> 00:26:01,330
+اللي احنا عايزينه، تمام، إن هذا برهان الجزء الثاني
+
+253
+00:26:01,330 --> 00:26:14,310
+لبرهان الجزء الثالث، الجزء
+
+254
+00:26:14,310 --> 00:26:14,730
+C
+
+255
+00:26:18,400 --> 00:26:23,700
+بنثبت أنه لأي عدد حقيقي موجب Z فيه عدد طبيعي بحيث
+
+256
+00:26:23,700 --> 00:26:30,940
+أن Z محصورة بين N سالب واحد و M تمام، نعرف الست EZ
+
+257
+00:26:30,940 --> 00:26:36,380
+على إنها كل الأعداد الطبيعية M اللي بتكون أكبر من
+
+258
+00:26:36,380 --> 00:26:46,880
+Z، الآن هذه المجموعة غير خالية، لأنه
+
+259
+00:26:51,070 --> 00:26:57,610
+لأن الـ Z هذا عدد موجب، وبالتالي في الآخر هو عدد
+
+260
+00:26:57,610 --> 00:27:01,950
+حقيقي، ف by Archimedean property
+
+261
+00:27:10,880 --> 00:27:17,220
+اللي هي 115 رقمها، نظرية 115 بتقول أي عدد حقيقي z
+
+262
+00:27:17,220 --> 00:27:26,880
+يوجد عدد .. يوجد عدد طبيعي، يوجد m في n بحيث أن z
+
+263
+00:27:26,880 --> 00:27:32,820
+أصغر من n، إذا
+
+264
+00:27:32,820 --> 00:27:42,120
+المجموعة هذه على الأقل فيها عنصر واحد اللي هو الـ
+
+265
+00:27:42,120 --> 00:27:49,100
+M هذا، أو خليني اسميه MZ تمام
+
+266
+00:27:49,100 --> 00:27:58,000
+الـ Archimedean property تضمن أنه للعدد Z هذا اللي
+
+267
+00:27:58,000 --> 00:28:05,100
+هو يعني احنا فرضين أن العدد موجب، الـ set هذه بقدر
+
+268
+00:28:05,100 --> 00:28:10,460
+ألاقي عدد طبيعي MZ أكبر من Z، وبالتالي المجموعة هذه
+
+269
+00:28:10,460 --> 00:28:15,580
+تحتوي تحتوي على العنصر هذا على الأقل، لأن هذه
+
+270
+00:28:15,580 --> 00:28:22,720
+مجموعة غير خالية، واضحة النقطة هذه؟ الآن في خاصية
+
+271
+00:28:22,720 --> 00:28:29,920
+الترتيب أو بنسميها ال well ordering property، وهذه
+
+272
+00:28:29,920 --> 00:28:34,400
+في الحقيقة بتدرسها في نهاية في آخر chapter في
+
+273
+00:28:34,400 --> 00:28:40,640
+مبادئ رياضيات، ال well ordering property بتقول إن
+
+274
+00:28:40,640 --> 00:28:46,240
+every non-empty subset of N has a least element
+
+275
+00:28:46,240 --> 00:28:51,020
+يعني أي مجموعة غير خالية من مجموعة الأعداد
+
+276
+00:28:51,020 --> 00:28:55,880
+الطبيعية لازم اللي جي لها least element، لازم يكون
+
+277
+00:28:55,880 --> 00:29:00,520
+لها أصغر عنصر، يعني خدي أنت على الجربة حتى خدي أي
+
+278
+00:29:00,520 --> 00:29:04,060
+مجموعة جزئية من العدالة الطبيعية هتجد أن فيها عنصر
+
+279
+00:29:04,060 --> 00:29:08,620
+فيها هو أصغر عنصر، فهذا طبعا حسب ال well ordering
+
+280
+00:29:08,620 --> 00:29:12,880
+property، يعني درس المبادئ، وأنا شخصيا لما بدرس 
+
+281
+00:29:12,880 --> 00:29:16,400
+مبادئ بحاول يعني أمر عليها أو أعطيها حتى لو يعني
+
+282
+00:29:16,400 --> 00:29:21,620
+بصورة مختصرة بقرابش الناس الثانية لما بدرسوا 
+
+283
+00:29:21,620 --> 00:29:25,340
+المبادئ بعتقد ممكن ما وصلوش إليها لكن مش مشكلة هاي
+
+284
+00:29:25,340 --> 00:29:26,400
+نحن بنحكيلكم عنها
+
+285
+00:29:29,700 --> 00:29:35,480
+إذا هي عندي هذه عبارة عن subset من مجموعة الأعداد
+
+286
+00:29:35,480 --> 00:29:40,060
+الطبيعية و non-empty إذا لازم يكون فيها least
+
+287
+00:29:40,060 --> 00:29:45,640
+element إذا بقدر ألاقي NZ في مجموعة الأعداد
+
+288
+00:29:45,640 --> 00:29:49,300
+الطبيعية و هذا ال NZ هو least element لل set هذه
+
+289
+00:29:49,300 --> 00:29:56,530
+الغير خالية okay تمام إذا هنا يوجد عنصر nz عدد
+
+290
+00:29:56,530 --> 00:30:02,390
+طبيعي وهذا العدد الطبيعي هو ال least element ل
+
+291
+00:30:02,390 --> 00:30:09,530
+easy طيب
+
+292
+00:30:09,530 --> 00:30:17,350
+الآن هذا أصغر عنصر في ال set هذه يعني معناه nz لو
+
+293
+00:30:17,350 --> 00:30:25,080
+طرحت من nz طرحت منها واحد فطبعا هذا أصغر من NZ هذا
+
+294
+00:30:25,080 --> 00:30:34,920
+أصغر من NZ صح؟ مظبوط؟ وهذا أصغر عنصر لل set easy
+
+295
+00:30:34,920 --> 00:30:41,700
+هذا أصغر عنصر وهذا أصغر منه إذا هذا العنصر مش
+
+296
+00:30:41,700 --> 00:30:49,690
+ممكن يكون موجود بال set easy صح؟ لأن هذا أصغر من
+
+297
+00:30:49,690 --> 00:30:53,370
+أصغر
+
+298
+00:30:53,370 --> 00:30:59,410
+عنصر في ال set طيب،
+
+299
+00:30:59,410 --> 00:31:04,290
+معناه أن هذا nz سالب واحد ما هوش في ez
+
+300
+00:31:09,210 --> 00:31:13,650
+يعني هذا العنصر مش موجود في set ez هذا هي
+
+301
+00:31:13,650 --> 00:31:21,730
+معناته بيحققش الصفة المميزة لل set ez متى
+
+302
+00:31:21,730 --> 00:31:27,210
+العنصر بيكون موجود هنا إذا بيحقق الصفة هذه أو
+
+303
+00:31:27,210 --> 00:31:30,390
+المتباينة هذه طب إذا كان العنصر لا ينتمي لل set
+
+304
+00:31:30,390 --> 00:31:36,240
+معناته بيحققش المتباينة دي بيحقق ما فيها إذا هي بيحقق
+
+305
+00:31:36,240 --> 00:31:43,740
+ما فيها هاي nz-1 بدل ما يكون أكبر بيصير أصغر من أو 
+
+306
+00:31:43,740 --> 00:31:47,900
+يساوي ال z إذا كون العنصر هذا مش موجود في ez
+
+307
+00:31:47,900 --> 00:31:56,560
+معناته بيطلع أصغر من أو يساوي ال z وال z هو أصغر
+
+308
+00:31:56,560 --> 00:31:59,440
+عنصر لل set ez
+
+309
+00:32:06,800 --> 00:32:16,820
+ف ال z أصغر من n احنا قلنا أنه ال ..
+
+310
+00:32:16,820 --> 00:32:18,760
+أو أصغر من ال nz
+
+311
+00:32:44,130 --> 00:32:50,890
+الآن زي هذا عنصر يعني
+
+312
+00:32:50,890 --> 00:32:57,270
+هذا بينتمي إلى ال set ez لأنه أصغر عنصر فيها
+
+313
+00:32:57,270 --> 00:33:06,070
+فينتمي إليها فإن زي ينتمي ل ez معناته العنصر زي
+
+314
+00:33:06,070 --> 00:33:11,050
+هذا أكبر من ال z العنصر زي أكبر من ال z ومن هنا أن
+
+315
+00:33:11,050 --> 00:33:17,910
+زي سالب واحد مش موجود في ez فهو أصغر من أو يساوي
+
+316
+00:33:17,910 --> 00:33:24,290
+ال z وبالتالي هيك بنكون أثبتنا المتباينة هذه اللي
+
+317
+00:33:24,290 --> 00:33:29,090
+هو اللي احنا عايزينه في الجزء c لأن هيك بنكون
+
+318
+00:33:29,090 --> 00:33:34,420
+كملنا برهان الجزء c الأقل بالنسبة للجزء الأخير هذا
+
+319
+00:33:34,420 --> 00:33:42,460
+يعني عبارة عن ليس مش alternative form لل
+
+320
+00:33:42,460 --> 00:33:46,180
+Archimedean property ليس صورة أخرى لخاصية
+
+321
+00:33:46,180 --> 00:33:51,500
+Archimedean بس مجرد مثال، مجرد مثال أعطى ست والست
+
+322
+00:33:51,500 --> 00:33:56,290
+هذه bounded bounded above by one bounded below by
+
+323
+00:33:56,290 --> 00:34:02,570
+zero لبرهان
+
+324
+00:34:02,570 --> 00:34:12,350
+ذلك البرهان سهل نشوف
+
+325
+00:34:12,350 --> 00:34:12,950
+البرهان
+
+326
+00:34:29,410 --> 00:34:34,370
+كمان مرة ال set هذه هي عبارة عن .. نكتبها إيش هي
+
+327
+00:34:34,370 --> 00:34:37,710
+ال
+
+328
+00:34:37,710 --> 00:34:44,490
+set is عبارة عن ال set of all واحد على n حيث n is
+
+329
+00:34:44,490 --> 00:34:45,650
+natural number
+
+330
+00:34:51,720 --> 00:34:59,580
+واضح أن العنصر أصغر من أو يساوي واحد على n لكل n 
+
+331
+00:34:59,580 --> 00:35:11,180
+ينتمي إلى n صح؟ وبالتالي إذا zero is lower lower
+
+332
+00:35:11,180 --> 00:35:22,090
+bound لمين of set s وبالتالي ال infimum إذا it has
+
+333
+00:35:22,090 --> 00:35:25,890
+an infimum by the infimum property ال infimum
+
+334
+00:35:25,890 --> 00:35:30,630
+property بتقول كل set bounded below بيكون ال في
+
+335
+00:35:30,630 --> 00:35:37,070
+إلها infimum say w بيساوي infimum s إذا هنا say
+
+336
+00:35:37,070 --> 00:35:41,290
+دعنا نسمي ال infimum هذا اللي إحنا ضمنين وجوده
+
+337
+00:35:41,290 --> 00:35:48,760
+باستخدام ال infimum property دعنا نسميه w تمام؟ إذا
+
+338
+00:35:48,760 --> 00:35:55,540
+الـ ال w هذا هو أكبر هو أكبر lower bound لست
+
+339
+00:35:55,540 --> 00:36:02,640
+s والعنصر lower bound إذا أكيد ال w أكبر من أو يساوي
+
+340
+00:36:02,640 --> 00:36:09,100
+والعنصر صح؟ العنصر قلنا هذه lower bound لست و ال w
+
+341
+00:36:09,100 --> 00:36:11,960
+هو ال infimum اللي هو أكبر lower bound إذا ال w
+
+342
+00:36:11,960 --> 00:36:16,830
+أكبر من أو أكبر من أو يساوي العنصر طب احنا عايزين
+
+343
+00:36:16,830 --> 00:36:22,630
+نثبت احنا عايزين في النهاية نثبت أن ال w هذا
+
+344
+00:36:22,630 --> 00:36:27,490
+اللي هو ال infimum بيساوي العنصر هذا اللي عايزين
+
+345
+00:36:27,490 --> 00:36:33,570
+نثبته أنا عندي w أكبر من أو يساوي العنصر لكن أنا بدي
+
+346
+00:36:33,570 --> 00:36:39,750
+أثبت أن ال w بيساوي العنصر، تمام؟
+
+347
+00:36:39,750 --> 00:36:41,510
+فلإثبات ذلك
+
+348
+00:36:47,400 --> 00:36:54,780
+خلّينا ناخد أي إبسلون أكبر من العنصر فحسب
+
+349
+00:36:54,780 --> 00:36:59,600
+ال Archimedean property اللي هو الجزء ب المكافئ
+
+350
+00:36:59,600 --> 00:37:04,640
+Archimedean property لأي عدد موجب إبسلون بقدر
+
+351
+00:37:04,640 --> 00:37:08,880
+ألاقي عدد طبيعي مقلوبه وأصغر من إبسلون، صح؟ هذا
+
+352
+00:37:08,880 --> 00:37:12,000
+الجزء ب من النتيجة
+
+353
+00:37:14,540 --> 00:37:18,960
+إن أنا في عندي هي 1 على n أصغر من epsilon يوجد
+
+354
+00:37:18,960 --> 00:37:24,760
+n هذا الطبيعي بحيث 1 على n أصغر من epsilon و 1
+
+355
+00:37:24,760 --> 00:37:30,700
+على n هذه عنصر ال 1 على n هذه عبارة عن عنصر في ال
+
+356
+00:37:30,700 --> 00:37:37,180
+set s و ال w هذه lower bound إلها ال w هذه هو ال
+
+357
+00:37:37,180 --> 00:37:44,890
+minimum لل set s و 1 على n عنصر في s إذا ال w بيطلع
+
+358
+00:37:44,890 --> 00:37:48,490
+أصغر من أو يساوي أي عنصر في ال set لأنه lower bound
+
+359
+00:37:48,490 --> 00:37:53,830
+صح؟ وقبل شوية قلنا إن ال w هي u بس نتجنا إن ال w
+
+360
+00:37:53,830 --> 00:37:57,990
+اللي هو ال infimum أكبر من أو يساوي العنصر اللي هو
+
+361
+00:37:57,990 --> 00:38:02,190
+lower bound وهذا أكبر lower bound الآن هذه ال
+
+362
+00:38:02,190 --> 00:38:06,850
+epsilon عشوائية إن الكلام هذا صحيح لكل epsilon
+
+363
+00:38:06,850 --> 00:38:13,170
+أكبر من العنصر إذا في عندي نظرية واحد ثمانية بتقول
+
+364
+00:38:13,170 --> 00:38:19,630
+ليه؟ كانت بتقول إن لو كان ال a عدد غير سالب و أصغر
+
+365
+00:38:19,630 --> 00:38:24,810
+من epsilon لكل epsilon أكبر من العنصر فهذا بيقود إلى أن
+
+366
+00:38:24,810 --> 00:38:33,630
+a بيساوي العنصر، صح؟ هذه نظرية واحد ثمانية، صح؟ هي ال
+
+367
+00:38:33,630 --> 00:38:39,230
+w التي هي ال a أكبر من أو يساوي العنصر وأصغر من
+
+368
+00:38:39,230 --> 00:38:44,590
+إبسلون لكل إبسلون عدد موجب فحسب النظرية هذه بيطلع
+
+369
+00:38:44,590 --> 00:38:50,590
+w بيساوي العنصر وهذا اللي احنا عايزينه نثبته، تمام؟ إذن
+
+370
+00:38:50,590 --> 00:38:56,050
+هذا بيثبت أن ال infimum للست دي أو لل sequence
+
+371
+00:38:56,050 --> 00:39:03,650
+واحد على n هو العنصر، تمام؟ وهنا استخدمنا في البرهان
+
+372
+00:39:03,650 --> 00:39:09,010
+ال Archimedean property الصورة بيه من ال
+
+373
+00:39:09,010 --> 00:39:24,610
+Archimedean property في
+
+374
+00:39:24,610 --> 00:39:27,390
+النظرية هذه احنا أثبتنا قبل هيك
+
+375
+00:39:32,670 --> 00:39:41,530
+احنا أثبتنا سابقا في
+
+376
+00:39:41,530 --> 00:39:51,490
+السابق أثبتنا أنه في كان نظرية أو مثال بتقول أن
+
+377
+00:39:51,490 --> 00:39:55,550
+جذر 2 is not a rational number
+
+378
+00:39:58,290 --> 00:40:04,470
+أو العدد جذر اثنين is irrational نعم مظبوط فطبعا
+
+379
+00:40:04,470 --> 00:40:08,730
+في البرهان هذا اعتمدنا في البرهان على أن جذر
+
+380
+00:40:08,730 --> 00:40:12,850
+اثنين هذا عدد حقيقي يعني exist هو أحد العداد
+
+381
+00:40:12,850 --> 00:40:20,950
+الحقيقية وفرضنا عملنا برهان غير مباشر فرضنا أنه
+
+382
+00:40:20,950 --> 00:40:26,450
+جذر اثنين ينتمي ل q أو عدد نسبي ووصلنا إلى تناقض
+
+383
+00:40:26,450 --> 00:40:32,380
+تمام اليوم بنرجع للوراء شوية وبنقول احنا هنا في
+
+384
+00:40:32,380 --> 00:40:36,220
+النظرية هذه في البرهان أو في النظرية هذه افترضنا
+
+385
+00:40:36,220 --> 00:40:42,140
+جدلا أو افترضنا مسبقا أن جذر اثنين هذا عدد حقيقي
+
+386
+00:40:42,140 --> 00:40:47,600
+اليوم هنرجع ونثبت أن existence of جذر اثنين يعني
+
+387
+00:40:47,600 --> 00:40:51,720
+جذر اثنين هذا بنثبت أن هو فعلا عدد حقيقي مش عدد
+
+388
+00:40:51,720 --> 00:40:53,040
+آخر مش عدد تخيّلي
+
+389
+00:40:55,660 --> 00:41:02,360
+فهذا يعني البرهان أو 
+
+390
+00:41:02,360 --> 00:41:05,560
+نظريها دي بالظبط بتقول انه جذر اثنين وعدد حقيقي
+
+391
+00:41:05,560 --> 00:41:14,760
+يعني يوجد عدد حقيقي موجب x ومربعه هو اثنين okay
+
+392
+00:41:16,030 --> 00:41:20,890
+فبرهان النظرية هذه يعني ممكن شوية طويل لكن موجود
+
+393
+00:41:20,890 --> 00:41:29,250
+عندكم بالتفصيل ويعني موجود إلى أعزاء ويعني مش صعب
+
+394
+00:41:29,250 --> 00:41:35,490
+أنكم يعني تقرؤوا بمجموعتهم و تفهموه فأرجو أنكم
+
+395
+00:41:35,490 --> 00:41:39,990
+تقرؤوا البرهان و تحاولوا تفهموه و ممكن يعني المرة
+
+396
+00:41:39,990 --> 00:41:45,510
+الجاية إن شاء الله نسأل نحاول نمر عليه أو نحاول
+
+397
+00:41:45,510 --> 00:41:52,090
+نبرهن نقصر عليه، طبعا؟ إذا نكتفي بهذا القدر ونكمل
+
+398
+00:41:52,090 --> 00:41:53,230
+إن شاء الله المرة الجاية
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg.srt
new file mode 100644
index 0000000000000000000000000000000000000000..f79321e81623331ac8baeecf77dc45a97f50a484
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg.srt
@@ -0,0 +1,1959 @@
+1
+00:00:21,630 --> 00:00:28,730
+Okay إن شاء الله اليوم هنعمل مناقشة لبعض المسائل
+
+2
+00:00:28,730 --> 00:00:34,230
+في section 2.3 و 2.4 زي ما وعدناكم 
+
+3
+00:00:34,230 --> 00:00:44,690
+سابقا ونشوف بعض الحلول لبعض المسائل المهمة ففي
+
+4
+00:00:44,690 --> 00:00:52,530
+بسألة سؤال خامس في section 2.3 بيقول لو في
+
+5
+00:00:52,530 --> 00:00:57,270
+عندي مجموعة غير خالية من الأعداد الحقيقية و
+
+6
+00:00:57,270 --> 00:01:03,550
+bounded below فالـ infimum للـ set S هو سالب الـ
+
+7
+00:01:03,550 --> 00:01:09,110
+supremum لـ سالب S هذا
+
+8
+00:01:09,110 --> 00:01:13,870
+التمرين حالة خاصة من التمرين رقم أربعة في section
+
+9
+00:01:13,870 --> 00:01:21,120
+2.4 و بالتحديد هو حالة خاصة من الجزء B من
+
+10
+00:01:21,120 --> 00:01:26,980
+التمرين هذا ففي الجزء B لو كان B .. إيش بقول هذا
+
+11
+00:01:26,980 --> 00:01:34,160
+الجزء؟ لو كان B عدد سالب فـ infimum لـ S بيساوي B
+
+12
+00:01:34,160 --> 00:01:42,620
+في supremum S فلو أخدت B بيساوي سالب واحد و هذا عدد
+
+13
+00:01:42,620 --> 00:01:50,580
+سالب فبطل عندي infimum  infimum 
+
+14
+00:01:50,580 --> 00:01:58,780
+سالب S لأ هذا عبارة عن حالة خاصة من الجزء الثاني
+
+15
+00:01:58,780 --> 00:02:05,560
+لو أخدنا B بيساوي سالب واحد في الجزء هذا اللي هنا
+
+16
+00:02:07,490 --> 00:02:14,150
+فبطلع عندي supremum سالب S بيساوي
+
+17
+00:02:14,150 --> 00:02:19,390
+سالب infimum S هاي سالب اضربك سالب واحد سالب
+
+18
+00:02:19,390 --> 00:02:23,390
+infimum S لأن هذا التمرين حالة خاصة من الجزء هذا
+
+19
+00:02:23,390 --> 00:02:30,450
+الثاني في الفرع B وبالتالي هذا التمرين تعميم لهذا
+
+20
+00:02:30,450 --> 00:02:37,140
+الجزء ولا جزء ثانيو لجزء ثاني اللي هو عبارة عن ال
+
+21
+00:02:37,140 --> 00:02:47,140
+supremum أو الـ infimum لـ سالب S بيساوي سالب الـ
+
+22
+00:02:47,140 --> 00:02:54,240
+supremum لـ S هذا تعميم لجزء اللي هان وهذا تعميم
+
+23
+00:02:54,240 --> 00:03:00,920
+لجزء اللي هان وذلك بـ taking B equals سالب
+
+24
+00:03:00,920 --> 00:03:12,510
+واحد خلينا نبرهن الجزء الأول من الفرع A والجزء 
+
+25
+00:03:12,510 --> 00:03:17,190
+الأول من الفرع B وبالمثل بإمكانكم تبرهن الجزء
+
+26
+00:03:17,190 --> 00:03:22,510
+الثاني من الـ part A والجزء الثاني من part B
+
+27
+00:03:22,510 --> 00:03:30,890
+فنبرهن الجزء A لبرهان الجزء A اللي 
+
+28
+00:03:30,890 --> 00:03:37,490
+هو هذا الجزء فأنا عندي a عدد موجب S is bounded
+
+29
+00:03:37,490 --> 00:03:42,390
+وبالتالي bounded below إذا الـ infimum لـ S exist سميه 
+
+30
+00:03:42,390 --> 00:03:47,110
+w طبعا الـ infimum عبارة عن lower bound لـ S إذا الـ w
+
+31
+00:03:47,110 --> 00:03:53,030
+أصغر من أو يساوي X لكل X ∈ S وبالتالي لو ضربت في عدد 
+
+32
+00:03:53,030 --> 00:03:57,510
+موجب a فبطلع aw أصغر من أو يساوي aX لكل S هذا 
+
+33
+00:03:57,510 --> 00:04:05,230
+معناه إن العدد هذا lower bound لـ aS أنا عايز أثبت
+
+34
+00:04:05,230 --> 00:04:10,670
+أن أي w هذا العدد مش بس lower bound هو أكبر lower
+
+35
+00:04:10,670 --> 00:04:19,690
+bound للـ set aS فباخد أي let V be any lower bound
+
+36
+00:04:19,690 --> 00:04:27,790
+any lower bound للـ set aS وبينا
+
+37
+00:04:27,790 --> 00:04:32,710
+نثبت أن هذا الـ V أصغر من أو يساوي aw عشان يكون هو
+
+38
+00:04:32,710 --> 00:04:33,390
+الـ infimum
+
+39
+00:04:35,910 --> 00:04:43,990
+طيب هذا معناه V lower bound للـ set aS معناه V أصغر
+
+40
+00:04:43,990 --> 00:04:52,010
+من أو يساوي aX لكل X في S طيب أنا عندي 1/a
+
+41
+00:04:52,010 --> 00:04:57,330
+عدد موجب إذا 1/a عدد موجب فلو ضربت المتباينة
+
+42
+00:04:57,330 --> 00:05:00,270
+هذه في العدد الموجب 1/a اشتغلت هنا
+
+43
+00:05:00,270 --> 00:05:07,900
+مابتتغيرش فبصير عندي V/a أصغر من أو يساوي X لكل
+
+44
+00:05:07,900 --> 00:05:12,300
+X ∈ S طب
+
+45
+00:05:12,300 --> 00:05:20,540
+ما هذا معناه أنه العدد الـ number V over A is a
+
+46
+00:05:20,540 --> 00:05:25,840
+lower bound لمن؟
+
+47
+00:05:25,840 --> 00:05:30,580
+لـ S وبالتالي
+
+48
+00:05:30,580 --> 00:05:38,490
+إذا الـ infimum .. إذا الـ V/a أصغر من أو يساوي الـ
+
+49
+00:05:38,490 --> 00:05:48,090
+infimum للـ set S صح؟ طب اضربي في a عدد موجب بطلع
+
+50
+00:05:48,090 --> 00:05:58,990
+عندي V أصغر من أو يساوي a في infimum S طب
+
+51
+00:05:58,990 --> 00:06:07,730
+infimum S هذا سميته w لأن هذا بيساوي aw إذن هين
+
+52
+00:06:07,730 --> 00:06:13,790
+أثبتنا إنه العدد aw هذا أبرع الـ lower bound للـ set
+
+53
+00:06:13,790 --> 00:06:20,390
+aS وأخدنا أي lower bound للـ set aS فوجدنا إن الـ
+
+54
+00:06:20,390 --> 00:06:27,770
+lower bound هذا أصغر من أو يساوي a في w فهذا معناه
+
+55
+00:06:27,770 --> 00:06:37,630
+إن aw هو الـ infimum لمن؟ للـ set aS كما هو موضح في الـ
+
+56
+00:06:37,630 --> 00:06:44,290
+claim أو في الإدعاء تمام؟ وهذا بيثبت الجزء الأول في
+
+57
+00:06:44,290 --> 00:06:51,650
+الـ part A هاي infimum aS بيساوي a في w اللي هو
+
+58
+00:06:51,650 --> 00:06:58,250
+infimum S إذن هذا بيثبت الجزء الأول في الفرع A
+
+59
+00:06:58,250 --> 00:07:01,850
+Similarly بالمثل ممكن
+
+60
+00:07:05,820 --> 00:07:12,760
+بالمثل ممكن نثبت الفرع الثاني أو
+
+61
+00:07:12,760 --> 00:07:20,060
+الجزء الثاني في الفرع A تمام؟ فهسيب هذا جزء لكم
+
+62
+00:07:20,060 --> 00:07:27,840
+لأن هذا مشابه للفرع اللي أنا واضح؟ في أي سؤال؟ طيب
+
+63
+00:07:27,840 --> 00:07:30,780
+نحاول نثبت الجزء الأول في الفرع B
+
+64
+00:07:35,110 --> 00:07:42,150
+بنثبت الجزء هذا في الفرع B لت
+
+65
+00:07:42,150 --> 00:07:53,770
+بـ أصغر من صفر، عدد حقيقي سالب وأنا عندي الـ set الـ
+
+66
+00:07:53,770 --> 00:07:58,230
+set since الـ set S is bounded
+
+67
+00:08:01,660 --> 00:08:10,440
+إذا الـ infimum w بيساوي الـ infimum لـ S exists in R
+
+68
+00:08:10,440 --> 00:08:13,460
+إذا
+
+69
+00:08:13,460 --> 00:08:18,240
+في عندي أنا الـ .. الـ infimum لـ S .. S bounded
+
+70
+00:08:18,240 --> 00:08:21,180
+below bounded وبالتالي bounded below إذا by
+
+71
+00:08:21,180 --> 00:08:26,460
+infimum property الـ infimum لـ S سميته w exist
+
+72
+00:08:30,860 --> 00:08:41,580
+هذا معناه .. أو هذا بقد .. إذا 
+
+73
+00:08:41,580 --> 00:08:46,180
+هذا معناه أن w lower bound لـ S و w أصغر من أو يساوي
+
+74
+00:08:46,180 --> 00:08:49,880
+X لكل X ∈ S
+
+75
+00:08:53,000 --> 00:08:58,980
+طيب وعندي أنا الـ B عدد سالب فلو ضربنا المتباينة
+
+76
+00:08:58,980 --> 00:09:06,840
+هذه في B عدد سالب فبصير bX أصغر من أو يساوي bW لكل
+
+77
+00:09:06,840 --> 00:09:18,890
+X ∈ S صح؟ إذن هذا معناه إنه العدد bW is an
+
+78
+00:09:18,890 --> 00:09:28,750
+upper is an upper bound لمين؟ للـ set bS للـ set b
+
+79
+00:09:28,750 --> 00:09:33,930
+في S اللي هي مجموعة كل العناصر b ضرب X b ضرب
+
+80
+00:09:33,930 --> 00:09:38,570
+X حيث X ينتمي للـ S هذا عبارة عن upper bound
+
+81
+00:09:38,570 --> 00:09:46,570
+طيب الـ set هذه الـ set هذه bounded لأن الـ set S bounded
+
+82
+00:09:46,570 --> 00:09:51,270
+فضربها في عدد بتظلها bounded وبالتالي bounded above
+
+83
+00:09:51,270 --> 00:09:57,250
+إذا الـ .. الـ .. إلها supremum by supremum property
+
+84
+00:09:57,250 --> 00:10:08,990
+وبالتالي إذا الـ bW هذا أو الـ supremum للـ set bS هذا
+
+85
+00:10:08,990 --> 00:10:14,330
+عبارة عن الـ least upper bound for the set bS هذا
+
+86
+00:10:14,330 --> 00:10:20,270
+بيطلع أصغر من أو يساوي أي upper bound وليه هو أصغر
+
+87
+00:10:20,270 --> 00:10:28,150
+من أو يساوي الـ upper bound bW للـ set bS طب
+
+88
+00:10:28,150 --> 00:10:29,610
+احنا عايزين نثبت
+
+89
+00:10:32,240 --> 00:10:38,840
+احنا عايزين نثبت أن bW هي الـ supremum لـ set b
+
+90
+00:10:38,840 --> 00:10:42,460
+في S فهين
+
+91
+00:10:42,460 --> 00:10:47,020
+أثبتنا أن العدد bW هذا upper bound للـ set هذه
+
+92
+00:10:47,020 --> 00:10:51,240
+bW هو upper bound للـ set الإثبات إنه هو الـ
+
+93
+00:10:51,240 --> 00:10:55,240
+supremum باقي إثبات إن أنا لو أخدت أي upper bound
+
+94
+00:10:55,240 --> 00:11:00,400
+للـ set هذه لازم يطلع أكبر من أو يساوي bW
+
+95
+00:11:04,070 --> 00:11:11,310
+any upper bound
+
+96
+00:11:11,310 --> 00:11:18,490
+of except bS هذا 
+
+97
+00:11:18,490 --> 00:11:28,090
+معناه أن b في x أصغر من أو يساوي v لكل x ∈ S تمام؟
+
+98
+00:11:29,920 --> 00:11:34,420
+طيب أنا عندي b عدد سالب إذا 1/b ايضا عدد
+
+99
+00:11:34,420 --> 00:11:38,960
+سالب فلو ضربت المتباينة هذه في عدد سالب اللي هو
+
+100
+00:11:38,960 --> 00:11:50,040
+1/b فهيطلع عندي v/b أصغر من أو
+
+101
+00:11:50,040 --> 00:11:52,340
+يساوي X لكل X ∈ S
+
+102
+00:11:55,350 --> 00:12:04,150
+هذا معناه أن العدد V/b is a lower bound لمن؟
+
+103
+00:12:04,150 --> 00:12:11,510
+لـ set S مضبوط صح؟ وبالتالي
+
+104
+00:12:11,510 --> 00:12:17,930
+إذا .. إذا 
+
+105
+00:12:17,930 --> 00:12:23,970
+الـ V/b اللي هو lower bound للـ set S أصغر من أو
+
+106
+00:12:23,970 --> 00:12:28,370
+يساوي الـ infimum للـ set S
+
+107
+00:12:54,340 --> 00:13:06,560
+احنا إيش قاعدين نثبت الـ ..
+
+108
+00:13:06,560 --> 00:13:12,960
+يبدو أن أنا يعني هنا بيثبت الجزء الثاني يعني، يلا 
+
+109
+00:13:12,960 --> 00:13:22,410
+من حظكم نحاول نثبت الجزء الثاني مش الأول فكمان مرة
+
+110
+00:13:22,410 --> 00:13:26,810
+نراجع B عدد سالب S is bounded وبالتالي bounded
+
+111
+00:13:26,810 --> 00:13:33,650
+below إذن الـ infimum لـ set S موجود وبالتالي 
+
+112
+00:13:33,650 --> 00:13:37,630
+المتباينة هذه بتتحقق وبالتالي هذه بتتحقق بعد ما
+
+113
+00:13:37,630 --> 00:13:42,070
+ضربنا في B عدد سالب إذن b وطلع upper bound لـ
+
+114
+00:13:42,070 --> 00:13:48,410
+set bS وبالتالي الـ supremum للـ set bS بيطلع أصغر
+
+115
+00:13:48,410 --> 00:13:52,510
+من أو يساوي bW الآن بدنا نثبت أن الـ b
+
+116
+00:13:52,510 --> 00:14:00,810
+W هذا هو الـ supremum لـ set bS تمام فأخدنا أي 
+
+117
+00:14:00,810 --> 00:14:05,550
+upper bound v .. أي upper bound لـ set bS فوجدنا
+
+118
+00:14:05,550 --> 00:14:09,930
+أن v/b is a lower bound لـ set S وبالتالي v على 
+
+119
+00:14:09,930 --> 00:14:14,290
+b أصغر من أو يساوي الـ greatest lower bound لـ set S
+
+120
+00:14:17,060 --> 00:14:27,860
+طب لو ضربنا في b و b عدد سالب فهيطلع عندي .. إذا 
+
+121
+00:14:27,860 --> 00:14:34,940
+لو ضربنا المتباينة هذه في b عدد سالب فهيطلع عندي
+
+122
+00:14:34,940 --> 00:14:43,120
+اللي هو b في infimum S هيطلع أصغر من أو يساوي الـ
+
+123
+00:14:43,120 --> 00:14:45,120
+v، مضبوط هيك؟
+
+124
+00:14:48,920 --> 00:14:56,120
+طب هذا هذا سميته w إذا b في w أصغر من أو يساوي الـ
+
+125
+00:14:56,120 --> 00:15:02,100
+v إذا البرهان هذا أثبتنا فيه حاجتين إنه أول شيء
+
+126
+00:15:02,100 --> 00:15:07,540
+العدد bW هذا upper bound للـ set bS وبعدين 
+
+127
+00:15:07,540 --> 00:15:14,350
+أخدنا أي upper bound v أي upper bound لـ set bS طلع
+
+128
+00:15:14,350 --> 00:15:19,910
+الـ v هذا أكبر من أو يساوي bW وبالتالي هذا
+
+129
+00:15:19,910 --> 00:15:29,650
+معناه إذا العدد bW هو عبارة عن الـ supremum
+
+130
+00:15:29,650 --> 00:15:40,970
+الـ supremum لـ set b في S لـ set b في S لأن هذا العدد
+
+131
+00:15:40,970 --> 00:15:45,570
+upper bound للـ set هذه وهو أصغر upper bound أخدنا أي 
+
+132
+00:15:45,570 --> 00:15:51,390
+upper bound للـ set هذه طلع bW أصغر من أو يساوي
+
+133
+00:15:51,390 --> 00:15:56,050
+إذن bW هو أصغر upper bound للـ set هذه والآن
+
+134
+00:15:56,050 --> 00:16:03,410
+بنعود عن w إذن الـ b في w اللي هو infimum of S
+
+135
+00:16:03,410 --> 00:16:12,590
+بتطلع بيساوي supremum لـ b في S وهذا بيبرهن الجزء
+
+136
+00:16:12,590 --> 00:16:18,330
+الثاني من الفرع B بالمثل ممكن برهان الجزء الأول
+
+137
+00:16:18,330 --> 00:16:24,850
+من الفرع B فأنا بأدعوكم إلى كتابة برهان الأجزاء
+
+138
+00:16:24,850 --> 00:16:30,330
+المشابهة هذه تمام؟ إذن هيك بنكون .. يعني أخدنا
+
+139
+00:16:30,330 --> 00:16:37,150
+حلول تقريبا شبه كاملة للتمرين 5 section 2.3 في
+
+140
+00:16:37,150 --> 00:16:41,530
+عندكم أي أسئلة ثانية في الـ section 2.3 أو 
+
+141
+00:16:41,530 --> 00:16:48,470
+اتنين أربعة؟ في
+
+142
+00:16:48,470 --> 00:16:54,190
+أي أسئلة ثانية؟ السؤال عشرة في section اتنين ثلاثة
+
+143
+00:17:28,800 --> 00:17:38,060
+سؤال عشرة section اتنين ثلاثة ملخص السؤال بيقول S
+
+144
+00:17:38,060 --> 00:17:52,000
+is a bounded bounded subset of R و Phi
+
+145
+00:17:52,000 --> 00:17:55,460
+لا يساوي S subset
+
+146
+00:18:00,440 --> 00:18:07,020
+فإن S0 non-empty subset من S مجموعة جزئية غير 
+
+147
+00:18:07,020 --> 00:18:17,280
+خالية من المجموعة S فبدنا نثبت شو برهني أن ال
+
+148
+00:18:17,280 --> 00:18:26,260
+infimum لـ S أصغر من أو يساوي ال infimum لـ S0
+
+149
+00:18:26,260 --> 00:18:32,540
+أصغر من أو يساوي ال supremum للـ S Zero أصغر من لو
+
+150
+00:18:32,540 --> 00:18:41,940
+يساوي ال supremum للـ S نشوف
+
+151
+00:18:41,940 --> 00:18:46,860
+البرهان مع بعض برهان سهل وبسيط يعتمد على تعريف ال
+
+152
+00:18:46,860 --> 00:18:52,760
+infimum وعلى تعريف ال supremum طيب
+
+153
+00:18:52,760 --> 00:18:57,900
+أنا عندي المجموعة S since
+
+154
+00:19:00,710 --> 00:19:08,790
+بما أن S مجموعة غير خالية و bounded is a bounded
+
+155
+00:19:08,790 --> 00:19:12,990
+then ال
+
+156
+00:19:12,990 --> 00:19:28,810
+infimum لـ S exist and supremum لـ S both exist
+
+157
+00:19:36,050 --> 00:19:44,310
+بعد الـ infimum property ست اس لإنفمام وكذلك ست اس
+
+158
+00:19:44,310 --> 00:19:52,290
+لسوبرمام هدول موجودين في R طيب
+
+159
+00:19:52,290 --> 00:19:56,150
+أنا عندي السوبرمام
+
+160
+00:19:56,150 --> 00:20:15,640
+للـ S السوبرمام للـ S is an upper bound فهي
+
+161
+00:20:15,640 --> 00:20:25,520
+أيضا it is also an upper bound لأي
+
+162
+00:20:25,520 --> 00:20:31,060
+subset لأي subset S0 من ال S
+
+163
+00:20:36,460 --> 00:20:44,900
+و بالتالي and therefore and
+
+164
+00:20:44,900 --> 00:20:52,600
+therefore ال
+
+165
+00:20:52,600 --> 00:20:57,540
+supremum لـ S0
+
+166
+00:20:57,540 --> 00:21:01,840
+أصغر من أو يساوي ال supremum لـ S
+
+167
+00:21:07,110 --> 00:21:15,710
+كمان مرة ال .. ال S هذه ال S0 سبسط من S فأي upper
+
+168
+00:21:15,710 --> 00:21:20,070
+bound ل S هو أيضا upper bound لأي مجموعة جزئية
+
+169
+00:21:20,070 --> 00:21:26,410
+منها طيب ال supremum ل S upper bound ل S
+
+170
+00:21:26,410 --> 00:21:32,830
+وبالتالي هو upper bound ل S0 طيب ال supremum ل S0
+
+171
+00:21:32,830 --> 00:21:39,130
+هذا أصغر upper bound ل S0وهذا upper bound ل S0 إذا
+
+172
+00:21:39,130 --> 00:21:42,550
+أصغر upper bound أصغر من لو يساوي أي upper bound
+
+173
+00:21:42,550 --> 00:21:51,650
+وبالتالي المتباينة هذه صحيحة كذلك by
+
+174
+00:21:51,650 --> 00:21:57,950
+definition حسب التعريفات ال
+
+175
+00:21:57,950 --> 00:22:06,790
+infimum للـ S0 أصغر من أو يساوي ال supremum للـ S0
+
+176
+00:22:06,790 --> 00:22:10,750
+الـ
+
+177
+00:22:10,750 --> 00:22:11,750
+S0 هذه
+
+178
+00:22:15,230 --> 00:22:21,930
+طبعا هذه ال set S0 subset من S و S bounded إلى S0
+
+179
+00:22:21,930 --> 00:22:26,710
+bounded ال infimum ل S0 exist و ال suprem ل S0
+
+180
+00:22:26,710 --> 00:22:32,770
+exist دائما لأي set S0 ال infimum دائما أصغر من أو
+
+181
+00:22:32,770 --> 00:22:39,250
+يساوي ال supremum نعمل رسمة نوضح الكلام هذا
+
+182
+00:22:44,850 --> 00:22:56,850
+نعتبر أن هذه هي الست اس وهي
+
+183
+00:22:56,850 --> 00:23:07,950
+ال .. ال .. ال supremum للست اس وهي ال infimum
+
+184
+00:23:11,090 --> 00:23:17,810
+للـ set S فدائما ال .. دائما
+
+185
+00:23:17,810 --> 00:23:24,050
+ال minimum لأي set هو lower bound لل set وبالتالي
+
+186
+00:23:24,050 --> 00:23:28,950
+أصغر من لو يساوي كل عناصرهاهو عبارة عن lower bound
+
+187
+00:23:28,950 --> 00:23:32,810
+للست ال supreme للست S هو عبارة عن upper bound
+
+188
+00:23:32,810 --> 00:23:37,650
+للست وبالتالي أكبر من أو يساوي كل عناصرها فواضح أن
+
+189
+00:23:37,650 --> 00:23:42,770
+ال infimum للست S لازم يكون أصغر من أو يساوي ال
+
+190
+00:23:42,770 --> 00:23:52,970
+supremum ونفس الشيء لو أخذنا أي مجموعة جزئية سمنها
+
+191
+00:23:52,970 --> 00:23:53,790
+S0
+
+192
+00:23:56,180 --> 00:24:02,200
+يعني هذه المجموعة اسمها S0 فبما أن ال set S
+
+193
+00:24:02,200 --> 00:24:10,400
+bounded إذن S0 bounded وبالتالي ال supremum ل S0
+
+194
+00:24:10,400 --> 00:24:16,220
+دايما أكبر من أو يساوي ال infimum ل S0 بنفس الطريقة
+
+195
+00:24:16,220 --> 00:24:23,710
+إذن هذا دايما .. هذا دايما صحيح عشان احنا نكمل
+
+196
+00:24:23,710 --> 00:24:30,150
+البرهان إذا احنا أثبتنا هذا واضح من التعريفات وهذا
+
+197
+00:24:30,150 --> 00:24:35,150
+الجزء أثبتناه باقي
+
+198
+00:24:35,150 --> 00:24:40,930
+إثبات الجزء الأخير هذا فإذا
+
+199
+00:24:40,930 --> 00:24:45,790
+بنقول finally أخيرا لإثبات الجزء الأخير هذا أنا
+
+200
+00:24:45,790 --> 00:24:49,570
+عندي ال inform ل S is a lower bound ل S
+
+201
+00:24:52,070 --> 00:24:57,350
+وبالتالي هو lower bound لأي مجموعة جزئية S0 من S
+
+202
+00:24:57,350 --> 00:25:00,890
+وبالتالي
+
+203
+00:25:00,890 --> 00:25:11,770
+إذا ال influence ل S0 هذا
+
+204
+00:25:11,770 --> 00:25:19,180
+أكبر lower bound ل S0 هذا أكبر lower bound ل S0 و
+
+205
+00:25:19,180 --> 00:25:25,960
+هذا lower bound ل S0 إذاً هذا بيطلع أكبر من أو
+
+206
+00:25:25,960 --> 00:25:33,500
+ساوي infimum ال S هذا lower bound ل S0 و هذا
+
+207
+00:25:33,500 --> 00:25:37,820
+أكبر lower bound ل S0 إذاً هذا أصغر من أو يساوي
+
+208
+00:25:37,820 --> 00:25:43,700
+هذا و هذا بيكمل برهان المتباينة اللى حاطين عليها
+
+209
+00:25:43,700 --> 00:25:48,380
+علامة استفهام إذا هيك بيكون برهاننا التمرين okay
+
+210
+00:25:48,380 --> 00:25:53,660
+تمام واضح؟
+
+211
+00:25:53,660 --> 00:26:03,660
+في أسئلة ثانية خلنا نحل كمان سؤال إذا بتحبه ممكن
+
+212
+00:26:03,660 --> 00:26:04,900
+نحل كمان سؤال
+
+213
+00:26:08,660 --> 00:26:16,040
+في section اتنين ثلاثة برضه؟ اه في أي section؟
+
+214
+00:26:16,040 --> 00:26:21,840
+اتنين ثلاثة ولا اتنين أربعة؟ اتنين ثلاثة؟ طيب نحل
+
+215
+00:26:21,840 --> 00:26:24,020
+هذا السؤال و بعد هيك يعني نوجد
+
+216
+00:26:43,630 --> 00:26:57,410
+هي السؤال الحادي عشر سيكشن اتنين ثلاثة بنشوف
+
+217
+00:26:57,410 --> 00:27:05,850
+السؤال شو بيقول S
+
+218
+00:27:05,850 --> 00:27:11,530
+subset من R و
+
+219
+00:27:11,530 --> 00:27:25,720
+S* بساوي ال supremum لـ S وهذا بينتمي لل S
+
+220
+00:27:25,720 --> 00:27:31,040
+belongs to S فإذا
+
+221
+00:27:31,040 --> 00:27:41,140
+كان U لا ينتمي لل S إذا كان U لا ينتمي لل S شو
+
+222
+00:27:42,390 --> 00:27:49,090
+عايزين نثبت أن ال superman لـ
+
+223
+00:27:49,090 --> 00:28:05,890
+S union singleton U بيطلع بيساوي ال superman لـ
+
+224
+00:28:05,890 --> 00:28:10,330
+اللي تتكون من عنصرين S* و U
+
+225
+00:28:13,540 --> 00:28:28,400
+where are you؟ طبعا في برهانين للسؤال هذا ال
+
+226
+00:28:28,400 --> 00:28:33,840
+proof one البرهان الأول we
+
+227
+00:28:33,840 --> 00:28:38,580
+use .. we use exercise
+
+228
+00:28:42,560 --> 00:28:51,600
+تسعة section اتنين ثلاثة وهذا ال exercise بيقول
+
+229
+00:28:51,600 --> 00:28:59,340
+إذا كانت لو
+
+230
+00:28:59,340 --> 00:29:03,380
+كان a و b bounded
+
+231
+00:29:09,480 --> 00:29:18,660
+فهذا بيؤدي أن a union b is bounded and
+
+232
+00:29:18,660 --> 00:29:32,360
+مش هيكوا بس و ال supremum .. ال supremum لإتحاد b
+
+233
+00:29:32,360 --> 00:29:36,980
+بساوي supremum
+
+234
+00:29:39,920 --> 00:29:44,900
+Supermom A و Supermom
+
+235
+00:29:44,900 --> 00:29:51,760
+B إذا
+
+236
+00:29:51,760 --> 00:29:57,440
+هذا تمرين رقم تسعة هناخده نستخدمه فلو استخدمنا هذا
+
+237
+00:29:57,440 --> 00:30:07,700
+التمرين فالنتيجة هذه بتطلع على طول مباشرة إذا
+
+238
+00:30:07,700 --> 00:30:08,540
+هنا take
+
+239
+00:30:11,570 --> 00:30:17,410
+A بساوي S و
+
+240
+00:30:17,410 --> 00:30:25,570
+طبعا هادي ال set bounded ال set هادي bounded و
+
+241
+00:30:25,570 --> 00:30:32,610
+عندي ال set B هاخدها singleton U و هادي bounded
+
+242
+00:30:32,610 --> 00:30:41,790
+set إذا by exercise 9 a hat b اللي هي ال S هذه
+
+243
+00:30:41,790 --> 00:30:47,650
+بتطلع bounded by
+
+244
+00:30:47,650 --> 00:30:56,490
+exercise 9 section 2 3 ال S union singleton u is
+
+245
+00:30:56,490 --> 00:31:00,750
+bounded and
+
+246
+00:31:00,750 --> 00:31:10,540
+مش هيكوا بس ال supremum لـ A اتحاد بالـ S union
+
+247
+00:31:10,540 --> 00:31:18,160
+هذا الـ A وهذا الـ Singleton U بتساوي الـ Supremum
+
+248
+00:31:18,160 --> 00:31:22,440
+لـ
+
+249
+00:31:22,440 --> 00:31:32,820
+Supremum A هذا عبارة عن S* و Supremum D هذا
+
+250
+00:31:32,820 --> 00:31:37,830
+عبارة عن Singleton U أنا عندي set فيها عنصر واحد
+
+251
+00:31:37,830 --> 00:31:42,510
+فال Supreme تبعها هو ال info تبعها هو نفس ال
+
+252
+00:31:42,510 --> 00:31:46,850
+answer يعني هذا واضح من تعريف ال suprem
+
+253
+00:31:54,620 --> 00:31:59,580
+و هذا هو المطلوب إذا هذا تطبيق مباشر على تمرين 9
+
+254
+00:31:59,580 --> 00:32:03,860
+إذا المعنى أن أنتم لازم تحلوا تمرين 9 و هذا
+
+255
+00:32:03,860 --> 00:32:11,260
+التمرين موجود في يعني في إرشاد له أو hint لحله في
+
+256
+00:32:11,260 --> 00:32:16,680
+خلف .. خلف الكتاب في حل تمرين اللي .. اللي الكتاب
+
+257
+00:32:16,680 --> 00:32:21,280
+بيحاول يعرضها عشان يساعد الطالب نعم تفضلي
+
+258
+00:32:28,890 --> 00:32:37,250
+آه صحيح نعم و
+
+259
+00:32:37,250 --> 00:32:45,170
+في السؤال تسعة و في السؤال الحادي عشر ال S
+
+260
+00:32:45,170 --> 00:32:51,010
+من المقطيات bounded صحيح لأنها احنا فرضين أن S
+
+261
+00:32:51,010 --> 00:32:56,370
+subset من R و ال supremum لل S اللي هو S* عدد
+
+262
+00:32:56,370 --> 00:33:06,050
+ينتمي ل S و S subset من R هذا بيؤدي أن ال S is
+
+263
+00:33:06,050 --> 00:33:12,750
+bounded above على الأقل bounded above تمام؟
+
+264
+00:33:16,370 --> 00:33:22,230
+تمام؟ فلو كانت ال A و ال B bounded above فهيطلع
+
+265
+00:33:22,230 --> 00:33:25,510
+الاتحاد تبعهم bounded above و هذا اللي احنا
+
+266
+00:33:25,510 --> 00:33:30,490
+عايزينه و ال supremum اللي لهم بساوي .. لاتحادهم
+
+267
+00:33:30,490 --> 00:33:37,540
+بساوي الكلام هذا فعلى الأقل .. آه؟ و نفس الكلام
+
+268
+00:33:37,540 --> 00:33:41,860
+للإنفمام ممكن نثبت حاجة مشابهة بالنسبة للإنفمام
+
+269
+00:33:41,860 --> 00:33:47,140
+يعني ممكن نثبت أن الإنفايم هنا يعني ها and ممكن
+
+270
+00:33:47,140 --> 00:33:58,820
+نضيف إنفمام ل a union b بساوي انفمام انف a و انف b
+
+271
+00:34:01,670 --> 00:34:06,630
+فاحنا بس أخدنا .. طبخنا الجزء هذا الجزء بيكون صحيح
+
+272
+00:34:06,630 --> 00:34:13,390
+إذا كانت a و b both are bounded above وبالتالي
+
+273
+00:34:13,390 --> 00:34:16,430
+اتحادهم بيطلع bounded below و ال infimum للاتحاد
+
+274
+00:34:16,430 --> 00:34:23,780
+بيطلع infimum ل infimum المجمعة الثانية فهذا متحقق
+
+275
+00:34:23,780 --> 00:34:28,640
+هنا متحقق أن هاي S* ينتمي ل S وبالتالي عدد
+
+276
+00:34:28,640 --> 00:34:32,420
+حقيقي أن S ال set هذه لها supremum وبالتالي
+
+277
+00:34:32,420 --> 00:34:37,360
+bounded above و single to new ما هي finite set و
+
+278
+00:34:37,360 --> 00:34:41,960
+كل finite set is bounded فهي bounded above و below
+
+279
+00:34:41,960 --> 00:34:47,530
+طبعا وبالتالي ممكن نطبق الجزء هذاهذا برهان برهان
+
+280
+00:34:47,530 --> 00:34:51,790
+ثاني ممكن أن احنا نعمل برهان مباشر يعني بلاش 
+
+281
+00:34:51,790 --> 00:35:00,970
+نستخدم exercise تسعة ثاني
+
+282
+00:35:00,970 --> 00:35:09,310
+ممكن we 
+
+283
+00:35:09,310 --> 00:35:13,450
+consider we
+
+284
+00:35:13,450 --> 00:35:15,230
+consider two cases
+
+285
+00:35:18,470 --> 00:35:24,390
+نعتبر حالتين الـ S star هذا من المعطيات عدد حقيقي و
+
+286
+00:35:24,390 --> 00:35:31,790
+U عدد حقيقي آخر لا ينتمي لـ S فممكن يكون عندي الـ U
+
+287
+00:35:31,790 --> 00:35:40,850
+أكبر من أو يساوي S star or الـ U أصغر من S star هذا
+
+288
+00:35:40,850 --> 00:35:46,750
+طبعا by trichotomy by trichotomy
+
+289
+00:35:50,710 --> 00:35:58,670
+property من الخاصية الثلاثية U, S*) أعداد حقيقية
+
+290
+00:35:58,670 --> 00:36:04,850
+ففي عندي تلت حالات أما U أصغر من S*) أو U أكبر من 
+
+291
+00:36:04,850 --> 00:36:10,450
+S*) أو U بيساوي S*) هدول حالتين وهذه الثالثة
+
+292
+00:36:10,450 --> 00:36:15,950
+فتعالوا في كل حالة نثبت هذا اللي هو المطلوب فإذا
+
+293
+00:36:15,950 --> 00:36:22,180
+في عندي في الحالة الأولى X أقل أو بيساوي من الـ Supremum
+
+294
+00:36:22,180 --> 00:36:27,400
+الموجود في الـ U أو
+
+295
+00:36:27,400 --> 00:36:33,000
+إيش الثانية؟ أو X أقل أو بيساوي الـ U، X أصغر من أو
+
+296
+00:36:33,000 --> 00:36:38,280
+بيساوي الـ U، صح؟ بعدها أنا هقول أكيد إن الـ X أقل
+
+297
+00:36:38,280 --> 00:36:45,360
+أو بيساوي من الـ .. إن الـ X lower bound is lower 
+
+298
+00:36:45,360 --> 00:36:45,960
+bound
+
+299
+00:36:49,050 --> 00:37:03,630
+للـ set اللي بتتكون من S star و U صح؟ وبالتالي لحظة
+
+300
+00:37:03,630 --> 00:37:09,490
+شوية لو سمحتني إذا
+
+301
+00:37:09,490 --> 00:37:14,830
+الـ X lower bound للـ set هذه إذا الـ infimum
+
+302
+00:37:22,180 --> 00:37:27,840
+الـ X أصغر
+
+303
+00:37:27,840 --> 00:37:36,400
+من أو ساوي الـ infimum لـ Sلأ ما هو هذا lower bound
+
+304
+00:37:36,400 --> 00:37:41,960
+لـ S star للمجموعة هذه وبالتالي هو أصغر من أو 
+
+305
+00:37:41,960 --> 00:37:45,700
+ساوي الـ infimum و الـ infimum دائما قولنا قبل شوية
+
+306
+00:37:45,700 --> 00:37:51,780
+أصغر من أو ساوي الـ supremum لنفس المجموعة لسه
+
+307
+00:37:51,780 --> 00:37:58,160
+متبتيلوا قبل شوية في التمرين السابق صح؟ طيب هيك
+
+308
+00:37:58,160 --> 00:37:59,260
+منكون أثبتنا
+
+309
+00:38:06,750 --> 00:38:17,210
+إذا هذا صحيح since this holds لكل
+
+310
+00:38:17,210 --> 00:38:26,130
+x ينتمي احنا خدنا x عشوائية فهي fix x مظبوط؟ x 
+
+311
+00:38:26,130 --> 00:38:33,700
+كانت عنصر عشوائي ف fix x ينتمي لـ S union Singleton
+
+312
+00:38:33,700 --> 00:38:39,260
+U فإذا هذه الأداء صحيح لكل X ينتمي للمجموعة هذه
+
+313
+00:38:39,260 --> 00:38:50,460
+وبالتالي إذا الـ supremum لـ S star و U is upper
+
+314
+00:38:50,460 --> 00:39:00,300
+bound Upper bound لمن؟ لـ S union singleton U
+
+315
+00:39:08,160 --> 00:39:23,180
+مظبوط؟ إذا الـ supremum لـ S union singleton U لأ
+
+316
+00:39:23,180 --> 00:39:28,280
+مش هيك لأ إذا هذا عبارة عن upper bound لـ set هذه
+
+317
+00:39:28,280 --> 00:39:34,830
+بنثبت إن هو الـ supremumيعني هيك بيطلع هذا .. هذا
+
+318
+00:39:34,830 --> 00:39:40,610
+upper bound لـ S هذه لأن هذا بيطلع أكبر من أو ساوي
+
+319
+00:39:40,610 --> 00:39:49,610
+.. هذا أصغر من أو ساوي الـ supremum لـ
+
+320
+00:39:49,610 --> 00:39:57,310
+S star و U احنا بدنا مساواة صح؟ فبقدرش أستنتج
+
+321
+00:39:57,310 --> 00:40:03,070
+مساواة هنا تمام؟ أما شو ممكن أما زي ما عملنا في
+
+322
+00:40:03,070 --> 00:40:07,430
+البراهين السابقة ممكن نثبت الـ claim ممكن نثبت 
+
+323
+00:40:07,430 --> 00:40:13,070
+المساواة كما يلي أنا عندي هذا .. هذا العدد .. هذا
+
+324
+00:40:13,070 --> 00:40:19,270
+العدد عبارة عن upper bound للـ set هذه احنا عايزين
+
+325
+00:40:19,270 --> 00:40:22,970
+نثبت إن هذا مش upper bound هو الـ least upper bound
+
+326
+00:40:22,970 --> 00:40:29,330
+إذا نـ claim إن الـ supremum
+
+327
+00:40:29,330 --> 00:40:36,590
+لـ S union لـ set هذه هو العدد هذا
+
+328
+00:40:49,020 --> 00:41:02,440
+انشوف let V be any upper bound لـ S union
+
+329
+00:41:02,440 --> 00:41:11,840
+singleton U هذا بيقدي ان X أصغر من أو بساوي او هذا 
+
+330
+00:41:11,840 --> 00:41:12,640
+بيقدي ان
+
+331
+00:41:25,690 --> 00:41:38,530
+هذا بيقدي أن x أصغر من أو يساوي S لكل x في S and
+
+332
+00:41:38,530 --> 00:41:43,990
+x أصغر من أو يساوي لأ
+
+333
+00:41:46,040 --> 00:41:53,780
+عفوا إيش هذا؟ X أصغر من أو ساوي V لكل X في S and U
+
+334
+00:41:53,780 --> 00:41:57,120
+أصغر من أو ساوي V صح؟
+
+335
+00:42:02,420 --> 00:42:05,840
+طيب، معناته هذا upper bound، الـ V upper bound للـ set
+
+336
+00:42:05,840 --> 00:42:13,880
+S إذن الـ supremum للـ set S اللي هو S star بطلع أصغر
+
+337
+00:42:13,880 --> 00:42:22,600
+من أو ساوى V and U أصغر من أو ساوى V معناته إن الـ
+
+338
+00:42:22,600 --> 00:42:30,660
+V is upper bound Upper bound لمين؟ للـ set
+
+339
+00:42:33,070 --> 00:42:39,670
+اللي هي S star و U صح؟ لأن هاي V أكبر من أو يساوي
+
+340
+00:42:39,670 --> 00:42:48,670
+S star و أكبر من أو يساوي الـ U فهذا
+
+341
+00:42:48,670 --> 00:42:55,990
+بيقدي إذا الـ supremum إذا كان الـ V upper bound للـ
+
+342
+00:42:55,990 --> 00:43:10,590
+S هذه فالـ supremum للـ set هذي اللي هي S star و U أصغر
+
+343
+00:43:10,590 --> 00:43:17,270
+من أو ساوي الـ V هذا أكبر upper bound للـ set وهذا
+
+344
+00:43:17,270 --> 00:43:21,490
+upper bound لنفس الـ set لأن أصغر upper bound أصغر من
+
+345
+00:43:21,490 --> 00:43:23,050
+أو ساوي أي upper bound
+
+346
+00:43:26,490 --> 00:43:33,690
+وبالتالي هين أثبتنا .. هين أثبتنا أنه الـ .. العدد
+
+347
+00:43:33,690 --> 00:43:40,890
+هذا .. العدد هذا .. هذا العدد أثبتنا حاجتين هذا
+
+348
+00:43:40,890 --> 00:43:46,470
+العدد هيه upper bound لمين للـ S هذه كذلك في الـ
+
+349
+00:43:46,470 --> 00:43:51,410
+claim هذا أثبتنا أنه لو أخدت أي upper bound للـ S
+
+350
+00:43:51,410 --> 00:43:57,370
+هذه وسميته V فهذا العدد أصغر من أو ساوى V، إذن
+
+351
+00:43:57,370 --> 00:44:04,550
+العدد هذا هو أصغر، إذن العدد هذا هو الـ supremum لـ set
+
+352
+00:44:04,550 --> 00:44:10,750
+هذه، إذن هذا this proves
+
+353
+00:44:10,750 --> 00:44:14,110
+the
+
+354
+00:44:14,110 --> 00:44:21,070
+claim الادعاء اللي احنا حكينا عنه وبالتالي هذا
+
+355
+00:44:21,070 --> 00:44:27,310
+بيكون برهان ثاني أو برهان آخر وزي ما زميلتكم اقترحت
+
+356
+00:44:27,310 --> 00:44:33,670
+مافيش داعي للـ cases هنا البرهان الثاني يبدأ بـ X
+
+357
+00:44:33,670 --> 00:44:43,180
+تنتمي للـ set هذه وهنا أثبتنا ان العدد هذا هو الـ
+
+358
+00:44:43,180 --> 00:44:48,440
+supremum للـ set هذه أو الـ supremum للـ set هذه اللي هي
+
+359
+00:44:48,440 --> 00:44:52,400
+S اتحاد single to new الـ supremum إليها exist
+
+360
+00:44:52,400 --> 00:45:00,900
+موجود و بيساوي العدد supremum S star و U هو هذا
+
+361
+00:45:00,900 --> 00:45:05,240
+العدد upper bound للـ set هذه و أي upper bound آخر 
+
+362
+00:45:05,240 --> 00:45:10,340
+للـ set طلع أصغر من .. أكبر من أو يساوي العدد هذا
+
+363
+00:45:10,340 --> 00:45:13,520
+وبالتالي هذا هو أصغر upper bound أو super bound
+
+364
+00:45:13,520 --> 00:45:19,780
+نعم هذي؟
+
+365
+00:45:19,780 --> 00:45:23,180
+اه
+
+366
+00:45:23,180 --> 00:45:24,260
+صح
+
+367
+00:45:32,010 --> 00:45:38,490
+عن؟ بينهم or مش end لأ من تعريف .. من تعريف
+
+368
+00:45:38,490 --> 00:45:43,710
+الاتحاد x ينتمي للاتحاد معناته x ينتمي للـ .. أو ..
+
+369
+00:45:43,710 --> 00:45:47,130
+مش هيك تعريف الاتحاد؟ اه sorry اه ف or مافيش end
+
+370
+00:45:47,130 --> 00:45:51,330
+ليش الـ end؟ معرفة إنها or بس احنا استنتجنا .. يعني
+
+371
+00:45:51,330 --> 00:45:54,730
+هنا مكان الـ end استنتجنا إنها upper bound لكن هنا 
+
+372
+00:45:54,730 --> 00:45:57,490
+or يعني مش end عشان نستنتج إنها x lower bound
+
+373
+00:46:05,960 --> 00:46:10,580
+صحيح يعني لو كانت x أقل من أم يساوي أس أسطر and x 
+
+374
+00:46:10,580 --> 00:46:13,860
+أقل من أم يساوي u فإنت صحيح احنا نستنتج إنه x
+
+375
+00:46:13,860 --> 00:46:18,340
+lower bound للمجموعة أه صحيح كلامك إذا عشان هيك
+
+376
+00:46:18,340 --> 00:46:25,920
+احنا لازم نحدد هل الـ u هو بالتالي كان لازم عشان
+
+377
+00:46:25,920 --> 00:46:32,760
+البرهنة ده فعلا يكون صح كان لازم نفصل حالتين فلو
+
+378
+00:46:32,760 --> 00:46:41,400
+كانت هنا الـ u لو كانت الـ .. الـ S star أصغر من أو
+
+379
+00:46:41,400 --> 00:46:45,420
+يساوي الـ U دكتور؟
+
+380
+00:46:45,420 --> 00:46:51,540
+نعم مش X هي أصغر أو يساوي الـ supremum للـ S أو إن
+
+381
+00:46:51,540 --> 00:46:56,060
+الـ X أصغر أو يساوي مجموعة الـ U الحالة هي كأنا خبرت
+
+382
+00:46:56,060 --> 00:46:59,460
+إن الـ X هتكون أصغر أو يساوي الـ supremum يا إما
+
+383
+00:46:59,460 --> 00:47:06,300
+supremum للـ S أو supremum للـ مجموعة الـ U يعني المهم
+
+384
+00:47:06,300 --> 00:47:14,460
+هي هتطلع الـ Supremum لواحدة من المجموعتين أنا
+
+385
+00:47:14,460 --> 00:47:19,900
+قبل جملة الـ X أزيدور أنا قصدي إن أكثر X أصغر أو
+
+386
+00:47:19,900 --> 00:47:28,380
+بيساوي الـ Supremum يعني بشكل مجمعة واحدة X أصغر
+
+387
+00:47:28,380 --> 00:47:35,770
+أو بيساوي الـ Supremum لـ S star يعني هي اللي هولأ
+
+388
+00:47:35,770 --> 00:47:43,570
+هاد أبراهين S أنها أصغر أو نسبة مجموعة بستار كمه
+
+389
+00:47:43,570 --> 00:47:50,620
+قلو يعني لو حضرتيهم المهم هتطلع للـ super أه صح لأن
+
+390
+00:47:50,620 --> 00:47:56,760
+الـ suprem هذا أكبر من أو ساوي S star و أكبر من أو
+
+391
+00:47:56,760 --> 00:48:02,960
+ساوي الـ U و X أصغر من أو ساوي .. لو كانت الـ X أصغر
+
+392
+00:48:02,960 --> 00:48:05,980
+من أو ساوي هذا فهي أكيد أصغر من أو ساوي الـ suprem
+
+393
+00:48:05,980 --> 00:48:10,780
+و لو كانت الـ X أصغر من أو ساوي الـ U فهي أكيد أصغر
+
+394
+00:48:10,780 --> 00:48:12,900
+من أو ساوي الـ suprem
+
+395
+00:48:17,590 --> 00:48:26,170
+وبالتالي هذا معناه إنه الصحيح
+
+396
+00:48:26,170 --> 00:48:34,450
+ففي الحالة هذه إذا الـ supremum لـ set الـ star و you 
+
+397
+00:48:34,450 --> 00:48:41,610
+is upper bound upper bound للإتحاد
+
+398
+00:48:44,300 --> 00:48:54,800
+bound of S union single to new لأن
+
+399
+00:48:54,800 --> 00:49:03,260
+هذا fixed ماشي الحال فهذا بحل إشكالية و بعديها
+
+400
+00:49:03,260 --> 00:49:07,380
+بنشطب كل الكلام هذا لأ ما هو هذا الكلام يعني هو
+
+401
+00:49:07,380 --> 00:49:15,430
+تقريبا تفسير ل .. بما أن الـ ..هذا مالوش داعي صار
+
+402
+00:49:15,430 --> 00:49:23,350
+هذا مالوش داعي وهذه الخطوة بدل ما نكتبها هنا هذا
+
+403
+00:49:23,350 --> 00:49:27,430
+هي إذا مرة ثانية إن أيد البرهان الآن يعني البرهان 
+
+404
+00:49:27,430 --> 00:49:33,170
+مافي مشكلة إن شاء الله هاي بنثبت X في الاتحاد تبع
+
+405
+00:49:33,170 --> 00:49:38,990
+المجموعتين هذول الآن X تنتمي للـ set هذه أو تنتمي للـ set
+
+406
+00:49:38,990 --> 00:49:52,140
+هذه يعني بتساوي LU وبالتالي الـ X تنتمي لـ S فهي
+
+407
+00:49:52,140 --> 00:49:56,180
+أصغر من أو ساوي الـ supremum لـ S اللي هو S الصغير
+
+408
+00:49:57,460 --> 00:50:04,020
+أو X أصغر من أو يساوي الـ U، X بالساوي الـ U بتقدي ان
+
+409
+00:50:04,020 --> 00:50:08,900
+X أصغر من أو يساوي الـ U الآن لو أخدت الـ supremum لـ S
+
+410
+00:50:08,900 --> 00:50:12,920
+أصغر و U طبعا هذه finite set of real numbers وفي
+
+411
+00:50:12,920 --> 00:50:16,780
+تمرين بيقول لو عندي finite set of real numbers فالـ
+
+412
+00:50:16,780 --> 00:50:21,390
+suprem تبعها موجود و ينتمي للـ set و الـ infimum
+
+413
+00:50:21,390 --> 00:50:24,630
+تبعها أيضا موجود و ينتمي لـ .. يعني يكون عنصر في الـ
+
+414
+00:50:24,630 --> 00:50:28,530
+set هذا أحد التمارين اللي طبعا ما عليناهوش لكن
+
+415
+00:50:28,530 --> 00:50:34,090
+بإمكانكم تثبتوه by induction فهذه finally الـ set
+
+416
+00:50:34,090 --> 00:50:37,390
+إذا الـ supremum تبعها exist إلا أن هذا الـ supremum
+
+417
+00:50:37,390 --> 00:50:41,990
+أكبر من أو ساوي S star وبالتالي أكبر من أو ساوي X
+
+418
+00:50:41,990 --> 00:50:46,790
+و هذا الـ supremum أكبر من أو ساوي U
+
+419
+00:50:50,610 --> 00:50:55,450
+وبالتالي أكبر من أو يساوي الـ X اللي هي U أكبر من
+
+420
+00:50:55,450 --> 00:51:01,150
+أو ساوي، إذا الآن هذا الكلام صحيح لكل X ينتمي 
+
+421
+00:51:01,150 --> 00:51:09,230
+للإتحاد هذا العدد الآن أكبر من أو يساوي كل عناصر ال
+
+422
+00:51:09,230 --> 00:51:13,350
+6 في الاتحاد فهو upper bound للـ 6 هذه فهو upper bound
+
+423
+00:51:13,350 --> 00:51:18,770
+العدد هذا upper bound للـ 6 هذه الآن أثبتنا أن هذا
+
+424
+00:51:18,770 --> 00:51:23,380
+الـ upper bound هو أصغر upper bound للاتحاد وهو
+
+425
+00:51:23,380 --> 00:51:29,160
+أخذنا أي upper bound عشوائي للاتحاد طلع هذا ال
+
+426
+00:51:29,160 --> 00:51:33,140
+upper bound العشوائي أكبر من أو يساوي العدد هذا
+
+427
+00:51:33,140 --> 00:51:36,720
+الذي نريد هو الـ supremum إذا هذا العدد هو الـ
+
+428
+00:51:36,720 --> 00:51:42,940
+supremum للست هذه تمام؟ okay؟ في أي سؤال آخر؟
+
+429
+00:51:42,940 --> 00:51:51,480
+فلنحلّ كمان سؤالين في الـ .. نحلّ مثلا خليني
+
+430
+00:51:51,480 --> 00:51:54,300
+أنا اخترت لكم بعض الأسئلة مادام أنتم يعني شاكلّكم
+
+431
+00:51:54,300 --> 00:51:59,300
+إلا طبعا إذا أحد سأل خليني أمسح اللوح الأول ونحلّ
+
+432
+00:51:59,300 --> 00:52:00,240
+كمان سؤالين
+
+433
+00:52:16,370 --> 00:52:21,990
+يعني قبل قليل ذكرنا التمرين
+
+434
+00:52:21,990 --> 00:52:34,770
+هذا التمرين 12 section 2 3 وهذا التمرين يقول let
+
+435
+00:52:34,770 --> 00:52:51,380
+S بيـ .. let S يساوي X1 إلى XN be any non
+
+436
+00:52:51,380 --> 00:52:58,260
+-empty finite finite
+
+437
+00:52:58,260 --> 00:53:12,080
+set أو subset من R فنثبت
+
+438
+00:53:12,080 --> 00:53:14,920
+أن الـ show
+
+439
+00:53:17,460 --> 00:53:34,980
+infimum from S و supremum S ينتمي لـ S وكذلك
+
+440
+00:53:34,980 --> 00:53:41,720
+الـ supremum لـ 6S موجود وهو عنصر في 6S
+
+441
+00:53:52,980 --> 00:53:59,400
+Okay إذا الـ finite set تبعتي هذه فرضنا أن عناصرها
+
+442
+00:53:59,400 --> 00:54:06,300
+سمينا عناصرها x1, x2 إلى xn لأن هذه set فيها n
+
+443
+00:54:06,300 --> 00:54:18,540
+elements طيب ممكن نرتب العناصر هذه by rearranging
+
+444
+00:54:18,540 --> 00:54:23,200
+indices
+
+445
+00:54:23,200 --> 00:54:27,220
+if
+
+446
+00:54:27,220 --> 00:54:36,520
+necessary إذا كان ضروري we
+
+447
+00:54:36,520 --> 00:54:50,310
+may and dowe may and do assume that
+
+448
+00:54:50,310 --> 00:54:53,890
+x1
+
+449
+00:54:53,890 --> 00:55:04,950
+less than x2 less than less than xn أنا
+
+450
+00:55:04,950 --> 00:55:13,580
+عندي finite set call it x1 إلى xn ممكن أن أعيد
+
+451
+00:55:13,580 --> 00:55:20,620
+ترتيب العناصر هذه هي طبعا أعداد حقيقية فممكن أن
+
+452
+00:55:20,620 --> 00:55:26,880
+أعيد .. وطبعا كلهم عناصر غير متساوية فممكن
+
+453
+00:55:26,880 --> 00:55:32,200
+أعيد ترتيب أو تسمية العناصر هذه المؤشرات تبعات هذه
+
+454
+00:55:32,200 --> 00:55:38,680
+ممكن أعيد ترتيبها بحيث أن يطلع x1 أصغر من x2 أصغر
+
+455
+00:55:38,680 --> 00:55:44,920
+من x3 أو هكذا الأكثر هذا ممكن نعمله ولا لا؟ ممكن
+
+456
+00:55:44,920 --> 00:55:48,380
+الآن
+
+457
+00:55:48,380 --> 00:55:54,640
+تعالوا نثبت claim
+
+458
+00:55:54,640 --> 00:56:01,120
+أنا أُدّعي أن الـ minimum للـ set S سيطلع يساوي X
+
+459
+00:56:01,120 --> 00:56:08,200
+واحد وهذا ينتمي لـ S يعني بعد ما رتبت العناصر عملت
+
+460
+00:56:08,200 --> 00:56:12,740
+ordering لهم بالطريقة دي فحسبت أن الـ infimum plus
+
+461
+00:56:12,740 --> 00:56:18,820
+set S يساوي أصغر عنصر في الـ set الذي هو X1 وهذا
+
+462
+00:56:18,820 --> 00:56:29,620
+طبعا ينتمي إلى S طيب لإثبات ذلك clearly واضح
+
+463
+00:56:29,620 --> 00:56:40,900
+أن X1 is a lower bound lower bound لـ set S نظراً لأن
+
+464
+00:56:40,900 --> 00:56:45,740
+X1 أصغر من أو يساوي كل العناصر التي في الـ set فهو
+
+465
+00:56:45,740 --> 00:56:51,000
+واضح أنه lower bound الآن أنا أُثبت أنه ليس فقط
+
+466
+00:56:51,000 --> 00:56:54,400
+lower bound هو الـ infimum هو الـ greatest lower
+
+467
+00:56:54,400 --> 00:57:01,620
+bound إذا هنا now if W is
+
+468
+00:57:04,400 --> 00:57:16,580
+any lower bound .. any lower bound of S فهذا
+
+469
+00:57:16,580 --> 00:57:25,780
+معناه أن W أصغر من أو يساوي Xi لكل I يساوي 1 2
+
+470
+00:57:25,780 --> 00:57:29,640
+إلى N صح؟
+
+471
+00:57:30,510 --> 00:57:38,370
+وأصغر من أو يساوي كل عناصرها وبالتالي therefore w
+
+472
+00:57:38,370 --> 00:57:44,970
+أصغر من أو يساوي x واحد لأن x واحد هو أحد عناصر
+
+473
+00:57:44,970 --> 00:57:54,350
+الـ set إذا أنا عندي الآن x واحد is lower bound للـ set و
+
+474
+00:57:54,350 --> 00:58:00,190
+أي lower bound للـ set يطلع أصغر من أو يساوي x واحد
+
+475
+00:58:00,190 --> 00:58:08,770
+إذا by definition الـ x واحد آه أو الـ infimum للـ set
+
+476
+00:58:08,770 --> 00:58:16,330
+s exist and يساوي x واحد تمام؟
+
+477
+00:58:16,330 --> 00:58:22,610
+بالمثل ممكن نثبت الـ .. آه هنا similarly
+
+478
+00:58:26,410 --> 00:58:33,190
+similarly show that أن أنا سأترككم بطريقة مشابهة
+
+479
+00:58:34,440 --> 00:58:39,920
+تثبتوا الـ claim الثاني وهو أن الـ supremum للـ set S
+
+480
+00:58:39,920 --> 00:58:47,620
+exist و يساوي XN وطبعا هذا ينتمي للـ set S وهو
+
+481
+00:58:47,620 --> 00:58:52,040
+المطلوب okay تمام إن هيك نكون أثبتنا أن أي finite
+
+482
+00:58:52,040 --> 00:58:56,920
+set لها supremum لها infimum وهذان يطلعان عناصر
+
+483
+00:58:56,920 --> 00:59:01,960
+فيها بالتحديد الـ infimum هو الـ least element أصغر
+
+484
+00:59:01,960 --> 00:59:07,600
+عنصر في الـ set والـ supremum هو الـ greatest element
+
+485
+00:59:07,600 --> 00:59:12,480
+الذي هو أكبر عنصر في الـ set هذا طبعا الكلام غير
+
+486
+00:59:12,480 --> 00:59:16,360
+صحيح إذا الـ set S كانت infinite هذا فقط صحيح في
+
+487
+00:59:16,360 --> 00:59:22,600
+حالة الـ finite set إذا الـ .. هذا يكون يكمل برهان
+
+488
+00:59:22,600 --> 00:59:30,220
+التمرين هذا وبالتالي نكتفي بحل أو بهذا القدر من
+
+489
+00:59:30,220 --> 00:59:34,260
+حل التمرين وإن شاء الله أسبوع القادم نكمل حلّ
+
+490
+00:59:34,260 --> 00:59:35,400
+تمارين أخرى 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..15583f134bed5c8a17e38b71103959e566c7d548
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_postprocess.srt
@@ -0,0 +1,1960 @@
+1
+00:00:21,630 --> 00:00:28,730
+Okay ان شاء الله اليوم هنعمل مناقشة لبعض المسائل
+
+2
+00:00:28,730 --> 00:00:34,230
+في section اتنين تلاتة و اتنين اربعة زي ما وعدناكم
+
+3
+00:00:34,230 --> 00:00:44,690
+سابقا و نشوف بعض الحلول لبعض المسائل المهمة ففي
+
+4
+00:00:44,690 --> 00:00:52,530
+بسألةسؤال خامسة في section اتنين تلاتة بيقول لو في
+
+5
+00:00:52,530 --> 00:00:57,270
+عندي مجموعة غير خالية من الأعداد الحقيقية و
+
+6
+00:00:57,270 --> 00:01:03,550
+bounded below فال infimum ل ال set S هو سالب ال
+
+7
+00:01:03,550 --> 00:01:09,110
+supremum ل سالب S هذا
+
+8
+00:01:09,110 --> 00:01:13,870
+التمرين حالة خاصة من التمرين رقم أربعة في section
+
+9
+00:01:13,870 --> 00:01:21,120
+اتنين أربعةو بالتحديد هو حالة خاصة من الجزء بي من
+
+10
+00:01:21,120 --> 00:01:26,980
+التمرين هذا ففي الجزء بي لو كان بي .. إيش بقول هذا
+
+11
+00:01:26,980 --> 00:01:34,160
+الجزء؟ لو كان بي عدد سالب ف infimum بي S بساوي بي
+
+12
+00:01:34,160 --> 00:01:42,620
+في suprem S فلو أخدت بي بساوي سالب واحد و هذا عدد
+
+13
+00:01:42,620 --> 00:01:50,580
+سالبفبطل عندى infimum infimum
+
+14
+00:01:50,580 --> 00:01:58,780
+سالب s لأ هذا عبارة عن حالة خاصة من الجزء التانى
+
+15
+00:01:58,780 --> 00:02:05,560
+لو أخدنا بيه بساوي سالب واحد في الجزء هذا اللى هنا
+
+16
+00:02:07,490 --> 00:02:14,150
+فبطلع عندي supremum سالب S بيساوي
+
+17
+00:02:14,150 --> 00:02:19,390
+سالب infimum S هاي سالب اضربك سالب واحد سالب
+
+18
+00:02:19,390 --> 00:02:23,390
+infimum S لأن هذا التمرين حالة خاصة من الجزء هذا
+
+19
+00:02:23,390 --> 00:02:30,450
+التاني في الفرع B وبالتالي هذا التمرين تعميم لهذا
+
+20
+00:02:30,450 --> 00:02:37,140
+الجزء ولا جزء تانيو لجزء تاني اللي هو عبارة عن ال
+
+21
+00:02:37,140 --> 00:02:47,140
+supremum او ال infimum ل سالب S بساوي سالب ال
+
+22
+00:02:47,140 --> 00:02:54,240
+supremum ل S هذا تعميم لجزء اللي هان وهذا تعميم
+
+23
+00:02:54,240 --> 00:03:00,920
+لجزء اللي هان و ذلك باخذ by taking B equals سالب
+
+24
+00:03:00,920 --> 00:03:12,510
+واحدخلّينا نبرهن الجزء الأول من الفرع A و الجزء
+
+25
+00:03:12,510 --> 00:03:17,190
+الأول من الفرع B و بالمثل بإمكانكم تبرهن الجزء
+
+26
+00:03:17,190 --> 00:03:22,510
+التاني من ال part A و الجزء التاني من part B
+
+27
+00:03:22,510 --> 00:03:30,890
+فنبرهن الجزء A لبرهان الجزء A اللي
+
+28
+00:03:30,890 --> 00:03:37,490
+هو هذا الجزءفانا عندي a عدد موجب S is bounded
+
+29
+00:03:37,490 --> 00:03:42,390
+وبالتالي bounded below إذا ال info ل S exist سميه
+
+30
+00:03:42,390 --> 00:03:47,110
+W طبعا ال info عبارة عن lower bound ل 6S إذا ال W
+
+31
+00:03:47,110 --> 00:03:53,030
+أزرر من أو ساوي X لكل X S وبالتالي لو ضربت في عدد
+
+32
+00:03:53,030 --> 00:03:57,510
+موجب A فبطلع AW أزرر من أو ساوي A X لكل S هذا
+
+33
+00:03:57,510 --> 00:04:05,230
+معناه إن العدد هذا lower bound ل 6ASانا عايز اثبت
+
+34
+00:04:05,230 --> 00:04:10,670
+ان اي w هذا العدد مش بس lower bound هو اكبر lower
+
+35
+00:04:10,670 --> 00:04:19,690
+bound للست AS فباخد اي let V be any lower bound
+
+36
+00:04:19,690 --> 00:04:27,790
+any lower bound للست AS وبينا
+
+37
+00:04:27,790 --> 00:04:32,710
+نثبت ان هذا ال V أصغر من أو ساوي AW عشان يكون هو
+
+38
+00:04:32,710 --> 00:04:33,390
+ال infimum
+
+39
+00:04:35,910 --> 00:04:43,990
+طيب هذا معناه V lower bound للست AS معناه V أصغر
+
+40
+00:04:43,990 --> 00:04:52,010
+من أوي ساوي A X لكل X في S طيب أنا عندي واحد على A
+
+41
+00:04:52,010 --> 00:04:57,330
+عدد موجب إذا واحد على A عدد موجب فلو ضربت المتباني
+
+42
+00:04:57,330 --> 00:05:00,270
+هذه في العدد الموجب واحد على A اشتريت هنا
+
+43
+00:05:00,270 --> 00:05:07,900
+مابتتغيرش فبصير عندي V على Aأصغر من أو ساوي X لكل
+
+44
+00:05:07,900 --> 00:05:12,300
+XS طب
+
+45
+00:05:12,300 --> 00:05:20,540
+ما هذا معناه أنه العدد ال number V over A is a
+
+46
+00:05:20,540 --> 00:05:25,840
+lower bound لمن؟
+
+47
+00:05:25,840 --> 00:05:30,580
+لل 6S وبالتالي
+
+48
+00:05:30,580 --> 00:05:38,490
+إذا ال infimum .. إذا ال V على Aأصغر من أو ساوي ال
+
+49
+00:05:38,490 --> 00:05:48,090
+infimum للست S صح؟ طب اضربي في A عدد موجب بطلع
+
+50
+00:05:48,090 --> 00:05:58,990
+عندي V أصغر من أو ساوي A في infimum S طب
+
+51
+00:05:58,990 --> 00:06:07,730
+infimum S هذا سمنها W لأن هذا بساوي AWإذن هين
+
+52
+00:06:07,730 --> 00:06:13,790
+أثبتنا إنه العدد AW هذا أبارع ال lower bound للست
+
+53
+00:06:13,790 --> 00:06:20,390
+AS واخدنا أي lower bound للست AS فوجدنا إن ال
+
+54
+00:06:20,390 --> 00:06:27,770
+lower bound هذا أصغر من أو ساوي A في W فهذا معناه
+
+55
+00:06:27,770 --> 00:06:37,630
+إن AW هو ال infimum لمن؟ للست AS كما هوموضح في الـ
+
+56
+00:06:37,630 --> 00:06:44,290
+claim أو في الإدعاء تمام؟ وهذا بثبت الجزء الأول في
+
+57
+00:06:44,290 --> 00:06:51,650
+ال part A هاي infimum AS بساوي A في W اللي هو
+
+58
+00:06:51,650 --> 00:06:58,250
+infimum S إذن هذا بثبت الجزء الأول في الفرع A
+
+59
+00:06:58,250 --> 00:07:01,850
+Similarly بالمثل ممكن
+
+60
+00:07:05,820 --> 00:07:12,760
+بالمثل ممكن نثبت الفرع التاني او
+
+61
+00:07:12,760 --> 00:07:20,060
+الجزء التاني في الفرع A تمام؟ فهسيب هذا جزء لكم
+
+62
+00:07:20,060 --> 00:07:27,840
+لأن هذا مشابه الفرع اللي انا واضح؟ في اي سؤال؟ طيب
+
+63
+00:07:27,840 --> 00:07:30,780
+نحاول نثبت الجزء الأول في الفرع B
+
+64
+00:07:35,110 --> 00:07:42,150
+بنثبت الجزء هذا في الفرق دي لت
+
+65
+00:07:42,150 --> 00:07:53,770
+بأصغر من سفر، عدد حقيقي سالب وأنا عندي ال set ال
+
+66
+00:07:53,770 --> 00:07:58,230
+set since ال set S is bounded
+
+67
+00:08:01,660 --> 00:08:10,440
+إذا الـ infimum w بساوي ال infimum ل S exists in R
+
+68
+00:08:10,440 --> 00:08:13,460
+إذا
+
+69
+00:08:13,460 --> 00:08:18,240
+في عندي أنا ال .. ال infimum ل 6S .. 6S bounded
+
+70
+00:08:18,240 --> 00:08:21,180
+below bounded وبالتالي bounded below إذا by
+
+71
+00:08:21,180 --> 00:08:26,460
+infimum property ال infimum ل S مي W exist
+
+72
+00:08:30,860 --> 00:08:41,580
+هذا معناه .. او هذا بقد .. اذا
+
+73
+00:08:41,580 --> 00:08:46,180
+هذا معناه ان w lower bound ل S و W أصغر من أو ساوي
+
+74
+00:08:46,180 --> 00:08:49,880
+X لكل X في S
+
+75
+00:08:53,000 --> 00:08:58,980
+طيب و أندي أنا ال B عدد سالب فلو ضربنا المتباينة
+
+76
+00:08:58,980 --> 00:09:06,840
+هذه في B عدد سالب فبصير BX أصغر من أو ساوي BW لكل
+
+77
+00:09:06,840 --> 00:09:18,890
+XS صح؟ إذن هذا معناهإنه العدد بي دابليو is an
+
+78
+00:09:18,890 --> 00:09:28,750
+upper is an upper bound لمين للست بي في اس للست بي
+
+79
+00:09:28,750 --> 00:09:33,930
+في اس اللي هي مجموعة كل العناصر بي ضرب اكس بي ضرب
+
+80
+00:09:33,930 --> 00:09:38,570
+اكس حيث اكس ينتمي الاس هذا عبارة عن upper bound
+
+81
+00:09:38,570 --> 00:09:46,570
+طيب الست هذي الست هذي boundedلأن ال set S bounded
+
+82
+00:09:46,570 --> 00:09:51,270
+فضربها تعدد بتظلها bounded وبالتالي bounded above
+
+83
+00:09:51,270 --> 00:09:57,250
+إذا ال .. ال .. إلها superman by superman property
+
+84
+00:09:57,250 --> 00:10:08,990
+ودلتالي إذا ال BW هذا أو ال supermanللست BS هذا
+
+85
+00:10:08,990 --> 00:10:14,330
+عبارة عن ال least upper bound for the set BS هذا
+
+86
+00:10:14,330 --> 00:10:20,270
+بيطلع أصغر من أو ساوي أي upper bound و ليه هو أصغر
+
+87
+00:10:20,270 --> 00:10:28,150
+من أو ساوي ال upper bound BW للست BS طب
+
+88
+00:10:28,150 --> 00:10:29,610
+احنا عايزين نثبت
+
+89
+00:10:32,240 --> 00:10:38,840
+احنا عايزين نثبت ان بي دابليو هي ال supreme لست بي
+
+90
+00:10:38,840 --> 00:10:42,460
+في اس فهين
+
+91
+00:10:42,460 --> 00:10:47,020
+اثبتنا ان العدد بي دابليو هذا upper bound للست هذي
+
+92
+00:10:47,020 --> 00:10:51,240
+بي دابليو هو upper bound للست الاثبات ان هو ال
+
+93
+00:10:51,240 --> 00:10:55,240
+supreme باقي اثبات ان انا لو اخدت اي upper bound
+
+94
+00:10:55,240 --> 00:11:00,400
+للست هذه لازم يطلع اكبر من او يساوي بي دابليو
+
+95
+00:11:04,070 --> 00:11:11,310
+any upper bound
+
+96
+00:11:11,310 --> 00:11:18,490
+of except bs هذا
+
+97
+00:11:18,490 --> 00:11:28,090
+معناه أن b في x أصغر من أوي سوى b لكل xs تمام؟
+
+98
+00:11:29,920 --> 00:11:34,420
+طيب انا عندي بي عدل سالب اذا واحد على بي ايضا عدل
+
+99
+00:11:34,420 --> 00:11:38,960
+سالب فلو ضربت المتباينة هذه في عدل سالب اللي هو
+
+100
+00:11:38,960 --> 00:11:50,040
+واحد على بي فهيطلع عندي بي .. بي على بي أصغر من أو
+
+101
+00:11:50,040 --> 00:11:52,340
+ساوي X لكل X في S
+
+102
+00:11:55,350 --> 00:12:04,150
+هذا معناه ان العدد V على B is a lower bound لمن؟
+
+103
+00:12:04,150 --> 00:12:11,510
+لست S مصبوط صح؟ وبالتالي
+
+104
+00:12:11,510 --> 00:12:17,930
+اذا .. اذا
+
+105
+00:12:17,930 --> 00:12:23,970
+ال V على Bاللي هو lower bound للست S أصغر من أو
+
+106
+00:12:23,970 --> 00:12:28,370
+ساوي ال infimum للست S
+
+107
+00:12:54,340 --> 00:13:06,560
+احنا ايش قاعدين نثبت ال ..
+
+108
+00:13:06,560 --> 00:13:12,960
+يبدو ان انا يعني هنا بثبت الجزء التاني يعنى، يالا
+
+109
+00:13:12,960 --> 00:13:22,410
+من حظكمحاول نثبت الجزء التاني مش الأول فكمان مرة
+
+110
+00:13:22,410 --> 00:13:26,810
+نراجع بي عدد سالم S is bounded وبالتالي bounded
+
+111
+00:13:26,810 --> 00:13:33,650
+below إذن ال inform ل set S موجود وبالتالي
+
+112
+00:13:33,650 --> 00:13:37,630
+المتابعين هذا بتتحقق وبالتالي هذا بتتحقق بعد ما
+
+113
+00:13:37,630 --> 00:13:42,070
+ضربنا في بي عدد سالم إذن بي و طلع upper bound ل
+
+114
+00:13:42,070 --> 00:13:48,410
+set بي S وبالتالي ال supermanللست بي اس بيطلع أصغر
+
+115
+00:13:48,410 --> 00:13:52,510
+من أو ساوي بي دابليو الان بدنا نثبت ان ال بي
+
+116
+00:13:52,510 --> 00:14:00,810
+دابليو هذا هو ال supremum لست بي اس تمام فأخدنا اي
+
+117
+00:14:00,810 --> 00:14:05,550
+upper bound بي .. اي upper bound لست بي اس فوجدنا
+
+118
+00:14:05,550 --> 00:14:09,930
+ان v على بي is a lower bound لست اس وبالتالي v على
+
+119
+00:14:09,930 --> 00:14:14,290
+بي أصغر من أو ساوي ال greatest lower bound لست اس
+
+120
+00:14:17,060 --> 00:14:27,860
+طب لو ضربنا في بي و بي عدد سالب فهيطلع عندي .. إذا
+
+121
+00:14:27,860 --> 00:14:34,940
+لو ضربنا المتباينة هذه في بي عدد سالب فهيطلع عندي
+
+122
+00:14:34,940 --> 00:14:43,120
+اللي هو بي في infimum S هيطلع أصغر من أو ساوي ال
+
+123
+00:14:43,120 --> 00:14:45,120
+V، مظبوط هيك؟
+
+124
+00:14:48,920 --> 00:14:56,120
+طب هذا هذا سمنها w إذا بي في w أصغر من أو ساوي ال
+
+125
+00:14:56,120 --> 00:15:02,100
+b إذا البرهان هذا أثبتنا فيه حاجتين إنه أول شيء
+
+126
+00:15:02,100 --> 00:15:07,540
+العدد بي دابليو هذا upper bound للست بي اس و بعدين
+
+127
+00:15:07,540 --> 00:15:14,350
+أخدنا أي upper boundV أي upper bound لست بي اس طلع
+
+128
+00:15:14,350 --> 00:15:19,910
+ال V هذا أكبر من أو ساوي بي دابليو وبالتالي هذا
+
+129
+00:15:19,910 --> 00:15:29,650
+معناه إذا العدد بي دابليو هو عبارة عن ال supremum
+
+130
+00:15:29,650 --> 00:15:40,970
+ال supremum لست بي في اس لست بي في اسلأن هذا العدد
+
+131
+00:15:40,970 --> 00:15:45,570
+upper bound للست هذه وهو أصغر upper bound أخدنا أي
+
+132
+00:15:45,570 --> 00:15:51,390
+upper bound للست هذه طلع بي دابليو أصغر من أو ساوي
+
+133
+00:15:51,390 --> 00:15:56,050
+إذن بي دابليو هو أصغر upper bound للست هذه والأن
+
+134
+00:15:56,050 --> 00:16:03,410
+بنعود عن w إذن ال b في w اللي هو infimum of s
+
+135
+00:16:03,410 --> 00:16:12,590
+بتطلع بساوي supremum ل b في sوهذا برهين الجزء
+
+136
+00:16:12,590 --> 00:16:18,330
+التاني من الفرع B بالمثل الممكن برهان الجزء الأول
+
+137
+00:16:18,330 --> 00:16:24,850
+من الفرع B فأنا بدأكم إلى كتابة برهين الأجزاء
+
+138
+00:16:24,850 --> 00:16:30,330
+المشابهة هذه تمام؟ إذن هيك بنكون .. يعني أخدنا
+
+139
+00:16:30,330 --> 00:16:37,150
+حلول تقريبا شبه كاملة للتمرين 5 section 2 تلاتةفي
+
+140
+00:16:37,150 --> 00:16:41,530
+عندكم أي أسئلة تانية في ال section اتنين تلاتة او
+
+141
+00:16:41,530 --> 00:16:48,470
+اتنين اربعة؟ في
+
+142
+00:16:48,470 --> 00:16:54,190
+أي أسئلة تانية؟ السؤال عشرة في section اتنين تلاتة
+
+143
+00:17:28,800 --> 00:17:38,060
+سؤال عشرة section اتنين تلاتة ملخص السؤال بيقول S
+
+144
+00:17:38,060 --> 00:17:52,000
+is bounded bounded subset of R و Phi
+
+145
+00:17:52,000 --> 00:17:55,460
+لا يساوي S subset
+
+146
+00:18:00,440 --> 00:18:07,020
+ف ال S0 non-empty subset من S مجموعة جزئية غير
+
+147
+00:18:07,020 --> 00:18:17,280
+خالية من المجموعة S فبدنا نثبت شو برهني ان ال
+
+148
+00:18:17,280 --> 00:18:26,260
+infimum لست S أصغر من أو ساوي ال infimum لست S0
+
+149
+00:18:26,260 --> 00:18:32,540
+أصغر من أو ساوي ال supremumللست S Zero أصغر من لو
+
+150
+00:18:32,540 --> 00:18:41,940
+يساوي ال supremum للست S نشوف
+
+151
+00:18:41,940 --> 00:18:46,860
+البرهان مع بعض برهان سهل وبسيط يعتمد على تعريف ال
+
+152
+00:18:46,860 --> 00:18:52,760
+infimum وعلى تعريف ال supremum طيب
+
+153
+00:18:52,760 --> 00:18:57,900
+أنا عندي المجموعة S since
+
+154
+00:19:00,710 --> 00:19:08,790
+بما أن S مجموعة غير خالية و bounded is bounded
+
+155
+00:19:08,790 --> 00:19:12,990
+then ال
+
+156
+00:19:12,990 --> 00:19:28,810
+infimum لست S exist and supremum لست S both exist
+
+157
+00:19:36,050 --> 00:19:44,310
+بعد الـ infimum property ست اس لإنفمام وكذلك ست اس
+
+158
+00:19:44,310 --> 00:19:52,290
+لسوبرمام هدول موجودين في R طيب
+
+159
+00:19:52,290 --> 00:19:56,150
+أنا عندي السوبرمام
+
+160
+00:19:56,150 --> 00:20:15,640
+للست اس السوبرمام للست اسis an upper bound فهي
+
+161
+00:20:15,640 --> 00:20:25,520
+أيضا it is also an upper bound لأي
+
+162
+00:20:25,520 --> 00:20:31,060
+subset لأي subset S0 من ال 6S
+
+163
+00:20:36,460 --> 00:20:44,900
+و بالتالي and therefore and
+
+164
+00:20:44,900 --> 00:20:52,600
+therefore ال
+
+165
+00:20:52,600 --> 00:20:57,540
+supremum لست S0
+
+166
+00:20:57,540 --> 00:21:01,840
+أصغر من أو ساوي ال supremum لست S
+
+167
+00:21:07,110 --> 00:21:15,710
+كمان مرة ال .. ال 6S هذه ال S0 سبسط من S فأي upper
+
+168
+00:21:15,710 --> 00:21:20,070
+bound ل S هو أيضا upper bound لأي مجموعة جزئية
+
+169
+00:21:20,070 --> 00:21:26,410
+منها طيب ال supremum ل 6S upper bound ل 6S
+
+170
+00:21:26,410 --> 00:21:32,830
+وبالتالي هو upper bound ل 6S0 طيب ال supremum ل S0
+
+171
+00:21:32,830 --> 00:21:39,130
+هذا أصغر upper bound ل S0وهذا upper bound ل S0 إذا
+
+172
+00:21:39,130 --> 00:21:42,550
+أصغر upper bound أصغر من لو ساوي أي upper bound
+
+173
+00:21:42,550 --> 00:21:51,650
+وبالتالي المتباينة هذه صحيحة كذلك by
+
+174
+00:21:51,650 --> 00:21:57,950
+definition حسب التعريفات ال
+
+175
+00:21:57,950 --> 00:22:06,790
+infimumللست S0 أصغر من أو ساوي ال supremum للست S0
+
+176
+00:22:06,790 --> 00:22:10,750
+الست
+
+177
+00:22:10,750 --> 00:22:11,750
+S0 هذه
+
+178
+00:22:15,230 --> 00:22:21,930
+طبعا هذه ال set S0 subset من S و S bounded إلى S0
+
+179
+00:22:21,930 --> 00:22:26,710
+bounded ال infimum ل S0 exist و ال suprem ل S0
+
+180
+00:22:26,710 --> 00:22:32,770
+exist دائما لأي set S0 ال infimum دائما أصغر من أو
+
+181
+00:22:32,770 --> 00:22:39,250
+يساوي ال supremum نعمل رسمة نوضح الكلام هذا
+
+182
+00:22:44,850 --> 00:22:56,850
+نعتبر أن هذه هي الست اس وهي
+
+183
+00:22:56,850 --> 00:23:07,950
+ال .. ال .. ال supremum للست اس وهي ال infimum
+
+184
+00:23:11,090 --> 00:23:17,810
+للـ set S فدائما ال .. دائما
+
+185
+00:23:17,810 --> 00:23:24,050
+ال minimum لأي set هو lower bound لل set وبالتالي
+
+186
+00:23:24,050 --> 00:23:28,950
+أصغر من لو ساوي كل عناصرهاهو عبارة عن lower bound
+
+187
+00:23:28,950 --> 00:23:32,810
+للست ال supreme للست S هو عبارة عن upper bound
+
+188
+00:23:32,810 --> 00:23:37,650
+للست وبالتالي أكبر من أو ساوي كل عناصرها فواضح أن
+
+189
+00:23:37,650 --> 00:23:42,770
+ال infimum للست S لازم يكون أصغر من أو ساوي ال
+
+190
+00:23:42,770 --> 00:23:52,970
+supremum ونفس الشيء لو أخذنا أي مجموعة جزئية سمنها
+
+191
+00:23:52,970 --> 00:23:53,790
+S0
+
+192
+00:23:56,180 --> 00:24:02,200
+يعني هذه المجموعة اسمها S0 فبما أن ال set S
+
+193
+00:24:02,200 --> 00:24:10,400
+bounded إذن S0 bounded وبالتالي ال supremum ل S0
+
+194
+00:24:10,400 --> 00:24:16,220
+دايما أكبر من أو ساوي ال infimum ل S0 بنفس الطريقة
+
+195
+00:24:16,220 --> 00:24:23,710
+إذن هذا دايما .. هذا دايما صحيحعشان احنا نكمل
+
+196
+00:24:23,710 --> 00:24:30,150
+البرهان اذا احنا أثبتنا هذا واضح من التعريفات وهذا
+
+197
+00:24:30,150 --> 00:24:35,150
+الجزء أثبتناه باقي
+
+198
+00:24:35,150 --> 00:24:40,930
+إثبات الجزء الأخير هذا فإذا
+
+199
+00:24:40,930 --> 00:24:45,790
+بنقول finally أخيرا لإثبات الجزء الأخير هذا أنا
+
+200
+00:24:45,790 --> 00:24:49,570
+عندي ال inform ل S is lower bound ل 6S
+
+201
+00:24:52,070 --> 00:24:57,350
+وبالتالي هو lower bound لأي مجموعة جزئية S0 من S
+
+202
+00:24:57,350 --> 00:25:00,890
+وبالتالي
+
+203
+00:25:00,890 --> 00:25:11,770
+إذا ال influence ل S0 هذا
+
+204
+00:25:11,770 --> 00:25:19,180
+أكبر lower bound ل S0 هذا أكبر lower bound ل S0و
+
+205
+00:25:19,180 --> 00:25:25,960
+هذا lower bound ل S0 إذاً هذا بيطلع أكبر من أو
+
+206
+00:25:25,960 --> 00:25:33,500
+ساوي infimum ال 6S هذا lower bound ل 6S0 و هذا
+
+207
+00:25:33,500 --> 00:25:37,820
+أكبر lower bound ل 6S0 إذاً هذا أصغر من أو ساوي
+
+208
+00:25:37,820 --> 00:25:43,700
+هذا و هذا بيكملبرهان المتباينة اللى حاطين عليها
+
+209
+00:25:43,700 --> 00:25:48,380
+علامة استفهام إذا هيك بيكون برهاننا التمرين okay
+
+210
+00:25:48,380 --> 00:25:53,660
+تمام واضح؟
+
+211
+00:25:53,660 --> 00:26:03,660
+فى أسئلة تانية خلنا نحل كمان سؤال إذا بتحبه ممكن
+
+212
+00:26:03,660 --> 00:26:04,900
+نحل كمان سؤال
+
+213
+00:26:08,660 --> 00:26:16,040
+في section اتنين تلاتة برضه؟ اه في اي section؟
+
+214
+00:26:16,040 --> 00:26:21,840
+اتنين تلاتة ولا اتنين اربعة؟ اتنين تلاتة؟ طيب نحل
+
+215
+00:26:21,840 --> 00:26:24,020
+هذا السؤال و بعد هيك يعني نوجد
+
+216
+00:26:43,630 --> 00:26:57,410
+هي السؤال الأحداش سيكشن اتنين تلاتة بنشوف
+
+217
+00:26:57,410 --> 00:27:05,850
+السؤال شو بيقول S
+
+218
+00:27:05,850 --> 00:27:11,530
+subset من R و
+
+219
+00:27:11,530 --> 00:27:25,720
+SS star بساوي ال supremum ل 6S وهذا بينتمي لل 6S
+
+220
+00:27:25,720 --> 00:27:31,040
+belongs to S فإذا
+
+221
+00:27:31,040 --> 00:27:41,140
+كان U لا ينتمي لل 6S إذا كان U لا ينتمي لل 6S شو
+
+222
+00:27:42,390 --> 00:27:49,090
+عايزين نثبت ان ال superman لست
+
+223
+00:27:49,090 --> 00:28:05,890
+S union singleton U بيطلع بيساوي ال superman لست
+
+224
+00:28:05,890 --> 00:28:10,330
+اللي تتكون من أنصرين S star و U
+
+225
+00:28:13,540 --> 00:28:28,400
+where are you؟ طبعا في برهانين للسؤال هذا ال
+
+226
+00:28:28,400 --> 00:28:33,840
+proof one البرهان الأول we
+
+227
+00:28:33,840 --> 00:28:38,580
+use .. we use exercise
+
+228
+00:28:42,560 --> 00:28:51,600
+تسعة section اتنين تلاتة وهذا ال exercise بيقول
+
+229
+00:28:51,600 --> 00:28:59,340
+إذا كانت لو
+
+230
+00:28:59,340 --> 00:29:03,380
+كان a و b bounded
+
+231
+00:29:09,480 --> 00:29:18,660
+فهذا بيقدي ان a union b is bounded and
+
+232
+00:29:18,660 --> 00:29:32,360
+مش هيكوا بس و ال supremum .. ال supremum لإتحاد b
+
+233
+00:29:32,360 --> 00:29:36,980
+بساوي supremum
+
+234
+00:29:39,920 --> 00:29:44,900
+Supermom A وSupermom
+
+235
+00:29:44,900 --> 00:29:51,760
+B إذا
+
+236
+00:29:51,760 --> 00:29:57,440
+هذا تمرين رقم تسعة هناخده نستخدمه فلو استخدمنا هذا
+
+237
+00:29:57,440 --> 00:30:07,700
+التمرين فالنتيجة هذه بتطلع على طول مباشرة إذا
+
+238
+00:30:07,700 --> 00:30:08,540
+هنا take
+
+239
+00:30:11,570 --> 00:30:17,410
+A بساوي S و
+
+240
+00:30:17,410 --> 00:30:25,570
+طبعا هادي ال set bounded ال set هادي bounded و
+
+241
+00:30:25,570 --> 00:30:32,610
+عندي ال set B هاخدها singleton euro و هادي bounded
+
+242
+00:30:32,610 --> 00:30:41,790
+setإذا by exercise 9 a hat b اللي هي ال 6 هذه
+
+243
+00:30:41,790 --> 00:30:47,650
+بتطلع bounded by
+
+244
+00:30:47,650 --> 00:30:56,490
+exercise 9 section 2 3 ال 6 a union singleton u is
+
+245
+00:30:56,490 --> 00:31:00,750
+bounded and
+
+246
+00:31:00,750 --> 00:31:10,540
+مش هيكوا بس ال supremumلـ A اتحاد بالـ 6S union
+
+247
+00:31:10,540 --> 00:31:18,160
+هذا الـ A وهذا الـ Singleton U بتساوي الـ Supremum
+
+248
+00:31:18,160 --> 00:31:22,440
+لـ
+
+249
+00:31:22,440 --> 00:31:32,820
+Supremum A هذا عبارة عن S star و Supremum D هذا
+
+250
+00:31:32,820 --> 00:31:37,830
+عبارة عن Singleton Uأنا عندي set فيها عنصر واحد
+
+251
+00:31:37,830 --> 00:31:42,510
+فال Supreme تبعها هو ال info تبعها هو نفس ال
+
+252
+00:31:42,510 --> 00:31:46,850
+answer يعني هذا واضح من تعريف ال suprem
+
+253
+00:31:54,620 --> 00:31:59,580
+و هذا هو المطلوب اذا هذا تطبيق مباشر على تمرين 9
+
+254
+00:31:59,580 --> 00:32:03,860
+اذا المعناه ان انتوا لازم تحلوا تمرين 9 و هذا
+
+255
+00:32:03,860 --> 00:32:11,260
+التمرين موجود في يعني في رشاد له او hint لحله في
+
+256
+00:32:11,260 --> 00:32:16,680
+خلف .. خلف الكتاب في حل تمرين اللي .. اللي الكتاب
+
+257
+00:32:16,680 --> 00:32:21,280
+بيحاول يعرضها عشان يساعد الطالب نعم تفضلي
+
+258
+00:32:28,890 --> 00:32:37,250
+أه صحيح نعم و
+
+259
+00:32:37,250 --> 00:32:45,170
+في السؤال تسعة و في السؤال إحداش ال 6S
+
+260
+00:32:45,170 --> 00:32:51,010
+من المقطيات bounded صحيح لإنهاحنا فرضين ان S
+
+261
+00:32:51,010 --> 00:32:56,370
+subset من R و ال supremum لل 6S اللي هو S star عدد
+
+262
+00:32:56,370 --> 00:33:06,050
+ينتمي ل S و S subset من R هذا بيقدي ان ال 6S is
+
+263
+00:33:06,050 --> 00:33:12,750
+bounded above على الأقل bounded above تمام؟
+
+264
+00:33:16,370 --> 00:33:22,230
+تمام؟ فلو كانت ال A و ال B bounded above فهيطلع
+
+265
+00:33:22,230 --> 00:33:25,510
+الاتحاد تبعهم bounded above و هذا اللي احنا
+
+266
+00:33:25,510 --> 00:33:30,490
+عايزينه و ال supremum اللي لهم بساوي .. لاتحادهم
+
+267
+00:33:30,490 --> 00:33:37,540
+بساوي الكلام هذا فعلى الأقل .. اه؟و نفس الكلام
+
+268
+00:33:37,540 --> 00:33:41,860
+للانفمام ممكن نثبت حاجة مشابه بالنسبة للانفمام
+
+269
+00:33:41,860 --> 00:33:47,140
+يعني ممكن نثبت ان الانفمام هنا يعني ها and ممكن
+
+270
+00:33:47,140 --> 00:33:58,820
+نضيف انفمام ل a union b بساوي انفمام انف a و انف b
+
+271
+00:34:01,670 --> 00:34:06,630
+فاحنا بس أخدنا .. طبخنا الجزء هذا الجزء بيكون صحيح
+
+272
+00:34:06,630 --> 00:34:13,390
+إذا كانت a و b both are bounded above وبالتالي
+
+273
+00:34:13,390 --> 00:34:16,430
+اتحادهم بيطلع bounded below و ال infimum للاتحاد
+
+274
+00:34:16,430 --> 00:34:23,780
+بيطلع infimum لinfimum المجمعة التانيةفهذا متحقق
+
+275
+00:34:23,780 --> 00:34:28,640
+هنا متحقق ان هاي S star ينتمي ل S وبالتالي عدد
+
+276
+00:34:28,640 --> 00:34:32,420
+حقيقي انها S ال set هذه لها supremum وبالتالي
+
+277
+00:34:32,420 --> 00:34:37,360
+bounded above و single to new ما هي finite set و
+
+278
+00:34:37,360 --> 00:34:41,960
+كل finite set is bounded فهي bounded above و below
+
+279
+00:34:41,960 --> 00:34:47,530
+طبعا وبالتالي ممكن نطبق الجزء هذاهذا برهان برهان
+
+280
+00:34:47,530 --> 00:34:51,790
+تاني ممكن ان احنا نعمل برهان مباشر يعني بلاش
+
+281
+00:34:51,790 --> 00:35:00,970
+نستخدم exercise تسعة تاني
+
+282
+00:35:00,970 --> 00:35:09,310
+ممكن we
+
+283
+00:35:09,310 --> 00:35:13,450
+consider we
+
+284
+00:35:13,450 --> 00:35:15,230
+consider two cases
+
+285
+00:35:18,470 --> 00:35:24,390
+نعتبر حالتين ال S star هذا من المعطيات عدد حقيقي و
+
+286
+00:35:24,390 --> 00:35:31,790
+U عدد حقيقي آخر لا ينتمي ل S فممكن يكون عندي ال U
+
+287
+00:35:31,790 --> 00:35:40,850
+أكبر من أو يساوي S star or ال U أصغر من S star هذا
+
+288
+00:35:40,850 --> 00:35:46,750
+طبعا by trichotomy by trichotomy
+
+289
+00:35:50,710 --> 00:35:58,670
+property من الخاصية الثلاثية U S*) أعداد حقيقية
+
+290
+00:35:58,670 --> 00:36:04,850
+ففي عندي تلت حالات أما U أصغر من S*) أو U أكبر من
+
+291
+00:36:04,850 --> 00:36:10,450
+S*) أو U بساوي S*) هدول حالتين وهذه التالتة
+
+292
+00:36:10,450 --> 00:36:15,950
+فتعالوا في كل حالة نثبت هذا اللي هو المطلوب فإذا
+
+293
+00:36:15,950 --> 00:36:22,180
+في عندي في الحالة الأولىX أقل أو بيساوي من السقر
+
+294
+00:36:22,180 --> 00:36:27,400
+الموجود في ال U أو
+
+295
+00:36:27,400 --> 00:36:33,000
+إيش التانية؟ أو X أقل أو بيساوي ال U X أصغر من أو
+
+296
+00:36:33,000 --> 00:36:38,280
+بيساوي ال U، صح؟ بعدها أنا هقول أكيد إن ال X أقل
+
+297
+00:36:38,280 --> 00:36:45,360
+أو بيساوي من ال .. إن ال X lower bound is lower
+
+298
+00:36:45,360 --> 00:36:45,960
+bound
+
+299
+00:36:49,050 --> 00:37:03,630
+لل set اللي بتتكون من S star و U صح؟ وبالتالي لحظة
+
+300
+00:37:03,630 --> 00:37:09,490
+شوية لو سمحتني اذا
+
+301
+00:37:09,490 --> 00:37:14,830
+ال X lower bound لل set هذي اذا ال infimum
+
+302
+00:37:22,180 --> 00:37:27,840
+الـ X أصغر
+
+303
+00:37:27,840 --> 00:37:36,400
+من أو ساوي الـ infimum ل Sلأ ما هو هذا lower bound
+
+304
+00:37:36,400 --> 00:37:41,960
+ل S star لسنا المجموعة هذه وبالتالي هو أصغر من أو
+
+305
+00:37:41,960 --> 00:37:45,700
+ساوي ال infimum و ال infimum دائما قولنا قبل شوية
+
+306
+00:37:45,700 --> 00:37:51,780
+أصغر من أو ساوي ال supremum لنفس المجموعة لسه
+
+307
+00:37:51,780 --> 00:37:58,160
+متبتيلوا قبل شوية في التمرين السابق صح؟ طيب هيك
+
+308
+00:37:58,160 --> 00:37:59,260
+منكون أثبتنا
+
+309
+00:38:06,750 --> 00:38:17,210
+إذا هذا صحيح since this holds لكل
+
+310
+00:38:17,210 --> 00:38:26,130
+x ينتمي احنا خدنا x عشوائية فهي fix x مظبوط؟ x
+
+311
+00:38:26,130 --> 00:38:33,700
+كانت عنصر عشوائي ف fix x ينتمي ل S unionSingleton
+
+312
+00:38:33,700 --> 00:38:39,260
+U فإذا هذه الأداء صحيح لكل X ينتمي للمجموعة هذه
+
+313
+00:38:39,260 --> 00:38:50,460
+وبالتالي إذا ال supreme ل S star و U is upper
+
+314
+00:38:50,460 --> 00:39:00,300
+bound Upper bound لمن؟ ل 6 S union singleton U
+
+315
+00:39:08,160 --> 00:39:23,180
+مظبوط؟ اذا ال supremum لست S union singleton U لأ
+
+316
+00:39:23,180 --> 00:39:28,280
+مش هيك لأ اذا هذا عبارة عن upper bound لست هذه
+
+317
+00:39:28,280 --> 00:39:34,830
+بنثبت ان هو ال supremumيعني هيك بيطلع هذا .. هذا
+
+318
+00:39:34,830 --> 00:39:40,610
+upper bound ل 6 هذه لأن هذا بيطلع أكبر من أو ساوي
+
+319
+00:39:40,610 --> 00:39:49,610
+.. هذا أصغر من أو ساوي ال supremum ل
+
+320
+00:39:49,610 --> 00:39:57,310
+S star و U احنا بدنا مساوية صح؟فبقدرش أستنتج
+
+321
+00:39:57,310 --> 00:40:03,070
+مساواة هنا تمام؟ أما شو ممكن أما زي ما عملنا في
+
+322
+00:40:03,070 --> 00:40:07,430
+البراهين السابقة ممكن نثبت ال claim ممكن نثبت
+
+323
+00:40:07,430 --> 00:40:13,070
+المساواة كما يليه أنا عندي هذا .. هذا العدد .. هذا
+
+324
+00:40:13,070 --> 00:40:19,270
+العدد عبارة عن upper bound لل set هذهأحنا عايزين
+
+325
+00:40:19,270 --> 00:40:22,970
+نثبت إن هذا مش upper bound هو ال least upper bound
+
+326
+00:40:22,970 --> 00:40:29,330
+إذا ن claim إن ال supremum
+
+327
+00:40:29,330 --> 00:40:36,590
+لست S union لست هذه هو العدد هذا
+
+328
+00:40:49,020 --> 00:41:02,440
+انشوف let V be any upper bound لست S union
+
+329
+00:41:02,440 --> 00:41:11,840
+singleton U هذا بيقدي ان X أصغر من أو بساوي او هذا
+
+330
+00:41:11,840 --> 00:41:12,640
+بيقدي ان
+
+331
+00:41:25,690 --> 00:41:38,530
+هذا بيقدي أن x أصغر من أو يساوي S لكل x في S and
+
+332
+00:41:38,530 --> 00:41:43,990
+x أصغر من أو يساوي لأ
+
+333
+00:41:46,040 --> 00:41:53,780
+عفوا إيش هذا؟ X أصغر من أو ساوي V لكل X في S and U
+
+334
+00:41:53,780 --> 00:41:57,120
+أصغر من أو ساوي V صح؟
+
+335
+00:42:02,420 --> 00:42:05,840
+طيب، معناته هذا upper bound، ال V upper bound للست
+
+336
+00:42:05,840 --> 00:42:13,880
+S إذن ال supremum للست S اللي هو S star بطلع أصغر
+
+337
+00:42:13,880 --> 00:42:22,600
+من أو ساوى V and U أصغر من أو ساوى V معناته إن ال
+
+338
+00:42:22,600 --> 00:42:30,660
+V is upper bound Upper bound لمين؟ للست
+
+339
+00:42:33,070 --> 00:42:39,670
+اللي هي S star و U صح؟ لأن هاي V أكبر من أو يساوي
+
+340
+00:42:39,670 --> 00:42:48,670
+S star و أكبر من أو يساوي ال U فهذا
+
+341
+00:42:48,670 --> 00:42:55,990
+بيقدي إذا ال supremum إذا كان ال V upper bound لل
+
+342
+00:42:55,990 --> 00:43:10,590
+6 هذه فال supremumللست هذي اللي هي S star و U أصغر
+
+343
+00:43:10,590 --> 00:43:17,270
+من أو ساوي ال V هذا أكبر upper bound للست وهذا
+
+344
+00:43:17,270 --> 00:43:21,490
+upper bound لنفس الست لأن أصغر upper bound أصغر من
+
+345
+00:43:21,490 --> 00:43:23,050
+أو ساوي أي upper bound
+
+346
+00:43:26,490 --> 00:43:33,690
+وبالتالي هين أثبتنا .. هين أثبتنا أنه ال .. العدد
+
+347
+00:43:33,690 --> 00:43:40,890
+هذا .. العدد هذا .. هذا العدد أثبتنا حاجتين هذا
+
+348
+00:43:40,890 --> 00:43:46,470
+العدد هيه upper bound لمين لل 6 هذه كذلك في ال
+
+349
+00:43:46,470 --> 00:43:51,410
+claim هذا أثبتنا أنه لو أخدت أي upper bound لل 6
+
+350
+00:43:51,410 --> 00:43:57,370
+هذه وسميته Vفهذا العدد أصغر من أو ساوى D، إذن
+
+351
+00:43:57,370 --> 00:44:04,550
+العدد هذا هو أصغر، إذن العدد هذا هو ال supreme لست
+
+352
+00:44:04,550 --> 00:44:10,750
+هذه، إذن هذا this proves
+
+353
+00:44:10,750 --> 00:44:14,110
+the
+
+354
+00:44:14,110 --> 00:44:21,070
+claim الادعاء اللي احنا حكينا عنه وبالتاليهذا
+
+355
+00:44:21,070 --> 00:44:27,310
+بيكون برهان تاني او برهان اخر وزي مزمرتكم اقترحت
+
+356
+00:44:27,310 --> 00:44:33,670
+مافيش داعي لل cases هنا البرهان التاني مبدأ ب X
+
+357
+00:44:33,670 --> 00:44:43,180
+تنتمي لل set هذه وهنا أثبتنا ان العدد هذاهو ال
+
+358
+00:44:43,180 --> 00:44:48,440
+supremum للست هذه او ال supremum للست هذه اللي هي
+
+359
+00:44:48,440 --> 00:44:52,400
+S إتحاد single to new ال supremum إليها exist
+
+360
+00:44:52,400 --> 00:45:00,900
+موجود و بساوي العدد supremum S star و Uهيو هذا
+
+361
+00:45:00,900 --> 00:45:05,240
+العدد upper bound للست هذه و أي upper bound أخر
+
+362
+00:45:05,240 --> 00:45:10,340
+للست طلع أصغر من .. أكبر من أو يساوي العدد هذا
+
+363
+00:45:10,340 --> 00:45:13,520
+وبالتالي هذا هو أصغر upper bound أو super bound
+
+364
+00:45:13,520 --> 00:45:19,780
+نعم هذي؟
+
+365
+00:45:19,780 --> 00:45:23,180
+اه
+
+366
+00:45:23,180 --> 00:45:24,260
+صح
+
+367
+00:45:32,010 --> 00:45:38,490
+عن؟ بينهم or مش end لأ من تعريف .. من تعريف
+
+368
+00:45:38,490 --> 00:45:43,710
+الاتحاد x ينتمي للاتحاد معناته x ينتمي لل .. او ..
+
+369
+00:45:43,710 --> 00:45:47,130
+مش هيك تعريف الاتحاد؟ اه sorry اه ف or مافيش end
+
+370
+00:45:47,130 --> 00:45:51,330
+ليش ال end؟ معرفة انها or بس احنا استنتجنا .. يعني
+
+371
+00:45:51,330 --> 00:45:54,730
+هنا مكان ال end استنتجنا انها upper bound لكن هنا
+
+372
+00:45:54,730 --> 00:45:57,490
+or يعني مش end عشان نستنتج انها x lower bound
+
+373
+00:46:05,960 --> 00:46:10,580
+صحيح يعني لو كانت x أقل من أم يساوي أس أسطر and x
+
+374
+00:46:10,580 --> 00:46:13,860
+أقل من أم يساوي u فإنت صحيح إحنا نستنتج إنه x
+
+375
+00:46:13,860 --> 00:46:18,340
+lower bound للمجموعة أه صحيح كلامك إذا عشان هيك
+
+376
+00:46:18,340 --> 00:46:25,920
+احنا لازم نحدد هل ال u هو بالتالي كان لازم عشان
+
+377
+00:46:25,920 --> 00:46:32,760
+البرهنة ده فعلا يكون صح كان لازم نفصل حالتين فلو
+
+378
+00:46:32,760 --> 00:46:41,400
+كانت هنا ال uلو كانت ال .. ال S star أصغر من أو
+
+379
+00:46:41,400 --> 00:46:45,420
+يساوي ال U دكتور؟
+
+380
+00:46:45,420 --> 00:46:51,540
+نعم مش X هي أصغر أو يساوي ال supremum لل S أو إن
+
+381
+00:46:51,540 --> 00:46:56,060
+ال X أصغر أو يساوي مجموعة ال U الحالة هي كأنا خبرت
+
+382
+00:46:56,060 --> 00:46:59,460
+إن ال X هتكون أصغر أو يساوي ال supremum يا إما
+
+383
+00:46:59,460 --> 00:47:06,300
+supremum لل S أو supremum لل مجموعة ال Uيعني المهم
+
+384
+00:47:06,300 --> 00:47:14,460
+هي هتطلع الـ Supremum لواحدة من المجموع التاني أنا
+
+385
+00:47:14,460 --> 00:47:19,900
+قبل جملة ال X أزيدور أنا قصدي إن أكتر X أصغر أو
+
+386
+00:47:19,900 --> 00:47:28,380
+بيساوي ال Supremum يعني بشكل مجموحة واحدة X أصغر
+
+387
+00:47:28,380 --> 00:47:35,770
+أو بيساوي ال Supremum لأسطر Star يعني هي اللي هولأ
+
+388
+00:47:35,770 --> 00:47:43,570
+هاد أبراهن S أنها أصغر أو نسبة مجموعة بستار كمه
+
+389
+00:47:43,570 --> 00:47:50,620
+قلو يعني لو حضرتيهم المهم هتطلع لل superأه صح لأن
+
+390
+00:47:50,620 --> 00:47:56,760
+ال suprem هذا أكبر من أو ساوي S star و أكبر من أو
+
+391
+00:47:56,760 --> 00:48:02,960
+ساوي ال U و X أصغر من أو ساوي .. لو كانت ال X أصغر
+
+392
+00:48:02,960 --> 00:48:05,980
+من أو ساوي هذا فهي أكيد أصغر من أو ساوي ال suprem
+
+393
+00:48:05,980 --> 00:48:10,780
+و لو كانت ال X أصغر من أو ساوي ال U فهي أكيد أصغر
+
+394
+00:48:10,780 --> 00:48:12,900
+من أو ساوي ال suprem
+
+395
+00:48:17,590 --> 00:48:26,170
+وبالتالي هذا معناه انه الصحيح
+
+396
+00:48:26,170 --> 00:48:34,450
+ففي الحالة هذه اذا ال supreman لست ال star و you
+
+397
+00:48:34,450 --> 00:48:41,610
+is upper bound upper bound للإتحاد
+
+398
+00:48:44,300 --> 00:48:54,800
+bound of S union single to new لأن
+
+399
+00:48:54,800 --> 00:49:03,260
+هذا fixed ماشي الحال فهذا بحل إشكالية و بعديها
+
+400
+00:49:03,260 --> 00:49:07,380
+بنشطب كل الكلام هذا لأ ما هو هذا الكلام يعني هو
+
+401
+00:49:07,380 --> 00:49:15,430
+تقريبا تفسير ل .. بما أن ال ..هذا مالوش داعي صار
+
+402
+00:49:15,430 --> 00:49:23,350
+هذا مالوش داعي وهذه الخطوة بدل ما نكتبها هنا هذا
+
+403
+00:49:23,350 --> 00:49:27,430
+هي إذا مرة تانية إن أيد البرهان الآن يعني البرهان
+
+404
+00:49:27,430 --> 00:49:33,170
+مافي مشكلة ان شاء الله هاي بنثبت X في الاتحاد تبع
+
+405
+00:49:33,170 --> 00:49:38,990
+المجمعتين هذول الآن X تنتمي للست هذه أو تنتمي للست
+
+406
+00:49:38,990 --> 00:49:52,140
+هذه يعني بتساوي LUوبالتالي ال X تنتمي ل S فهي
+
+407
+00:49:52,140 --> 00:49:56,180
+أصغر من أو ساوي ال supremum ل 6S اللي هو S الصغير
+
+408
+00:49:57,460 --> 00:50:04,020
+أو X أصغر من أو يساوي ال U X بالساوي ال U بتقدي ان
+
+409
+00:50:04,020 --> 00:50:08,900
+X أصغر من أو يساوي ال U الان لو أخدت ال suprem ل S
+
+410
+00:50:08,900 --> 00:50:12,920
+أصغر و U طبعا هذه finite set of real numbers وفي
+
+411
+00:50:12,920 --> 00:50:16,780
+تمرين بيقول لو عندي finite set of real numbers فال
+
+412
+00:50:16,780 --> 00:50:21,390
+suprem تبعها موجودو ينتمي لل set و ال infimum
+
+413
+00:50:21,390 --> 00:50:24,630
+تبعها أيضا موجود و ينتمي ل .. يعني يكون عنصر في ال
+
+414
+00:50:24,630 --> 00:50:28,530
+set هذا أحد التمارين اللي طبعا ما عليناهوش لكن
+
+415
+00:50:28,530 --> 00:50:34,090
+بإمكانكم تثبتوه by induction فهذه finally ال set
+
+416
+00:50:34,090 --> 00:50:37,390
+إذا ال suprem تبعها exist إلا أن هذا ال suprem
+
+417
+00:50:37,390 --> 00:50:41,990
+أكبر من أو ساوي S star وبالتالي أكبر من أو ساوي X
+
+418
+00:50:41,990 --> 00:50:46,790
+و هذا ال suprem أكبر من أو ساوي U
+
+419
+00:50:50,610 --> 00:50:55,450
+وبالتالي أكبر من أو يساوي ال X اللي هي U أكبر من
+
+420
+00:50:55,450 --> 00:51:01,150
+أو ساوي، إذا الأن هذا الكلام صحيح لكل X ينتمي
+
+421
+00:51:01,150 --> 00:51:09,230
+للإتحادهذا العدد الان أكبر من أو ساوي كل عناصر ال
+
+422
+00:51:09,230 --> 00:51:13,350
+6 في الاتحاد فهو upper bound لل 6 هذه فهو upper ان
+
+423
+00:51:13,350 --> 00:51:18,770
+العدد هذا upper bound لل 6 هذه الان أثبتنا ان هذا
+
+424
+00:51:18,770 --> 00:51:23,380
+ال upper bound هو أصغر upper bound للاتحادو هي
+
+425
+00:51:23,380 --> 00:51:29,160
+أخدنا أي upper bound عشوائي للاتحاد طلع هذا ال
+
+426
+00:51:29,160 --> 00:51:33,140
+upper bound العشوائي أكبر من أو ساوي العدد هذا
+
+427
+00:51:33,140 --> 00:51:36,720
+اللي بدنا إياه هو ال supremum إذا هذا العدد هو ال
+
+428
+00:51:36,720 --> 00:51:42,940
+supremum للست هذه تمام؟ okay؟ في أي سؤال تاني؟
+
+429
+00:51:42,940 --> 00:51:51,480
+فخلينا نحللنا كمان سؤالينفي ال .. نحل مثلا خليني
+
+430
+00:51:51,480 --> 00:51:54,300
+انا اختارلكم بعض الأسئلة مدام انتوا يعني شاكلكم
+
+431
+00:51:54,300 --> 00:51:59,300
+الا طبعا اذا حد سائل خليني امسح اللوح الأول و نحل
+
+432
+00:51:59,300 --> 00:52:00,240
+كمان سؤالين
+
+433
+00:52:16,370 --> 00:52:21,990
+يعني قبل شوية ذكرنا التمرين
+
+434
+00:52:21,990 --> 00:52:34,770
+هذا التمرين 12 section 2 3 وهذا التمرين بيقول let
+
+435
+00:52:34,770 --> 00:52:51,380
+S بي .. let S بالساوي X1 إلى XNbe any non
+
+436
+00:52:51,380 --> 00:52:58,260
+-empty finite finite
+
+437
+00:52:58,260 --> 00:53:12,080
+set أو subset من R فبنثبت
+
+438
+00:53:12,080 --> 00:53:14,920
+ان ال show
+
+439
+00:53:17,460 --> 00:53:34,980
+in from S و supreme S ينتمي ل S وكذلك
+
+440
+00:53:34,980 --> 00:53:41,720
+ال supreme ل 6S موجود و هو عنصر في 6S
+
+441
+00:53:52,980 --> 00:53:59,400
+Okay إذا ال finite set تبعتي هذه فرضنا أن عناصرها
+
+442
+00:53:59,400 --> 00:54:06,300
+سمينا عناصرها x1, x2 إلى xn لأن هذه set فيها n
+
+443
+00:54:06,300 --> 00:54:18,540
+elements طيب ممكن نرتب العناصر هذهby rearranging
+
+444
+00:54:18,540 --> 00:54:23,200
+indices
+
+445
+00:54:23,200 --> 00:54:27,220
+if
+
+446
+00:54:27,220 --> 00:54:36,520
+necessary اذا كان ضروري we
+
+447
+00:54:36,520 --> 00:54:50,310
+may and dowe may and do assume that
+
+448
+00:54:50,310 --> 00:54:53,890
+x1
+
+449
+00:54:53,890 --> 00:55:04,950
+less than x2 less than less than xn أنا
+
+450
+00:55:04,950 --> 00:55:13,580
+عندي finite set call it x1 إلى xnممكن ان اعيد
+
+451
+00:55:13,580 --> 00:55:20,620
+ترتيب العناصر هذه هى طبعا عداد حقيقية فممكن ان
+
+452
+00:55:20,620 --> 00:55:26,880
+اعيد .. و طبعا كلهم عناصر مش متساوية فممكن
+
+453
+00:55:26,880 --> 00:55:32,200
+اعيد ترتيب او تسمية العناصر هذه المؤشرات تبعات هذه
+
+454
+00:55:32,200 --> 00:55:38,680
+ممكن اعيد ترتيبها بحيث انه يطلع x1 اصغر من x2 اصغر
+
+455
+00:55:38,680 --> 00:55:44,920
+من x3 او هكذا الاكثرهذا ممكن نعمله ولا لأ؟ ممكن
+
+456
+00:55:44,920 --> 00:55:48,380
+الان
+
+457
+00:55:48,380 --> 00:55:54,640
+تعالوا نثبت claim
+
+458
+00:55:54,640 --> 00:56:01,120
+انا بتدعي ان ال minimum لل set S هيطلع بساوي X
+
+459
+00:56:01,120 --> 00:56:08,200
+واحد وهذا ينتمي ل Sيعني بعد ما رتبت العناصر عملت
+
+460
+00:56:08,200 --> 00:56:12,740
+ordering لهم بالطريقة دي فحثبت أن الinfant plus
+
+461
+00:56:12,740 --> 00:56:18,820
+set S بساوي أصغر عنصر في ال set اللي هو X1 و هذا
+
+462
+00:56:18,820 --> 00:56:29,620
+طبعا ينتمي إلى S طيب لبرهان ذلك clearly واضح
+
+463
+00:56:29,620 --> 00:56:40,900
+أن X1 is a lower boundlower bound لست S نظبط لأن
+
+464
+00:56:40,900 --> 00:56:45,740
+X1 أصغر من أو ساوي كل العناصر اللي في الست فهو
+
+465
+00:56:45,740 --> 00:56:51,000
+واضح انه lower bound الان انا بتثبت انه مش بس
+
+466
+00:56:51,000 --> 00:56:54,400
+lower bound هو ال infimum هو ال greatest lower
+
+467
+00:56:54,400 --> 00:57:01,620
+bound اذا هنا now if W is
+
+468
+00:57:04,400 --> 00:57:16,580
+any lower bound .. any lower bound of S فهذا
+
+469
+00:57:16,580 --> 00:57:25,780
+معناه أن W أصغر من أو يساوي Xi لكل I بيساوي 1 2
+
+470
+00:57:25,780 --> 00:57:29,640
+إلى N صح؟
+
+471
+00:57:30,510 --> 00:57:38,370
+و أصغر من أو ساوي كل عناصرها و بالتالي therefore w
+
+472
+00:57:38,370 --> 00:57:44,970
+أصغر من أو ساوي x واحد لأن x واحد هو واحد من عناصر
+
+473
+00:57:44,970 --> 00:57:54,350
+الست إذا أنا عندي الان x واحد is lower bound للستو
+
+474
+00:57:54,350 --> 00:58:00,190
+أي lower bound للست بيطلع أصغر من أو يساوي x واحد
+
+475
+00:58:00,190 --> 00:58:08,770
+اذا by definition ال x واحد اه او ال infimum للست
+
+476
+00:58:08,770 --> 00:58:16,330
+s exist and بيساوي x واحد تمام؟
+
+477
+00:58:16,330 --> 00:58:22,610
+بالمثل ممكن نثبت ال .. اه هنا similarly
+
+478
+00:58:26,410 --> 00:58:33,190
+similarly show that ان انا هاسيبكم بطريقة مشابعة
+
+479
+00:58:34,440 --> 00:58:39,920
+تثبتوا ال claim التاني وهو ان ال supremum لل set S
+
+480
+00:58:39,920 --> 00:58:47,620
+exist و بساوي XN و طبعا هذا بينتمي لل set S و هو
+
+481
+00:58:47,620 --> 00:58:52,040
+المطلوب okay تمام ان هيك بنكون أثبتنا ان اي finite
+
+482
+00:58:52,040 --> 00:58:56,920
+set لها supremum لها infimum و هدولة بيطلعوا عناصر
+
+483
+00:58:56,920 --> 00:59:01,960
+فيها بالتحديد ال infimum هو ال least element اصغر
+
+484
+00:59:01,960 --> 00:59:07,600
+عنصرفي ال set و ال supremum هو ال greatest element
+
+485
+00:59:07,600 --> 00:59:12,480
+اللي هو أكبر أنصار في ال setهذا طبعا الكلام مش
+
+486
+00:59:12,480 --> 00:59:16,360
+صحيح إذا ال set S كانت infinite هذا بس صحيح في
+
+487
+00:59:16,360 --> 00:59:22,600
+حالة ال finite set إذا ال .. هذا بيكون بيكمل برهان
+
+488
+00:59:22,600 --> 00:59:30,220
+التمرين هذا و بالتالي بنكتفي بحل أو بهذا القدر من
+
+489
+00:59:30,220 --> 00:59:34,260
+حل التمرين و ان شاء الله أسبوع الجاي بنكمل حل
+
+490
+00:59:34,260 --> 00:59:35,400
+تمرين أخرى
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..adfa7ac728bb97c4f1941c427001cd1a42225a29
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8N3n8lL04hg_raw.json
@@ -0,0 +1 @@
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{"start": 124.86, "end": 125.16, "word": " هذا", "probability": 0.82470703125}, {"start": 125.16, "end": 125.36, "word": " اللى", "probability": 0.85107421875}, {"start": 125.36, "end": 125.56, "word": " هنا", "probability": 0.9228515625}], "temperature": 1.0}, {"id": 5, "seek": 15237, "start": 127.49, "end": 152.37, "text": "فبطلع عندي supremum سالب S بيساوي سالب infimum S هاي سالب اضربك سالب واحد سالب infimum S لأن هذا التمرين حالة خاصة من الجزء هذا التاني في الفرع B وبالتالي هذا التمرين تعميم لهذا الجزء ولا جزء تاني", "tokens": [5172, 3555, 9566, 1211, 3615, 18871, 16254, 23710, 449, 8608, 6027, 3555, 318, 4724, 1829, 3794, 995, 45865, 8608, 6027, 3555, 1536, 332, 449, 318, 8032, 47302, 8608, 6027, 3555, 1975, 11242, 25513, 4117, 8608, 6027, 3555, 36764, 24401, 8608, 6027, 3555, 1536, 332, 449, 318, 5296, 33456, 23758, 16712, 29973, 9957, 11331, 6027, 3660, 16490, 33546, 3660, 9154, 25724, 11622, 38207, 23758, 16712, 7649, 1829, 8978, 27188, 2288, 3615, 363, 46599, 6027, 2655, 6027, 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136.37, "word": " هاي", "probability": 0.5096435546875}, {"start": 136.37, "end": 137.07, "word": " سالب", "probability": 0.9459635416666666}, {"start": 137.07, "end": 137.91, "word": " اضربك", "probability": 0.71234130859375}, {"start": 137.91, "end": 138.27, "word": " سالب", "probability": 0.9637044270833334}, {"start": 138.27, "end": 138.69, "word": " واحد", "probability": 0.9560546875}, {"start": 138.69, "end": 139.39, "word": " سالب", "probability": 0.9142252604166666}, {"start": 139.39, "end": 139.81, "word": " infimum", "probability": 0.9415690104166666}, {"start": 139.81, "end": 140.09, "word": " S", "probability": 0.91162109375}, {"start": 140.09, "end": 140.65, "word": " لأن", "probability": 0.295166015625}, {"start": 140.65, "end": 140.93, "word": " هذا", "probability": 0.87841796875}, {"start": 140.93, "end": 141.43, "word": " التمرين", "probability": 0.98779296875}, {"start": 141.43, "end": 141.81, "word": " حالة", "probability": 0.89208984375}, {"start": 141.81, "end": 142.23, "word": " خاصة", "probability": 0.9873046875}, {"start": 142.23, "end": 142.43, "word": " من", "probability": 0.96875}, {"start": 142.43, "end": 142.93, "word": " الجزء", "probability": 0.974609375}, {"start": 142.93, "end": 143.39, "word": " هذا", "probability": 0.86962890625}, {"start": 143.39, "end": 144.73, "word": " التاني", "probability": 0.95263671875}, {"start": 144.73, "end": 144.95, "word": " في", "probability": 0.92041015625}, {"start": 144.95, "end": 145.33, "word": " الفرع", "probability": 0.7020670572916666}, {"start": 145.33, "end": 145.59, "word": " B", "probability": 0.70263671875}, {"start": 145.59, "end": 147.89, "word": " وبالتالي", "probability": 0.91533203125}, {"start": 147.89, "end": 148.11, "word": " هذا", "probability": 0.9189453125}, {"start": 148.11, "end": 148.77, "word": " التمرين", "probability": 0.99560546875}, {"start": 148.77, "end": 150.03, "word": " تعميم", "probability": 0.9405924479166666}, {"start": 150.03, "end": 150.45, "word": " لهذا", 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في الـ claim أو في الإدعاء تمام؟ وهذا بثبت الجزء الأول في ال part A هاي infimum AS بساوي A في W اللي هو infimum S إذن هذا بثبت الجزء الأول في الفرع A Similarly بالمثل ممكن", "tokens": [2304, 2407, 11242, 5016, 8978, 2423, 39184, 3932, 34051, 8978, 33688, 3215, 3615, 16606, 46811, 10943, 22807, 37037, 15730, 4724, 12984, 3555, 2655, 25724, 11622, 38207, 16247, 12610, 8978, 2423, 644, 316, 8032, 47302, 1536, 332, 449, 7469, 4724, 3794, 995, 45865, 316, 8978, 343, 13672, 1829, 31439, 1536, 332, 449, 318, 11933, 8848, 1863, 23758, 4724, 12984, 3555, 2655, 25724, 11622, 38207, 16247, 12610, 8978, 27188, 2288, 3615, 316, 13157, 20666, 2304, 12984, 1211, 3714, 43020], "avg_logprob": -0.19821714017635736, "compression_ratio": 1.6235294117647059, "no_speech_prob": 0.0, "words": [{"start": 396.63, "end": 397.31, "word": "موضح", "probability": 0.8367919921875}, {"start": 397.31, "end": 397.49, "word": " في", "probability": 0.93017578125}, {"start": 397.49, "end": 397.63, "word": " الـ", 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X في S", "tokens": [9566, 1829, 3555, 1975, 8315, 18871, 16254, 4724, 1829, 6225, 3215, 1211, 8608, 6027, 3555, 1975, 15730, 36764, 24401, 15844, 4724, 1829, 1975, 1829, 11242, 995, 6225, 3215, 1211, 8608, 6027, 3555, 6156, 1211, 2407, 48812, 25513, 2655, 9673, 2655, 3555, 995, 9957, 3660, 29538, 8978, 6225, 3215, 1211, 8608, 6027, 3555, 13672, 1829, 31439, 36764, 24401, 15844, 4724, 1829, 6156, 3224, 1829, 9566, 1211, 3615, 18871, 16254, 4724, 1829, 4386, 4724, 1829, 15844, 4724, 1829, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 1783, 5296, 28820, 1783, 8978, 318], "avg_logprob": -0.18206522127856378, "compression_ratio": 1.8516129032258064, "no_speech_prob": 0.0, "words": [{"start": 689.92, "end": 690.38, "word": "طيب", "probability": 0.6832275390625}, {"start": 690.38, "end": 690.6, "word": " انا", "probability": 0.640869140625}, {"start": 690.6, "end": 691.06, "word": " عندي", "probability": 0.786376953125}, {"start": 691.06, "end": 691.56, "word": " بي", "probability": 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دابليو هو أصغر upper bound للست هذه والأن بنعود عن w إذن ال b في w اللي هو infimum of s بتطلع بساوي supremum ل b في s", "tokens": [1211, 33456, 23758, 18863, 3215, 3215, 6597, 5472, 24976, 14851, 29538, 37037, 2407, 5551, 9381, 17082, 2288, 6597, 5472, 5551, 9778, 3215, 8315, 36632, 6597, 5472, 24976, 14851, 29538, 23032, 1211, 3615, 4724, 1829, 11778, 16758, 20292, 2407, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 11933, 8848, 1863, 4724, 1829, 11778, 16758, 20292, 2407, 31439, 5551, 9381, 17082, 2288, 6597, 5472, 24976, 14851, 29538, 4032, 6027, 33456, 44945, 3615, 23328, 18871, 261, 11933, 8848, 1863, 2423, 272, 8978, 261, 13672, 1829, 31439, 1536, 332, 449, 295, 262, 39894, 9566, 1211, 3615, 4724, 3794, 995, 45865, 23710, 449, 5296, 272, 8978, 262], "avg_logprob": -0.24263821427638715, "compression_ratio": 1.989071038251366, "no_speech_prob": 0.0, "words": [{"start": 939.99, "end": 940.33, "word": "لأن", "probability": 0.5487060546875}, {"start": 940.33, "end": 940.57, 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"probability": 0.9737955729166666}, {"start": 1308.51, "end": 1311.65, "word": " by", "probability": 0.73095703125}, {"start": 1311.65, "end": 1312.33, "word": " definition", "probability": 0.951171875}, {"start": 1312.33, "end": 1312.95, "word": " حسب", "probability": 0.969970703125}, {"start": 1312.95, "end": 1314.35, "word": " التعريفات", "probability": 0.97314453125}, {"start": 1314.35, "end": 1317.95, "word": " ال", "probability": 0.890625}, {"start": 1317.95, "end": 1318.67, "word": " infimum", "probability": 0.598388671875}], "temperature": 1.0}, {"id": 46, "seek": 133175, "start": 1319.99, "end": 1331.75, "text": "للست S0 أصغر من أو ساوي ال supremum للست S0 الست S0 هذه", "tokens": [1211, 1211, 14851, 318, 15, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 23710, 449, 24976, 14851, 318, 15, 2423, 14851, 318, 15, 29538], "avg_logprob": -0.24811921958570127, "compression_ratio": 1.1506849315068493, "no_speech_prob": 0.0, "words": [{"start": 1319.99, "end": 1320.83, "word": "للست", "probability": 0.5049641927083334}, {"start": 1320.83, "end": 1321.77, "word": " S0", "probability": 0.78515625}, {"start": 1321.77, "end": 1322.49, "word": " أصغر", "probability": 0.90673828125}, {"start": 1322.49, "end": 1322.67, "word": " من", "probability": 0.9765625}, {"start": 1322.67, "end": 1322.89, "word": " أو", "probability": 0.97119140625}, {"start": 1322.89, "end": 1323.61, "word": " ساوي", "probability": 0.9202473958333334}, {"start": 1323.61, "end": 1324.85, "word": " ال", "probability": 0.7685546875}, {"start": 1324.85, "end": 1325.49, "word": " supremum", "probability": 0.6038818359375}, {"start": 1325.49, "end": 1326.07, "word": " للست", "probability": 0.829833984375}, {"start": 1326.07, "end": 1326.79, "word": " S0", "probability": 0.96044921875}, {"start": 1326.79, "end": 1330.75, "word": " الست", "probability": 0.764892578125}, {"start": 1330.75, "end": 1331.39, "word": " S0", "probability": 0.961181640625}, {"start": 1331.39, "end": 1331.75, "word": " هذه", "probability": 0.61572265625}], "temperature": 1.0}, {"id": 47, "seek": 135925, "start": 1335.23, "end": 1359.25, "text": "طبعا هذه ال set S0 subset من S و S bounded إلى S0 bounded ال infimum ل S0 exist و ال suprem ل S0 exist دائما لأي set S0 ال infimum دائما أصغر من أو يساوي ال supremum نعمل رسمة نوضح الكلام هذا", "tokens": [9566, 3555, 3615, 995, 29538, 2423, 992, 318, 15, 25993, 9154, 318, 4032, 318, 37498, 30731, 318, 15, 37498, 2423, 1536, 332, 449, 5296, 318, 15, 2514, 4032, 2423, 23710, 5296, 318, 15, 2514, 11778, 16373, 15042, 5296, 10721, 1829, 992, 318, 15, 2423, 1536, 332, 449, 11778, 16373, 15042, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 2423, 23710, 449, 8717, 25957, 1211, 12602, 38251, 3660, 8717, 2407, 11242, 5016, 2423, 28820, 10943, 23758], "avg_logprob": -0.2049278825139388, "compression_ratio": 1.543859649122807, "no_speech_prob": 0.0, "words": [{"start": 1335.23, "end": 1336.15, "word": "طبعا", "probability": 0.769195556640625}, 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" دائما", "probability": 0.7999674479166666}, {"start": 1348.61, "end": 1349.03, "word": " لأي", "probability": 0.81787109375}, {"start": 1349.03, "end": 1349.47, "word": " set", "probability": 0.98095703125}, {"start": 1349.47, "end": 1350.31, "word": " S0", "probability": 0.98974609375}, {"start": 1350.31, "end": 1350.99, "word": " ال", "probability": 0.90478515625}, {"start": 1350.99, "end": 1351.45, "word": " infimum", "probability": 0.97314453125}, {"start": 1351.45, "end": 1351.89, "word": " دائما", "probability": 0.8854166666666666}, {"start": 1351.89, "end": 1352.43, "word": " أصغر", "probability": 0.9620361328125}, {"start": 1352.43, "end": 1352.61, "word": " من", "probability": 0.94775390625}, {"start": 1352.61, "end": 1352.77, "word": " أو", "probability": 0.98193359375}, {"start": 1352.77, "end": 1353.15, "word": " يساوي", "probability": 0.8641357421875}, {"start": 1353.15, "end": 1353.31, "word": " ال", "probability": 0.82080078125}, {"start": 1353.31, "end": 1354.91, "word": " supremum", "probability": 0.6375732421875}, {"start": 1354.91, "end": 1357.21, "word": " نعمل", "probability": 0.9026692708333334}, {"start": 1357.21, "end": 1357.81, "word": " رسمة", "probability": 0.9435221354166666}, {"start": 1357.81, "end": 1358.33, "word": " نوضح", "probability": 0.9464111328125}, {"start": 1358.33, "end": 1358.81, "word": " الكلام", "probability": 0.9518229166666666}, {"start": 1358.81, "end": 1359.25, "word": " هذا", "probability": 0.8876953125}], "temperature": 1.0}, {"id": 48, "seek": 138795, "start": 1364.85, "end": 1387.95, "text": "نعتبر أن هذه هي الست اس وهي ال .. ال .. ال supremum للست اس وهي ال infimum", "tokens": [1863, 34268, 26890, 14739, 29538, 39896, 2423, 14851, 24525, 37037, 1829, 2423, 4386, 2423, 4386, 2423, 23710, 449, 24976, 14851, 24525, 37037, 1829, 2423, 1536, 332, 449], "avg_logprob": -0.39536828973463606, "compression_ratio": 1.435897435897436, "no_speech_prob": 0.0, "words": [{"start": 1364.8500000000001, "end": 1365.65, "word": "نعتبر", "probability": 0.8885091145833334}, {"start": 1365.65, "end": 1365.81, "word": " أن", "probability": 0.308349609375}, {"start": 1365.81, "end": 1366.11, "word": " هذه", "probability": 0.73876953125}, {"start": 1366.11, "end": 1366.41, "word": " هي", "probability": 0.5625}, {"start": 1366.41, "end": 1366.85, "word": " الست", "probability": 0.557861328125}, {"start": 1366.85, "end": 1367.25, "word": " اس", "probability": 0.60009765625}, {"start": 1367.25, "end": 1376.85, "word": " وهي", "probability": 0.7333984375}, {"start": 1376.85, "end": 1378.31, "word": " ال", "probability": 0.9208984375}, {"start": 1378.31, "end": 1378.31, "word": " ..", "probability": 0.56005859375}, {"start": 1378.31, "end": 1380.39, "word": " ال", "probability": 0.8173828125}, {"start": 1380.39, "end": 1380.51, "word": " ..", "probability": 0.407958984375}, {"start": 1380.51, "end": 1380.83, "word": " ال", "probability": 0.640625}, {"start": 1380.83, "end": 1383.37, "word": " supremum", "probability": 0.473388671875}, {"start": 1383.37, "end": 1386.37, "word": " للست", "probability": 0.71923828125}, {"start": 1386.37, "end": 1386.77, "word": " اس", "probability": 0.89306640625}, {"start": 1386.77, "end": 1387.17, "word": " وهي", "probability": 0.85400390625}, {"start": 1387.17, "end": 1387.35, "word": " ال", "probability": 0.974609375}, {"start": 1387.35, "end": 1387.95, "word": " infimum", "probability": 0.7086588541666666}], "temperature": 1.0}, {"id": 49, "seek": 140607, "start": 1391.09, "end": 1406.07, "text": "للـ set S فدائما ال .. دائما ال minimum لأي set هو lower bound لل set وبالتالي أصغر من لو ساوي كل عناصرها", "tokens": [1211, 1211, 39184, 992, 318, 6156, 3215, 16373, 15042, 2423, 4386, 11778, 16373, 15042, 2423, 7285, 5296, 10721, 1829, 992, 31439, 3126, 5472, 24976, 992, 46599, 6027, 2655, 6027, 1829, 5551, 9381, 17082, 2288, 9154, 45164, 8608, 995, 45865, 28242, 18871, 33546, 2288, 11296], "avg_logprob": -0.28038193384806315, "compression_ratio": 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"word": " هو", "probability": 0.93603515625}, {"start": 1400.73, "end": 1401.09, "word": " lower", "probability": 0.93798828125}, {"start": 1401.09, "end": 1401.49, "word": " bound", "probability": 0.94873046875}, {"start": 1401.49, "end": 1401.71, "word": " لل", "probability": 0.84423828125}, {"start": 1401.71, "end": 1402.09, "word": " set", "probability": 0.9072265625}, {"start": 1402.09, "end": 1404.05, "word": " وبالتالي", "probability": 0.91318359375}, {"start": 1404.05, "end": 1404.45, "word": " أصغر", "probability": 0.9752197265625}, {"start": 1404.45, "end": 1404.57, "word": " من", "probability": 0.912109375}, {"start": 1404.57, "end": 1404.73, "word": " لو", "probability": 0.59423828125}, {"start": 1404.73, "end": 1405.15, "word": " ساوي", "probability": 0.7265625}, {"start": 1405.15, "end": 1405.45, "word": " كل", "probability": 0.98828125}, {"start": 1405.45, "end": 1406.07, "word": " عناصرها", "probability": 0.9625244140625}], "temperature": 1.0}, {"id": 50, "seek": 143379, "start": 1407.63, "end": 1433.79, "text": "هو عبارة عن lower bound للست ال supreme للست S هو عبارة عن upper bound للست وبالتالي أكبر من أو ساوي كل عناصرها فواضح أن ال infimum للست S لازم يكون أصغر من أو ساوي ال supremum ونفس الشيء لو أخذنا أي مجموعة جزئية سمنها S0", "tokens": [3224, 2407, 6225, 3555, 9640, 3660, 18871, 3126, 5472, 24976, 14851, 2423, 27756, 24976, 14851, 318, 31439, 6225, 3555, 9640, 3660, 18871, 6597, 5472, 24976, 14851, 46599, 6027, 2655, 6027, 1829, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 28242, 18871, 33546, 2288, 11296, 6156, 2407, 46958, 5016, 14739, 2423, 1536, 332, 449, 24976, 14851, 318, 5296, 31377, 2304, 7251, 30544, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 23710, 449, 4032, 1863, 36178, 25124, 1829, 38207, 45164, 5551, 9778, 8848, 8315, 36632, 3714, 7435, 2304, 2407, 27884, 10874, 11622, 19986, 10632, 8608, 27842, 11296, 318, 15], "avg_logprob": -0.15046875402331353, "compression_ratio": 1.703883495145631, 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"word": " فواضح", "probability": 0.9693603515625}, {"start": 1417.47, "end": 1417.65, "word": " أن", "probability": 0.4775390625}, {"start": 1417.65, "end": 1417.77, "word": " ال", "probability": 0.9365234375}, {"start": 1417.77, "end": 1418.27, "word": " infimum", "probability": 0.7821451822916666}, {"start": 1418.27, "end": 1419.41, "word": " للست", "probability": 0.944580078125}, {"start": 1419.41, "end": 1419.75, "word": " S", "probability": 0.93603515625}, {"start": 1419.75, "end": 1420.21, "word": " لازم", "probability": 0.9744466145833334}, {"start": 1420.21, "end": 1420.61, "word": " يكون", "probability": 0.9931640625}, {"start": 1420.61, "end": 1421.13, "word": " أصغر", "probability": 0.9661865234375}, {"start": 1421.13, "end": 1421.29, "word": " من", "probability": 0.9921875}, {"start": 1421.29, "end": 1421.47, "word": " أو", "probability": 0.99462890625}, {"start": 1421.47, "end": 1422.03, "word": " ساوي", "probability": 0.9811197916666666}, {"start": 1422.03, "end": 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المجموعة اسمها S0 فبما أن ال set S bounded إذن S0 bounded وبالتالي ال supremum ل S0 دايما أكبر من أو ساوي ال infimum ل S0 بنفس الطريقة إذن هذا دايما .. هذا دايما صحيح", "tokens": [40228, 22653, 29538, 9673, 7435, 2304, 2407, 27884, 24525, 2304, 11296, 318, 15, 6156, 3555, 15042, 14739, 2423, 992, 318, 37498, 11933, 8848, 1863, 318, 15, 37498, 46599, 6027, 2655, 6027, 1829, 2423, 23710, 449, 5296, 318, 15, 11778, 47302, 15042, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2423, 1536, 332, 449, 5296, 318, 15, 44945, 36178, 41950, 16572, 28671, 11933, 8848, 1863, 23758, 11778, 47302, 15042, 4386, 23758, 11778, 47302, 15042, 20328, 5016, 1829, 5016], "avg_logprob": -0.1938100909957519, "compression_ratio": 1.5664739884393064, "no_speech_prob": 0.0, "words": [{"start": 1436.18, "end": 1436.58, "word": "يعني", "probability": 0.593505859375}, {"start": 1436.58, "end": 1436.86, "word": " هذه", "probability": 0.75439453125}, {"start": 1436.86, "end": 1437.42, "word": " المجموعة", 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"probability": 0.89931640625}, {"start": 1445.64, "end": 1446.98, "word": " ال", "probability": 0.7783203125}, {"start": 1446.98, "end": 1448.16, "word": " supremum", "probability": 0.6937255859375}, {"start": 1448.16, "end": 1449.38, "word": " ل", "probability": 0.94580078125}, {"start": 1449.38, "end": 1450.4, "word": " S0", "probability": 0.958984375}, {"start": 1450.4, "end": 1451.8, "word": " دايما", "probability": 0.8863932291666666}, {"start": 1451.8, "end": 1452.32, "word": " أكبر", "probability": 0.9039713541666666}, {"start": 1452.32, "end": 1452.46, "word": " من", "probability": 0.78564453125}, {"start": 1452.46, "end": 1452.62, "word": " أو", "probability": 0.90869140625}, {"start": 1452.62, "end": 1453.04, "word": " ساوي", "probability": 0.7916666666666666}, {"start": 1453.04, "end": 1453.2, "word": " ال", "probability": 0.9404296875}, {"start": 1453.2, "end": 1453.8, "word": " infimum", "probability": 0.9021809895833334}, {"start": 1453.8, "end": 1454.0, "word": " ل", 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واضح من التعريفات وهذا الجزء أثبتناه باقي إثبات الجزء الأخير هذا فإذا بنقول finally أخيرا لإثبات الجزء الأخير هذا أنا عندي ال inform ل S is lower bound ل 6S", "tokens": [3615, 8592, 7649, 1975, 5016, 8315, 8717, 24793, 1211, 2423, 26890, 3224, 7649, 1975, 15730, 1975, 5016, 8315, 5551, 12984, 3555, 2655, 8315, 23758, 4032, 46958, 5016, 9154, 16712, 3615, 16572, 5172, 9307, 37037, 15730, 25724, 11622, 38207, 5551, 12984, 3555, 2655, 8315, 3224, 4724, 995, 38436, 11933, 12984, 3555, 9307, 25724, 11622, 38207, 16247, 9778, 13546, 23758, 6156, 28814, 15730, 44945, 39648, 2721, 5551, 9778, 13546, 995, 5296, 28814, 12984, 3555, 9307, 25724, 11622, 38207, 16247, 9778, 13546, 23758, 41850, 18871, 16254, 2423, 1356, 5296, 318, 307, 3126, 5472, 5296, 1386, 50], "avg_logprob": -0.18284574721721894, "compression_ratio": 1.7712765957446808, "no_speech_prob": 0.0, "words": [{"start": 1461.95, "end": 1462.85, "word": "عشان", "probability": 0.9606119791666666}, {"start": 1462.85, "end": 1463.13, 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"end": 1471.31, "word": " أثبتناه", "probability": 0.9703776041666666}, {"start": 1471.31, "end": 1475.15, "word": " باقي", "probability": 0.7610677083333334}, {"start": 1475.15, "end": 1475.79, "word": " إثبات", "probability": 0.85345458984375}, {"start": 1475.79, "end": 1476.35, "word": " الجزء", "probability": 0.9881184895833334}, {"start": 1476.35, "end": 1476.97, "word": " الأخير", "probability": 0.89990234375}, {"start": 1476.97, "end": 1477.67, "word": " هذا", "probability": 0.88671875}, {"start": 1477.67, "end": 1480.93, "word": " فإذا", "probability": 0.7384440104166666}, {"start": 1480.93, "end": 1481.51, "word": " بنقول", "probability": 0.800537109375}, {"start": 1481.51, "end": 1482.09, "word": " finally", "probability": 0.79931640625}, {"start": 1482.09, "end": 1483.39, "word": " أخيرا", "probability": 0.9298095703125}, {"start": 1483.39, "end": 1484.45, "word": " لإثبات", "probability": 0.899609375}, {"start": 1484.45, "end": 1484.79, "word": " الجزء", "probability": 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"probability": 0.8046875}, {"start": 1493.43, "end": 1493.95, "word": " bound", "probability": 0.919921875}, {"start": 1493.95, "end": 1494.41, "word": " لأي", "probability": 0.91357421875}, {"start": 1494.41, "end": 1494.99, "word": " مجموعة", "probability": 0.9314453125}, {"start": 1494.99, "end": 1495.67, "word": " جزئية", "probability": 0.9178466796875}, {"start": 1495.67, "end": 1496.51, "word": " S0", "probability": 0.634521484375}, {"start": 1496.51, "end": 1496.95, "word": " من", "probability": 0.9697265625}, {"start": 1496.95, "end": 1497.35, "word": " S", "probability": 0.96435546875}, {"start": 1497.35, "end": 1500.89, "word": " وبالتالي", "probability": 0.882861328125}, {"start": 1500.89, "end": 1501.53, "word": " إذا", "probability": 0.54693603515625}, {"start": 1501.53, "end": 1502.25, "word": " ال", "probability": 0.97705078125}, {"start": 1502.25, "end": 1502.67, "word": " influence", "probability": 0.068359375}, {"start": 1502.67, "end": 1503.05, "word": " ل", "probability": 0.90185546875}, {"start": 1503.05, "end": 1505.69, "word": " S0", "probability": 0.931884765625}, {"start": 1505.69, "end": 1511.77, "word": " هذا", "probability": 0.29833984375}, {"start": 1511.77, "end": 1512.33, "word": " أكبر", "probability": 0.9684244791666666}, {"start": 1512.33, "end": 1513.85, "word": " lower", "probability": 0.65087890625}, {"start": 1513.85, "end": 1514.39, "word": " bound", "probability": 0.9287109375}, {"start": 1514.39, "end": 1514.57, "word": " ل", "probability": 0.9677734375}, {"start": 1514.57, "end": 1515.27, "word": " S0", "probability": 0.973388671875}, {"start": 1515.27, "end": 1515.97, "word": " هذا", "probability": 0.6552734375}, {"start": 1515.97, "end": 1516.35, "word": " أكبر", "probability": 0.9781901041666666}, {"start": 1516.35, "end": 1516.63, "word": " lower", "probability": 0.92138671875}, {"start": 1516.63, "end": 1516.95, "word": " bound", "probability": 0.9296875}, {"start": 1516.95, "end": 1517.17, "word": " ل", "probability": 0.98388671875}, {"start": 1517.17, "end": 1517.77, "word": " S0", "probability": 0.988525390625}], "temperature": 1.0}, {"id": 54, "seek": 153974, "start": 1518.9, "end": 1539.74, "text": "و هذا lower bound ل S0 إذاً هذا بيطلع أكبر من أو ساوي infimum ال 6S هذا lower bound ل 6S0 و هذا أكبر lower bound ل 6S0 إذاً هذا أصغر من أو ساوي هذا و هذا بيكمل", "tokens": [2407, 23758, 3126, 5472, 5296, 318, 15, 11933, 15730, 14111, 23758, 4724, 1829, 9566, 1211, 3615, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 1536, 332, 449, 2423, 1386, 50, 23758, 3126, 5472, 5296, 1386, 50, 15, 4032, 23758, 5551, 4117, 26890, 3126, 5472, 5296, 1386, 50, 15, 11933, 15730, 14111, 23758, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 23758, 4032, 23758, 4724, 1829, 24793, 1211], "avg_logprob": -0.19916213854499484, "compression_ratio": 1.8, "no_speech_prob": 0.0, "words": [{"start": 1518.9, "end": 1519.18, "word": "و", "probability": 0.818359375}, {"start": 1519.18, "end": 1519.66, 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infimum", "probability": 0.7501627604166666}, {"start": 1528.18, "end": 1528.76, "word": " ال", "probability": 0.64453125}, {"start": 1528.76, "end": 1529.52, "word": " 6S", "probability": 0.724853515625}, {"start": 1529.52, "end": 1530.62, "word": " هذا", "probability": 0.86181640625}, {"start": 1530.62, "end": 1530.96, "word": " lower", "probability": 0.90869140625}, {"start": 1530.96, "end": 1531.46, "word": " bound", "probability": 0.9150390625}, {"start": 1531.46, "end": 1531.68, "word": " ل", "probability": 0.8759765625}, {"start": 1531.68, "end": 1532.7, "word": " 6S0", "probability": 0.9270833333333334}, {"start": 1532.7, "end": 1533.24, "word": " و", "probability": 0.4921875}, {"start": 1533.24, "end": 1533.5, "word": " هذا", "probability": 0.96875}, {"start": 1533.5, "end": 1533.9, "word": " أكبر", "probability": 0.9812825520833334}, {"start": 1533.9, "end": 1534.24, "word": " lower", "probability": 0.908203125}, {"start": 1534.24, "end": 1534.62, "word": " bound", 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0.97314453125}, {"start": 2084.38, "end": 2084.72, "word": " الجزء", "probability": 0.6194661458333334}, {"start": 2084.72, "end": 2085.0, "word": " هذا", "probability": 0.5283203125}], "temperature": 1.0}, {"id": 75, "seek": 211523, "start": 2086.31, "end": 2115.23, "text": "هذا برهان برهان تاني ممكن ان احنا نعمل برهان مباشر يعني بلاش نستخدم exercise تسعة تاني ممكن we consider we consider two cases", "tokens": [3224, 15730, 4724, 2288, 3224, 7649, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 3714, 43020, 16472, 1975, 5016, 8315, 8717, 25957, 1211, 4724, 2288, 3224, 7649, 3714, 3555, 33599, 2288, 37495, 22653, 4724, 1211, 33599, 8717, 14851, 9778, 40448, 5380, 6055, 3794, 27884, 6055, 7649, 1829, 3714, 43020, 321, 1949, 321, 1949, 732, 3331], "avg_logprob": -0.1262784101746299, "compression_ratio": 1.5238095238095237, "no_speech_prob": 0.0, "words": [{"start": 2086.31, "end": 2086.65, "word": "هذا", "probability": 0.917724609375}, {"start": 2086.65, "end": 2087.05, "word": " برهان", "probability": 0.924072265625}, {"start": 2087.05, "end": 2087.53, "word": " برهان", "probability": 0.8345947265625}, {"start": 2087.53, "end": 2088.03, "word": " تاني", "probability": 0.95361328125}, {"start": 2088.03, "end": 2088.65, "word": " ممكن", "probability": 0.986083984375}, {"start": 2088.65, "end": 2088.85, "word": " ان", "probability": 0.50732421875}, {"start": 2088.85, "end": 2089.21, "word": " احنا", "probability": 0.9480794270833334}, {"start": 2089.21, "end": 2090.33, "word": " نعمل", "probability": 0.9851888020833334}, {"start": 2090.33, "end": 2090.75, "word": " برهان", "probability": 0.98828125}, {"start": 2090.75, "end": 2091.29, "word": " مباشر", "probability": 0.993408203125}, {"start": 2091.29, "end": 2091.45, "word": " يعني", "probability": 0.816650390625}, {"start": 2091.45, "end": 2091.79, "word": " بلاش", "probability": 0.7978515625}, {"start": 2091.79, "end": 2092.49, "word": " نستخدم", "probability": 0.994384765625}, {"start": 2092.49, "end": 2093.55, 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"word": " ل", "probability": 0.78564453125}, {"start": 2375.67, "end": 2375.95, "word": " 6", "probability": 0.43017578125}, {"start": 2375.95, "end": 2376.33, "word": " هذه", "probability": 0.67822265625}, {"start": 2376.33, "end": 2377.93, "word": " لأن", "probability": 0.533203125}, {"start": 2377.93, "end": 2378.19, "word": " هذا", "probability": 0.9619140625}, {"start": 2378.19, "end": 2378.63, "word": " بيطلع", "probability": 0.919140625}, {"start": 2378.63, "end": 2379.21, "word": " أكبر", "probability": 0.9200846354166666}, {"start": 2379.21, "end": 2379.61, "word": " من", "probability": 0.9931640625}, {"start": 2379.61, "end": 2379.91, "word": " أو", "probability": 0.9912109375}, {"start": 2379.91, "end": 2380.61, "word": " ساوي", "probability": 0.7976888020833334}, {"start": 2380.61, "end": 2381.79, "word": " ..", "probability": 0.7373046875}, {"start": 2381.79, "end": 2382.27, "word": " هذا", "probability": 0.86962890625}, {"start": 2382.27, "end": 2382.83, "word": " أصغر", "probability": 0.9715576171875}, {"start": 2382.83, "end": 2383.01, "word": " من", "probability": 0.9970703125}, {"start": 2383.01, "end": 2383.17, "word": " أو", "probability": 0.98876953125}, {"start": 2383.17, "end": 2383.65, "word": " ساوي", "probability": 0.9703776041666666}, {"start": 2383.65, "end": 2383.83, "word": " ال", "probability": 0.60888671875}, {"start": 2383.83, "end": 2385.79, "word": " supremum", "probability": 0.719970703125}, {"start": 2385.79, "end": 2389.61, "word": " ل", "probability": 0.91650390625}, {"start": 2389.61, "end": 2390.03, "word": " S", "probability": 0.75439453125}, {"start": 2390.03, "end": 2390.69, "word": " star", "probability": 0.1995849609375}, {"start": 2390.69, "end": 2390.95, "word": " و", "probability": 0.55712890625}, {"start": 2390.95, "end": 2391.21, "word": " U", "probability": 0.87158203125}, {"start": 2391.21, "end": 2393.53, "word": " احنا", "probability": 0.826171875}, {"start": 2393.53, "end": 2393.83, "word": " بدنا", "probability": 0.61480712890625}, {"start": 2393.83, "end": 2394.39, "word": " مساوية", "probability": 0.8363037109375}, {"start": 2394.39, "end": 2395.05, "word": " صح؟", "probability": 0.900390625}], "temperature": 1.0}, {"id": 86, "seek": 241696, "start": 2396.21, "end": 2416.97, "text": "فبقدرش أستنتج مساواة هنا تمام؟ أما شو ممكن أما زي ما عملنا في البراهين السابقة ممكن نثبت ال claim ممكن نثبت المساواة كما يليه أنا عندي هذا .. هذا العدد .. هذا العدد عبارة عن upper bound لل set هذه", "tokens": [5172, 3555, 28543, 2288, 8592, 5551, 14851, 29399, 7435, 47524, 995, 14407, 3660, 34105, 46811, 10943, 22807, 5551, 15042, 13412, 2407, 3714, 43020, 5551, 15042, 30767, 1829, 19446, 6225, 42213, 8315, 8978, 2423, 26890, 40294, 9957, 21136, 16758, 28671, 3714, 43020, 8717, 12984, 3555, 2655, 2423, 3932, 3714, 43020, 8717, 12984, 3555, 2655, 9673, 3794, 995, 14407, 3660, 9122, 15042, 7251, 20292, 3224, 41850, 18871, 16254, 23758, 4386, 23758, 18863, 3215, 3215, 4386, 23758, 18863, 3215, 3215, 6225, 3555, 9640, 3660, 18871, 6597, 5472, 24976, 992, 29538], "avg_logprob": -0.21857244656844574, "compression_ratio": 1.7513227513227514, "no_speech_prob": 0.0, "words": [{"start": 2396.21, "end": 2396.85, "word": "فبقدرش", "probability": 0.6226806640625}, {"start": 2396.85, "end": 2397.31, "word": " أستنتج", "probability": 0.56085205078125}, {"start": 2397.31, "end": 2397.83, "word": " مساواة", "probability": 0.8321533203125}, {"start": 2397.83, "end": 2398.13, "word": " هنا", "probability": 0.91064453125}, {"start": 2398.13, "end": 2400.33, "word": " تمام؟", "probability": 0.5243733723958334}, {"start": 2400.33, "end": 2401.09, "word": " أما", "probability": 0.8720703125}, {"start": 2401.09, "end": 2401.39, "word": " شو", "probability": 0.914794921875}, {"start": 2401.39, "end": 2401.73, "word": " ممكن", "probability": 0.960693359375}, {"start": 2401.73, "end": 2402.09, "word": " أما", "probability": 0.628173828125}, {"start": 2402.09, "end": 2402.39, "word": " زي", "probability": 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"probability": 0.96298828125}, {"start": 2408.29, "end": 2408.77, "word": " كما", "probability": 0.9423828125}, {"start": 2408.77, "end": 2409.75, "word": " يليه", "probability": 0.84130859375}, {"start": 2409.75, "end": 2410.89, "word": " أنا", "probability": 0.68115234375}, {"start": 2410.89, "end": 2411.29, "word": " عندي", "probability": 0.88427734375}, {"start": 2411.29, "end": 2411.75, "word": " هذا", "probability": 0.8984375}, {"start": 2411.75, "end": 2411.83, "word": " ..", "probability": 0.320556640625}, {"start": 2411.83, "end": 2412.19, "word": " هذا", "probability": 0.9208984375}, {"start": 2412.19, "end": 2412.75, "word": " العدد", "probability": 0.958984375}, {"start": 2412.75, "end": 2412.83, "word": " ..", "probability": 0.640625}, {"start": 2412.83, "end": 2413.07, "word": " هذا", "probability": 0.95751953125}, {"start": 2413.07, "end": 2413.67, "word": " العدد", "probability": 0.9894205729166666}, {"start": 2413.67, "end": 2414.19, "word": " عبارة", "probability": 0.939697265625}, {"start": 2414.19, "end": 2414.39, "word": " عن", "probability": 0.9970703125}, {"start": 2414.39, "end": 2414.79, "word": " upper", "probability": 0.9072265625}, {"start": 2414.79, "end": 2415.43, "word": " bound", "probability": 0.88525390625}, {"start": 2415.43, "end": 2416.31, "word": " لل", "probability": 0.80078125}, {"start": 2416.31, "end": 2416.63, "word": " set", "probability": 0.50390625}, {"start": 2416.63, "end": 2416.97, "word": " هذه", "probability": 0.6787109375}], "temperature": 1.0}, {"id": 87, "seek": 243659, "start": 2418.47, "end": 2436.59, "text": "أحنا عايزين نثبت إن هذا مش upper bound هو ال least upper bound إذا ن claim إن ال supremum لست S union لست هذه هو العدد هذا", "tokens": [10721, 5016, 8315, 6225, 47302, 11622, 9957, 8717, 12984, 3555, 2655, 36145, 23758, 37893, 6597, 5472, 31439, 2423, 1935, 6597, 5472, 11933, 15730, 8717, 3932, 36145, 2423, 23710, 449, 5296, 14851, 318, 11671, 5296, 14851, 29538, 31439, 18863, 3215, 3215, 23758], "avg_logprob": -0.290922632529622, "compression_ratio": 1.359375, "no_speech_prob": 0.0, "words": [{"start": 2418.47, "end": 2418.83, "word": "أحنا", "probability": 0.7503255208333334}, {"start": 2418.83, "end": 2419.27, "word": " عايزين", "probability": 0.952880859375}, {"start": 2419.27, "end": 2419.77, "word": " نثبت", "probability": 0.991943359375}, {"start": 2419.77, "end": 2419.95, "word": " إن", "probability": 0.337158203125}, {"start": 2419.95, "end": 2420.31, "word": " هذا", "probability": 0.736328125}, {"start": 2420.31, "end": 2420.63, "word": " مش", "probability": 0.69580078125}, {"start": 2420.63, "end": 2420.95, "word": " upper", "probability": 0.7138671875}, {"start": 2420.95, "end": 2421.29, "word": " bound", "probability": 0.8779296875}, {"start": 2421.29, "end": 2421.63, "word": " هو", "probability": 0.869140625}, {"start": 2421.63, "end": 2422.05, "word": " ال", "probability": 0.779296875}, {"start": 2422.05, "end": 2422.27, "word": " least", "probability": 0.60205078125}, {"start": 2422.27, "end": 2422.59, "word": " upper", "probability": 0.9228515625}, {"start": 2422.59, "end": 2422.97, "word": " bound", "probability": 0.91845703125}, {"start": 2422.97, "end": 2423.69, "word": " إذا", "probability": 0.5987548828125}, {"start": 2423.69, "end": 2423.81, "word": " ن", "probability": 0.98779296875}, {"start": 2423.81, "end": 2424.25, "word": " claim", "probability": 0.755859375}, {"start": 2424.25, "end": 2424.83, "word": " إن", "probability": 0.8046875}, {"start": 2424.83, "end": 2425.37, "word": " ال", "probability": 0.79052734375}, {"start": 2425.37, "end": 2429.33, "word": " supremum", "probability": 0.500457763671875}, {"start": 2429.33, "end": 2430.79, "word": " لست", "probability": 0.719970703125}, {"start": 2430.79, "end": 2431.23, "word": " S", "probability": 0.5361328125}, {"start": 2431.23, "end": 2431.93, "word": " union", "probability": 0.59814453125}, {"start": 2431.93, "end": 2434.85, "word": " لست", "probability": 0.855224609375}, {"start": 2434.85, "end": 2435.29, "word": " هذه", "probability": 0.474853515625}, {"start": 2435.29, "end": 2435.73, "word": " هو", "probability": 0.89453125}, {"start": 2435.73, "end": 2436.23, "word": " العدد", "probability": 0.9754231770833334}, {"start": 2436.23, "end": 2436.59, "word": " هذا", "probability": 0.8984375}], "temperature": 1.0}, {"id": 88, "seek": 247264, "start": 2449.02, "end": 2472.64, "text": "انشوف let V be any upper bound لست S union singleton U هذا بيقدي ان X أصغر من أو بساوي او هذا بيقدي ان", "tokens": [7649, 8592, 38688, 718, 691, 312, 604, 6597, 5472, 5296, 14851, 318, 11671, 1522, 14806, 624, 23758, 4724, 1829, 4587, 16254, 16472, 1783, 5551, 9381, 17082, 2288, 9154, 34051, 4724, 3794, 995, 45865, 1975, 2407, 23758, 4724, 1829, 4587, 16254, 16472], "avg_logprob": -0.28720236888953615, "compression_ratio": 1.1983471074380165, "no_speech_prob": 0.0, "words": [{"start": 2449.02, "end": 2449.84, "word": "انشوف", "probability": 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0.75927734375}, {"start": 2466.72, "end": 2467.32, "word": " بيقدي", "probability": 0.6817626953125}, {"start": 2467.32, "end": 2469.16, "word": " ان", "probability": 0.7626953125}, {"start": 2469.16, "end": 2469.66, "word": " X", "probability": 0.8388671875}, {"start": 2469.66, "end": 2470.32, "word": " أصغر", "probability": 0.915771484375}, {"start": 2470.32, "end": 2470.56, "word": " من", "probability": 0.970703125}, {"start": 2470.56, "end": 2470.76, "word": " أو", "probability": 0.84130859375}, {"start": 2470.76, "end": 2471.34, "word": " بساوي", "probability": 0.766357421875}, {"start": 2471.34, "end": 2471.6, "word": " او", "probability": 0.7205810546875}, {"start": 2471.6, "end": 2471.84, "word": " هذا", "probability": 0.9091796875}, {"start": 2471.84, "end": 2472.32, "word": " بيقدي", "probability": 0.905517578125}, {"start": 2472.32, "end": 2472.64, "word": " ان", "probability": 0.84716796875}], "temperature": 1.0}, {"id": 89, "seek": 250398, "start": 2485.69, "end": 2503.99, "text": "هذا بيقدي أن x أصغر من أو يساوي S لكل x في S and x أصغر من أو يساوي لأ", "tokens": [3224, 15730, 4724, 1829, 4587, 16254, 14739, 2031, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 318, 5296, 28820, 2031, 8978, 318, 293, 2031, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 5296, 10721], "avg_logprob": -0.14873798688252768, "compression_ratio": 1.3452380952380953, "no_speech_prob": 0.0, "words": [{"start": 2485.69, "end": 2486.19, "word": "هذا", "probability": 0.9150390625}, {"start": 2486.19, "end": 2486.63, "word": " بيقدي", "probability": 0.72509765625}, {"start": 2486.63, "end": 2486.85, "word": " أن", "probability": 0.309814453125}, {"start": 2486.85, "end": 2487.33, "word": " x", "probability": 0.62451171875}, {"start": 2487.33, "end": 2488.03, "word": " أصغر", "probability": 0.9703369140625}, {"start": 2488.03, "end": 2488.29, "word": " من", "probability": 0.9814453125}, {"start": 2488.29, "end": 2488.55, "word": " أو", "probability": 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X أصغر أو بيساوي ال Supremum لأسطر Star يعني هي اللي هو", "tokens": [40228, 22653, 9673, 16095, 39896, 8032, 2655, 9566, 1211, 3615, 2423, 39184, 9141, 2579, 449, 5296, 14407, 24401, 3660, 9154, 9673, 7435, 2304, 45367, 16712, 7649, 1829, 41850, 12174, 36150, 10874, 2304, 37977, 2423, 1783, 5551, 11622, 25708, 13063, 41850, 12174, 9381, 16254, 36145, 5551, 4117, 2655, 2288, 1783, 5551, 9381, 17082, 2288, 34051, 4724, 1829, 3794, 995, 45865, 2423, 9141, 2579, 449, 37495, 22653, 4724, 8592, 28820, 3714, 7435, 2304, 2407, 5016, 3660, 36764, 24401, 3660, 1783, 5551, 9381, 17082, 2288, 34051, 4724, 1829, 3794, 995, 45865, 2423, 9141, 2579, 449, 5296, 10721, 3794, 9566, 2288, 5705, 37495, 22653, 39896, 13672, 1829, 31439], "avg_logprob": -0.3077380884261358, "compression_ratio": 1.8895027624309393, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2825.7, "end": 2825.94, "word": "يعني", "probability": 0.90478515625}, {"start": 2825.94, "end": 2826.3, "word": " المهم", 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3009.7, "end": 3010.1, "word": " طبعا", "probability": 0.9053955078125}, {"start": 3010.1, "end": 3010.36, "word": " هذه", "probability": 0.301025390625}, {"start": 3010.36, "end": 3011.22, "word": " finite", "probability": 0.96337890625}, {"start": 3011.22, "end": 3011.78, "word": " set", "probability": 0.9755859375}, {"start": 3011.78, "end": 3012.0, "word": " of", "probability": 0.96630859375}, {"start": 3012.0, "end": 3012.24, "word": " real", "probability": 0.9677734375}, {"start": 3012.24, "end": 3012.66, "word": " numbers", "probability": 0.7685546875}, {"start": 3012.66, "end": 3012.92, "word": " وفي", "probability": 0.6453857421875}, {"start": 3012.92, "end": 3013.4, "word": " تمرين", "probability": 0.8448893229166666}, {"start": 3013.4, "end": 3013.8, "word": " بيقول", "probability": 0.8997395833333334}, {"start": 3013.8, "end": 3014.56, "word": " لو", "probability": 0.880859375}, {"start": 3014.56, "end": 3014.86, "word": " عندي", "probability": 0.7398681640625}, {"start": 3014.86, "end": 3015.2, "word": " finite", "probability": 0.98828125}, {"start": 3015.2, "end": 3015.64, "word": " set", "probability": 0.9736328125}, {"start": 3015.64, "end": 3015.8, "word": " of", "probability": 0.97314453125}, {"start": 3015.8, "end": 3015.98, "word": " real", "probability": 0.966796875}, {"start": 3015.98, "end": 3016.48, "word": " numbers", "probability": 0.91015625}, {"start": 3016.48, "end": 3016.78, "word": " فال", "probability": 0.755859375}, {"start": 3016.78, "end": 3017.06, "word": " suprem", "probability": 0.783203125}, {"start": 3017.06, "end": 3017.68, "word": " تبعها", "probability": 0.928466796875}, {"start": 3017.68, "end": 3018.4, "word": " موجود", "probability": 0.9874674479166666}], "temperature": 1.0}, {"id": 110, "seek": 304679, "start": 3018.97, "end": 3046.79, "text": "و ينتمي لل set و ال infimum تبعها أيضا موجود و ينتمي ل .. يعني يكون عنصر في ال set هذا أحد التمارين اللي طبعا ما عليناهوش لكن بإمكانكم تثبتوه by induction فهذه finally ال set إذا ال suprem تبعها exist إلا أن هذا ال suprem أكبر من أو ساوي S star وبالتالي أكبر من أو ساوي X و هذا ال suprem أكبر من أو ساوي U", "tokens": [2407, 7251, 29399, 2304, 1829, 24976, 992, 4032, 2423, 1536, 332, 449, 6055, 3555, 3615, 11296, 36632, 11242, 995, 3714, 29245, 23328, 4032, 7251, 29399, 2304, 1829, 5296, 4386, 37495, 22653, 7251, 30544, 18871, 9381, 2288, 8978, 2423, 992, 23758, 5551, 24401, 16712, 2304, 9640, 9957, 13672, 1829, 23032, 3555, 3615, 995, 19446, 11203, 9957, 40294, 2407, 8592, 44381, 4724, 28814, 2304, 41361, 24793, 6055, 12984, 3555, 2655, 2407, 3224, 538, 33371, 6156, 3224, 24192, 2721, 2423, 992, 11933, 15730, 2423, 23710, 6055, 3555, 3615, 11296, 2514, 11933, 15040, 14739, 23758, 2423, 23710, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 318, 3543, 46599, 6027, 2655, 6027, 1829, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 1783, 4032, 23758, 2423, 23710, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 624], "avg_logprob": -0.2149332127043309, 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{"start": 3033.85, "end": 3034.09, "word": " set", "probability": 0.9873046875}, {"start": 3034.09, "end": 3034.31, "word": " إذا", "probability": 0.69189453125}, {"start": 3034.31, "end": 3034.47, "word": " ال", "probability": 0.865234375}, {"start": 3034.47, "end": 3034.73, "word": " suprem", "probability": 0.828125}, {"start": 3034.73, "end": 3035.25, "word": " تبعها", "probability": 0.9371337890625}, {"start": 3035.25, "end": 3035.77, "word": " exist", "probability": 0.90625}, {"start": 3035.77, "end": 3036.55, "word": " إلا", "probability": 0.504638671875}, {"start": 3036.55, "end": 3036.71, "word": " أن", "probability": 0.68408203125}, {"start": 3036.71, "end": 3036.93, "word": " هذا", "probability": 0.93212890625}, {"start": 3036.93, "end": 3037.09, "word": " ال", "probability": 0.8896484375}, {"start": 3037.09, "end": 3037.39, "word": " suprem", "probability": 0.876953125}, {"start": 3037.39, "end": 3038.01, "word": " أكبر", "probability": 0.9737955729166666}, {"start": 3038.01, "end": 3038.19, "word": " من", "probability": 0.9736328125}, {"start": 3038.19, "end": 3038.39, "word": " أو", "probability": 0.98046875}, {"start": 3038.39, "end": 3039.01, "word": " ساوي", "probability": 0.7952473958333334}, {"start": 3039.01, "end": 3039.35, "word": " S", "probability": 0.59423828125}, {"start": 3039.35, "end": 3039.85, "word": " star", "probability": 0.051971435546875}, {"start": 3039.85, "end": 3040.39, "word": " وبالتالي", "probability": 0.88828125}, {"start": 3040.39, "end": 3040.75, "word": " أكبر", "probability": 0.9723307291666666}, {"start": 3040.75, "end": 3040.87, "word": " من", "probability": 0.9931640625}, {"start": 3040.87, "end": 3041.07, "word": " أو", "probability": 0.9892578125}, {"start": 3041.07, "end": 3041.57, "word": " ساوي", "probability": 0.9703776041666666}, {"start": 3041.57, "end": 3041.99, "word": " X", "probability": 0.94775390625}, {"start": 3041.99, "end": 3042.91, "word": " و", "probability": 0.54443359375}, {"start": 3042.91, "end": 3043.15, "word": " هذا", "probability": 0.9248046875}, {"start": 3043.15, "end": 3043.35, "word": " ال", "probability": 0.91357421875}, {"start": 3043.35, "end": 3043.65, "word": " suprem", "probability": 0.9169921875}, {"start": 3043.65, "end": 3045.61, "word": " أكبر", "probability": 0.9775390625}, {"start": 3045.61, "end": 3045.81, "word": " من", "probability": 0.9970703125}, {"start": 3045.81, "end": 3046.01, "word": " أو", "probability": 0.99609375}, {"start": 3046.01, "end": 3046.47, "word": " ساوي", "probability": 0.95361328125}, {"start": 3046.47, "end": 3046.79, "word": " U", "probability": 0.52734375}], "temperature": 1.0}, {"id": 111, "seek": 306187, "start": 3050.61, "end": 3061.87, "text": "وبالتالي أكبر من أو يساوي ال X اللي هي U أكبر من أو ساوي، إذا الأن هذا الكلام صحيح لكل X ينتمي للإتحاد", "tokens": [37746, 6027, 2655, 6027, 1829, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 2423, 1783, 13672, 1829, 39896, 624, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 12399, 11933, 15730, 16247, 1863, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 1783, 7251, 29399, 2304, 1829, 24976, 28814, 2655, 5016, 18513], "avg_logprob": -0.2093460703337634, "compression_ratio": 1.6422018348623852, "no_speech_prob": 0.0, "words": [{"start": 3050.61, "end": 3051.57, "word": "وبالتالي", "probability": 0.839794921875}, {"start": 3051.57, "end": 3052.05, "word": " أكبر", "probability": 0.8052571614583334}, {"start": 3052.05, "end": 3052.25, "word": " من", "probability": 0.96923828125}, {"start": 3052.25, "end": 3052.43, "word": " أو", "probability": 0.974609375}, {"start": 3052.43, "end": 3053.11, "word": " يساوي", "probability": 0.8826904296875}, {"start": 3053.11, "end": 3053.89, "word": " ال", "probability": 0.77099609375}, {"start": 3053.89, "end": 3054.21, "word": " X", "probability": 0.619140625}, {"start": 3054.21, "end": 3054.45, "word": " اللي", "probability": 0.508544921875}, {"start": 3054.45, "end": 3054.73, "word": " هي", "probability": 0.83251953125}, {"start": 3054.73, "end": 3054.93, "word": " U", "probability": 0.8984375}, {"start": 3054.93, "end": 3055.29, "word": " أكبر", "probability": 0.9567057291666666}, {"start": 3055.29, "end": 3055.45, "word": " من", "probability": 0.99072265625}, {"start": 3055.45, "end": 3055.61, "word": " أو", "probability": 0.97998046875}, {"start": 3055.61, "end": 3056.19, "word": " ساوي،", "probability": 0.647796630859375}, {"start": 3056.19, "end": 3057.35, "word": " إذا", "probability": 0.65625}, {"start": 3057.35, "end": 3058.11, "word": " الأن", "probability": 0.736083984375}, {"start": 3058.11, "end": 3058.43, "word": " هذا", "probability": 0.93359375}, {"start": 3058.43, "end": 3058.87, "word": " الكلام", "probability": 0.9508463541666666}, {"start": 3058.87, "end": 3059.51, "word": " صحيح", "probability": 0.9951171875}, {"start": 3059.51, "end": 3060.09, "word": " لكل", "probability": 0.97314453125}, {"start": 3060.09, "end": 3060.57, "word": " X", "probability": 0.9892578125}, {"start": 3060.57, "end": 3061.15, "word": " ينتمي", "probability": 0.919921875}, {"start": 3061.15, "end": 3061.87, "word": " للإتحاد", "probability": 0.81923828125}], "temperature": 1.0}, {"id": 112, "seek": 308175, "start": 3063.67, "end": 3081.75, "text": "هذا العدد الان أكبر من أو ساوي كل عناصر ال 6 في الاتحاد فهو upper bound لل 6 هذه فهو upper ان العدد هذا upper bound لل 6 هذه الان أثبتنا ان هذا ال upper bound هو أصغر upper bound للاتحاد", "tokens": [3224, 15730, 18863, 3215, 3215, 2423, 7649, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 28242, 18871, 33546, 2288, 2423, 1386, 8978, 2423, 9307, 5016, 18513, 6156, 3224, 2407, 6597, 5472, 24976, 1386, 29538, 6156, 3224, 2407, 6597, 16472, 18863, 3215, 3215, 23758, 6597, 5472, 24976, 1386, 29538, 2423, 7649, 5551, 12984, 3555, 2655, 8315, 16472, 23758, 2423, 6597, 5472, 31439, 5551, 9381, 17082, 2288, 6597, 5472, 24976, 9307, 5016, 18513], "avg_logprob": -0.2074652732246452, "compression_ratio": 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"end": 3069.53, "word": " في", "probability": 0.83837890625}, {"start": 3069.53, "end": 3070.11, "word": " الاتحاد", "probability": 0.8594970703125}, {"start": 3070.11, "end": 3070.91, "word": " فهو", "probability": 0.9388020833333334}, {"start": 3070.91, "end": 3071.19, "word": " upper", "probability": 0.91259765625}, {"start": 3071.19, "end": 3071.65, "word": " bound", "probability": 0.91845703125}, {"start": 3071.65, "end": 3071.87, "word": " لل", "probability": 0.77685546875}, {"start": 3071.87, "end": 3072.17, "word": " 6", "probability": 0.78515625}, {"start": 3072.17, "end": 3072.47, "word": " هذه", "probability": 0.65673828125}, {"start": 3072.47, "end": 3072.89, "word": " فهو", "probability": 0.8525390625}, {"start": 3072.89, "end": 3073.15, "word": " upper", "probability": 0.73388671875}, {"start": 3073.15, "end": 3073.35, "word": " ان", "probability": 0.072998046875}, {"start": 3073.35, "end": 3073.75, "word": " العدد", "probability": 0.9765625}, {"start": 3073.75, "end": 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bound هو ال infimum هو ال greatest lower bound اذا هنا now if W is", "tokens": [34598, 5472, 5296, 14851, 318, 8717, 19913, 3555, 9566, 5296, 33456, 1783, 16, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 28242, 18863, 1863, 33546, 2288, 13672, 1829, 8978, 2423, 14851, 6156, 3224, 2407, 4032, 46958, 5016, 16472, 3224, 3126, 5472, 2423, 7649, 1975, 8315, 39894, 12984, 3555, 2655, 16472, 3224, 37893, 4724, 3794, 3126, 5472, 31439, 2423, 1536, 332, 449, 31439, 2423, 6636, 3126, 5472, 1975, 15730, 34105, 586, 498, 343, 307], "avg_logprob": -0.3018750047683716, "compression_ratio": 1.5593220338983051, "no_speech_prob": 0.0, "words": [{"start": 3395.4, "end": 3395.98, "word": "lower", "probability": 0.3759765625}, {"start": 3395.98, "end": 3396.62, "word": " bound", "probability": 0.88916015625}, {"start": 3396.62, "end": 3399.08, "word": " لست", "probability": 0.54638671875}, {"start": 3399.08, "end": 3399.54, "word": " S", "probability": 0.57275390625}, {"start": 3399.54, "end": 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0.720703125}, {"start": 3419.94, "end": 3420.52, "word": " W", "probability": 0.65283203125}, {"start": 3420.52, "end": 3421.62, "word": " is", "probability": 0.9541015625}], "temperature": 1.0}, {"id": 125, "seek": 344964, "start": 3424.4, "end": 3449.64, "text": "any lower bound .. any lower bound of S فهذا معناه أن W أصغر من أو يساوي Xi لكل I بيساوي 1 2 إلى N صح؟", "tokens": [1325, 3126, 5472, 4386, 604, 3126, 5472, 295, 318, 6156, 3224, 15730, 20449, 8315, 3224, 14739, 343, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 15712, 5296, 28820, 286, 4724, 1829, 3794, 995, 45865, 502, 568, 30731, 426, 20328, 5016, 22807], "avg_logprob": -0.25035512379624625, "compression_ratio": 1.146341463414634, "no_speech_prob": 0.0, "words": [{"start": 3424.4000000000005, "end": 3425.6400000000003, "word": "any", "probability": 0.496826171875}, {"start": 3425.6400000000003, "end": 3426.88, "word": " lower", "probability": 0.9208984375}, {"start": 3426.88, "end": 3427.52, "word": " 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أو ساوي كل عناصرها و بالتالي therefore w أصغر من أو ساوي x واحد لأن x واحد هو واحد من عناصر الست إذا أنا عندي الان x واحد is lower bound للست", "tokens": [2407, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 28242, 18871, 33546, 2288, 11296, 4032, 20666, 2655, 6027, 1829, 4412, 261, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2031, 36764, 24401, 5296, 33456, 2031, 36764, 24401, 31439, 36764, 24401, 9154, 18871, 33546, 2288, 2423, 14851, 11933, 15730, 41850, 18871, 16254, 2423, 7649, 2031, 36764, 24401, 307, 3126, 5472, 24976, 14851], "avg_logprob": -0.25903321243822575, "compression_ratio": 1.653061224489796, "no_speech_prob": 0.0, "words": [{"start": 3450.51, "end": 3451.25, "word": "و", "probability": 0.316650390625}, {"start": 3451.25, "end": 3453.83, "word": " أصغر", "probability": 0.733306884765625}, {"start": 3453.83, "end": 3453.95, "word": " من", "probability": 0.97900390625}, {"start": 3453.95, "end": 3454.07, "word": " أو", "probability": 0.962890625}, 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3486.45, "word": " او", "probability": 0.86865234375}, {"start": 3486.45, "end": 3486.61, "word": " ال", "probability": 0.96533203125}, {"start": 3486.61, "end": 3487.27, "word": " infimum", "probability": 0.82177734375}, {"start": 3487.27, "end": 3488.77, "word": " للست", "probability": 0.9306640625}, {"start": 3488.77, "end": 3489.27, "word": " s", "probability": 0.45751953125}, {"start": 3489.27, "end": 3490.29, "word": " exist", "probability": 0.8173828125}, {"start": 3490.29, "end": 3491.53, "word": " and", "probability": 0.92138671875}, {"start": 3491.53, "end": 3492.33, "word": " بيساوي", "probability": 0.919140625}, {"start": 3492.33, "end": 3492.65, "word": " x", "probability": 0.98486328125}, {"start": 3492.65, "end": 3493.17, "word": " واحد", "probability": 0.9853515625}, {"start": 3493.17, "end": 3496.33, "word": " تمام؟", "probability": 0.7903645833333334}, {"start": 3496.33, "end": 3496.93, "word": " بالمثل", "probability": 0.954833984375}, {"start": 3496.93, "end": 3497.33, "word": " ممكن", "probability": 0.98193359375}, {"start": 3497.33, "end": 3498.61, "word": " نثبت", "probability": 0.9869384765625}, {"start": 3498.61, "end": 3499.13, "word": " ال", "probability": 0.402099609375}, {"start": 3499.13, "end": 3500.39, "word": " ..", "probability": 0.74755859375}, {"start": 3500.39, "end": 3500.93, "word": " اه", "probability": 0.85986328125}, {"start": 3500.93, "end": 3501.93, "word": " هنا", "probability": 0.42529296875}, {"start": 3501.93, "end": 3502.61, "word": " similarly", "probability": 0.5166015625}], "temperature": 1.0}, {"id": 128, "seek": 351319, "start": 3506.41, "end": 3513.19, "text": "similarly show that ان انا هاسيبكم بطريقة مشابعة", "tokens": [30937, 2202, 356, 855, 300, 16472, 1975, 8315, 8032, 32277, 1829, 3555, 24793, 4724, 9566, 16572, 28671, 37893, 16758, 27884], "avg_logprob": -0.22172618479955764, "compression_ratio": 0.9113924050632911, "no_speech_prob": 0.0, "words": [{"start": 3506.41, "end": 3507.81, "word": "similarly", "probability": 0.7784016927083334}, {"start": 3507.81, "end": 3509.21, "word": " show", "probability": 0.77294921875}, {"start": 3509.21, "end": 3509.79, "word": " that", "probability": 0.90625}, {"start": 3509.79, "end": 3510.69, "word": " ان", "probability": 0.35595703125}, {"start": 3510.69, "end": 3510.79, "word": " انا", "probability": 0.804443359375}, {"start": 3510.79, "end": 3511.31, "word": " هاسيبكم", "probability": 0.80888671875}, {"start": 3511.31, "end": 3512.03, "word": " بطريقة", "probability": 0.9713134765625}, {"start": 3512.03, "end": 3513.19, "word": " مشابعة", "probability": 0.8595377604166666}], "temperature": 1.0}, {"id": 129, "seek": 354242, "start": 3514.44, "end": 3542.42, "text": "تثبتوا ال claim التاني وهو ان ال supremum لل set S exist و بساوي XN و طبعا هذا بينتمي لل set S و هو المطلوب okay تمام ان هيك بنكون أثبتنا ان اي finite set لها supremum لها infimum و هدولة بيطلعوا عناصر فيها بالتحديد ال infimum هو ال least element اصغر عنصر", "tokens": [2655, 12984, 3555, 2655, 14407, 2423, 3932, 16712, 7649, 1829, 37037, 2407, 16472, 2423, 23710, 449, 24976, 992, 318, 2514, 4032, 4724, 3794, 995, 45865, 1783, 45, 4032, 23032, 3555, 3615, 995, 23758, 4724, 9957, 39237, 1829, 24976, 992, 318, 4032, 31439, 9673, 9566, 1211, 37746, 1392, 46811, 10943, 16472, 39896, 4117, 44945, 30544, 5551, 12984, 3555, 2655, 8315, 16472, 1975, 1829, 19362, 992, 5296, 11296, 23710, 449, 5296, 11296, 1536, 332, 449, 4032, 8032, 3215, 12610, 3660, 4724, 1829, 9566, 1211, 3615, 14407, 18871, 33546, 2288, 8978, 11296, 20666, 2655, 5016, 16254, 3215, 2423, 1536, 332, 449, 31439, 2423, 1935, 4478, 1975, 9381, 17082, 2288, 18871, 9381, 2288], "avg_logprob": -0.19062500365755775, "compression_ratio": 1.6176470588235294, "no_speech_prob": 0.0, "words": [{"start": 3514.44, "end": 3515.06, "word": "تثبتوا", "probability": 0.91015625}, {"start": 3515.06, "end": 3515.16, "word": " ال", "probability": 0.76220703125}, {"start": 3515.16, "end": 3515.5, "word": " claim", "probability": 0.595703125}, {"start": 3515.5, "end": 3516.16, "word": " التاني", "probability": 0.8694661458333334}, {"start": 3516.16, "end": 3517.32, "word": " وهو", "probability": 0.851318359375}, {"start": 3517.32, "end": 3517.6, "word": " ان", "probability": 0.358642578125}, {"start": 3517.6, "end": 3517.86, "word": " ال", "probability": 0.74072265625}, {"start": 3517.86, "end": 3518.52, "word": " supremum", "probability": 0.80908203125}, {"start": 3518.52, "end": 3518.9, "word": " لل", "probability": 0.5966796875}, {"start": 3518.9, "end": 3519.28, "word": " set", "probability": 0.63916015625}, {"start": 3519.28, "end": 3519.92, "word": " S", "probability": 0.8740234375}, {"start": 3519.92, "end": 3521.28, "word": " exist", "probability": 0.6806640625}, {"start": 3521.28, "end": 3521.62, "word": " و", "probability": 0.796875}, {"start": 3521.62, "end": 3522.28, "word": " بساوي", "probability": 0.84375}, {"start": 3522.28, "end": 3523.16, "word": " XN", "probability": 0.47802734375}, {"start": 3523.16, "end": 3524.32, "word": " و", "probability": 0.84423828125}, {"start": 3524.32, "end": 3524.66, "word": " طبعا", "probability": 0.9144287109375}, {"start": 3524.66, "end": 3525.06, "word": " هذا", "probability": 0.84130859375}, {"start": 3525.06, "end": 3525.86, "word": " بينتمي", "probability": 0.734375}, {"start": 3525.86, "end": 3526.1, "word": " لل", "probability": 0.72119140625}, {"start": 3526.1, "end": 3526.38, "word": " set", "probability": 0.97119140625}, {"start": 3526.38, "end": 3526.86, "word": " S", "probability": 0.97998046875}, {"start": 3526.86, "end": 3527.42, "word": " و", "probability": 0.64794921875}, {"start": 3527.42, "end": 3527.62, "word": " هو", "probability": 0.96484375}, {"start": 3527.62, "end": 3528.2, "word": " المطلوب", "probability": 0.9930419921875}, {"start": 3528.2, "end": 3528.64, "word": " okay", "probability": 0.46435546875}, {"start": 3528.64, "end": 3529.14, "word": " تمام", "probability": 0.9208984375}, {"start": 3529.14, "end": 3530.08, "word": " ان", "probability": 0.50927734375}, {"start": 3530.08, "end": 3530.32, "word": " هيك", "probability": 0.738037109375}, {"start": 3530.32, "end": 3530.62, "word": " بنكون", "probability": 0.90185546875}, {"start": 3530.62, "end": 3531.2, "word": " أثبتنا", "probability": 0.94892578125}, {"start": 3531.2, "end": 3531.42, "word": " ان", "probability": 0.65576171875}, {"start": 3531.42, "end": 3531.68, "word": " اي", "probability": 0.760986328125}, {"start": 3531.68, "end": 3532.04, "word": " finite", "probability": 0.99658203125}, {"start": 3532.04, "end": 3532.56, "word": " set", "probability": 0.97802734375}, {"start": 3532.56, "end": 3533.92, "word": " لها", "probability": 0.561676025390625}, {"start": 3533.92, "end": 3534.48, "word": " supremum", "probability": 0.893310546875}, {"start": 3534.48, "end": 3534.82, "word": " لها", "probability": 0.806884765625}, {"start": 3534.82, "end": 3535.34, "word": " infimum", "probability": 0.9508463541666666}, {"start": 3535.34, "end": 3535.58, "word": " و", "probability": 0.7783203125}, {"start": 3535.58, "end": 3536.06, "word": " هدولة", "probability": 0.8963623046875}, {"start": 3536.06, "end": 3536.46, "word": " بيطلعوا", "probability": 0.9052734375}, {"start": 3536.46, "end": 3536.92, "word": " عناصر", "probability": 0.9860026041666666}, {"start": 3536.92, "end": 3537.3, "word": " فيها", "probability": 0.982177734375}, {"start": 3537.3, "end": 3538.72, "word": " بالتحديد", "probability": 0.96484375}, {"start": 3538.72, "end": 3538.92, "word": " ال", "probability": 0.9765625}, {"start": 3538.92, "end": 3539.44, "word": " infimum", "probability": 0.8653971354166666}, {"start": 3539.44, "end": 3539.78, "word": " هو", "probability": 0.98388671875}, {"start": 3539.78, "end": 3540.64, "word": " ال", "probability": 0.8857421875}, {"start": 3540.64, "end": 3540.98, "word": " least", "probability": 0.98193359375}, {"start": 3540.98, "end": 3541.48, "word": " element", "probability": 0.9716796875}, {"start": 3541.48, "end": 3541.96, "word": " اصغر", "probability": 0.86181640625}, {"start": 3541.96, "end": 3542.42, "word": " عنصر", "probability": 0.9923502604166666}], "temperature": 1.0}, {"id": 130, "seek": 355006, "start": 3543.14, "end": 3550.06, "text": "في ال set و ال supremum هو ال greatest element اللي هو أكبر أنصار في ال set", "tokens": [41185, 2423, 992, 4032, 2423, 23710, 449, 31439, 2423, 6636, 4478, 13672, 1829, 31439, 5551, 4117, 26890, 14739, 9381, 9640, 8978, 2423, 992], "avg_logprob": -0.24772135044137636, "compression_ratio": 1.2962962962962963, "no_speech_prob": 0.0, "words": [{"start": 3543.14, "end": 3543.32, "word": "في", "probability": 0.6103515625}, {"start": 3543.32, "end": 3543.48, "word": " ال", "probability": 0.85107421875}, {"start": 3543.48, "end": 3543.74, "word": " set", "probability": 0.279296875}, {"start": 3543.74, "end": 3544.64, "word": " و", "probability": 0.54150390625}, {"start": 3544.64, "end": 3544.94, "word": " ال", "probability": 0.82177734375}, {"start": 3544.94, "end": 3545.54, "word": " supremum", "probability": 0.39764404296875}, {"start": 3545.54, "end": 3545.78, "word": " هو", "probability": 0.98779296875}, {"start": 3545.78, "end": 3546.02, "word": " ال", "probability": 0.923828125}, {"start": 3546.02, "end": 3546.88, "word": " greatest", "probability": 0.95068359375}, {"start": 3546.88, "end": 3547.6, "word": " element", "probability": 0.9912109375}, {"start": 3547.6, "end": 3548.58, "word": " اللي", "probability": 0.922607421875}, {"start": 3548.58, "end": 3548.74, "word": " هو", "probability": 0.98974609375}, {"start": 3548.74, "end": 3549.1, "word": " أكبر", "probability": 0.9138997395833334}, {"start": 3549.1, "end": 3549.5, "word": " أنصار", "probability": 0.8917643229166666}, {"start": 3549.5, "end": 3549.62, "word": " في", "probability": 0.9833984375}, {"start": 3549.62, "end": 3549.76, "word": " ال", "probability": 0.97412109375}, {"start": 3549.76, "end": 3550.06, "word": " set", "probability": 0.99072265625}], "temperature": 1.0}, {"id": 131, "seek": 357540, "start": 3551.06, "end": 3575.4, "text": "هذا طبعا الكلام مش صحيح إذا ال set S كانت infinite هذا بس صحيح في حالة ال finite set إذا ال .. هذا بيكون بيكمل برهان التمرين هذا و بالتالي بنكتفي بحل أو بهذا القدر من حل التمرين و ان شاء الله أسبوع الجاي بنكمل حل تمرين أخرى", "tokens": [3224, 15730, 23032, 3555, 3615, 995, 2423, 28820, 10943, 37893, 20328, 5016, 1829, 5016, 11933, 15730, 2423, 992, 318, 25961, 2655, 13785, 23758, 4724, 3794, 20328, 5016, 1829, 5016, 8978, 11331, 6027, 3660, 2423, 19362, 992, 11933, 15730, 2423, 4386, 23758, 4724, 1829, 30544, 4724, 1829, 24793, 1211, 4724, 2288, 3224, 7649, 16712, 29973, 9957, 23758, 4032, 20666, 2655, 6027, 1829, 44945, 4117, 2655, 41185, 4724, 5016, 1211, 34051, 39627, 15730, 25062, 3215, 2288, 9154, 11331, 1211, 16712, 29973, 9957, 4032, 16472, 13412, 16606, 21984, 5551, 35457, 45367, 25724, 47302, 44945, 24793, 1211, 11331, 1211, 6055, 29973, 9957, 5551, 34740, 7578], "avg_logprob": -0.12262561354859203, "compression_ratio": 1.8613861386138615, "no_speech_prob": 0.0, "words": [{"start": 3551.06, "end": 3551.42, "word": "هذا", "probability": 0.92919921875}, {"start": 3551.42, "end": 3551.74, "word": " طبعا", "probability": 0.85107421875}, {"start": 3551.74, "end": 3552.18, "word": " الكلام", "probability": 0.9293619791666666}, {"start": 3552.18, "end": 3552.48, "word": " مش", "probability": 0.966796875}, {"start": 3552.48, "end": 3552.94, "word": " صحيح", "probability": 0.9952392578125}, {"start": 3552.94, "end": 3553.18, "word": " إذا", "probability": 0.797607421875}, {"start": 3553.18, "end": 3553.32, "word": " ال", "probability": 0.7294921875}, {"start": 3553.32, "end": 3553.54, "word": " set", "probability": 0.9482421875}, {"start": 3553.54, "end": 3553.8, "word": " S", "probability": 0.43310546875}, {"start": 3553.8, "end": 3554.26, "word": " كانت", "probability": 0.901123046875}, {"start": 3554.26, "end": 3554.62, "word": " infinite", "probability": 0.8466796875}, {"start": 3554.62, "end": 3555.56, "word": " هذا", "probability": 0.55224609375}, {"start": 3555.56, "end": 3555.84, "word": " بس", "probability": 0.966796875}, {"start": 3555.84, "end": 3556.22, "word": " صحيح", "probability": 0.99658203125}, {"start": 3556.22, "end": 3556.36, "word": " في", "probability": 0.9375}, {"start": 3556.36, "end": 3556.86, "word": " حالة", "probability": 0.9978841145833334}, {"start": 3556.86, "end": 3557.44, "word": " ال", "probability": 0.90185546875}, {"start": 3557.44, "end": 3557.74, "word": " finite", "probability": 0.984375}, {"start": 3557.74, "end": 3558.22, "word": " set", "probability": 0.98095703125}, {"start": 3558.22, "end": 3559.6, "word": " إذا", "probability": 0.697265625}, {"start": 3559.6, "end": 3560.08, "word": " ال", "probability": 0.6279296875}, {"start": 3560.08, "end": 3560.36, "word": " ..", "probability": 0.634765625}, {"start": 3560.36, "end": 3560.82, "word": " هذا", "probability": 0.7880859375}, {"start": 3560.82, "end": 3561.32, "word": " بيكون", "probability": 0.9451497395833334}, {"start": 3561.32, "end": 3562.0, "word": " بيكمل", "probability": 0.84527587890625}, {"start": 3562.0, "end": 3562.6, "word": " برهان", "probability": 0.9677734375}, {"start": 3562.6, "end": 3564.32, "word": " التمرين", "probability": 0.9052734375}, {"start": 3564.32, "end": 3564.68, "word": " هذا", "probability": 0.97998046875}, {"start": 3564.68, "end": 3565.14, "word": " و", "probability": 0.55615234375}, {"start": 3565.14, "end": 3565.66, "word": " بالتالي", "probability": 0.90625}, {"start": 3565.66, "end": 3566.28, "word": " بنكتفي", "probability": 0.9171142578125}, {"start": 3566.28, "end": 3568.04, "word": " بحل", "probability": 0.9715169270833334}, {"start": 3568.04, "end": 3569.22, "word": " أو", "probability": 0.75537109375}, {"start": 3569.22, "end": 3569.6, "word": " بهذا", "probability": 0.9716796875}, {"start": 3569.6, "end": 3570.02, "word": " القدر", "probability": 0.9943033854166666}, {"start": 3570.02, "end": 3570.22, "word": " من", "probability": 0.994140625}, {"start": 3570.22, "end": 3570.5, "word": " حل", "probability": 0.992431640625}, {"start": 3570.5, "end": 3571.12, "word": " التمرين", "probability": 0.9938151041666666}, {"start": 3571.12, "end": 3571.28, "word": " و", "probability": 0.98974609375}, {"start": 3571.28, "end": 3571.36, "word": " ان", "probability": 0.54931640625}, {"start": 3571.36, "end": 3571.58, "word": " شاء", "probability": 0.9873046875}, {"start": 3571.58, "end": 3571.7, "word": " الله", "probability": 0.95849609375}, {"start": 3571.7, "end": 3572.1, "word": " أسبوع", "probability": 0.8404947916666666}, {"start": 3572.1, "end": 3572.54, "word": " الجاي", "probability": 0.568603515625}, {"start": 3572.54, "end": 3573.94, "word": " بنكمل", "probability": 0.9462890625}, {"start": 3573.94, "end": 3574.26, "word": " حل", "probability": 0.993896484375}, {"start": 3574.26, "end": 3574.88, "word": " تمرين", "probability": 0.9461263020833334}, {"start": 3574.88, "end": 3575.4, "word": " أخرى", "probability": 0.97705078125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3576.15175, "duration_after_vad": 3214.81812499998}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e611318aa053aee6e767730ecf81d8fef433fd01
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/8Xs3EWM1_9g_postprocess.srt
@@ -0,0 +1,896 @@
+1
+00:00:21,620 --> 00:00:25,660
+طيب ناخد أمثلة
+
+2
+00:00:25,660 --> 00:00:31,280
+كيف نجيب ال supremum و ال infimum لمجموعات جزئية
+
+3
+00:00:31,280 --> 00:00:36,300
+من مجموعة الأعداد الحقيقية فلو أخدت الفترة المغلقة
+
+4
+00:00:36,300 --> 00:00:42,660
+من سفر لواحد فعايز أفبت claim هنا ادعي ان ال
+
+5
+00:00:42,660 --> 00:00:48,540
+supremum لست اسم سار واحدلبرهان ذلك حسب تعريف ال
+
+6
+00:00:48,540 --> 00:00:53,320
+supremum اللي هو least upper bound لازم أثبت شرطين
+
+7
+00:00:53,320 --> 00:00:59,140
+أول شي الواحد upper bound ل S وهذا صحيح واضح واحد
+
+8
+00:00:59,140 --> 00:01:03,860
+is upper bound لمجموع S لأن الواحد أكبر من أو
+
+9
+00:01:03,860 --> 00:01:08,930
+يساوي كل العناصر اللي في الفترة صح؟إذاً واحد upper
+
+10
+00:01:08,930 --> 00:01:13,170
+bound الآن لإثبات أن واحد هو أصغر upper bound ال
+
+11
+00:01:13,170 --> 00:01:16,950
+supremum يعني لازم أثبته أن واحد أصغر من أو ساوي
+
+12
+00:01:16,950 --> 00:01:25,170
+أي upper bound فلو خدنا V V any upper bound فال V
+
+13
+00:01:25,170 --> 00:01:28,310
+أكبر من أو ساوي كل العناصر اللي هنا من ضمنها
+
+14
+00:01:28,310 --> 00:01:33,530
+الواحدإذن ال V أكبر من أو ساوي ال واحد الان واحد
+
+15
+00:01:33,530 --> 00:01:38,230
+upper bound والواحد أصغر من أو ساوي أي upper bound
+
+16
+00:01:38,230 --> 00:01:43,910
+V إذن ال واحد هو ال supremum إذن هيك أثبتنا إن
+
+17
+00:01:43,910 --> 00:01:49,390
+واحد هو ال supremum بالمثل ممكن أثبات إن العنصر أو
+
+18
+00:01:49,390 --> 00:01:54,170
+العدد سفر هو ال infimum للفترة المغلقة من سفر إلى
+
+19
+00:01:54,170 --> 00:02:00,850
+واحدطيب مثال تاني لو أخدت T هي الفترة المفتوحة من
+
+20
+00:02:00,850 --> 00:02:11,950
+0 ل1 فبرضه كمان لو
+
+21
+00:02:11,950 --> 00:02:18,030
+أخدت T هي الفترة المفتوحة من 0 ل1 فممكن أثبات أن
+
+22
+00:02:18,030 --> 00:02:23,970
+ال supremum ل T هو 1واضح ان الواحد upper bound
+
+23
+00:02:23,970 --> 00:02:29,030
+للست للفترة المفتوحة لأن واحد أكبر من أو ساوي كل
+
+24
+00:02:29,030 --> 00:02:34,390
+ال X اللي هنا هذا واضح الان لإثبات أن الواحد هذا
+
+25
+00:02:34,390 --> 00:02:37,310
+هو ال supremum في لمّة واحد اتناش خدناها المرة
+
+26
+00:02:37,310 --> 00:02:42,070
+اللي فاتت بتقول عشان ال upper bound واحد يكون هو
+
+27
+00:02:42,070 --> 00:02:47,310
+ال supremum لازم أثبت أنه في شرط لكل ابسلون أكبر
+
+28
+00:02:47,310 --> 00:02:56,120
+من السفر يوجدعنصر S Y في السفر S أو T هنا بحيث أنه
+
+29
+00:02:56,120 --> 00:03:02,300
+واحد سالب ال epsilon أصغر من S epsilon فهنثبت
+
+30
+00:03:02,300 --> 00:03:07,900
+الكلام هذا إذن هنا هينبدأ let epsilon أكبر من
+
+31
+00:03:07,900 --> 00:03:11,940
+السفر be given لأن ال epsilon هذا ممكن يكون أصغر
+
+32
+00:03:11,940 --> 00:03:17,980
+من أو ساوي الواحد أو أكبر من أو أكبر من الواحد
+
+33
+00:03:20,030 --> 00:03:22,970
+الإبسلون هذا عدد موجب ممكن جدا يكون أصغر من أو
+
+34
+00:03:22,970 --> 00:03:26,170
+ساوي الواحد أو أكبر من واحد ناخد الحالة الأولى، لو
+
+35
+00:03:26,170 --> 00:03:30,770
+إبسلون أصغر من أو ساوي الواحد فحاخد S إبسلون، أعرف
+
+36
+00:03:30,770 --> 00:03:36,330
+S إبسلون واحد سالب إبسلون على اتنين هذا العدد
+
+37
+00:03:36,330 --> 00:03:41,350
+بيطلع عدد أكبر من سفر وأصغر من واحد وبالتالي ينتمي
+
+38
+00:03:41,350 --> 00:03:45,510
+لتين الآن
+
+39
+00:03:45,510 --> 00:03:53,380
+لو أخدت واحد وطرحت منها إبسلونفهذا بيطلع أصغر يعني
+
+40
+00:03:53,380 --> 00:03:59,840
+لو أخدت واحد و طرحت منها epsilon فهذا أصغر من واحد
+
+41
+00:03:59,840 --> 00:04:06,500
+سالب epsilon ع اتنين هذا طرحت منه عدد أكبر من هذا
+
+42
+00:04:06,500 --> 00:04:17,080
+لذا هذا أصغر من التاني و بعدين ليش يقصر؟ طب
+
+43
+00:04:17,080 --> 00:04:25,100
+ما هذا هو S epsilonهذا هو سإبسلون إذا
+
+44
+00:04:25,100 --> 00:04:30,160
+في الحالة هذه لأي إبسلون أكبر من السفر هين أثبتت
+
+45
+00:04:30,160 --> 00:04:36,740
+إن يوجد سإبسلون في T وهذا الـ S إبسلون أكبر من
+
+46
+00:04:36,740 --> 00:04:40,600
+واحد سالب إبسلون أو واحد سالب إبسلون أصغر من S
+
+47
+00:04:40,600 --> 00:04:47,480
+إبسلون هذا هو الشرط اللي في لمبة واحد اتناش هينتقل
+
+48
+00:04:48,090 --> 00:04:52,170
+الحالة التانية، لو كان إمسنان أكبر من واحد فأكيد
+
+49
+00:04:52,170 --> 00:04:56,050
+واحد سالب إمسنان هيطلع عدد سالب، يعني أصغر من سفر،
+
+50
+00:04:56,050 --> 00:05:01,930
+وال X هذا .. ال X هذا لو أخدت أي X في T فأي X في T
+
+51
+00:05:01,930 --> 00:05:06,300
+موجب، أي X في T موجبإذن هين أثبتنا في الحالة
+
+52
+00:05:06,300 --> 00:05:13,160
+التانية إنه لو كان epsilon أكبر من واحد فبطلع مش
+
+53
+00:05:13,160 --> 00:05:18,620
+يوجد S epsilon واحد في T كل عناصر ال T بتحقق إنه
+
+54
+00:05:18,620 --> 00:05:24,120
+واحد سالب epsilon أصغر من S أو S epsilon وبالتالي
+
+55
+00:05:24,120 --> 00:05:28,420
+في كلتال حالتين ال both cases الشرط تبع لما واحد
+
+56
+00:05:28,420 --> 00:05:33,490
+اتناشر تبع ال supremum اللي بكافئ ال supremumمتحقق
+
+57
+00:05:33,490 --> 00:05:39,810
+وبالتالي واحد هو ال supremum لتين مثال
+
+58
+00:05:39,810 --> 00:05:46,710
+تالت احنا شفنا قبل شوية في بداية المحاضرة ان كل
+
+59
+00:05:46,710 --> 00:05:51,510
+عدد حقيقي هو upper bound و كذلك lower bound
+
+60
+00:05:51,510 --> 00:05:57,070
+للمجموع الخالي Phi و بناء على ذلك Phi does not
+
+61
+00:05:57,070 --> 00:06:00,730
+have a supremum ولا infimum
+
+62
+00:06:03,600 --> 00:06:14,960
+هي برهان فاي has no .. فاي has no supremum البرهان
+
+63
+00:06:14,960 --> 00:06:19,380
+proof assume
+
+64
+00:06:19,380 --> 00:06:24,240
+you
+
+65
+00:06:24,240 --> 00:06:32,620
+belong to R is supremum فاي ال least upper bound
+
+66
+00:06:32,620 --> 00:06:33,120
+لفاي
+
+67
+00:06:40,890 --> 00:06:53,830
+then u سالب واحد أصغر من u and u سالب واحد هاد عدد
+
+68
+00:06:53,830 --> 00:07:00,610
+حقيقي is upper bound
+
+69
+00:07:00,610 --> 00:07:13,110
+of ال fiveكمان مرة نفرض ان U جد U نفرض
+
+70
+00:07:13,110 --> 00:07:21,590
+ان U جد U جد U بالنمط R و هو Supremum ل Phi طيب U
+
+71
+00:07:21,590 --> 00:07:27,000
+سالب واحد أصغر من Uو قبل شوية كنا ملاحظة ان اي عدد
+
+72
+00:07:27,000 --> 00:07:32,440
+حقيقي زي هذا عبارة عن upper bound لفائي ف K في ال
+
+73
+00:07:32,440 --> 00:07:37,080
+U .. K في ال U هو ال supremum K في ال U هو ال
+
+74
+00:07:37,080 --> 00:07:40,580
+supremum هو أصغر upper bound و في upper bound أصغر
+
+75
+00:07:40,580 --> 00:07:47,260
+منه هذا بدي تناقض which
+
+76
+00:07:47,260 --> 00:07:52,340
+.. which is a contradiction
+
+77
+00:07:59,520 --> 00:08:04,320
+إن هذا بدّيني تناقض وبالتالي هذا أثبات أن الـ Fi
+
+78
+00:08:04,320 --> 00:08:10,700
+مالهاش Supremum بالمثل ممكن أثبات أن الـ Fi أو
+
+79
+00:08:10,700 --> 00:08:20,420
+المجموعة الخالية ليس لها Supremum طيب
+
+80
+00:08:20,420 --> 00:08:22,620
+نيجي لل completeness property
+
+81
+00:08:29,610 --> 00:08:34,370
+الـ completeness property of R بتنص على إنه كل
+
+82
+00:08:34,370 --> 00:08:40,990
+مجموعة غير خالية .. كل مجموعة غير خالية S من R و
+
+83
+00:08:40,990 --> 00:08:45,010
+bounded above .. و bounded above محدودة من أعلى
+
+84
+00:08:45,010 --> 00:08:50,430
+has supremum لازم يكون فيه لها supremum يعني مثال
+
+85
+00:08:50,430 --> 00:08:57,580
+على ذلك لو أخدنا S بسبب الفترة المغلقة 01 أوالفترة
+
+86
+00:08:57,580 --> 00:09:04,960
+مفتوحة من صفر واحد فهي هذي set و bounded above اذا
+
+87
+00:09:04,960 --> 00:09:10,960
+ال property بتقولي بتضمنلي تضمن ان هذي ال set لها
+
+88
+00:09:10,960 --> 00:09:15,840
+soprano اللي هو الواحد اللي اثبتناه قبل شوية اذا
+
+89
+00:09:15,840 --> 00:09:19,700
+ال property بتضمن وجود soprano لكن ما بتجيبليها
+
+90
+00:09:19,700 --> 00:09:26,050
+ولا بتقوليإيش هو؟ عشان نجيبه لازم نعمل برهان زي ما
+
+91
+00:09:26,050 --> 00:09:30,310
+شوفنا في الأمثلة السابقة هد هي ال supremum أو ال
+
+92
+00:09:30,310 --> 00:09:33,790
+completeness property خاصية التمام للأعداد
+
+93
+00:09:33,790 --> 00:09:38,510
+الحقيقية الآن زي ما قلتلكم قبل هيك في توقع ما بين
+
+94
+00:09:38,510 --> 00:09:42,130
+ال upper bounds و ال lower bounds ال supremums و
+
+95
+00:09:42,130 --> 00:09:52,510
+ال infimumsفال .. ال .. اي خاصية صحيحة لل supreme
+
+96
+00:09:52,510 --> 00:09:58,170
+بتكون في بقابلها خاصية صحيحة لل infimum ففي نتيجة
+
+97
+00:09:58,170 --> 00:10:03,640
+هنا على completeness property corollaryبنسميها الـ
+
+98
+00:10:03,640 --> 00:10:07,580
+infimum property of R لإن في supremum property of
+
+99
+00:10:07,580 --> 00:10:12,260
+R وفي بقبلها infimum property of R فال infimum
+
+100
+00:10:12,260 --> 00:10:16,160
+property of R بتقول ان every non-empty subset S of
+
+101
+00:10:16,160 --> 00:10:21,160
+R which is bounded below has an infimum يعني كل
+
+102
+00:10:21,160 --> 00:10:26,440
+مجموعة غير خالية من العداد الحقيقية ومحصورة من
+
+103
+00:10:26,440 --> 00:10:30,460
+أسفل لازم يكون لها infimum أو أكبر حد أدنى
+
+104
+00:10:38,820 --> 00:10:45,060
+وهي البرهان .. نشوف البرهان تبع ال .. ال corollary
+
+105
+00:10:45,060 --> 00:10:54,520
+أو النتيجة هذه بنعرف set .. بنعرف ال set E علي
+
+106
+00:10:54,520 --> 00:10:59,120
+أنها كل العناصر W اللي بتكون lower bound للمجموعة
+
+107
+00:10:59,120 --> 00:11:06,510
+S طيب by hypothesis حسب الفرضالـ E مجموعة غير
+
+108
+00:11:06,510 --> 00:11:09,610
+خالية، يعني فيها على الأقل عنصر، ليه؟ لإن احنا
+
+109
+00:11:09,610 --> 00:11:16,090
+فرضين إن المجموعة S، المجموعة S هذه bounded below،
+
+110
+00:11:16,090 --> 00:11:19,710
+يعني إلها lower bound وبالتالي إذا في على الأقل
+
+111
+00:11:19,710 --> 00:11:24,350
+عنصر واحد، W في E، إذا الـ E مجموعة غير خالية،
+
+112
+00:11:24,350 --> 00:11:25,990
+تمام؟ هذا من الفرض
+
+113
+00:11:29,380 --> 00:11:34,720
+كذلك من الفرض أي X في S ثبار عن upper bound لـ E
+
+114
+00:11:34,720 --> 00:11:49,760
+لو كان X ينتمي إلى S فهذا بيقدّي انه W أصغر من أو
+
+115
+00:11:49,760 --> 00:11:56,160
+يساوي X لكل W في E
+
+116
+00:12:04,760 --> 00:12:11,300
+ليش هذا الكلام صحيح؟ لأن كل W في E عبارة عن lower
+
+117
+00:12:11,300 --> 00:12:17,300
+bound ل S وبما أن W lower bound ل S فأي أنصر في S
+
+118
+00:12:17,300 --> 00:12:23,480
+بيكون أكبر من أو ساوي ال lower bound، صح؟ إذن هذا
+
+119
+00:12:23,480 --> 00:12:28,360
+معناه إن X upper bound هي X أكبر من أو ساوي كل
+
+120
+00:12:28,360 --> 00:12:33,820
+عناصر ال E وبالتالي أي X في S هو عبارة عن
+
+121
+00:12:40,550 --> 00:12:45,910
+أي x في s هو upper bound للست
+
+122
+00:12:51,680 --> 00:12:57,900
+خاصية التمام، إذا ال .. ال set E هذه is bounded
+
+123
+00:12:57,900 --> 00:13:02,580
+above وبالتالي يوجد إلها suprem، ال suprem تبعها
+
+124
+00:13:02,580 --> 00:13:08,100
+لو سميته small s exists in R هذا .. وجود ال suprem
+
+125
+00:13:08,100 --> 00:13:14,560
+مضمون باستخدام ال suprem propertyالان بدنا نثبت ان
+
+126
+00:13:14,560 --> 00:13:21,000
+هذا العدد small s هو الـ infimum هو الـ infimum
+
+127
+00:13:21,000 --> 00:13:27,100
+للست S وهيك بنكون كملنا البرهان إذا الإثبات
+
+128
+00:13:27,100 --> 00:13:33,580
+للادعاء هذا ان عندي ال S هنا بساوي supremum E
+
+129
+00:13:33,580 --> 00:13:40,780
+وبالتالي ال S هذا upper bound ل E يعني S أكبر من
+
+130
+00:13:40,780 --> 00:13:42,340
+أو ساوي كل ال X في E
+
+131
+00:13:46,050 --> 00:13:52,070
+الأن بناء على المتباينة هذه أو الجملة هذه لإثبات
+
+132
+00:13:52,070 --> 00:13:58,610
+أن S هي الـ infimum لcapital S يبقى إثبات أن S
+
+133
+00:13:58,610 --> 00:14:06,830
+عبارة عن lower bound S is a lower bound of S ليش
+
+134
+00:14:06,830 --> 00:14:11,350
+هذا يكفي لإثبات أن S هو الinfimum لS؟
+
+135
+00:14:15,610 --> 00:14:20,590
+تعالى نشوف ليش هذا يكفي يكفي
+
+136
+00:14:20,590 --> 00:14:28,850
+اثبات ان ال S is a lower bound لل 6S يعني بدنا
+
+137
+00:14:28,850 --> 00:14:34,830
+نثبت ان ال X عفوا
+
+138
+00:14:34,830 --> 00:14:43,410
+ال S أصغر من أو ساوي كل العناصر Y
+
+139
+00:14:58,200 --> 00:15:03,540
+يعني بدنا نثبت أن S ينتمي
+
+140
+00:15:03,540 --> 00:15:09,980
+للset E يعني
+
+141
+00:15:09,980 --> 00:15:17,320
+لإثبات أن S is the lower bound of S معناه بد أثبت
+
+142
+00:15:17,320 --> 00:15:20,560
+أن S عنصر في E لأن E is the set of all lower
+
+143
+00:15:20,560 --> 00:15:25,380
+bounds of S صح؟ فلو أثبتت أن S تنتمي إلى E
+
+144
+00:15:34,100 --> 00:15:41,300
+فالمفروض هذا معناه ان ال S .. اه هايه .. لو هذا ال
+
+145
+00:15:41,300 --> 00:15:47,680
+S .. لو هذا ال S أثبتت انه .. لو أثبتت ان ال S هذا
+
+146
+00:15:47,680 --> 00:15:49,380
+ينتمي إلى ايه؟
+
+147
+00:15:52,900 --> 00:15:58,420
+فمعناه ان كل العناصر اللي في E أصغر من أو يساوي ال
+
+148
+00:15:58,420 --> 00:16:04,900
+S طيب كل العناصر X اللي في E هي عبارة عن lower
+
+149
+00:16:04,900 --> 00:16:11,330
+bounds ل Sواذا كان S موجود في E بيكون أيضا lower
+
+150
+00:16:11,330 --> 00:16:17,350
+bound ل S لكن ال S هذا بتمتع بالخاصية أنه أكبر من
+
+151
+00:16:17,350 --> 00:16:22,970
+أو ساوي كل عناصر ال set A إذا هو أكبر lower bound
+
+152
+00:16:22,970 --> 00:16:29,560
+يعني هو ال infimum صح؟ تمام؟مرة تانية احنا وصلنا
+
+153
+00:16:29,560 --> 00:16:35,780
+ان ال X كل العناصر X في E اصغر من او ساوي S الان
+
+154
+00:16:35,780 --> 00:16:42,800
+لو اثبتت ان ال S هذا ينتمي ل E يعني lower bound ل
+
+155
+00:16:42,800 --> 00:16:50,130
+Sمعناته ال S هدى اكبر من او ساوي كل عناصر ال 6E
+
+156
+00:16:50,130 --> 00:16:54,890
+وبالتالي هو اكبر lower
+
+157
+00:16:54,890 --> 00:17:02,450
+bound يعني هو ال infimum اذا فعلا يكفي او يبقى
+
+158
+00:17:02,450 --> 00:17:06,990
+اثبات ان ال S اسمه ال S lower bound لل 6S فلبرهان
+
+159
+00:17:06,990 --> 00:17:11,770
+ذلك بنعمل برهان بالتناقض افرضى انه اللي احنا
+
+160
+00:17:11,770 --> 00:17:18,960
+بنلثبته خطأيعني اسمه ال S ليس lower bound للست S
+
+161
+00:17:18,960 --> 00:17:23,500
+هذا معناه بقدر ألاجي أنصر Y في S و هذا ال Y أصغر
+
+162
+00:17:23,500 --> 00:17:30,600
+من S لأن S ليس lower bound فهذا بيقدي .. لاحظوا أن
+
+163
+00:17:30,600 --> 00:17:35,400
+ال S هو ال supremum ل E .. S هو ال supremum ل E و
+
+164
+00:17:35,400 --> 00:17:42,980
+Y أصغر منه إذن Y هذا مش ممكن يكون upper bound للست
+
+165
+00:17:42,980 --> 00:17:49,920
+Eال Y أصغر من S و S بساوي supremum E إذا Y مش ممكن
+
+166
+00:17:49,920 --> 00:17:54,740
+يكون upper bound ل E لأنه بجوزش هذا يكون upper
+
+167
+00:17:54,740 --> 00:18:00,320
+bound ل E و هذا أصغر upper bound ل E صح؟ طيب إذا
+
+168
+00:18:00,320 --> 00:18:05,980
+ال Y مش ممكن يكون upper bound ل E إذا بقدر ألاقي X
+
+169
+00:18:05,980 --> 00:18:12,160
+في E و هذا ال X أكبر من ال Y هذه المتباينة بتعطيني
+
+170
+00:18:12,160 --> 00:18:12,840
+تناقض
+
+171
+00:18:16,450 --> 00:18:23,870
+تتناقض مع تعريف ال set E كيف X تنتمي ل E كيف ال X
+
+172
+00:18:23,870 --> 00:18:29,510
+تنتمي ل E و في نفس الوجهة X أكبر من عنصر ما اللي
+
+173
+00:18:29,510 --> 00:18:35,010
+هو Y في S يعني ال X هذا ليس lower bound هذا تناقض
+
+174
+00:18:35,010 --> 00:18:40,130
+okay إذا نصل إلى تناقض وبالتالي هذا التناقض بيقول
+
+175
+00:18:40,130 --> 00:18:42,990
+لي أن الفرض الفرض تبعنا هذا
+
+176
+00:18:45,580 --> 00:18:50,800
+إن small s is not lower bound كان فرض خطأ إذا لازم
+
+177
+00:18:50,800 --> 00:19:01,520
+يكون s lower bound وهذا بيكمل برهان ال claim تمام؟
+
+178
+00:19:01,520 --> 00:19:08,040
+في
+
+179
+00:19:08,040 --> 00:19:09,500
+ال section القادم
+
+180
+00:19:12,270 --> 00:19:18,530
+هناخد تطبيقات على الـ supreme property و ال infame
+
+181
+00:19:18,530 --> 00:19:24,410
+property فالتطبيقات
+
+182
+00:19:24,410 --> 00:19:35,230
+هذه هتكون على شكل أمثلة فمثلا
+
+183
+00:19:35,230 --> 00:19:43,410
+أول تطبيقلو أخدت أي subset من R و bounded above و
+
+184
+00:19:43,410 --> 00:19:49,510
+A أي عدد حقيقي فمنعرف A زائد capital S على أنه
+
+185
+00:19:49,510 --> 00:19:54,110
+مجموعة كل العناصر على الصورة A plus X حيث X ينتمي
+
+186
+00:19:54,110 --> 00:20:00,890
+لS الآن ممكن أثبت أن ال supremum للمجموعة هذه هو
+
+187
+00:20:00,890 --> 00:20:04,870
+عبارة عن A زائد ال supremum لS
+
+188
+00:20:07,460 --> 00:20:16,840
+و هذا يعني البرهان مش صعب أيه بسيط وسهل نشوف مع
+
+189
+00:20:16,840 --> 00:20:22,540
+بعض نفرض ان U هو ال suprem ل S ال set S is bounded
+
+190
+00:20:22,540 --> 00:20:28,980
+above، إذن إلها suprem هذا مضمون حسب ال suprem
+
+191
+00:20:28,980 --> 00:20:33,920
+propertyوبالتالي الـ U هذا اللي هو ال supreme هو
+
+192
+00:20:33,920 --> 00:20:38,520
+upper bound ل S إذا U أكبر من أو ساوي كل عناصر ال
+
+193
+00:20:38,520 --> 00:20:45,800
+S إذا لو ضفت A على الطرفين فبطلع A زاد X أصغر من
+
+194
+00:20:45,800 --> 00:20:54,270
+أو ساوي A زاد U لكل X في S وبالتالي العدد هذاعبارة
+
+195
+00:20:54,270 --> 00:20:59,830
+عن upper bound لمن؟ لست a زاد s اللي عرفناها قبل
+
+196
+00:20:59,830 --> 00:21:04,310
+شوية لأن هذا العدد أكبر من أو ساوي كل عناصر الست
+
+197
+00:21:04,310 --> 00:21:08,850
+هذه اللي على الصورة a زاد x لذلك هي اللي أثبتت أن
+
+198
+00:21:08,850 --> 00:21:13,110
+a زاد u is upper bound للست هذه لأن نريد أن نثبت
+
+199
+00:21:13,110 --> 00:21:18,510
+أن a زاد u هو أصغر upper bound للست هذه فبناخد أي
+
+200
+00:21:18,510 --> 00:21:24,550
+upper bound آخر للست a plus sفطبعا ال V Upper
+
+201
+00:21:24,550 --> 00:21:30,410
+Bound للست هي U أكبر من أو ساوي كل عناصرها الان
+
+202
+00:21:30,410 --> 00:21:34,430
+انجل ال A عن ناحية التانية فبصير X أصغر من أو ساوي
+
+203
+00:21:34,430 --> 00:21:40,710
+V minus A لكل X في S طيب
+
+204
+00:21:40,710 --> 00:21:47,410
+الان احنا عندنا ال U هو ال supremum ل S ال U هو ال
+
+205
+00:21:47,410 --> 00:21:52,800
+supremum ل S والان هذا العددهذا عبارة عن upper
+
+206
+00:21:52,800 --> 00:22:00,200
+bound of S لأن U أكبر من أو ساوي كل عناصر الـ S
+
+207
+00:22:00,200 --> 00:22:07,400
+وهذا أصغر upper bound لـ S إذن ال superman بيطلع
+
+208
+00:22:07,400 --> 00:22:13,240
+أصغر من أو ساوي ال upper bound V minus A ل S إذن
+
+209
+00:22:13,240 --> 00:22:16,080
+بيطلع عند U أصغر من أو ساوي
+
+210
+00:22:19,910 --> 00:22:26,350
+إن أنا بطلع عندي U أصغر من أو ساوي V minus A ودي A
+
+211
+00:22:26,350 --> 00:22:30,290
+عن ناحية التانية فبصير A زاد U أصغر من أو ساوي V
+
+212
+00:22:30,290 --> 00:22:35,870
+إذا هين أثبتنا حاجتين أول شيء إنه العدد هذا upper
+
+213
+00:22:35,870 --> 00:22:40,590
+bound للست هذه أخدنا أي upper bound عشوائي للست
+
+214
+00:22:40,590 --> 00:22:47,640
+هذهفطلع العدد a زاد u اصغر من او ساوي اي upper
+
+215
+00:22:47,640 --> 00:22:52,880
+bound لست a زاد s اذا من تعريف ال supremum بطلع ال
+
+216
+00:22:52,880 --> 00:23:00,520
+supremum لست a زاد s exist و بساوي a زاد uأن الـ
+
+217
+00:23:00,520 --> 00:23:05,380
+supremum للست هذي هو a زيد u وبالتالي و هذا بساوي
+
+218
+00:23:05,380 --> 00:23:08,720
+a و ال u هي ال supremum ل S أننا هيك بنكون أثبتنا
+
+219
+00:23:08,720 --> 00:23:15,900
+أن supremum الست a زيد s هو a زاد supremum S،
+
+220
+00:23:15,900 --> 00:23:21,540
+تمام؟ لو كانت الست هذي bounded below فممكن أيضا
+
+221
+00:23:21,540 --> 00:23:26,960
+نثبت أن ال infimum ل a زاد s بساوي a زاد infimum
+
+222
+00:23:26,960 --> 00:23:33,430
+S، تمام؟طبعا في أمثلة أخرى هنا ممكن تقرؤوها و
+
+223
+00:23:33,430 --> 00:23:39,650
+تحضروها و نوقف هنا نكتفي بهذا القدر و بنكمل ان شاء
+
+224
+00:23:39,650 --> 00:23:42,170
+الله يوم السبت المحاضرة القادمة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/BdWUrxEOLII_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/BdWUrxEOLII_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..76e56f6766a6e2e1014ef201a0a07eaa97a61a0d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/BdWUrxEOLII_raw.srt
@@ -0,0 +1,1276 @@
+1
+00:00:21,430 --> 00:00:27,610
+بسم الله الرحمن الرحيم أول شي بنحب يعني نرحب فيكم
+
+2
+00:00:27,610 --> 00:00:31,770
+بمناسبة
+
+3
+00:00:31,770 --> 00:00:38,690
+بداية العالم الدراسي الجديد و نسأل الله تعالى أنه
+
+4
+00:00:38,690 --> 00:00:46,510
+يكون الفصل هذا فصل يعني متميز و يعني فيه ان شاء
+
+5
+00:00:46,510 --> 00:00:51,860
+الله الخير الكتيرلكم خاصة بعد أجواء الحرب اللى
+
+6
+00:00:51,860 --> 00:00:57,020
+عشناها فى الفترة اللى فاتت وربنا
+
+7
+00:00:57,020 --> 00:01:03,860
+يعني يكلل جهدكم بال .. بالنجاح والتفوق يمكن
+
+8
+00:01:03,860 --> 00:01:09,640
+أول مرة يمكن تشوفونى او يمكن ما درستكم مش قبل هيك
+
+9
+00:01:09,640 --> 00:01:14,260
+فإذا مابتعرفوش مين أنا فأنا الدكتور أيسى اللى
+
+10
+00:01:14,260 --> 00:01:18,840
+هبيلىطبعا كان المفروض ان الدكتور Asad .. Asad هو
+
+11
+00:01:18,840 --> 00:01:23,280
+اللي درسكم ال course هذا لكن حصل في يعني الجداول
+
+12
+00:01:23,280 --> 00:01:26,020
+زي ما انتوا عارفين في .. بيصير فيها تغيرات في آخر
+
+13
+00:01:26,020 --> 00:01:35,060
+لحظة فانا ان شاء الله اللي هدرسكم المادة هذه ف ..
+
+14
+00:01:35,060 --> 00:01:38,380
+يعني ال ..
+
+15
+00:01:41,030 --> 00:01:46,430
+أهم حاجة في المادة هذه و في كل مواد رياضيات أن
+
+16
+00:01:46,430 --> 00:01:55,150
+الطالب يعني يواظب على الحضور يحاول يحضر المحاضرات
+
+17
+00:01:55,150 --> 00:02:02,710
+ي .. يقرأ المحاضرات أول بأول يحاول يشتغل في ال
+
+18
+00:02:02,710 --> 00:02:07,770
+homework برضه أول بأول مايجزلش ال .. الدراسة أو حل
+
+19
+00:02:07,770 --> 00:02:16,610
+المثالوما تتركمش عليه كمان يعني زي أي مادة في
+
+20
+00:02:16,610 --> 00:02:20,150
+رياضيات عشان الواحد يفهمها ويقدر يعني يستوعبها
+
+21
+00:02:20,150 --> 00:02:25,730
+لازم يحاول يحل أكبر عدد ممكن من المسائل أو بنسمي
+
+22
+00:02:25,730 --> 00:02:30,690
+ال homework assignment طبعا احنا هنعطيلكم syllabus
+
+23
+00:02:30,690 --> 00:02:37,010
+زي هذافيه كل البيانات اللازمة اللي هو بنسميه
+
+24
+00:02:37,010 --> 00:02:45,490
+course outline أو ملخص لcourse و syllabus فيه كل
+
+25
+00:02:45,490 --> 00:02:50,130
+المعلومات عن المدرس عن المساقة عن ال textbook عن
+
+26
+00:02:50,130 --> 00:02:54,190
+كتاب المقرر عن المراجع الإضافية اللي ممكن لاستعانى
+
+27
+00:02:54,190 --> 00:03:00,630
+بيها بالإضافة للمرجع الأساسي أيه المادة العلمية
+
+28
+00:03:00,630 --> 00:03:08,940
+اللي هناخدهاو كيف توزيعها على أسابيع أو على ال ..
+
+29
+00:03:08,940 --> 00:03:16,560
+اه ممكن توزيعها على أسابيع توزيع
+
+30
+00:03:16,560 --> 00:03:20,220
+الدرجات ال evaluation policy أو تقييم ال course
+
+31
+00:03:20,220 --> 00:03:27,720
+برضه هذا بيكون موجود عادة في ال syllabusو في
+
+32
+00:03:27,720 --> 00:03:32,320
+النهاية بنضع اللي هو ال homework assignments اللي
+
+33
+00:03:32,320 --> 00:03:36,680
+هو مسائل ال homework اللي المفروض تحلوها من الكتاب
+
+34
+00:03:36,680 --> 00:03:42,340
+ففي نهاية كل section هيكون في عدد من المسائل و هذه
+
+35
+00:03:42,340 --> 00:03:46,200
+المسائل احنا بنختار يعني جزء منها مش كلها على أساس
+
+36
+00:03:46,200 --> 00:03:51,860
+الطالب بيحاول يحلهاالمسائل طبعا يعني الطالبة اللي
+
+37
+00:03:51,860 --> 00:03:57,500
+يعني مستواها متواضع او متوسط المفروض تحاول تحل
+
+38
+00:03:57,500 --> 00:04:01,680
+يعني مش اقل من خمسين الى سبعين في المية من المسائل
+
+39
+00:04:01,680 --> 00:04:05,940
+لوحدها اذا حضرت المحاضرة ودرست المحاضرة كويس
+
+40
+00:04:05,940 --> 00:04:09,640
+المفروض انها يعني يكون عندك مقدرة انها تحل على
+
+41
+00:04:09,640 --> 00:04:13,580
+الاقل بين خمسين الى سبعين في المية اذا ماكانش اكتر
+
+42
+00:04:14,270 --> 00:04:17,750
+ال .. طبعا باقي المسائل الصعبة بيكون في اما بيكون
+
+43
+00:04:17,750 --> 00:04:21,630
+في .. بيكون دايما بنحاول نحلها في او نحل بعضها
+
+44
+00:04:21,630 --> 00:04:28,510
+المسائل الصعبة من خلال مناقشة فبنعمل مناقشة المادة
+
+45
+00:04:28,510 --> 00:04:32,690
+دي فيها اربع ساعات ممكن نخصص تلت ساعات محاضرة و
+
+46
+00:04:32,690 --> 00:04:38,650
+ساعة مناقشة او حسب يعني ال .. تطور ال course لكن
+
+47
+00:04:38,650 --> 00:04:42,660
+في عندنا يعني الساعة من الوقت ممكن ان احنايعني
+
+48
+00:04:42,660 --> 00:04:47,280
+أخصص أنا من وقت لآخر ساعة مناقشة و نتفق عليها يعني
+
+49
+00:04:47,280 --> 00:04:52,760
+قبل ما ناخدها، فعشان هيك الحضور يعني كتير ضروري
+
+50
+00:04:52,760 --> 00:04:54,640
+جدا و ..
+
+51
+00:04:56,630 --> 00:05:00,370
+طبعا بمكانكم منكم أنتوا يعني تستغلوا الساعات
+
+52
+00:05:00,370 --> 00:05:06,330
+المكتبية و أي واحد عنده استفسار، سؤال، أي شيء يعني
+
+53
+00:05:06,330 --> 00:05:12,350
+بتعلق بالمادة ممكن تجيلي على المكتب و تتناقش معاه،
+
+54
+00:05:12,350 --> 00:05:19,010
+تسألني و ممكن أساعدها ممكن برضه تسأل المهدين أو
+
+55
+00:05:19,010 --> 00:05:24,050
+المعيدات، الأخوات اللي هنا عندكم، ماعرفش .. ضايلين
+
+56
+00:05:24,050 --> 00:05:29,530
+مكان هم اللي غيروابرضه كمان هذا يعني وسيلة تانية
+
+57
+00:05:29,530 --> 00:05:33,270
+للمساعدة ممكن تستعينوا بالمراجعة اللي احنا بنكتبها
+
+58
+00:05:33,270 --> 00:05:37,450
+في ال syllabusهذه برضه بتساعدكم ممكن تستعملوا ال
+
+59
+00:05:37,450 --> 00:05:41,910
+internet ممكن تستعملوا المكتبة يعني في وسائل
+
+60
+00:05:41,910 --> 00:05:45,670
+مساعدة كتيرة لكن يعني أهم شيء .. أهم شيء في المادة
+
+61
+00:05:45,670 --> 00:05:51,010
+هذه بتحضروا المحاضرة و تحاولوا تحلوا المسائل و
+
+62
+00:05:51,010 --> 00:05:55,910
+تتناقشوا مع المدرس أكتر .. أكتر واحد بفيدكم مدرس
+
+63
+00:05:55,910 --> 00:06:00,660
+المادةو احنا مش هنبخل عليكم يعني في ان احنا نجاوب
+
+64
+00:06:00,660 --> 00:06:05,520
+على أسئلتكم و الصفصاراتكم سواء .. سواء الأسئلة ده
+
+65
+00:06:05,520 --> 00:06:10,800
+أو الصفصارات كانت بتتعلق بال homework أو بالمادة
+
+66
+00:06:10,800 --> 00:06:17,640
+ال material اللي احنا هناخدها okay تمام؟ عشان شوية
+
+67
+00:06:17,640 --> 00:06:24,860
+هيك احنا يعني حالنا زي حال ال ..ال .. البلد ال ..
+
+68
+00:06:24,860 --> 00:06:27,880
+انتوا عارفين مكاتبنا كلها كانت مبنى الإدارة و
+
+69
+00:06:27,880 --> 00:06:33,980
+بالتالي مكاتبنا يعني في عملية نزوح او نقل من
+
+70
+00:06:33,980 --> 00:06:39,940
+المبنى الإدارة لمبنى جديد فلسه مكاتبنا يعني ما
+
+71
+00:06:39,940 --> 00:06:43,900
+استقرناش ف .. لكن انا بحاول ان شاء الله مرة جاية
+
+72
+00:06:43,900 --> 00:06:49,040
+اجهزلكم ال syllabus تبع ال course و هحطه على
+
+73
+00:06:49,040 --> 00:06:53,990
+الصفحه تبعتيو بالتالي ممكن أنكم تاخدوا نسخة منه ..
+
+74
+00:06:53,990 --> 00:07:01,310
+من الصفحة كذلك بإمكانكم تروحوا على صفحة المدرس في
+
+75
+00:07:01,310 --> 00:07:05,450
+امتحانات أنا بضعها نصفية سابقة و امتحانات نهائية
+
+76
+00:07:05,450 --> 00:07:10,970
+برضه ممكن تلاجوا على صفحة المدرس ممكن لو في حاجات
+
+77
+00:07:10,970 --> 00:07:15,230
+معينة مهمة ممكن ادرس .. ا .. انزلها على الصفحة و
+
+78
+00:07:15,230 --> 00:07:19,190
+بعدين انتوا يعني تعملولها copy و paste و إش زي ذلك
+
+79
+00:07:20,970 --> 00:07:26,830
+إذا عشان احنا يعني ما نضيعش الوجد كتير خليني بس
+
+80
+00:07:26,830 --> 00:07:34,410
+أكتبلكم ال .. ال .. ال .. موقع الصفحة تبعتي عشان
+
+81
+00:07:34,410 --> 00:07:38,750
+إذا حد يعني .. و هترب ممكن برضه تخشوا على كلية
+
+82
+00:07:38,750 --> 00:07:43,080
+العلوم خاصة الرياضيات و المدرسين و تطلع الصفحةأو
+
+83
+00:07:43,080 --> 00:07:55,920
+إذا كان ممكن تستخدمه بالرابط اللي هو http://www
+
+84
+00:07:55,920 --> 00:08:00,440
+.iogaza
+
+85
+00:08:00,440 --> 00:08:04,800
+.edu
+
+86
+00:08:04,800 --> 00:08:09,400
+.ps
+
+87
+00:08:14,490 --> 00:08:20,790
+backslash employee habil
+
+88
+00:08:20,790 --> 00:08:26,830
+الكتاب
+
+89
+00:08:26,830 --> 00:08:35,790
+المقرر اللي هو introduction text الكتاب المقرر هو
+
+90
+00:08:35,790 --> 00:08:38,590
+عبارة عن introduction
+
+91
+00:08:43,370 --> 00:08:54,990
+introduction to real analysis by
+
+92
+00:08:54,990 --> 00:09:01,770
+bartel sherbert
+
+93
+00:09:01,770 --> 00:09:08,830
+or bartel and
+
+94
+00:09:08,830 --> 00:09:09,590
+sherbert
+
+95
+00:09:13,410 --> 00:09:22,990
+وهذا الطبع التالتة third edition اذا
+
+96
+00:09:22,990 --> 00:09:27,270
+هذا الكتاب المخرر اللي احنا هنعتمد عليه طبعا هذا
+
+97
+00:09:27,270 --> 00:09:30,670
+الكتاب موجود في مكتبة الطالب او الطالبة ويمكنكم
+
+98
+00:09:30,670 --> 00:09:32,610
+يعني تشتروا
+
+99
+00:09:34,470 --> 00:09:39,090
+Okay إذا يعني هذه معظم الشغلات، احنا ال .. بالنسبة
+
+100
+00:09:39,090 --> 00:09:44,970
+لل .. لل course يعني ممكن احنا حسب ما ال .. الكلية
+
+101
+00:09:44,970 --> 00:09:48,690
+شوية غيرت سياستها، كنا في الأول نعطي امتحانين
+
+102
+00:09:48,690 --> 00:09:53,950
+نصفيين و امتحان نهائيلكن إذا الكلية غيرت و رجعت
+
+103
+00:09:53,950 --> 00:09:59,450
+لامتحان نصف واحد و نهائي فهنحكيلكم
+
+104
+00:09:59,450 --> 00:10:04,270
+المرة الجاية يعني انحدد بالظبط بعدين الدكتور عشان
+
+105
+00:10:04,270 --> 00:10:07,350
+أسعد أنا وياه و ببدرس الطلاب و أنا بدرسكم فعشان
+
+106
+00:10:07,350 --> 00:10:11,990
+نعمل امتحانات موحدة فلازم السياسة تكون موحدة فيعني
+
+107
+00:10:11,990 --> 00:10:16,390
+يوم المحاضرة الجاية نتفق على قليل يعني عدد
+
+108
+00:10:16,390 --> 00:10:20,610
+الامتحانات و توزيها الدرجات هنتفق عليه ان شاء الله
+
+109
+00:10:20,610 --> 00:10:26,270
+المرة الجايةأنا يعني عامل زي ما أنتوا شايفين ملخص
+
+110
+00:10:26,270 --> 00:10:29,890
+يعني طبعا هذا الملخص لا يغني عن الكتاب المقرر يعني
+
+111
+00:10:29,890 --> 00:10:34,510
+المفروض الطالب ي .. أو الطالبة يعني .. يعني تفلي
+
+112
+00:10:34,510 --> 00:10:38,210
+الكتاب المقرر أو يعني تدرس من الكتاب المقرر أو
+
+113
+00:10:38,210 --> 00:10:43,870
+تشوفهلكن انا بحاول يعني الكتاب المقرر بحاول يعني
+
+114
+00:10:43,870 --> 00:10:49,770
+انا اخد الصفوة تبعته و احاول ألخص يعني ال material
+
+115
+00:10:49,770 --> 00:10:53,690
+بالطريقة و بالاسلوب تبعي انا .. انا اللي بقرا ..
+
+116
+00:10:53,690 --> 00:10:58,550
+بقرا مناسب فبرضه لو اعتمدتوا على الملخص هذا او
+
+117
+00:10:58,550 --> 00:11:02,130
+حليته المسائل برضه هذا شئ يعني كتير كويس وطيب جدا
+
+118
+00:11:04,800 --> 00:11:11,100
+أنا هحاول أن أشوف هل يعني عن طريق العرض زي هيك،
+
+119
+00:11:11,100 --> 00:11:15,880
+هشتغلكم كل شيء، إذا في أي شيء مش واضح أو مش
+
+120
+00:11:15,880 --> 00:11:21,640
+فاهمينه ممكن نحاول نكتب و نوضحه بالكتابة، لكن أنا
+
+121
+00:11:21,640 --> 00:11:28,280
+مش ه .. مش ه .. مش ه .. يعني مش هعديعن نقطة من
+
+122
+00:11:28,280 --> 00:11:31,880
+نقطة لنقطة تانية إلا إذا كانت أقل من انتوا
+
+123
+00:11:31,880 --> 00:11:37,440
+فاهمينها المادة هذه يعني حساسة وفيها عمق رياضي
+
+124
+00:11:37,440 --> 00:11:43,200
+ومادة كتير مهمة ماعناش نقول صعبة مش صعبة لكن بدها
+
+125
+00:11:43,200 --> 00:11:48,500
+يعني تركيز وبدها اهتمام وبدها جهود فحنحاول ان
+
+126
+00:11:48,500 --> 00:11:50,840
+ساعدكم ان شاء الله تفهموها بقدر الممكن
+
+127
+00:11:53,810 --> 00:11:58,050
+انا بحب دائما اعطي يعني material او اعطي محاضرة من
+
+128
+00:11:58,050 --> 00:12:03,170
+اول يوم فهنبدأ نشرح و بعدين المحاضرة جاية بنحكي
+
+129
+00:12:03,170 --> 00:12:06,730
+شوية عن ال evaluation و عن الامتحانات و العلامات
+
+130
+00:12:06,730 --> 00:12:12,630
+ماشي الحال فيمكن انتوا مش مستعدين لكن انا مستعد ان
+
+131
+00:12:12,630 --> 00:12:19,570
+انا يعني ناخد شوية ولو انه وجدت كتير يعني نراها لأ
+
+132
+00:12:19,570 --> 00:12:24,620
+في معانا وجدت ان احنا ناخد شويةOkay فيعني هذا يعني
+
+133
+00:12:24,620 --> 00:12:31,880
+مش يعني سيء ومش غلط فهنبدأ
+
+134
+00:12:31,880 --> 00:12:39,120
+.. احنا هناخد أربع شباتر في المادة هذه لكن الشباتر
+
+135
+00:12:39,120 --> 00:12:45,080
+مش متساوية الشبتر الأول بتحدث عن ال real number
+
+136
+00:12:45,080 --> 00:12:50,570
+system أو نظام الأعداد الحقيقيةوهذا أساس حاجات
+
+137
+00:12:50,570 --> 00:12:58,390
+كتيرة في رياضيات فأول شيء بنا ان نتحدث
+
+138
+00:12:58,390 --> 00:13:01,450
+في أول band في ال chapter هذا أو في أول section
+
+139
+00:13:01,450 --> 00:13:05,890
+بنا نتحدث عن ال algebraic properties of R أو
+
+140
+00:13:05,890 --> 00:13:12,070
+الصفات الجابرية لنظام الأعداد الحقيقية فما هو نظام
+
+141
+00:13:12,070 --> 00:13:19,180
+الأعداد الحقيقية؟نرمزه بالرمز هذا real number
+
+142
+00:13:19,180 --> 00:13:22,940
+system أو نظام الأعداد الحقيقية نرمزه بالرمز bold
+
+143
+00:13:22,940 --> 00:13:30,440
+في SR اللي هو الرمز هذا هذا يرمز لمجموعة الأعداد
+
+144
+00:13:30,440 --> 00:13:34,160
+الحقيقية الآن هذه مجموعة الأعداد الحقيقية بنعرف
+
+145
+00:13:34,160 --> 00:13:38,380
+عليها عمليتين جبريتين two algebraic operations
+
+146
+00:13:39,430 --> 00:13:43,270
+عملية جمع يعني باخد عددين حقيقيين زوج مرتب من
+
+147
+00:13:43,270 --> 00:13:47,530
+العداد الحقيقية و بعرف عملية الجمع على .. على
+
+148
+00:13:47,530 --> 00:13:53,050
+الزوج هذا فعملية الجمع بتجمعهم بعرف عملية تانية
+
+149
+00:13:53,050 --> 00:13:57,570
+binary operation جديدة باخد عددين حقيقيين او زوج
+
+150
+00:13:57,570 --> 00:14:01,910
+مرتب من العداد الحقيقية و بحاول اعرف عليهم عملية
+
+151
+00:14:01,910 --> 00:14:07,630
+جديدة عملية ضرب او multiplicationفهذا يعتبر
+
+152
+00:14:07,630 --> 00:14:10,830
+function هذا وهذا يعتبر function من الـ Cartesian
+
+153
+00:14:10,830 --> 00:14:15,490
+product الـ R مع نفسها إلى R فهدول بنسميهم binary
+
+154
+00:14:15,490 --> 00:14:20,210
+operations الان نظام العداد الحقيقية هو مجموعة
+
+155
+00:14:20,210 --> 00:14:24,190
+العداد الحقيقية R boldface R هذه الـ R الكبيرة
+
+156
+00:14:24,190 --> 00:14:30,370
+المغمخة مع العمليتين الجبرياتين هدول الان العمليات
+
+157
+00:14:30,370 --> 00:14:34,510
+هذه لازم تحقق خمس قواص
+
+158
+00:14:37,700 --> 00:14:43,380
+فالخواص هذه الخمسة أول خاصية فيهم هي ال
+
+159
+00:14:43,380 --> 00:14:47,200
+commutative laws قوانين الإبدال يعني عملية الجامعة
+
+160
+00:14:47,200 --> 00:14:51,580
+اللي اتحدثنا عنها قبل شوية هي عملية إبدالية بقدر
+
+161
+00:14:51,580 --> 00:14:57,180
+أبدل العناد الحقيقية في الجامعة كذلك عملية الضرب
+
+162
+00:14:58,760 --> 00:15:04,100
+برضه عملية إبدالية competitive فإذا عملية عمليات
+
+163
+00:15:04,100 --> 00:15:09,640
+الجامعة والضرب هي عمليات إبدالية كذلك العمليات
+
+164
+00:15:09,640 --> 00:15:15,960
+الجامعة والضرب عمليات ال associative laws قوانين
+
+165
+00:15:15,960 --> 00:15:21,460
+الدمج يعني الأقواص عملية الجامعة عملية دمج
+
+166
+00:15:21,460 --> 00:15:26,710
+associativeيعني بقدر لما أجمع X و Y و Z تلت أعداد
+
+167
+00:15:26,710 --> 00:15:31,410
+حقيقية بقدر أحط القواس حوالين هنا أو ممكن أحطهم
+
+168
+00:15:31,410 --> 00:15:35,870
+هنا سيا مابتفرجش هاي عملية الدمج أو ال associative
+
+169
+00:15:35,870 --> 00:15:44,050
+law نفس الحاجة نفس الشيء عملية الضرب عملية دمج
+
+170
+00:15:44,050 --> 00:15:45,170
+associative
+
+171
+00:15:48,760 --> 00:15:53,220
+الخاصية التالتة الـ distributive laws أو قوانين
+
+172
+00:15:53,220 --> 00:16:02,130
+التوزيع عملية الضرب تتوزع على عملية الجامعةهذه X
+
+173
+00:16:02,130 --> 00:16:09,170
+لما أضربها في مجموعة Y و Z فبوزع الضرب X على Y و
+
+174
+00:16:09,170 --> 00:16:15,470
+بوزع X على Z نفس الشيء برضه في قانون توزيع لما
+
+175
+00:16:15,470 --> 00:16:20,330
+أضرب من اليسار برضه بوزع الضرب على المجموعة من
+
+176
+00:16:20,330 --> 00:16:23,590
+اليسار إذا أنا دول القانونين بسميهم distributive
+
+177
+00:16:23,590 --> 00:16:27,830
+laws أو قوانين التوزيع عملية الضرب توزيعية على
+
+178
+00:16:27,830 --> 00:16:32,830
+عملية الجمعفيه برضه خاصية رابعة ال identity
+
+179
+00:16:32,830 --> 00:16:39,310
+elements وجود العناصر المحايدة ففي الأعداد
+
+180
+00:16:39,310 --> 00:16:40,110
+الحقيقية
+
+181
+00:16:42,360 --> 00:16:47,040
+في عددين او عنصرين واحد نرمزله بالـ 0 و واحد
+
+182
+00:16:47,040 --> 00:16:51,760
+نرمزله بالرمز 1 و طبعا هدول عنصرين مختلفين غير
+
+183
+00:16:51,760 --> 00:16:58,660
+متساوين اذا هنا نفترض ان في يوجد عنصرين متميزين في
+
+184
+00:16:58,660 --> 00:17:05,280
+R في مجموعة الاعداد الحقيقية بحيث ان الانصر السفر
+
+185
+00:17:05,280 --> 00:17:10,340
+هذا المتميز لما اجمعه على اي عدد حقيقي X بيعطيه X
+
+186
+00:17:12,550 --> 00:17:16,290
+فهذا بنسميه الانصار صفر هذا بنسميه ال additive
+
+187
+00:17:16,290 --> 00:17:25,750
+identity أو المحايد الجامعيكذلك واحد ضرب X لو ضربت
+
+188
+00:17:25,750 --> 00:17:30,370
+هذا العنصر المتميز في أي عدد حقيقي X هيطلع عندي
+
+189
+00:17:30,370 --> 00:17:35,290
+الناتج X نفس العنصر هذا صحيح لكل عداد الحقيقية اذا
+
+190
+00:17:35,290 --> 00:17:39,470
+هنا بنسمي الواحد multiplicative identity او
+
+191
+00:17:39,470 --> 00:17:45,070
+المحايد الضربي okay اذا هي اربع خواص في كمان خاصية
+
+192
+00:17:45,070 --> 00:17:46,890
+خامسة
+
+193
+00:17:52,510 --> 00:17:59,710
+اللي هي وجود العناصر أو
+
+194
+00:17:59,710 --> 00:18:04,590
+ال inverse .. وجود ال inverse elements أو اللي هو
+
+195
+00:18:04,590 --> 00:18:12,220
+بيسموها النظارة أو العناصر المعاكسةفلأي عدد حقيقي
+
+196
+00:18:12,220 --> 00:18:17,620
+X يوجد
+
+197
+00:18:17,620 --> 00:18:25,860
+أنصر وحيد سالب X ينتمي ل R بحيث لو جمعت X مع سالبه
+
+198
+00:18:25,860 --> 00:18:32,040
+بيطلع المحايد يجمع 0في الحالة هذه بنسمي negative x
+
+199
+00:18:32,040 --> 00:18:37,680
+هذا العنصر negative x بنسميه ال additive inverse ل
+
+200
+00:18:37,680 --> 00:18:45,900
+x ال additive inverse النظير الجمعي ل x كذلك
+
+201
+00:18:45,900 --> 00:18:53,400
+في حالة الضرب في حالة الضرب مش كل عنصر له نظير
+
+202
+00:18:53,400 --> 00:18:57,320
+ضربي عشان x يكون له نظير ضربي لازم يكون مختلف عن
+
+203
+00:18:57,320 --> 00:19:03,630
+السفريعني السفر مستفن السفر إذا كان X غير لا مختلف
+
+204
+00:19:03,630 --> 00:19:08,390
+عن السفر ففي عنصر واحد there exist unique element
+
+205
+00:19:08,390 --> 00:19:13,390
+نرمزه بالرمز X to negative one ينتمي لR بحيث لو
+
+206
+00:19:13,390 --> 00:19:18,530
+ضربت ال X هذا مع العنصر هذا بيطلع عندي المظير
+
+207
+00:19:18,530 --> 00:19:23,450
+الضربي او multiplicative identity
+
+208
+00:19:26,280 --> 00:19:33,100
+العنصر هذا بنسميه النظير الضربي أو multiplicative
+
+209
+00:19:33,100 --> 00:19:33,800
+inverse
+
+210
+00:19:36,420 --> 00:19:41,260
+إذا ما هو ال real number system هو عبارة عن مجموعة
+
+211
+00:19:41,260 --> 00:19:46,260
+الأعداد الحقيقية هذه امعرف عليها two binary
+
+212
+00:19:46,260 --> 00:19:50,880
+operations عمليتين جبريتين واحدة بنسميها الجامعة
+
+213
+00:19:50,880 --> 00:19:55,400
+واحدة بنسميها الضرب والعمليتين هدول بيحققوا خمس
+
+214
+00:19:55,400 --> 00:20:00,660
+خواص مهمة اللي هي الخمس خواص اللي سردناها قبل شويه
+
+215
+00:20:02,340 --> 00:20:07,060
+Okay تمام هذا هو نظام الأعداد الحقيقية احنا الآن
+
+216
+00:20:07,060 --> 00:20:16,660
+بدنا ندرس خواص الأعداد الحقيقية هذه فأول
+
+217
+00:20:16,660 --> 00:20:22,320
+خاصية وهذه الخواص كلها خواص طبيعية ومعروفة وانتوا
+
+218
+00:20:22,320 --> 00:20:26,340
+عارفينها قبل هيك بس ماحدش كان بيعطيلها أسماءها
+
+219
+00:20:26,340 --> 00:20:30,660
+الآن بدنا نسمي الأشياءبنعطي الأشياء أسماء أو
+
+220
+00:20:30,660 --> 00:20:38,100
+مسميات فأول نظرية في ال section هذا بتعطيني
+
+221
+00:20:38,100 --> 00:20:43,340
+cancellation laws او قوانين الحذف قوانين الحذف ايه
+
+222
+00:20:43,340 --> 00:20:46,580
+يعني قوانين الحذف النظرية هذه بتقول لو كان في عندي
+
+223
+00:20:46,580 --> 00:20:51,880
+x و y و z و w أعداد حقيقية و w مختلف عن الصفر
+
+224
+00:20:51,880 --> 00:20:55,320
+فالنتائج
+
+225
+00:20:55,320 --> 00:21:03,040
+التالية بتكون صحيحةلو كان x زائد z بساوي y plus z
+
+226
+00:21:03,040 --> 00:21:11,540
+فبقدر أنا أجيب الجلم و أشطب ال z مع ال z و أقول
+
+227
+00:21:11,540 --> 00:21:15,220
+أستنتج أن x لازم تطلع بالساوي y إذا أنا إيش عملت
+
+228
+00:21:15,220 --> 00:21:20,020
+حدفت فهذا cancellation أحد ال cancellation الوزر
+
+229
+00:21:20,020 --> 00:21:24,730
+أحد قوانين الحدفةالخانون التاني بيقول لو كان عندي
+
+230
+00:21:24,730 --> 00:21:31,150
+X ضرب W بساوي Y ضرب W فممكن و طبعا لازم W مايسويش
+
+231
+00:21:31,150 --> 00:21:37,070
+0 عشان القسم على 0 غير معرفة فبقدر انا اجسم ع W او
+
+232
+00:21:37,070 --> 00:21:43,350
+اشطب W او cancelling W و اقول انه لو كان هذا صحيح
+
+233
+00:21:43,350 --> 00:21:48,890
+فاكيد لازم يطلع X بساوي Y بشرط ان W مايسويش 0 اما
+
+234
+00:21:48,890 --> 00:21:55,290
+لو W بساوي 0 فهذا الكلامnonsense يعني هراء ليس له
+
+235
+00:21:55,290 --> 00:22:02,610
+أساس رياضي طيب هذه القوانين بدنا نثبتها شو عرفناها
+
+236
+00:22:02,610 --> 00:22:08,990
+صح فبدأ استخدم الآن تعريف نظام الأعداد الحقيقية
+
+237
+00:22:08,990 --> 00:22:15,810
+انه عبارة عن مجموعة R وعمليتين جبريتين بحقق خمس
+
+238
+00:22:15,810 --> 00:22:22,060
+قواصمن خلال الخمس خواصة دول بدي أسهل أحاول أثبت
+
+239
+00:22:22,060 --> 00:22:25,760
+صحة القوانين هذه اللي هي cancellation laws أنا
+
+240
+00:22:25,760 --> 00:22:29,260
+أختارتلكم أثبت التاني لأنه الأول أسهل دايما
+
+241
+00:22:29,260 --> 00:22:35,710
+التعامل مع الجامعة أسهل من الضربفنشوف برهان الجزء
+
+242
+00:22:35,710 --> 00:22:39,090
+التاني و طبعا برهان الجزء الأول هيكون بالمثل مماثل
+
+243
+00:22:39,090 --> 00:22:43,110
+فاحنا في رياضيات ما نحبش اتقرار و الحكي كتير لما
+
+244
+00:22:43,110 --> 00:22:47,570
+يكون في حاجة مماثلة فنقول ممكن برهانها بالمثل و
+
+245
+00:22:47,570 --> 00:22:52,950
+بنسيب الطالب يتدرب عليها او يعني يحاول يتمرن عليها
+
+246
+00:22:52,950 --> 00:22:57,970
+او يثبتها بنفسه لأنها هتكون مماثلة و نفس الفكرة و
+
+247
+00:22:57,970 --> 00:23:00,210
+في الرياضيات انت عارفين افكار اذا احنا عارفنا
+
+248
+00:23:00,210 --> 00:23:07,380
+الأفكار يعني ملكنا الحلOkay نشوف برهان الجزء
+
+249
+00:23:07,380 --> 00:23:11,640
+التاني انا بدي اثبت ايش ال .. ايش الفرض انتوا كلكم
+
+250
+00:23:11,640 --> 00:23:15,720
+درستوا مبادئ رياضيات هذا conditional statement هي
+
+251
+00:23:15,720 --> 00:23:20,860
+المقدم وهي التالي هي الفرض وهي النتيجة فبنفرض انه
+
+252
+00:23:20,860 --> 00:23:25,860
+الفرض هذا صحيح يعني هذا صح الان بنثبت النتيجة نعمل
+
+253
+00:23:25,860 --> 00:23:30,880
+direct proof برهان مباشر صح؟ يعني بدي اثبت النتيجة
+
+254
+00:23:30,880 --> 00:23:35,560
+فهي النتيجة بدي اثبت X بساوي Y هي ال Xالان ال X
+
+255
+00:23:35,560 --> 00:23:39,840
+هذه من الخواص الخمسة ممكن ابدل X بواحد في X لان
+
+256
+00:23:39,840 --> 00:23:45,620
+واحد في X عبارة عن X الان هذا عملية الضرب
+
+257
+00:23:45,620 --> 00:23:51,060
+commutative ابدالية فممكن ابدل الان هذا الواحد
+
+258
+00:23:51,060 --> 00:24:00,470
+هبدله W ضرب W انفرس وهذا صحيحطيب الان انا في عند
+
+259
+00:24:00,470 --> 00:24:04,490
+عملية ضرب عملية associative فبحاول ان انا ايه
+
+260
+00:24:04,490 --> 00:24:11,230
+استخدم ال associative law هين استخدمته الان xw من
+
+261
+00:24:11,230 --> 00:24:15,550
+المعطيات انا عندي x ضرب w بساوي y ضرب w اذا انا
+
+262
+00:24:15,550 --> 00:24:23,640
+هشيل xw و اضع مكانها ywوبالتالي صار عندي الكلام
+
+263
+00:24:23,640 --> 00:24:28,360
+هذا لان بستخدم ال associative law تغير ترتيب
+
+264
+00:24:28,360 --> 00:24:33,180
+الأقواص وهذا برجعه بساوي واحد و بستخدم ال
+
+265
+00:24:33,180 --> 00:24:37,240
+commutative law فبطلع عندي في النهاية Y إذن هين
+
+266
+00:24:37,240 --> 00:24:45,200
+أثبتت إن X بساوي Y و هنا في البرهان استخدمت بعض
+
+267
+00:24:45,200 --> 00:24:49,170
+الخواص الخمسة تبقىالـ real number system أو نظام
+
+268
+00:24:49,170 --> 00:24:54,350
+الأعداد الحقيقية بظبط زاد ال logic المنطق أو
+
+269
+00:24:54,350 --> 00:24:59,010
+أساسيات الرياضيات إذا هذا هو إيه البرهان برهان
+
+270
+00:24:59,010 --> 00:25:03,050
+الجزء الأول مماثل فهي اللي كتبلكم exercise يعني
+
+271
+00:25:03,050 --> 00:25:07,290
+اتمرنوا عليه اتمرنوا عليه يعني حاولوا تكتبوه بنفس
+
+272
+00:25:07,290 --> 00:25:12,250
+الطريقة تمام واضح البرهان واضح في أي سؤال
+
+273
+00:25:14,950 --> 00:25:19,530
+Okay تمام طيب نشوف كمان نظرية تانية نظريات لسه
+
+274
+00:25:19,530 --> 00:25:25,270
+حاجات بسيطة نشوف النص تبع النظرية انا عندي ضايل
+
+275
+00:25:25,270 --> 00:25:30,910
+دقيقتين ممكن احنا تبعين رياضيات يعني بنحب نستغل
+
+276
+00:25:30,910 --> 00:25:38,470
+وجتنا كتير وكل دقيقة انا اه هذه نظرية طويلة طيب مش
+
+277
+00:25:38,470 --> 00:25:43,530
+مشلا بس هنحاول نشوف النص تبعها و بعدين المرة
+
+278
+00:25:43,530 --> 00:25:49,920
+الجاية بنبرهنهاهذه النظرية فيها عشر أجزاء النظرية
+
+279
+00:25:49,920 --> 00:25:55,180
+هذه بتقول إذا أخدت أي أربع عداد حقيقية XYZW وإذا
+
+280
+00:25:55,180 --> 00:26:00,960
+كان ال Z و ال W مختلفين عن السفر فالخواص
+
+281
+00:26:00,960 --> 00:26:06,650
+التالية كلها صحيحةو هي اول خاصية لو ضربت اي عدد
+
+282
+00:26:06,650 --> 00:26:10,030
+حقيقي في سفر المفروض يطلع لعدد السفر ال additive
+
+283
+00:26:10,030 --> 00:26:16,050
+ال additive identity الان هذا ال additive inverse
+
+284
+00:26:16,050 --> 00:26:21,510
+ل X لما اخد ال additive inverse ل X مرتين كأن ايه
+
+285
+00:26:21,510 --> 00:26:25,390
+ارجعت ل X زي المصفوفة خد ال inverse ل المصفوفة
+
+286
+00:26:25,390 --> 00:26:30,980
+مرتين تطلع المصفوفة نفسها شبيها فيها صحيح؟طيب برضه
+
+287
+00:26:30,980 --> 00:26:35,360
+نفس الحاجة لما أخد ال multiplicative inverse مرتين
+
+288
+00:26:35,360 --> 00:26:39,940
+هذا بيساوي العنصر نفسه هذا صحيح طبعا هنا بشرط w
+
+289
+00:26:39,940 --> 00:26:45,080
+مايساويش سفر في الضرب دايما بنكون .. حاول نكون
+
+290
+00:26:45,080 --> 00:26:48,440
+careful حريصين أنه إيه الحاجة اللي بدنا نجيبلها
+
+291
+00:26:48,440 --> 00:26:52,910
+multiplicative inverse ماتكونش بتساوي سفرلو ضربت
+
+292
+00:26:52,910 --> 00:26:56,430
+العدد سالب واحد هذا real number في X كأن ضربت X في
+
+293
+00:26:56,430 --> 00:27:02,790
+سالب فهذا نفس الشيء لو ضربت X في سالب Y نفس الشيء
+
+294
+00:27:02,790 --> 00:27:07,030
+كما لو أني ضربت X في Y وضربت الكل في سالب واحد أو
+
+295
+00:27:07,030 --> 00:27:12,850
+هيك كل هذا صح لو أخدت negative X و جمعتها على
+
+296
+00:27:12,850 --> 00:27:18,470
+negative Y كأن أخدت X زاد Y وضربت في سالب هذا كله
+
+297
+00:27:18,470 --> 00:27:23,680
+صحطيب لو ضربت negative x في negative y كأنني ضربت
+
+298
+00:27:23,680 --> 00:27:31,880
+x في y هذا برضه صحيح لو جسمت x على z و y على w و
+
+299
+00:27:31,880 --> 00:27:42,120
+جمعتهم فهيطلع عندى ال .. من واحد المقامات و
+
+300
+00:27:42,120 --> 00:27:47,350
+العملية الجبرية هذه المعروفةأخيرا الخاصية الأخيرة
+
+301
+00:27:47,350 --> 00:27:51,990
+هذه لو أنا فيها عندي عددين حقيقيين كان حصل ضربهم
+
+302
+00:27:51,990 --> 00:27:58,090
+بساوي سفر فلازم واحد على أقل منهم بساوي سفر فاما x
+
+303
+00:27:58,090 --> 00:28:02,830
+بساوي سفر او y بساوي سفر وهذه ممكن مرت معاكم في
+
+304
+00:28:02,830 --> 00:28:08,430
+المبادئ عفوا هذا أكيد مرت معاكم في المبادئ كمثال
+
+305
+00:28:08,430 --> 00:28:15,100
+على indirect proofعلى برهان غير مباشر حاولوا انكم
+
+306
+00:28:15,100 --> 00:28:21,660
+انتوا تفكروا في براهين الحاجات هذه و المرة الجاية
+
+307
+00:28:21,660 --> 00:28:28,520
+ان شاء الله نحاول نبرهن بعض الأجزاء okay تمام ال
+
+308
+00:28:28,520 --> 00:28:34,310
+.. ال material هذه هحطها على الصفحه تبعتيو ممكنكم
+
+309
+00:28:34,310 --> 00:28:39,670
+أنكم تنسخوها و تاخدوها و تشوفوها فبالتالي مافيش
+
+310
+00:28:39,670 --> 00:28:44,670
+داعي أنكم تكتبوا لإن ممكن تنسخوها و تحطوها على ال
+
+311
+00:28:44,670 --> 00:28:49,250
+laptop تبعكم أو على ال computer okay تمام هنوقف
+
+312
+00:28:49,250 --> 00:28:53,090
+هنا و ان شاء الله المرة الجاية بنكمل و بنجيبلكم
+
+313
+00:28:53,090 --> 00:28:56,710
+معلومات جديدة عن توزيع الدرجات و الامتحانات فى حد
+
+314
+00:28:56,710 --> 00:28:58,090
+عنده اي سؤال او استفسار
+
+315
+00:29:03,220 --> 00:29:06,820
+اه من .. منحطلكم يعني اه منحطلكم يعني ان شاء الله
+
+316
+00:29:06,820 --> 00:29:12,660
+يعني كام كبير ليه يعني يكون يكفيكي من هالشهر okay
+
+317
+00:29:12,660 --> 00:29:16,980
+تمام؟ في اي سؤال تاني؟ okay شكرا لكم و ان شاء الله
+
+318
+00:29:16,980 --> 00:29:22,500
+نشوفكم المرة الجاية و نلتقي يوم الأتنين ان شاء
+
+319
+00:29:22,500 --> 00:29:22,600
+الله
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM.srt
new file mode 100644
index 0000000000000000000000000000000000000000..500bb5a4514e22b0bc6f34d00c79bbe172a39fbd
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM.srt
@@ -0,0 +1,1727 @@
+1
+00:00:21,320 --> 00:00:25,400
+هنبدأ إن شاء الله اليوم chapter جديد وهو ال
+
+2
+00:00:25,400 --> 00:00:30,060
+chapter الثاني عنوان الـ chapter sequences and 
+
+3
+00:00:30,060 --> 00:00:35,960
+series المتتاليات والمتسلسلات طبعًا الموضوع هذا
+
+4
+00:00:35,960 --> 00:00:43,220
+مرّ معكم في تفاضل ألف .. تفاضل باء عفوا ودرسنا 
+
+5
+00:00:43,220 --> 00:00:46,860
+خواص الـ sequences بطريقة مختصرة والـ series
+
+6
+00:00:46,860 --> 00:00:53,710
+توسعنا فيها، المرة هذه سنتوسع في الـ sequences و
+
+7
+00:00:53,710 --> 00:00:58,750
+سنختصر في الـ series العكس يعني وسنتناول دراسة كل
+
+8
+00:00:58,750 --> 00:01:06,130
+منهم بطريقة تحليلية وطريقة موضعية أكثر يعني من 
+
+9
+00:01:06,130 --> 00:01:07,270
+وجهة نظر رياضية
+
+10
+00:01:10,330 --> 00:01:13,590
+فأول section في هذا الـ chapter سيكون عنوانه 
+
+11
+00:01:13,590 --> 00:01:17,610
+sequences and their limits المتتاليات ونهاياتهم
+
+12
+00:01:22,470 --> 00:01:28,630
+فنشوف تعريف الـ sequence الـ sequence in X ما معنى
+
+13
+00:01:28,630 --> 00:01:33,110
+sequence in X، X مجموعة، أي مجموعة ممكن طبعًا هناخد
+
+14
+00:01:33,110 --> 00:01:37,470
+هنا X مجموعة الأعداد الحقيقية، هذه المجموعة التي
+
+15
+00:01:37,470 --> 00:01:42,450
+نحن نهتم فيها في الـ course هذا فـ sequence in X
+
+16
+00:01:42,450 --> 00:01:47,410
+يعني الـ sequence عناصرها تنتمي للمجموعة X، فلو أخذت
+
+17
+00:01:47,410 --> 00:01:52,610
+أي مجموعة x فعشان أعرف sequence عناصرها في x فما
+
+18
+00:01:52,610 --> 00:01:55,470
+هي الـ sequence في المجموعة x؟ هي عبارة مجرد
+
+19
+00:01:55,470 --> 00:02:00,970
+function دالة المجال تبعها الأعداد الطبيعية أو أي 
+
+20
+00:02:00,970 --> 00:02:04,970
+مجموعة جزئية منها، والمجال المقابل تبعها هي 
+
+21
+00:02:04,970 --> 00:02:09,820
+المجموعة x التي الـ sequence تنتمي إليها، وفي الحالة
+
+22
+00:02:09,820 --> 00:02:13,360
+هذه إذا الـ sequence هي function دالة بس دالة من 
+
+23
+00:02:13,360 --> 00:02:19,320
+نوع خاص مجالها مجموعة الأعداد الحقيقية، وعادة نحن 
+
+24
+00:02:19,320 --> 00:02:23,320
+نهتم بالـ sequences of real numbers أو المتتاليات
+
+25
+00:02:23,320 --> 00:02:27,280
+التي عناصرها أعداد حقيقية، وبالتالي X هذه ستكون
+
+26
+00:02:27,280 --> 00:02:31,460
+التي هو مجموعة الأعداد الحقيقية، طيب هذه الـ
+
+27
+00:02:31,460 --> 00:02:35,410
+sequence function مجالها العداد الطبيعي وبالتالي
+
+28
+00:02:35,410 --> 00:02:40,350
+ممكن نعرفها F هي عند أي عدد طبيعي N هي عبارة عن XN
+
+29
+00:02:40,350 --> 00:02:47,030
+XN طبعًا هذا ينتمي للمجموعة X وبالتالي الـ .. الـ ..
+
+30
+00:02:47,030 --> 00:02:52,910
+الـ sequence FN هذه نحن نحاول نعرفها بدلالة الـ 
+
+31
+00:02:52,910 --> 00:02:56,720
+range تبعها، يعني بدل ما أقول الـ sequence هي
+
+32
+00:02:56,720 --> 00:03:01,800
+function جرت العادة أن نحن نحذف رمز الـ function
+
+33
+00:03:01,800 --> 00:03:05,980
+ونستبدله بالـ range تبع الـ function الذي هو y الـ
+
+34
+00:03:05,980 --> 00:03:09,960
+range تبع الـ function كل الـ xn حيث n عدد طبيعي
+
+35
+00:03:09,960 --> 00:03:13,980
+يبدأ من واحد من ثم إلى نهاية، إذا الـ sequence 
+
+36
+00:03:13,980 --> 00:03:18,600
+بدل ما نكتبها على صورة function سنكتبها على الصورة
+
+37
+00:03:18,600 --> 00:03:24,340
+هذه أو الصورة هذه أو الصورة هذه أو الصورة هذه، okay
+
+38
+00:03:26,550 --> 00:03:30,070
+وطبعًا الـ sequence هذه يعني عناصرها هذه أو أي واحدة
+
+39
+00:03:30,070 --> 00:03:37,350
+منهم ممكن نكتبها برضه على الصورة x1, x2, x3 وهكذا
+
+40
+00:03:40,840 --> 00:03:45,180
+فكل الرموز هذه ترمز إلى الـ sequence هذه التي هي الـ
+
+41
+00:03:45,180 --> 00:03:53,400
+function f التي هي الـ function f، okay إذن أهم شيء 
+
+42
+00:03:53,400 --> 00:03:56,480
+في تعريفنا أن الـ sequence هي function دالة
+
+43
+00:03:56,480 --> 00:04:00,400
+وبالتالي لها مجال، مجالها العداد الطبيعي، المجال 
+
+44
+00:04:00,400 --> 00:04:04,420
+المقابل هي المجموعة التي عناصر الـ sequence تنتمي 
+
+45
+00:04:04,420 --> 00:04:10,950
+لها، الـ sequences ممكن أعرفهم بطريقتين، إذا في 
+
+46
+00:04:10,950 --> 00:04:15,970
+الملاحظة هذه sequences can be defined explicitly
+
+47
+00:04:15,970 --> 00:04:19,910
+هذه أحد الطرق، ممكن يعرف الـ sequence بطريقة صريحة
+
+48
+00:04:19,910 --> 00:04:27,890
+بطريقة بقانون، فمثلا الـ sequence if بالساوية عناصرها
+
+49
+00:04:27,890 --> 00:04:31,670
+اثنين أربعة ستة ثمانية، الأخرى هذه عبارة عن
+
+50
+00:04:31,670 --> 00:04:38,130
+sequence وهي معرفة بطريقة صريحة، فهذه عبارة عن
+
+51
+00:04:38,130 --> 00:04:42,630
+sequence of even natural numbers الأعداد الطبيعية 
+
+52
+00:04:42,630 --> 00:04:47,790
+الزوجية، ممكن نكتب الحد العام، الآن هذا نسميه الآن
+
+53
+00:04:47,790 --> 00:04:53,710
+term xn هذا هنا نسميه الـ term الحد النوني
+
+54
+00:04:53,710 --> 00:04:59,190
+الحد النوني أو الحد العام، فالـ term هنا هو
+
+55
+00:04:59,190 --> 00:05:08,180
+اثنين n، xn بساوي اثنين n حيث n عدد طبيعي، أو
+
+56
+00:05:08,180 --> 00:05:12,620
+ممكن نكتب الـ sequence على صورة 2n من n بساوي 
+
+57
+00:05:12,620 --> 00:05:16,740
+واحد إلى ما لا نهاية، إذا هنا أنا أعرف الـ sequence
+
+58
+00:05:16,740 --> 00:05:22,960
+برص حدودها، أول تلات حدود إلى وهكذا، أو بكتب قاعدة
+
+59
+00:05:22,960 --> 00:05:27,880
+لحد العام xn وطبعًا n عدد طبيعي، فمقدر من القاعدة 
+
+60
+00:05:27,880 --> 00:05:32,740
+هذه أجيب كل الحدود، إذا هذا explicit definition of 
+
+61
+00:05:32,740 --> 00:05:39,150
+a sequence هذا تعريف صريح للـ sequence، في طريقة 
+
+62
+00:05:39,150 --> 00:05:44,870
+ثانية لتعريف الـ sequence وهي الطريقة الاستقرائية، 
+
+63
+00:05:44,870 --> 00:05:49,330
+إذا الـ sequences can be defined inductively أو
+
+64
+00:05:49,330 --> 00:05:55,970
+recursively بطريقة استقرائية أو بطريقة تكرارية، كيف
+
+65
+00:05:55,970 --> 00:06:02,290
+هذه الطريقة؟ بأجي للـ sequence وبأخد أول حد فيها زي
+
+66
+00:06:02,290 --> 00:06:07,250
+x1 أو أول حدين أو أول تلات حدود وبعطيهم قيم
+
+67
+00:06:07,250 --> 00:06:16,010
+أحددهم، قيم محددة، أعطيهم قيم محددة، بعدين بأجي بأجي
+
+68
+00:06:16,010 --> 00:06:21,990
+بعبر عن الحد xn زائد واحد أو xn بدلالة الحدود
+
+69
+00:06:21,990 --> 00:06:27,850
+التي قبله وبستخدم طبعًا لهذه formula نسميها
+
+70
+00:06:27,850 --> 00:06:32,070
+recursive formula أو inductive formula كما في
+
+71
+00:06:32,070 --> 00:06:39,550
+المثال التالي، يعني أنا عند الـ sequence 2n هذه أنا 
+
+72
+00:06:39,550 --> 00:06:48,000
+عند الـ sequence xn بساوي 2n هذه ممكن أعرفها بطريقة 
+
+73
+00:06:48,000 --> 00:06:57,140
+استقرائية، كيف؟ بأخد بعطي أول حد فيه x1 بعطيله قيمة
+
+74
+00:06:57,140 --> 00:07:01,220
+محددة وهي 2، طبعًا أول حد في الـ sequence هذه هو 2
+
+75
+00:07:01,220 --> 00:07:06,760
+صح؟ لأن هنا أخذت x1 وعطيته قيمة محددة، ممكن في بعض
+
+76
+00:07:06,760 --> 00:07:12,140
+الأمثلة أعطي قيمة قيمة محددة لـ x1 وx2 وx3، بعدين 
+
+77
+00:07:12,140 --> 00:07:19,100
+بأجي إلى الحد رقم n زيادة واحد وبعبر عنه بـ
+
+78
+00:07:19,100 --> 00:07:23,000
+recursive formula بعبر عنه بدلالة الحد الذي قبله
+
+79
+00:07:23,000 --> 00:07:26,760
+أو الحد الذي قبله مباشرة والذي قبله و
+
+80
+00:07:26,760 --> 00:07:32,510
+هكذا، فهذه نسميها recursive أو inductive formula
+
+81
+00:07:32,510 --> 00:07:37,150
+تعطيني لحد رقم n زيادة واحد بدالة الحد الذي قبله xn
+
+82
+00:07:37,150 --> 00:07:43,870
+فمثلا لو بده أحسب x2 فبأخد n بساوي واحد هنا صح
+
+83
+00:07:43,870 --> 00:07:50,110
+فبطلع عند x2 بساوي x1 زائد اثنين، x1 بساوي اثنين زائد
+
+84
+00:07:50,110 --> 00:07:56,400
+اثنين بطلع أربعة، x3 برضه عشان أجيب x3 بستخدم الـ
+
+85
+00:07:56,400 --> 00:08:00,480
+recursive formula وبأخد N بساوي 2 فبطلع عند x3
+
+86
+00:08:00,480 --> 00:08:06,600
+بساوي x2 زائد 2، x2 أربعة واثنين بطلع ستة وهكذا
+
+87
+00:08:06,600 --> 00:08:13,340
+إذا هيك بحصل على الـ sequence 2N التي حدودها 2 4 6
+
+88
+00:08:13,340 --> 00:08:20,460
+8 وهكذا، آه okay تمام الـ
+
+89
+00:08:20,460 --> 00:08:30,520
+.. طيب الآن بدي أعرف ما معنى أن الـ sequence تكون
+
+90
+00:08:30,520 --> 00:08:36,500
+convergent أو لها limit لو في عندي sequence من
+
+91
+00:08:36,500 --> 00:08:37,720
+الأعداد الحقيقية
+
+92
+00:08:41,200 --> 00:08:45,480
+فبقول إن الـ sequence converge
+
+93
+00:08:45,480 --> 00:08:51,860
+الـ sequence of real numbers بتكون converge أو
+
+94
+00:08:51,860 --> 00:08:59,940
+convergent إذا قدرت ألاقي X ينتمي لـ R بحيث إنه لكل
+
+95
+00:08:59,940 --> 00:09:06,200
+neighborhood V لـ X لكل جوار V لـ X بقدر أو أجد أو 
+
+96
+00:09:06,200 --> 00:09:12,250
+ألاقي عدد طبيعي capital N يعتمد على الجوار V ينتمي 
+
+97
+00:09:12,250 --> 00:09:17,030
+لأعداد الطبيعية بحيث إنه لكل small n أكبر من أو يساوي
+
+98
+00:09:17,030 --> 00:09:21,770
+capital N، Xn ينتمي إلى V، يعني الجوار V هذا يحتوي
+
+99
+00:09:21,770 --> 00:09:29,100
+كل عناصر الـ sequence من capital N وأنت طالع، فلو هذا
+
+100
+00:09:29,100 --> 00:09:34,020
+الشرط تحقق فبنقول أن الـ sequence converge والـ 
+
+101
+00:09:34,020 --> 00:09:38,040
+limit تبعتها هي العدد X، في الحالة هذه بنقول أن X
+
+102
+00:09:38,040 --> 00:09:46,080
+is the limit of sequence X in  و
+
+103
+00:09:46,080 --> 00:09:51,180
+بنكتب limit Xn بساوي X أو نكتب Xn tends to
+
+104
+00:09:51,180 --> 00:09:57,750
+X as N tends to infinity، هذا التعريف نسميه الـ 
+
+105
+00:09:57,750 --> 00:10:05,170
+neighborhood neighborhood definition neighborhood
+
+106
+00:10:05,170 --> 00:10:16,710
+definition of convergence تعريف
+
+107
+00:10:16,710 --> 00:10:18,210
+الجوار للتقارب
+
+108
+00:10:22,960 --> 00:10:28,200
+طيب لو الـ sequence ما كانش لها limit يعني ما فيش لا
+
+109
+00:10:28,200 --> 00:10:34,560
+يوجد x ينتمي لـ r يحقق الشرط هذا فبنقول أن الـ
+
+110
+00:10:34,560 --> 00:10:40,060
+sequence ليست not convergent أو divergent إذا لو
+
+111
+00:10:40,060 --> 00:10:45,220
+الـ sequence مالهاش has no limit فبنسميها divergent
+
+112
+00:10:45,220 --> 00:10:50,820
+إذا مثلًا بتكون الـ sequence convergent إذا كان في 
+
+113
+00:10:50,820 --> 00:10:54,560
+لها limit، طب ما معنى أن الـ sequence يكون لها 
+
+114
+00:10:54,560 --> 00:11:01,680
+limit؟ معناه أن يوجد عدد حقيقي X بحيث لكل جوار V لـ 
+
+115
+00:11:01,680 --> 00:11:08,260
+X في عدد طبيعي capital N يعتمد على الجوار بحيث أن 
+
+116
+00:11:08,260 --> 00:11:14,120
+كل حدود الـ sequence تنتمي للجوار هذا، والمؤشر تبعها
+
+117
+00:11:14,120 --> 00:11:20,130
+يبدأ من capital N وأنت طالع، يعني معنى الكلام هذا ..
+
+118
+00:11:20,130 --> 00:11:28,290
+هذا الكلام معناه أن X capital N وX capital N زائد
+
+119
+00:11:28,290 --> 00:11:35,990
+واحد وX capital N زائد اثنين وهكذا كل هذول
+
+120
+00:11:35,990 --> 00:11:38,630
+بينتموا للجوار دي
+
+121
+00:11:44,830 --> 00:11:48,590
+لو الـ sequence مالهاش limit فبنسميها divergent
+
+122
+00:11:48,590 --> 00:11:56,190
+okay طبعًا؟ V جوار .. جوار يعني .. مجموعة .. آه
+
+123
+00:11:56,190 --> 00:12:01,410
+جوار لـ X يعني مجموعة تحتوي الـ X والجوار عشان V
+
+124
+00:12:01,410 --> 00:12:05,710
+يكون جوار لازم يكون داخله .. لازم نلاقي داخله 
+
+125
+00:12:05,710 --> 00:12:10,010
+epsilon نبرهنه، كل جوار لازم يحتوي epsilon نبرهنه 
+
+126
+00:12:15,360 --> 00:12:23,300
+يعني مش أي مجموعة، طيب
+
+127
+00:12:23,300 --> 00:12:27,780
+الـ .. أن لو
+
+128
+00:12:27,780 --> 00:12:32,800
+في أي sequence والسيكوانس هذا convergent فالـ
+
+129
+00:12:32,800 --> 00:12:34,600
+limit تبعتها بتطلع unique
+
+130
+00:12:41,740 --> 00:12:45,620
+النظرية الأولى بتقول لو كانت xn sequence of real
+
+131
+00:12:45,620 --> 00:12:51,320
+numbers  وتconverge لـ x وتconverge لـ y يعني لها two
+
+132
+00:12:51,320 --> 00:12:55,740
+limits فلازم الـ limits يكونوا متساويتين يعني ممنوع
+
+133
+00:12:55,740 --> 00:12:59,940
+الـ convergence sequence يكون لها أكثر من limit
+
+134
+00:12:59,940 --> 00:13:05,400
+يعني معناه بعبارة أخرى a convergent sequence has a
+
+135
+00:13:05,400 --> 00:13:06,140
+unique limit
+
+136
+00:13:09,340 --> 00:13:13,560
+خلّينا نبرهن الكلام هذا، افرض إنه في عندي sequence
+
+137
+00:13:13,560 --> 00:13:20,440
+xn converge لـ x وأيضًا converge لـ y، المطلوب
+
+138
+00:13:20,440 --> 00:13:25,540
+إثبات أن x بساوي y، لبرهان ذلك نعمل برهان بالتناقض
+
+139
+00:13:25,540 --> 00:13:30,680
+assume on contrary أن x لا تساوي y الذي هو نفي 
+
+140
+00:13:30,680 --> 00:13:36,600
+النتيجة وبينصل لتناقض في exercise 15 في section 2
+
+141
+00:13:36,600 --> 00:13:41,810
+أخذناها في ال chapter السابق بقول لو في عندي أي
+
+142
+00:13:41,810 --> 00:13:49,130
+عددين حقيقيين x و y فبقدر 
+
+143
+00:13:49,130 --> 00:13:57,250
+ألاقي v1 جوار ل x و
+
+144
+00:13:57,250 --> 00:14:05,390
+بقدر ألاقي v2 لـ v2
+
+145
+00:14:05,390 --> 00:14:06,610
+جوار ل y
+
+146
+00:14:09,920 --> 00:14:17,120
+بحيث أن تقاطعهم بساوي five يعني اثنين disjoint
+
+147
+00:14:19,260 --> 00:14:24,660
+تمام؟ لو كان في عندي عددين حقيقيين x لا يساوي y بقدر
+
+148
+00:14:24,660 --> 00:14:31,280
+ألاقي جوار v1 ل x و جوار v2 ل y والجوارين هدول
+
+149
+00:14:31,280 --> 00:14:36,660
+منفصلين بعتقد حلنا السؤال هذا آه فقلنا خدي
+
+150
+00:14:36,660 --> 00:14:45,290
+epsilon بساوي نص المسافة بين x و y وهد خلي x زائد
+
+151
+00:14:45,290 --> 00:14:50,410
+y والنقطة هد x سالب y هد عبارة عن y neighborhood
+
+152
+00:14:50,410 --> 00:14:55,570
+لـ x وبالتالي neighborhood لـ x وخدي هنا برضه هد
+
+153
+00:14:55,570 --> 00:15:01,030
+عبارة عن y سالب y والنقطة هد y زائد y
+
+154
+00:15:03,680 --> 00:15:09,460
+فالـ .. واضح أن الجوارين هدول متقاطعوش لأن أنا أخدت 
+
+155
+00:15:09,460 --> 00:15:13,180
+epsilon نص المسافة هذه وهذه فترة مفتوحة وهذه
+
+156
+00:15:13,180 --> 00:15:18,560
+مفتوحة فمافيش بينهم نقاط مشتركة okay إذا هذا
+
+157
+00:15:18,560 --> 00:15:23,620
+الكلام موجود إذا هذا صحيح exercise 15 بيقول لي إذا 
+
+158
+00:15:23,620 --> 00:15:30,310
+كان x لا يساوي y فطبعا ممكن نفرض أن x أصغر من y أو
+
+159
+00:15:30,310 --> 00:15:35,170
+y أصغر من x وبالتالي بقدر ألاقي this joint this
+
+160
+00:15:35,170 --> 00:15:43,630
+joint neighborhoods v1 ل x وv2 ل y على التوالي و 2
+
+161
+00:15:43,630 --> 00:15:50,910
+منفصلين الآن احنا فرضين أن x in converge ل x حسب
+
+162
+00:15:50,910 --> 00:15:54,790
+الـ Neighborhood Definition لـ Convergence لما أن
+
+163
+00:15:54,790 --> 00:16:00,550
+المتتالي Xn converge ل X و V1 جوار ل X إذا يوجد 
+
+164
+00:16:00,550 --> 00:16:07,710
+عدد طبيعي N1 يعتمد على الجوار V1 بحيث أن Xn تنتمي
+
+165
+00:16:07,710 --> 00:16:13,260
+للجوار V1 لكل N أكبر من أو يساوي N1 كذلك احنا فرضين
+
+166
+00:16:13,260 --> 00:16:18,320
+في النظرية أن sequence xn converge ل y والآن v2
+
+167
+00:16:18,320 --> 00:16:23,660
+neighborhood ل y، إذا حسب تعريف ال convergence بما
+
+168
+00:16:23,660 --> 00:16:27,680
+أن xn converge ل y و v2 neighborhood ل y، إذا
+
+169
+00:16:27,680 --> 00:16:32,440
+بنقدر نلاقي عدد طبيعي n2 يعتمد على v2، بحيث أن xn
+
+170
+00:16:32,440 --> 00:16:38,840
+ينتمي لv2 لكل n أكبر من أو يساوي n2 الآن لو عرفت
+
+171
+00:16:38,840 --> 00:16:42,320
+capital N على Nها ال maximum الأكبر بين N واحد و N
+
+172
+00:16:42,320 --> 00:16:47,360
+اثنين هذا معناه أن capital N عدد طبيعي لأن الأكبر
+
+173
+00:16:47,360 --> 00:16:52,320
+بين هدول هيكون واحد منهم فهو عدد طبيعي و capital N
+
+174
+00:16:52,320 --> 00:16:55,640
+أكبر من أو يساوي N واحد وأكبر من أو يساوي N اثنين
+
+175
+00:16:55,640 --> 00:16:59,820
+لأن الكبير فيهم الآن
+
+176
+00:16:59,820 --> 00:17:04,120
+لو أخدت small n أكبر من أو يساوي capital N فمن
+
+177
+00:17:04,120 --> 00:17:09,540
+تعريف capital N هذا بيقودى أن capital N أكبر من أو 
+
+178
+00:17:09,540 --> 00:17:14,760
+يساوي N واحد إذا الآن أنا عندي small n أكبر من أو
+
+179
+00:17:14,760 --> 00:17:23,820
+يساوي N واحد وبالتالي إذا Xn تنتمي لـ D واحد كذلك 
+
+180
+00:17:23,820 --> 00:17:29,560
+أنا عندي من تعريف capital N capital N أكبر من أو
+
+181
+00:17:29,560 --> 00:17:34,950
+يساوي N اثنين وبالتالي small n أكبر من أو يساوي
+
+182
+00:17:34,950 --> 00:17:38,970
+capital N اثنين لما تكون small n أكبر من أو يساوي
+
+183
+00:17:38,970 --> 00:17:45,450
+capital N اثنين فبطلع xn ينتمي إلى v2 إذا الآن أنا 
+
+184
+00:17:45,450 --> 00:17:49,110
+أثبتت أنه لو كانت small n أكبر من أو يساوي capital
+
+185
+00:17:49,110 --> 00:17:57,090
+N فبطلع xn ينتمي إلىV1 وإلى V2 وبالتالي تنتمي
+
+186
+00:17:57,090 --> 00:18:01,290
+لتقاطعهم إذا المعنى أن التقاطع هذا لا يساوي الـ فاي
+
+187
+00:18:01,290 --> 00:18:05,810
+وهذا بيديني contradiction لأنه exercise 15 بيقول
+
+188
+00:18:05,810 --> 00:18:10,450
+لي أن V1 و V2 هدول disjoint فكيف طلع مش disjoint
+
+189
+00:18:10,450 --> 00:18:16,070
+تناقض تناقض هذا بيقول لي أن ال assumption تبعي إن X
+
+190
+00:18:16,070 --> 00:18:20,390
+لا تساوي Y كان خطأ إذن الصح إن X بساوي Y
+
+191
+00:18:20,390 --> 00:18:25,430
+وبالتالي ال limit لل sequence لازم تكون واحدة
+
+192
+00:18:25,430 --> 00:18:33,990
+unique تمام؟ واضح البرهان؟ في أي استفسار؟
+
+193
+00:18:33,990 --> 00:18:37,510
+في أي سؤال؟
+
+194
+00:18:50,080 --> 00:19:02,120
+النظرية الثانية تعطيني
+
+195
+00:19:02,120 --> 00:19:09,740
+شروط متكافئة لتعريف ال convergence للسيكوينس فلو
+
+196
+00:19:09,740 --> 00:19:12,840
+في عندي سيكوينس of real numbers وعندي real number
+
+197
+00:19:12,840 --> 00:19:17,630
+x the following are equivalent هذا اختصار الكلمات
+
+198
+00:19:17,630 --> 00:19:21,530
+the following are equivalent العبارات التالية
+
+199
+00:19:21,530 --> 00:19:27,670
+متكافئة أول عبارة x in converge ل x هذا معناه حسب
+
+200
+00:19:27,670 --> 00:19:31,070
+تعريف ال convergence ال neighborhood definition أن 
+
+201
+00:19:31,070 --> 00:19:42,150
+for every neighborhood V of X of X there exists
+
+202
+00:19:42,150 --> 00:19:50,590
+capital N يعتمد على V عدد طبيعي بحيث أنه لو كان n
+
+203
+00:19:50,590 --> 00:19:56,150
+أكبر من أو يساوي capital N هذا بيقودى أن xn ينتمي
+
+204
+00:19:56,150 --> 00:20:03,390
+إلى V هاي معناه xn converge ل x الآن هذا ال
+
+205
+00:20:03,390 --> 00:20:06,990
+neighborhood definition لل convergence بيكافئ 
+
+206
+00:20:06,990 --> 00:20:11,770
+العبارة بي وهذا بنسميه ال epsilon neighborhood 
+
+207
+00:20:11,770 --> 00:20:16,150
+definition لل convergence هذا بقى بنسميه epsilon
+
+208
+00:20:16,150 --> 00:20:20,210
+neighborhood definition of convergence ليه؟
+
+209
+00:20:20,210 --> 00:20:22,850
+العبارة دي بتقول لكل for every epsilon
+
+210
+00:20:22,850 --> 00:20:27,930
+neighborhood V epsilon ل X يعني بدل لكل 
+
+211
+00:20:27,930 --> 00:20:32,550
+neighborhood بدلناها لكل epsilon neighborhood ل X
+
+212
+00:20:32,550 --> 00:20:35,630
+يوجد capital N يعتمد على ال epsilon neighborhood
+
+213
+00:20:35,630 --> 00:20:42,160
+وبالتالي يعتمد على ال epsilon عدد طبيعي بحيث أنه
+
+214
+00:20:42,160 --> 00:20:46,200
+لكل N أكبر من أوسعه capital N بطلع XN ينتمي لـ V
+
+215
+00:20:46,200 --> 00:20:52,820
+نفس العادى العبارة الثالثة بتقول لكل إبسلون لأي عدد
+
+216
+00:20:52,820 --> 00:20:56,260
+إبسلون موجبة بنقدر نلاقي عدد طبيعي يعتمد على إبسلون
+
+217
+00:20:56,260 --> 00:21:01,500
+بحيث لو كان n أكبر من أو يساوي capital N فالمسافة
+
+218
+00:21:01,500 --> 00:21:07,800
+بين x and x تطلع أصغر من إبسلون هذا بنسميه الجزء C
+
+219
+00:21:07,800 --> 00:21:13,180
+وهذا الجزء الأكثر جزء هنستخدمه في إثبات ال
+
+220
+00:21:13,180 --> 00:21:18,080
+convergence لـ sequences معينة هذا بيسميه epsilon
+
+221
+00:21:18,080 --> 00:21:25,600
+capital N definition of
+
+222
+00:21:25,600 --> 00:21:26,500
+convergence
+
+223
+00:21:30,350 --> 00:21:34,970
+أنا في عندي أنا الفرق A هذا عبارة عن epsilon عبارة
+
+224
+00:21:34,970 --> 00:21:38,530
+عن neighborhood definition of convergence الفرق B
+
+225
+00:21:38,530 --> 00:21:42,230
+بنسميه ال epsilon neighborhood definition لل
+
+226
+00:21:42,230 --> 00:21:46,210
+convergence الفرق C بنسميه epsilon capital N
+
+227
+00:21:46,210 --> 00:21:49,770
+definition of convergence هذا هيكون استعماله شائع
+
+228
+00:21:49,770 --> 00:21:57,370
+أكثر من العبارات السابقة البرهان أن هذا ال ثلاثة 
+
+229
+00:21:57,370 --> 00:22:02,490
+إبراهيم بتكافئ بعض هنثبت أن a implies b و b
+
+230
+00:22:02,490 --> 00:22:10,610
+implies c وبعد هيك هنثبت أن c implies a وبالتالي
+
+231
+00:22:10,610 --> 00:22:14,370
+هيك بيطلع الثلاثة متكافئة حسب قوانين ال logic
+
+232
+00:22:14,370 --> 00:22:21,830
+مظبوط صح؟ طيب نشوف الأول a implies b افرض أن x in
+
+233
+00:22:21,830 --> 00:22:28,010
+converge ل x يعني هذا الكلام صحيح حسب تعريف ال
+
+234
+00:22:28,010 --> 00:22:34,510
+neighborhood definition لل convergence طيب .. طيب
+
+235
+00:22:34,510 --> 00:22:39,150
+احنا عارفين أن كل epsilon .. طيب لإثبات أن b صحيح
+
+236
+00:22:39,150 --> 00:22:45,130
+ناخد أي epsilon neighborhood ل x طب احنا لما درسنا 
+
+237
+00:22:45,130 --> 00:22:48,990
+ال neighborhoods قلنا أن كل epsilon neighborhood
+
+238
+00:22:48,990 --> 00:22:52,130
+.. every epsilon neighborhood على الصورة هذه ل X
+
+239
+00:22:52,130 --> 00:22:57,490
+هو أيضا neighborhood ل X صح؟ هذه حقيقة معروفة ..
+
+240
+00:22:57,490 --> 00:23:02,570
+كل epsilon neighborhood ل X is also a neighborhood
+
+241
+00:23:02,570 --> 00:23:09,280
+of X وبالتالي إذا هنا لو أخدت أي إبسلون
+
+242
+00:23:09,280 --> 00:23:13,140
+neighborhood ل X فهذا neighborhood ل X وبالتالي
+
+243
+00:23:13,140 --> 00:23:15,820
+يوجد capital N يعتمد على الإبسلون neighborhood
+
+244
+00:23:15,820 --> 00:23:24,080
+وهذا الكلام صح وبالتالي A بيؤدي ل B نشوف
+
+245
+00:23:24,080 --> 00:23:27,460
+الآن B بيؤدي العبارة B بيؤدي إلى C
+
+246
+00:23:42,950 --> 00:23:55,970
+طيب العبارة P هذا هي لو كان P صحيح فبنثبت
+
+247
+00:23:55,970 --> 00:24:05,490
+أن C صحيح فخلينا ناخد خلينا 
+
+248
+00:24:05,490 --> 00:24:09,250
+ناخد أبسلون أكبر من الصفر ناخد أبسلون أكبر من
+
+249
+00:24:09,250 --> 00:24:09,730
+الصفر
+
+250
+00:24:13,900 --> 00:24:22,140
+لو أخدت أي epsilon أكبر من الصفر for any epsilon
+
+251
+00:24:22,140 --> 00:24:30,140
+أكبر من الصفر take v epsilon of x اللي هو عبارة عن
+
+252
+00:24:30,140 --> 00:24:36,040
+ال epsilon neighborhood ل x فهذا
+
+253
+00:24:36,040 --> 00:24:44,530
+is epsilon neighborhood of x صح؟ وبالتالي حسب B
+
+254
+00:24:44,530 --> 00:24:50,890
+لأي إبسلون neighborhood لهذا يوجد capital N إذا 
+
+255
+00:24:50,890 --> 00:24:56,350
+يوجد capital N by
+
+256
+00:24:56,350 --> 00:25:02,930
+B يوجد capital N يعتمد على الإبسلون neighborhood
+
+257
+00:25:02,930 --> 00:25:09,630
+وبالتالي يعتمد على إبسلون هذا عدد طبيعي بحيث
+
+258
+00:25:13,530 --> 00:25:19,590
+بحيث أنه لو كان n أكبر من أو يساوي n of epsilon
+
+259
+00:25:19,590 --> 00:25:28,030
+فهذا بيقودى أن xn ينتمي لـ v epsilon ل x اللي هو x
+
+260
+00:25:28,030 --> 00:25:35,630
+سالب epsilon وx زائد epsilon طب وهذا معناه أن ال
+
+261
+00:25:35,630 --> 00:25:44,930
+xn أكبر من x سالب epsilon أصغر من x زائد epsilon هذا
+
+262
+00:25:44,930 --> 00:25:50,630
+الـ xn ينتمي للفترة المفتوحة هذه معناته هذا الكلام
+
+263
+00:25:50,630 --> 00:25:56,670
+صح هذا معناه xn minus x أصغر من epsilon أكبر من
+
+264
+00:25:56,670 --> 00:26:01,950
+سالب epsilon هذا معناه absolute xn minus x أصغر من
+
+265
+00:26:01,950 --> 00:26:10,800
+epsilon إذن هنا أثبتنا إن لو كان b صحيح فلأي يبسلون
+
+266
+00:26:10,800 --> 00:26:18,300
+أكبر من الصفر يوجد capital N يعتمد على يبسلون بحيث
+
+267
+00:26:18,300 --> 00:26:23,160
+لكل N أكبر من أو يساوي capital N طلع absolute xn
+
+268
+00:26:23,160 --> 00:26:29,920
+minus x أصغر من يبسلون وبالتالي العبارة C صحيحة
+
+269
+00:26:29,920 --> 00:26:38,500
+متحققة okay تمام؟ الآن بقى نثبت أن العبارة
+
+270
+00:26:38,500 --> 00:26:59,280
+C بتقودى إلى العبارة A فأفرضي 
+
+271
+00:26:59,280 --> 00:27:08,370
+أن العبارة C متحققة suppose C holds بعدين، بدنا 
+
+272
+00:27:08,370 --> 00:27:12,250
+نثبت أن x in converge ل x أو ال neighborhood
+
+273
+00:27:12,250 --> 00:27:17,730
+definition ل x بتحقق فبناخد أي let v be any
+
+274
+00:27:17,730 --> 00:27:24,590
+neighborhood of x فمن تعريف ال neighborhood لأي
+
+275
+00:27:24,590 --> 00:27:28,910
+neighborhood كل neighborhood v ل x يحتوي داخله
+
+276
+00:27:28,910 --> 00:27:32,030
+epsilon neighborhood ل x هذا ما قلناه قبل هيك 
+
+277
+00:27:32,030 --> 00:27:37,430
+وبالتالي يوجد epsilon عدد موجب بحيث أن ال epsilon
+
+278
+00:27:37,430 --> 00:27:44,890
+neighborhood هذه الفترة عبارة عن x in .. هذه
+
+279
+00:27:44,890 --> 00:27:51,090
+المفروضة تكون عفوا هذه المفروضة تكون x مش x in
+
+280
+00:27:51,090 --> 00:28:01,600
+وهذه x سلب epsilon هذا عبارة عن v epsilon ل x هذا 
+
+281
+00:28:01,600 --> 00:28:08,880
+المفروض تكون x مش xm، إذا لو كان v epsilon 
+
+282
+00:28:08,880 --> 00:28:15,740
+neighborhood ففي عندي بقدر ألاقي جواته epsilon 
+
+283
+00:28:15,740 --> 00:28:20,520
+neighborhood للـ x اللي هو v epsilon للـ x الآن من 
+
+284
+00:28:20,520 --> 00:28:21,400
+الجزء c
+
+285
+00:28:25,470 --> 00:28:29,650
+لأي أبسلون من الجزء C، لأي أبسلون، لأي بما أن هذا
+
+286
+00:28:29,650 --> 00:28:33,170
+أبسلون أكبر من الصفر، إذا بنقدر نلاقي capital N
+
+287
+00:28:33,170 --> 00:28:36,310
+يعتمد على أبسلون، بحيث لكل N أكبر من أو يساوي
+
+288
+00:28:36,310 --> 00:28:40,230
+capital N، الـ absolute value هذه أصغر من أبسلون هذا
+
+289
+00:28:40,230 --> 00:28:45,660
+من الجزء C، طب ما هذا معناه الـ implication هذه
+
+290
+00:28:45,660 --> 00:28:50,920
+معناها لكل n أكبر من أو يساوي capital N، لو فكيت
+
+291
+00:28:50,920 --> 00:28:58,800
+المتباينة هذه، معناها xn ينتمي، هذا عبارة عن x ينتمي
+
+292
+00:28:58,800 --> 00:29:06,480
+للـ فترة المفتوحة x minus y و x زائد epsilon اللي هو الـ
+
+293
+00:29:06,480 --> 00:29:09,720
+epsilon neighborhood للـ x اللي هو subset من V
+
+294
+00:29:11,670 --> 00:29:19,650
+وبالتالي هيك بنكون أثبتنا أن الـ XIN ينتمي إلى الـ
+
+295
+00:29:19,650 --> 00:29:24,530
+neighborhood V كمان
+
+296
+00:29:24,530 --> 00:29:30,830
+مرة أنا بدي أثبت أن العبارة C بتأدي لـ a، افرض أن
+
+297
+00:29:30,830 --> 00:29:36,610
+العبارة C صحيحة، الآن لإثبات a اللي هي x in converge
+
+298
+00:29:36,610 --> 00:29:40,790
+للـ x، بتثبت أنه الـ neighborhood definition للـ
+
+299
+00:29:40,790 --> 00:29:45,750
+convergence بتحقق، يعني x عبارة عن limit للـ
+
+300
+00:29:45,750 --> 00:29:48,650
+sequence xn، فنرجع لتعريف الـ neighborhood
+
+301
+00:29:48,650 --> 00:29:53,190
+definition of convergence، نبدأ بـ neighborhood للـ x
+
+302
+00:29:53,190 --> 00:29:57,910
+ونستخدم الحقيقة أن كل neighborhood للـ x يحتوي
+
+303
+00:29:57,910 --> 00:30:04,160
+epsilon neighborhood، الآن من C.. C بيقول لي إذا في
+
+304
+00:30:04,160 --> 00:30:08,400
+عندك إبسلون موجبة، تقدر تلاقي capital N يعتمد عليها
+
+305
+00:30:08,400 --> 00:30:12,940
+بحيث أنه لكل N أكبر من أو يساوي capital N، المسافة
+
+306
+00:30:12,940 --> 00:30:17,660
+هذه أصغر من إبسلون، طب هذه الـ implication الأخيرة هي
+
+307
+00:30:17,660 --> 00:30:22,380
+N أكبر من أو يساوي capital N بتقدي في حل المتباينة
+
+308
+00:30:22,380 --> 00:30:28,640
+هذه في Xn، فبطلع Xn ينتمي إلى X سالب Y و X زائد epsilon اللي هو
+
+309
+00:30:28,640 --> 00:30:33,320
+هذا الـ epsilon neighborhood اللي هو داخل V وبالتالي
+
+310
+00:30:33,320 --> 00:30:37,660
+لكل N أكبر من أو يساوي capital N، طلع Xn ينتمي للـ
+
+311
+00:30:37,660 --> 00:30:42,300
+neighborhood V، هذا من التعريف معناه Xn converge لـ
+
+312
+00:30:42,300 --> 00:30:48,820
+X، وبالتالي اللي هي عبارة a صحيحة تمام؟ إذا هيك
+
+313
+00:30:48,820 --> 00:30:53,940
+بنكون أثبتنا النظرية، أن التلات تعريفات هذه كلها
+
+314
+00:30:53,940 --> 00:30:54,840
+متكافئة
+
+315
+00:31:02,750 --> 00:31:06,990
+في تعريف الـ tail of a sequence أو الـ M tail of a
+
+316
+00:31:06,990 --> 00:31:11,070
+sequence، احنا عارفين أن لو في عندي أي.. لأي
+
+317
+00:31:11,070 --> 00:31:18,570
+sequence Xn، لو خدت M عدد طبيعي أي عدد طبيعي
+
+318
+00:31:18,570 --> 00:31:24,210
+natural number، و Xn أي sequence of real numbers
+
+319
+00:31:24,210 --> 00:31:31,330
+فالـ Xn هذه ممكن انفرفتها نكتب حدودها X1 X2 وهكذا
+
+320
+00:31:32,450 --> 00:31:41,130
+إلى x رقم m، الآن الحد اللي بعد xm عبارة عن xm زائد
+
+321
+00:31:41,130 --> 00:31:50,010
+واحد واللي بعده xm زائد اتنين وهكذا إذا
+
+322
+00:31:50,010 --> 00:31:53,130
+الـ sequence هذه ممكن أكتبها على الصورة هذه حيث m
+
+323
+00:31:53,130 --> 00:31:57,770
+هنا عدد طبيعي ما ثابت
+
+324
+00:31:59,680 --> 00:32:10,460
+الآن لو أنا ركزت على الجزء هذا من الـ sequence و
+
+325
+00:32:10,460 --> 00:32:20,440
+الجزء هذا هو أول m من حدود الـ sequence، حذفتها، فإذا
+
+326
+00:32:20,440 --> 00:32:22,400
+هذا بنسميه m tail
+
+327
+00:32:28,870 --> 00:32:37,630
+مثل الـ sequence xn، الدنب m دنب m، مش هذا دنب يعني تصور
+
+328
+00:32:37,630 --> 00:32:42,110
+إنها دي أفع هي الرأس تبعها أول m من الحدود ده هي
+
+329
+00:32:42,110 --> 00:32:47,570
+الرأس، جاطعة الرأس تبعها فبقى الدنب، مش هيك بيقولوا
+
+330
+00:32:47,570 --> 00:32:50,870
+الدنب
+
+331
+00:32:50,870 --> 00:32:56,090
+هذا طويل، بنبدأ يعني في عدد لانهائي من الحدود، الرأس
+
+332
+00:32:56,090 --> 00:33:02,470
+محدود، هي عدد منتهي من الحدود، إذا الـ sequence لو
+
+333
+00:33:02,470 --> 00:33:08,250
+أنا حدفت أول M من حدودها، فباقي الجزء المتبقي من الـ
+
+334
+00:33:08,250 --> 00:33:16,070
+sequence بنسميه M tail، واضح؟ طيب إذا الآن في نظرية
+
+335
+00:33:16,070 --> 00:33:18,250
+اتنين تلاتة أو نظرية تالتة
+
+336
+00:33:20,720 --> 00:33:23,800
+ما هي هذه النظرية اللي بتقول؟ بتقول لو أنا في عندي
+
+337
+00:33:23,800 --> 00:33:29,500
+إذا هاي الـ m tail هذا الـ m tail ممكن كتابته على
+
+338
+00:33:29,500 --> 00:33:35,820
+صورة sequence هاي x المؤشر، الحد العام تبع الـ m
+
+339
+00:33:35,820 --> 00:33:40,660
+tail، m زائد n حيث n العداد الطبيعي، m ثابت و n 
+
+340
+00:33:40,660 --> 00:33:43,980
+العداد الطبيعي، وبالتالي هنا لو كانت n بـالساوية
+
+341
+00:33:43,980 --> 00:33:50,800
+واحد، أول حد xm زائد واحد وهكذا، طيب الآن النظرية
+
+342
+00:33:50,800 --> 00:33:57,980
+التالية بتقول لي أنه لو كان الـ M tail convergent
+
+343
+00:34:02,380 --> 00:34:07,760
+فالـ sequence نفسها الـ M بتكون convergent والعكس،
+
+344
+00:34:07,760 --> 00:34:12,020
+لو كانت الـ sequence convergent فأي M tail منها
+
+345
+00:34:12,020 --> 00:34:15,940
+هيكون convergent واثنين لهم نفس الـ limit، اثنين
+
+346
+00:34:15,940 --> 00:34:20,020
+لهم نفس الـ limit، إذا مرة ثانية لو كان في عندك
+
+347
+00:34:20,020 --> 00:34:27,500
+sequence Xn، M fixed natural number، فالـ M tail اللي
+
+348
+00:34:27,500 --> 00:34:32,350
+هو الـ sequence هذه، converges if and only if
+
+349
+00:34:32,350 --> 00:34:39,210
+الـ sequence نفسها converges، وهي البرهان هذا الـ part
+
+350
+00:34:39,210 --> 00:34:43,750
+f، افرض
+
+351
+00:34:43,750 --> 00:34:48,290
+أن xn convergent، نثبت أن الـ m tail convergent
+
+352
+00:34:48,290 --> 00:34:54,540
+ماشي الحال؟ طيب إذا كانت xn convergent للـ x، يعني الـ
+
+353
+00:34:54,540 --> 00:34:57,620
+limit تبعها، إذا كانت convergent فلازم يكون لها
+
+354
+00:34:57,620 --> 00:35:02,020
+limit، فأفرض أن الـ limit تبعها x، الآن حسب epsilon
+
+355
+00:35:02,020 --> 00:35:06,080
+capital N definition للـ limit أو للـ convergence
+
+356
+00:35:06,080 --> 00:35:11,140
+إذا لأي epsilon أكبر من 0، نقدر نلاقي N يعتمد على
+
+357
+00:35:11,140 --> 00:35:15,860
+epsilon، عدد طبيعي كبير وممكن ناخده يكون أكبر من
+
+358
+00:35:15,860 --> 00:35:22,040
+العدد الثابت، العدد الطبيعي الثابت M بحيث أنه لكل N
+
+359
+00:35:22,040 --> 00:35:25,900
+أكبر من أو يساوي capital N، المسافة بين X و N اللي هو X
+
+360
+00:35:25,900 --> 00:35:31,410
+أصغر من epsilon، هذا من تعريف الـ epsilon capital N
+
+361
+00:35:31,410 --> 00:35:37,590
+definition للـ convergence، طيب اللي أنا بقدر أعرف
+
+362
+00:35:37,590 --> 00:35:43,930
+capital N prime على أنه capital N مطروح منها
+
+363
+00:35:43,930 --> 00:35:50,060
+capital M، طبعا هنا capital N احنا اختارناها أكبر من
+
+364
+00:35:50,060 --> 00:35:54,220
+M، فالفرق هذا موجب وهذا عدد طبيعي وهذا عدد طبيعي
+
+365
+00:35:54,220 --> 00:35:59,500
+إذا الفرق عدد صحيح موجب يعني عدد طبيعي، هذا عدد
+
+366
+00:35:59,500 --> 00:36:03,220
+ثابت وهذا يعتمد على epsilon، إذا N prime الفرق
+
+367
+00:36:03,220 --> 00:36:09,000
+بينهم يعتمد على epsilon، تمام؟ إذا هنا عرفنا N' عدد
+
+368
+00:36:09,000 --> 00:36:14,320
+طبيعي ويعتمد على epsilon، الآن لو أخدت أي M عدد
+
+369
+00:36:14,320 --> 00:36:16,960
+طبيعي أكبر من أو يساوي N'
+
+370
+00:36:20,020 --> 00:36:25,520
+فنجمع capital M للطرفين فبطلع capital M زائد small
+
+371
+00:36:25,520 --> 00:36:29,980
+m أكبر من أو يساوي N prime زائد capital M، طب N prime
+
+372
+00:36:29,980 --> 00:36:34,540
+زائد capital M بيساوي N epsilon وبالتالي هذا أكبر من
+
+373
+00:36:34,540 --> 00:36:40,860
+أو يساوي N ل epsilon، إذا حسب الـ implication 1، الـ
+
+374
+00:36:40,860 --> 00:36:45,260
+implication 1 بتقول لي لأي عدد طبيعي.. لأي عدد
+
+375
+00:36:45,260 --> 00:36:50,560
+طبيعي أكبر من أو يساوي capital N لازم يطلع الـ
+
+376
+00:36:50,560 --> 00:36:56,900
+absolute value لـ X sub العدد الطبيعي اللي هو M زائد
+
+377
+00:36:56,900 --> 00:36:59,320
+M ناقص X أصغر من epsilon
+
+378
+00:37:03,110 --> 00:37:08,470
+وهذا بيدّي أن الـ tail.. الـ tail of the sequence
+
+379
+00:37:08,470 --> 00:37:13,110
+converge للـ X حسب التعريف، ما معناه أن الـ tail هذا
+
+380
+00:37:13,110 --> 00:37:18,470
+convergent؟ معناه أن لأي epsilon أكبر من الصفر..
+
+381
+00:37:18,470 --> 00:37:25,050
+لأي epsilon أكبر من الصفر هيوجد N prime.. هيوجد N
+
+382
+00:37:25,050 --> 00:37:29,130
+prime عدد طبيعي يعتمد على epsilon
+
+383
+00:37:31,850 --> 00:37:38,290
+يوجد عدد طبيعي N' يعتمد على إبسلون، بحيث لكل M أكبر
+
+384
+00:37:38,290 --> 00:37:44,350
+من أو يساوي N'، طلع المسافة بين الحد رقم capital M
+
+385
+00:37:44,350 --> 00:37:47,690
+زائد small m ناقص X أصغر من إبسلون، هذا بالضبط
+
+386
+00:37:47,690 --> 00:37:53,310
+معناه إن الـ sequence هذه converge لـ X as M tends
+
+387
+00:37:53,310 --> 00:37:59,580
+to infinity، إذاً هيك بنكون أثبتنا إنه لو كانت الـ
+
+388
+00:37:59,580 --> 00:38:03,240
+sequence xn converge للـ x، فالتالت تبعها converge
+
+389
+00:38:03,240 --> 00:38:10,720
+للـ x، okay، تمام، العكس، العكس يعني ضايق، ممكن يعني
+
+390
+00:38:10,720 --> 00:38:20,220
+نبرهن العكس في دقيقة أو دقيقتين، العكس
+
+391
+00:38:20,220 --> 00:38:26,390
+يعني هذا العكس اللي هو الـ only if part، نفرض المرة
+
+392
+00:38:26,390 --> 00:38:30,450
+هذه أن الـ sequence الـ tail of a sequence الـ
+
+393
+00:38:30,450 --> 00:38:34,770
+tail of the sequence converged للـ X وبينما نثبت أن
+
+394
+00:38:34,770 --> 00:38:40,170
+الـ sequence نفسها convergent للـ X برضه، فنستخدم
+
+395
+00:38:40,170 --> 00:38:42,930
+تعريف epsilon capital N definition للـ convergence
+
+396
+00:38:42,930 --> 00:38:48,710
+اللي هو الجزء C من نظرية 2 2، فناخد given epsilon
+
+397
+00:38:48,710 --> 00:38:53,080
+أو let epsilon أكبر من الصفر، بـ given، بما أن الـ
+
+398
+00:38:53,080 --> 00:38:56,560
+sequence هذه converge للـ X، إذا يوجد capital N يعتمد
+
+399
+00:38:56,560 --> 00:39:00,740
+على إبسلون، بحيث لكل N أكبر من أو يساوي capital N
+
+400
+00:39:00,740 --> 00:39:04,560
+المسافة بين الحد العام للـ sequence هذه و X أصغر
+
+401
+00:39:04,560 --> 00:39:12,790
+من إبسلون، الآن بنعرف capital K على أنه العدد
+
+402
+00:39:12,790 --> 00:39:18,250
+الطبيعي الثابت M زائد العدد الطبيعي capital N، فطبعا
+
+403
+00:39:18,250 --> 00:39:22,490
+مجموعة الأعداد الطبيعيين، عدد طبيعي capital N يعتمد على
+
+404
+00:39:22,490 --> 00:39:26,670
+epsilon، إذا المجموعة تبعهم بيطلع يعتمد على epsilon
+
+405
+00:39:26,670 --> 00:39:32,330
+إذا هنا أنا وجدت أو جدت أو عرفت عدد طبيعي capital
+
+406
+00:39:32,330 --> 00:39:37,610
+K يعتمد على epsilon، الآن لو أخدت أي N أكبر من أو
+
+407
+00:39:37,610 --> 00:39:43,170
+يساوي الـ capital K فاترحي.. اترحي N من هنا و اترحي
+
+408
+00:39:43,170 --> 00:39:50,350
+N من هنا، M عفوا، M، لو طرحنا M من الطرفين المتباينة
+
+409
+00:39:50,350 --> 00:39:55,330
+هذه فبطلع N ناقص capital M أكبر من أو يساوي K
+
+410
+00:39:55,330 --> 00:40:01,170
+ناقص M، طب هاي K اطرحي منها M بيساوي N وبالتالي
+
+411
+00:40:01,170 --> 00:40:05,950
+بطلع N ناقص M أكبر من أو يساوي N، الآن من الـ
+
+412
+00:40:05,950 --> 00:40:11,550
+implication 2، الـ implication 2 بتقول لأي N
+
+413
+00:40:11,550 --> 00:40:15,650
+أكبر من أو يساوي capital، أي عدد طبيعيلو كان العدد
+
+414
+00:40:15,650 --> 00:40:20,950
+الطبيعي هذا أكبر من أو يساوي capital N، فالمسافة بين
+
+415
+00:40:20,950 --> 00:40:27,390
+X للعدد الطبيعي، وأضيف عليه M، إذا بدي أضيف على هذا
+
+416
+00:40:27,390 --> 00:40:32,230
+M، المسافة بين X اللي المؤشر تبعها العدد الطبيعي
+
+417
+00:40:32,230 --> 00:40:37,770
+هذا زائد M اللي هو بيطلع N والمسافة بينه وبين X
+
+418
+00:40:37,770 --> 00:40:42,770
+بيطلع أصغر من Epsilon، إذاً هيك احنا أثبتنا أنه لأي
+
+419
+00:40:42,770 --> 00:40:46,970
+إبسلون أكبر من الصفر يوجد capital N يعتمد على
+
+420
+00:40:46,970 --> 00:40:53,790
+إبسلون بحيث أنه أو يوجد capital K لأي إبسلون أكبر 
+
+421
+00:40:53,790 --> 00:40:57,570
+من الصفر يوجد عدد طبيعي K يعتمد على إبسلون
+
+422
+00:40:57,570 --> 00:41:06,250
+بحيث أنه لكل N أكبر من أو يساوي K لكل n
+
+423
+00:41:06,250 --> 00:41:10,590
+أكبر من أو يساوي K تطلع المسافة بين xn و x
+
+424
+00:41:10,590 --> 00:41:15,370
+أصغر من إبسلون إذن هذا بالضبط معناه أن ال sequence
+
+425
+00:41:15,370 --> 00:41:22,590
+xn converge لـ x زي ما هو مطلوب وهذا يكمل برهان
+
+426
+00:41:22,590 --> 00:41:26,410
+النظرية okay تمام واضح
+
+427
+00:41:31,150 --> 00:41:37,130
+طيب احنا بنكتفي بهذا القدر وإن شاء الله في
+
+428
+00:41:37,130 --> 00:41:42,010
+المحاضرة القادمة هناخد برضه بعض النظريات وناخد 
+
+429
+00:41:42,010 --> 00:41:46,350
+أمثلة كيف نثبت أن ال limit لـ sequence لـ
+
+430
+00:41:46,350 --> 00:41:51,090
+convergence sequence بالساوي عدد معين وهكذا طبعا
+
+431
+00:41:51,090 --> 00:41:54,130
+كل الأجزاء هذه موجودة عندكم ممكن تقرؤوها وتحضروها
+
+432
+00:41:54,130 --> 00:41:56,010
+للمحاضرة الجاية 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a415a1cddca99fb771f4ecdd977029dfebb63a5d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_postprocess.srt
@@ -0,0 +1,1728 @@
+1
+00:00:21,320 --> 00:00:25,400
+هنبدأ ان شاء الله اليوم chapter جديد و هو ال
+
+2
+00:00:25,400 --> 00:00:30,060
+chapter التاني عنوان ال chapter sequences and
+
+3
+00:00:30,060 --> 00:00:35,960
+series المتتاليات و المتسلسلات طبعا الموضوع هذا
+
+4
+00:00:35,960 --> 00:00:43,220
+مار معاكم في تفاضل ألف .. تفاضل با عفوا و درسنا
+
+5
+00:00:43,220 --> 00:00:46,860
+خواص ال sequences بطريقة مختصرة و ال series
+
+6
+00:00:46,860 --> 00:00:53,710
+اتوسعنا فيهاالمرة هذه هنتوسع في ال sequences و
+
+7
+00:00:53,710 --> 00:00:58,750
+هنختصر في ال series العكس يعني و هنتناول دراسة كل
+
+8
+00:00:58,750 --> 00:01:06,130
+منهم بطريقة تحليلية و طريقة موضعية أكتر يعني من
+
+9
+00:01:06,130 --> 00:01:07,270
+وجه اتناظر رياضية
+
+10
+00:01:10,330 --> 00:01:13,590
+فأول section في هذا ال chapter هيكون عنوانه
+
+11
+00:01:13,590 --> 00:01:17,610
+sequences and their limits المتتاليات و نهاياتهم
+
+12
+00:01:22,470 --> 00:01:28,630
+فنشوف تعريف ال sequence ال sequence in X ما معنى
+
+13
+00:01:28,630 --> 00:01:33,110
+sequence in X، X مجموعة، أي مجموعة ممكن طبعا هناخد
+
+14
+00:01:33,110 --> 00:01:37,470
+هنا X مجموعة الأعداد الحقيقية، هذه المجموعة اللي
+
+15
+00:01:37,470 --> 00:01:42,450
+احنا بنهتم فيها في ال course هذا ف sequence in X
+
+16
+00:01:42,450 --> 00:01:47,410
+يعني ال sequence على سرها تنتمي للمجموعة Xفلو أخدت
+
+17
+00:01:47,410 --> 00:01:52,610
+أي مجموعة x فعشان أعرف sequence عناصرها في x فما
+
+18
+00:01:52,610 --> 00:01:55,470
+هي ال sequence في المجموعة x؟ هي عبارة مجرد
+
+19
+00:01:55,470 --> 00:02:00,970
+function دالة المجال تبعها الأعداد الطبيعية أو أي
+
+20
+00:02:00,970 --> 00:02:04,970
+مجموعة جزئية منها والمجال المقابل تبعها هي
+
+21
+00:02:04,970 --> 00:02:09,820
+المجموعة x اللي ال sequence تنتمي إليهاو في الحالة
+
+22
+00:02:09,820 --> 00:02:13,360
+هذه إذا ال sequence هي function دالة بس دالة من
+
+23
+00:02:13,360 --> 00:02:19,320
+نوع خاص مجالها مجموعة الأعداد الحقيقية و عادة احنا
+
+24
+00:02:19,320 --> 00:02:23,320
+بنهتم بال sequences of real numbers او المتتاليات
+
+25
+00:02:23,320 --> 00:02:27,280
+اللي عناصرها أعداد حقيقية وبالتالي X هذه هتكون
+
+26
+00:02:27,280 --> 00:02:31,460
+اللي هو مجموعة الأعداد الحقيقية طيب هذه ال
+
+27
+00:02:31,460 --> 00:02:35,410
+sequence functionمجالها العداد الطبيعي وبالتالي
+
+28
+00:02:35,410 --> 00:02:40,350
+ممكن نعرفها F هي عند أي عدد طبيعي N هي عبارة عن XN
+
+29
+00:02:40,350 --> 00:02:47,030
+XN طبعا هذا ينتمي للمجموعة X وبالتالي ال .. ال ..
+
+30
+00:02:47,030 --> 00:02:52,910
+ال sequence FN هذه احنا بنحاول نعرفها بدلالة ال
+
+31
+00:02:52,910 --> 00:02:56,720
+range تبعهايعني بدل ما اقول ال sequence هي
+
+32
+00:02:56,720 --> 00:03:01,800
+function جرّت العادة ان احنا نحذف رمز ال function
+
+33
+00:03:01,800 --> 00:03:05,980
+و نستبدله بال range تبع ال function اللي هو y ال
+
+34
+00:03:05,980 --> 00:03:09,960
+range تبع ال function كل ال x n حيث n عدد طبيعي
+
+35
+00:03:09,960 --> 00:03:13,980
+ببدأ من واحد من ت أنما إلى نهاية اذا ال sequence
+
+36
+00:03:13,980 --> 00:03:18,600
+بدل ما نكتبها على صورة function هنكتبها على الصورة
+
+37
+00:03:18,600 --> 00:03:24,340
+هذه او الصورة هذه او الصورة هذه او الصورة هذه okay
+
+38
+00:03:26,550 --> 00:03:30,070
+و طبعا ال sequence هذه يعني أسرها هذه أو أي واحدة
+
+39
+00:03:30,070 --> 00:03:37,350
+منهم ممكن نكتبها برضه على الصورة x1, x2, x3 و هكذا
+
+40
+00:03:40,840 --> 00:03:45,180
+فكل الرموز هذه ترمز إلى ال sequence هذه اللي هي ال
+
+41
+00:03:45,180 --> 00:03:53,400
+function f اللي هي ال function f okay إذن أهم شيء
+
+42
+00:03:53,400 --> 00:03:56,480
+في تعريفنا أن ال sequence هي function دلنا
+
+43
+00:03:56,480 --> 00:04:00,400
+وبالتالي لها مجال مجالها العداد الطبيعي المجال
+
+44
+00:04:00,400 --> 00:04:04,420
+المقابل هي المجموعة اللي عناصر ال sequence تنتمي
+
+45
+00:04:04,420 --> 00:04:10,950
+لها ال sequences ممكن أعرفهم بطريقتينإذا في
+
+46
+00:04:10,950 --> 00:04:15,970
+الملاحظة هذه sequences can be defined explicitly
+
+47
+00:04:15,970 --> 00:04:19,910
+هذه أحد الطرق ممكن يعرف ال sequence بطريقة صريحة
+
+48
+00:04:19,910 --> 00:04:27,890
+بطريقة بقانونفمثلا ال sequence if بالساوية عناصرها
+
+49
+00:04:27,890 --> 00:04:31,670
+اتنين اربعة ستة تمانية الاخرى هذه عبارة عن
+
+50
+00:04:31,670 --> 00:04:38,130
+sequence وهي معرفة بطريقة صريحة فهذه عبارة عن
+
+51
+00:04:38,130 --> 00:04:42,630
+sequence of even natural members العداد الطبيعية
+
+52
+00:04:42,630 --> 00:04:47,790
+الزوجيةممكن نكتب الحد العام الانف هذا بنسميه الانف
+
+53
+00:04:47,790 --> 00:04:53,710
+term اكس ان هذا هنا بنسميه الانف term الحد النوني
+
+54
+00:04:53,710 --> 00:04:59,190
+الحد النوني او الحد العام فال انف term هنا هو
+
+55
+00:04:59,190 --> 00:05:08,180
+اتنين ان اكس ان بساوي اتنين ان حيث ان عدد طبيعيأو
+
+56
+00:05:08,180 --> 00:05:12,620
+ممكن نكتب ال sequence على صورة 2n من n بالساعة
+
+57
+00:05:12,620 --> 00:05:16,740
+واحد إلى ملا نهائية إذا هنا أنا بعرف ال sequence
+
+58
+00:05:16,740 --> 00:05:22,960
+برص حدودها أول تلات حدود إلى و هكذا أو بكتب قاعدة
+
+59
+00:05:22,960 --> 00:05:27,880
+لحد العام xn و طبعا n أدى الطبيعي فمقدر من القاعدة
+
+60
+00:05:27,880 --> 00:05:32,740
+هذه أجيب كل الحدود إذا هذا explicit definition of
+
+61
+00:05:32,740 --> 00:05:39,150
+a sequence هذا تعريف صريح لل sequenceفي طريقة
+
+62
+00:05:39,150 --> 00:05:44,870
+تانية لتعريف ال sequence وهي الطريقة الاستقرائية،
+
+63
+00:05:44,870 --> 00:05:49,330
+إذا ال sequences can be defined inductively أو
+
+64
+00:05:49,330 --> 00:05:55,970
+recursivelyبطريقة استقرائية او بطريقة تكرارية كيف
+
+65
+00:05:55,970 --> 00:06:02,290
+هذه الطريقة باجي لل sequence و باخد اول حد فيها زي
+
+66
+00:06:02,290 --> 00:06:07,250
+X1 او اول حدين او اول تلات حدود و بعطيهم قيم
+
+67
+00:06:07,250 --> 00:06:16,010
+بحددهم قيم محددة بعطيهم قيم محددة بعدين باجيبباجي
+
+68
+00:06:16,010 --> 00:06:21,990
+بعبّر عن الحد xn زايد واحد او xn بدلالة الحدود
+
+69
+00:06:21,990 --> 00:06:27,850
+اللي جابله وبستخدم طبعا لهذا formula بنسميها
+
+70
+00:06:27,850 --> 00:06:32,070
+recursive formula او inductive formula كما في
+
+71
+00:06:32,070 --> 00:06:39,550
+المثال التالي يعني انا عند ال sequence 2n هذه انا
+
+72
+00:06:39,550 --> 00:06:48,000
+عند ال sequence xn بساوة 2nهذه ممكن أعرفها بطريقة
+
+73
+00:06:48,000 --> 00:06:57,140
+استقرائية كيف باخد بعطي أول حد فيه x1 بعطيله قيمة
+
+74
+00:06:57,140 --> 00:07:01,220
+محددة وهي 2 طبعا أول حد في ال sequence هذه هو 2
+
+75
+00:07:01,220 --> 00:07:06,760
+صح؟ لأن هنا أخدت x1 وعطيته قيمة محددة ممكن في بعض
+
+76
+00:07:06,760 --> 00:07:12,140
+الأمثلة أعطي قيمة قيمة محددة ل x1 و x2 و x3بعدين
+
+77
+00:07:12,140 --> 00:07:19,100
+باجي إلى الحد رقم n زياد واحد و بعبر عنه ب
+
+78
+00:07:19,100 --> 00:07:23,000
+recursive formula بعبر عنه بدلالة الحد اللي جابله
+
+79
+00:07:23,000 --> 00:07:26,760
+او الحد اللي جابله مباشرة و الجاب اللي جابله و
+
+80
+00:07:26,760 --> 00:07:32,510
+هكذافهذه بنسميها recursive أو inductive formula
+
+81
+00:07:32,510 --> 00:07:37,150
+تعطيني لحد رقم n زاد واحد بدالة الحد اللي جابله xn
+
+82
+00:07:37,150 --> 00:07:43,870
+فمثلا لو بده أحسب x2 فباخد n بساوي واحد هنا صح
+
+83
+00:07:43,870 --> 00:07:50,110
+فبطل عند x2 بساوي x1 زاد اتنين x1 بساوي اتنين زاد
+
+84
+00:07:50,110 --> 00:07:56,400
+اتنين بطلع أربعةX3 برضه عشان اجيب X3 بستخدم ال
+
+85
+00:07:56,400 --> 00:08:00,480
+recursive formula و باخد N بساوي 2 فبطلع عند X3
+
+86
+00:08:00,480 --> 00:08:06,600
+بساوي X2 زائد 2 X2 أربعة و اتنين بطلع ستة و هكذا
+
+87
+00:08:06,600 --> 00:08:13,340
+اذا هيك بحصل على ال sequence 2N اللي حدودها 2 4 6
+
+88
+00:08:13,340 --> 00:08:20,460
+8 و هكذا اه okay تمام ال
+
+89
+00:08:20,460 --> 00:08:30,520
+..طيب الان بدي اعرف ما معناه ان ال sequence تكون
+
+90
+00:08:30,520 --> 00:08:36,500
+convergent او لها limit لو في عندى sequence من
+
+91
+00:08:36,500 --> 00:08:37,720
+العداد الحقيقية
+
+92
+00:08:41,200 --> 00:08:45,480
+فبقول إن ال sequence converge
+
+93
+00:08:45,480 --> 00:08:51,860
+ال sequence of real numbers بتكون converge أو
+
+94
+00:08:51,860 --> 00:08:59,940
+convergent إذا قدرت ألاقي X ينتمي ل R بحيث إنه لكل
+
+95
+00:08:59,940 --> 00:09:06,200
+neighborhood V ل X لكل جوار V ل X بقدر أو جد أو
+
+96
+00:09:06,200 --> 00:09:12,250
+ألاقيعدد طبيعي capital N يعتمد على الجوار V ينتمي
+
+97
+00:09:12,250 --> 00:09:17,030
+لعداد الطبيعية بحيث أنه لكل small n أكبر من أو سوى
+
+98
+00:09:17,030 --> 00:09:21,770
+capital N، Xn ينتمي إلى V يعني الجوار V هذا يحتوي
+
+99
+00:09:21,770 --> 00:09:29,100
+كل عناصر ال sequence من capital N وانت طالعفلو هذا
+
+100
+00:09:29,100 --> 00:09:34,020
+الشرط اتحقق فبنقول ان الـ sequence converge و ال
+
+101
+00:09:34,020 --> 00:09:38,040
+limit تبعتها هي العدد X في الحالة هذه بنقول ان X
+
+102
+00:09:38,040 --> 00:09:46,080
+is the limit of sequence X in و
+
+103
+00:09:46,080 --> 00:09:51,180
+بنكتب limit X in بالساوية X او نكتب X in tends to
+
+104
+00:09:51,180 --> 00:09:57,750
+X as N tends to infinityهذا التعريف بنسميه ال
+
+105
+00:09:57,750 --> 00:10:05,170
+neighborhood neighborhood definition neighborhood
+
+106
+00:10:05,170 --> 00:10:16,710
+definition of convergence تعريف
+
+107
+00:10:16,710 --> 00:10:18,210
+الجوار للتقارب
+
+108
+00:10:22,960 --> 00:10:28,200
+طيب لو ال sequence ماكانش لها limit يعني مافيش لا
+
+109
+00:10:28,200 --> 00:10:34,560
+يوجد x ينتمي ل r بحقق الشرط هذا فبنقول ان ال
+
+110
+00:10:34,560 --> 00:10:40,060
+sequence ليست not convergent او divergent اذا لو
+
+111
+00:10:40,060 --> 00:10:45,220
+ال sequence مالهاش has no limit فبنسميها divergent
+
+112
+00:10:45,220 --> 00:10:50,820
+اذا مثلا بتكون ال sequence convergent اذا كان في
+
+113
+00:10:50,820 --> 00:10:54,560
+لها limitطب ما معناه ان ال sequence يكون لها
+
+114
+00:10:54,560 --> 00:11:01,680
+limit؟ معناه ان يوجد عدد حقيقي X بحيث لكل جوار V ل
+
+115
+00:11:01,680 --> 00:11:08,260
+X في عدد طبيعي capital N يعتمد على الجوار بحيث ان
+
+116
+00:11:08,260 --> 00:11:14,120
+كل حدود ال sequence تنتمي للجوار هذا والمؤشر تبعها
+
+117
+00:11:14,120 --> 00:11:20,130
+ببدأ من capital N وانت طالعيعني معنى الكلام هذا ..
+
+118
+00:11:20,130 --> 00:11:28,290
+هذا الكلام معناه ان X capital N و X capital N زائد
+
+119
+00:11:28,290 --> 00:11:35,990
+واحد و X capital N زائد اتنين و هكذا كل هدول
+
+120
+00:11:35,990 --> 00:11:38,630
+بينتموا الى الجوار دي
+
+121
+00:11:44,830 --> 00:11:48,590
+لو ال sequence مالهاش limit فبنسميها divergent
+
+122
+00:11:48,590 --> 00:11:56,190
+okay طبعا؟ V جوار .. جوار يعني .. مجموعة .. اه
+
+123
+00:11:56,190 --> 00:12:01,410
+جوار ل X يعني مجموعة تحتوي ال X و الجوار عشان V
+
+124
+00:12:01,410 --> 00:12:05,710
+يكون جوار لازم يكون داخله .. لازم نلاقي داخله
+
+125
+00:12:05,710 --> 00:12:10,010
+epsilon نبرهون كل جوار لازم يحتوي epsilon نبرهون
+
+126
+00:12:15,360 --> 00:12:23,300
+يعني مش اي مجموعة طيب
+
+127
+00:12:23,300 --> 00:12:27,780
+ال .. ان لو
+
+128
+00:12:27,780 --> 00:12:32,800
+في اندي سيكوانس و السيكوانس هاد convergent ف ال
+
+129
+00:12:32,800 --> 00:12:34,600
+limit تبعتها بتطلع unique
+
+130
+00:12:41,740 --> 00:12:45,620
+النظرية الأولى بتقول لو كانت x in sequence of real
+
+131
+00:12:45,620 --> 00:12:51,320
+numbers و converge ل x و converge ل y يعني لها two
+
+132
+00:12:51,320 --> 00:12:55,740
+limits فلازم ال limits يكونوا متساويتين يعني ممنوع
+
+133
+00:12:55,740 --> 00:12:59,940
+ال convergence sequence يكون لها أكتر من limit
+
+134
+00:12:59,940 --> 00:13:05,400
+يعني معناه بعبارة أخرى a convergent sequence has a
+
+135
+00:13:05,400 --> 00:13:06,140
+unique limit
+
+136
+00:13:09,340 --> 00:13:13,560
+خلّينا نبرهن الكلام هذا، افرض إنه في عندي sequence
+
+137
+00:13:13,560 --> 00:13:20,440
+x in converge ل x و أيضا converge ل y المطلوب
+
+138
+00:13:20,440 --> 00:13:25,540
+إثبات إن x بساوي y لبرهان ذلك نعمل برهان بالتناقض
+
+139
+00:13:25,540 --> 00:13:30,680
+assume on contrary إن x لا تساوي y اللي هو نفي
+
+140
+00:13:30,680 --> 00:13:36,600
+النتيجة و بينصل لتناقض في exercise 15 في section 2
+
+141
+00:13:36,600 --> 00:13:41,810
+.2أخذناها في ال chapter السابق بقول لو في عندي أي
+
+142
+00:13:41,810 --> 00:13:49,130
+عددين حقيقيين x و y فبقدر
+
+143
+00:13:49,130 --> 00:13:57,250
+ألاقي v1 جوار ل x و
+
+144
+00:13:57,250 --> 00:14:05,390
+بقدر ألاقي v2 v2
+
+145
+00:14:05,390 --> 00:14:06,610
+جوار ل y
+
+146
+00:14:09,920 --> 00:14:17,120
+بحيث ان تقاطعهم بساوي five يعني اثنين disjoint
+
+147
+00:14:19,260 --> 00:14:24,660
+تمام؟ لو كان في عندي عددين حققين x لا يساوي y بقدر
+
+148
+00:14:24,660 --> 00:14:31,280
+ألاقي جوار v1 ل x و جوار v2 ل y و الجوارين هدول
+
+149
+00:14:31,280 --> 00:14:36,660
+منفصلين بعتقد حلنا السؤال هذا اه فقولنا خدي
+
+150
+00:14:36,660 --> 00:14:45,290
+epsilon بساوي نص المسافة بين x و yو هد خلّي x زاد
+
+151
+00:14:45,290 --> 00:14:50,410
+y و النقطة هد x سالب y هد عبارة عن y neighborhood
+
+152
+00:14:50,410 --> 00:14:55,570
+ل x وبالتالي neighborhood ل x و خدي هنا برضه هد
+
+153
+00:14:55,570 --> 00:15:01,030
+عبارة عن y سالب y و النقطة هد y زاد y
+
+154
+00:15:03,680 --> 00:15:09,460
+فال .. واضح أن الجوارين هدول متقاطعوش لأن أنا أخدت
+
+155
+00:15:09,460 --> 00:15:13,180
+epsilon نص المسافة هذه و هذه فترة مفتوعة و هذه
+
+156
+00:15:13,180 --> 00:15:18,560
+مفتوعة فمافيش بينهم نقاط مشتركة okay إذا هذا
+
+157
+00:15:18,560 --> 00:15:23,620
+الكلام موجود إذا هذا صحيح exercise 15 بيقول لي إذا
+
+158
+00:15:23,620 --> 00:15:30,310
+كان x لا يساوي yفطبعا ممكن نفرض ان x أصغر من y أو
+
+159
+00:15:30,310 --> 00:15:35,170
+y أصغر من x وبالتالي بقدر ألاقي this joint this
+
+160
+00:15:35,170 --> 00:15:43,630
+joint neighborhoods v1 ل x وv2 ل y على التوالي و 2
+
+161
+00:15:43,630 --> 00:15:50,910
+منفصلين الان احنا فرضين ان x in converge ل xحسب
+
+162
+00:15:50,910 --> 00:15:54,790
+الـ Neighborhood Definition لـ Convergence لما أن
+
+163
+00:15:54,790 --> 00:16:00,550
+المتتالي Xn converge ل X و V1 جوار ل X إذا يوجد
+
+164
+00:16:00,550 --> 00:16:07,710
+عدد طبيعي N1 يعتمد على الجوار V1 بحيث أن Xn تنتمي
+
+165
+00:16:07,710 --> 00:16:13,260
+للجوار V1 لكل N أكبر من أو ساوى N1كذلك احنا فرضين
+
+166
+00:16:13,260 --> 00:16:18,320
+في النظرية ان sequence xn converge ل y و الان v2
+
+167
+00:16:18,320 --> 00:16:23,660
+neighborhood ل y، اذا حسب تعريف ال convergence بما
+
+168
+00:16:23,660 --> 00:16:27,680
+ان xn converge ل y و v2 neighborhood ل y، اذا
+
+169
+00:16:27,680 --> 00:16:32,440
+بنقدر نلاقي عدد طبيعي n2 يعتمد على v2، بحيث ان xn
+
+170
+00:16:32,440 --> 00:16:38,840
+ينتمي لv2 لكل n أكبر من أو ساوي n2الان لو عرفت
+
+171
+00:16:38,840 --> 00:16:42,320
+capital N على Nها ال maximum الاكبر بين N واحد و N
+
+172
+00:16:42,320 --> 00:16:47,360
+اتنين هذا معناه ان capital N عدد طبيعي لان الاكبر
+
+173
+00:16:47,360 --> 00:16:52,320
+بين هدول هيكون واحد منهم فهو عدد طبيعي و capital N
+
+174
+00:16:52,320 --> 00:16:55,640
+اكبر من او ساوي N واحد و اكبر من او ساوي N اتنين
+
+175
+00:16:55,640 --> 00:16:59,820
+لان الكبير فيهم الان
+
+176
+00:16:59,820 --> 00:17:04,120
+لو اخدت small n اكبر من او ساوي capital N فمن
+
+177
+00:17:04,120 --> 00:17:09,540
+تعريف capital Nهذا بيقدي ان capital N أكبر من أو
+
+178
+00:17:09,540 --> 00:17:14,760
+ساوي N واحد اذا الان انا عندي small n أكبر من أو
+
+179
+00:17:14,760 --> 00:17:23,820
+ساوي N واحد وبالتالي اذا Xn تنتمي ل D واحد كذلك
+
+180
+00:17:23,820 --> 00:17:29,560
+انا عندي من تعريف capital N capital N أكبر من أو
+
+181
+00:17:29,560 --> 00:17:34,950
+ساوي N اتنينوبالتالي small n أكبر من أو ساوي
+
+182
+00:17:34,950 --> 00:17:38,970
+capital N اتنين لما تكون small n أكبر من أو ساوي
+
+183
+00:17:38,970 --> 00:17:45,450
+capital N اتنين فبطلع xn ينتمي إلى v2 إذا الأن أنا
+
+184
+00:17:45,450 --> 00:17:49,110
+أثبتت أنه لو كانت small n أكبر من أو ساوي capital
+
+185
+00:17:49,110 --> 00:17:57,090
+N فبطلع xn ينتمي إلىV1 و الى V2 وبالتالي تنتمي
+
+186
+00:17:57,090 --> 00:18:01,290
+لتقاطعهم إذا المعنى أن التقاطع هذا لا يساوي فيه
+
+187
+00:18:01,290 --> 00:18:05,810
+وهذا بيديني contradiction لأنه exercise 15 بيقول
+
+188
+00:18:05,810 --> 00:18:10,450
+لي أن V1 و V2 هدول disjoint فكيف طلع مش disjoint
+
+189
+00:18:10,450 --> 00:18:16,070
+تناقض تناقض هذا بيقول لي أن ال assumption تبعيإن X
+
+190
+00:18:16,070 --> 00:18:20,390
+لا تساوي Y كان خطأ إذن الصح إن X بالساوي Y
+
+191
+00:18:20,390 --> 00:18:25,430
+وبالتالي ال limit لل sequence لازم تكون واحدة
+
+192
+00:18:25,430 --> 00:18:33,990
+unique تمام؟ واضح البرهان؟ في أي استفسار؟
+
+193
+00:18:33,990 --> 00:18:37,510
+في أي سؤال؟
+
+194
+00:18:50,080 --> 00:19:02,120
+النظرية التانية تعطيني
+
+195
+00:19:02,120 --> 00:19:09,740
+شروط متكافئة لتعريف ال convergence للسيكوينس فلو
+
+196
+00:19:09,740 --> 00:19:12,840
+في عندي سيكوينس of real numbers وعندي real number
+
+197
+00:19:12,840 --> 00:19:17,630
+x the following are equivalentهذا اختصار الكلمات
+
+198
+00:19:17,630 --> 00:19:21,530
+the following are equivalent الاعبارات التالية
+
+199
+00:19:21,530 --> 00:19:27,670
+متكافئة اول عبارة x in converge ل x هذا معناه حسب
+
+200
+00:19:27,670 --> 00:19:31,070
+تعريف ال convergence ال neighborhood definition ان
+
+201
+00:19:31,070 --> 00:19:42,150
+for every neighborhood V of X of X there exists
+
+202
+00:19:42,150 --> 00:19:50,590
+capital N يعتمد على Vعدد طبيعي بحيث أنه لو كان n
+
+203
+00:19:50,590 --> 00:19:56,150
+أكبر من أو ساوي capital N هذا بيقدر ان xn ينتمي
+
+204
+00:19:56,150 --> 00:20:03,390
+إلى b هاي معناه xn converge ل x الان هذا ال
+
+205
+00:20:03,390 --> 00:20:06,990
+neighborhood definition لل convergence بيكافئ
+
+206
+00:20:06,990 --> 00:20:11,770
+العبارة بي وهذا بنسميها ال epsilon neighborhood
+
+207
+00:20:11,770 --> 00:20:16,150
+definition لل convergenceهذا بقى بنسميه epsilon
+
+208
+00:20:16,150 --> 00:20:20,210
+neighborhood definition of convergence ليه؟
+
+209
+00:20:20,210 --> 00:20:22,850
+العبارة دي بتقول لكل for every epsilon
+
+210
+00:20:22,850 --> 00:20:27,930
+neighborhood V epsilon ل X يعني بدل لكل
+
+211
+00:20:27,930 --> 00:20:32,550
+neighborhood بدلناها لكل epsilon neighborhood ل X
+
+212
+00:20:32,550 --> 00:20:35,630
+يوجد capital N يعتمد على ال epsilon neighborhood
+
+213
+00:20:35,630 --> 00:20:42,160
+وبالتالي يعتمد على ال epsilon عدد طبيعيبحيث أنه
+
+214
+00:20:42,160 --> 00:20:46,200
+لكل N أكبر من أوسعه capital N بطلع XN ينتمي لبي
+
+215
+00:20:46,200 --> 00:20:52,820
+نفس العادلالعبارة التالتة بتقول لكل إبسلون لأي عدد
+
+216
+00:20:52,820 --> 00:20:56,260
+إبسلون موجة بنقدر نلاقي عدد طبيعي يعتمد على إبسلون
+
+217
+00:20:56,260 --> 00:21:01,500
+بحيث لو كان n أكبر من أو ساوي capital N فالمسافة
+
+218
+00:21:01,500 --> 00:21:07,800
+بين x and x تطلع أصغر من إبسلون هذا بنسميه الجزء C
+
+219
+00:21:07,800 --> 00:21:13,180
+وهذا الجزء الأكتر جزء هنستخدمه في إثبات ال
+
+220
+00:21:13,180 --> 00:21:18,080
+convergence لsequences معينةهذا بيسميه epsilon
+
+221
+00:21:18,080 --> 00:21:25,600
+capital N definition of
+
+222
+00:21:25,600 --> 00:21:26,500
+convergence
+
+223
+00:21:30,350 --> 00:21:34,970
+انا في عندى انا الفرق A هذا عبارة عن epsilon عبارة
+
+224
+00:21:34,970 --> 00:21:38,530
+عن neighborhood definition of convergence الفرق B
+
+225
+00:21:38,530 --> 00:21:42,230
+بنسميه ال epsilon neighborhood definition لل
+
+226
+00:21:42,230 --> 00:21:46,210
+convergence الفرق C بنسميه epsilon capital N
+
+227
+00:21:46,210 --> 00:21:49,770
+definition of convergence هذا هيكون استعماله شائع
+
+228
+00:21:49,770 --> 00:21:57,370
+اكتر من العبارات السابقةالبرهان ان هذا ال تلاتة
+
+229
+00:21:57,370 --> 00:22:02,490
+إبراهيم بتكافئ بعض هنثبت ان a implies b و b
+
+230
+00:22:02,490 --> 00:22:10,610
+implies c و بعد هيك هنثبت ان c implies a وبالتالي
+
+231
+00:22:10,610 --> 00:22:14,370
+هيك بيطلع التلاتة متكافئة حسب قوانين ال logic
+
+232
+00:22:14,370 --> 00:22:21,830
+مظبوط صح؟طيب نشوف الأول a implies b افرض ان x in
+
+233
+00:22:21,830 --> 00:22:28,010
+converge ل x يعني هذا الكلام صحيح حسب تعريف ال
+
+234
+00:22:28,010 --> 00:22:34,510
+neighborhood definition لل convergence طيب .. طيب
+
+235
+00:22:34,510 --> 00:22:39,150
+احنا عارفين ان كل epsilon .. طيب لإثبات ان b صحيح
+
+236
+00:22:39,150 --> 00:22:45,130
+ناخد أي epsilon neighborhood ل xطب احنا لما درسنا
+
+237
+00:22:45,130 --> 00:22:48,990
+ال neighborhoods قلنا ان كل epsilon neighborhood
+
+238
+00:22:48,990 --> 00:22:52,130
+.. every epsilon neighborhood على الصورة هذه ل X
+
+239
+00:22:52,130 --> 00:22:57,490
+هو ايضا neighborhood ل X صح؟ هذه حقيقة معروفة ..
+
+240
+00:22:57,490 --> 00:23:02,570
+كل epsilon neighborhood ل X is also a neighborhood
+
+241
+00:23:02,570 --> 00:23:09,280
+of Xوبالتالي إذا هنا لو أخدت أي إبسلون
+
+242
+00:23:09,280 --> 00:23:13,140
+neighborhood ل X فهذا neighborhood ل X وبالتالي
+
+243
+00:23:13,140 --> 00:23:15,820
+يوجد capital N يعتمد على الإبسلون neighborhood
+
+244
+00:23:15,820 --> 00:23:24,080
+وهذا الكلام صح وبالتالي A بيؤدي ل B نشوف
+
+245
+00:23:24,080 --> 00:23:27,460
+الآن بيؤدي العبارة بيؤدي إلى C
+
+246
+00:23:42,950 --> 00:23:55,970
+طيب العبارة P هذا هي لو كان P صحيح فبنثبت
+
+247
+00:23:55,970 --> 00:24:05,490
+ان C صحيح فخلينا ناخد خلينا
+
+248
+00:24:05,490 --> 00:24:09,250
+ناخد أبسلون أكبر من السفر ناخد أبسلون أكبر من
+
+249
+00:24:09,250 --> 00:24:09,730
+السفر
+
+250
+00:24:13,900 --> 00:24:22,140
+لو أخدت أي epsilon أكبر من السفر for any epsilon
+
+251
+00:24:22,140 --> 00:24:30,140
+أكبر من السفر take v epsilon of x اللي هو عبارة عن
+
+252
+00:24:30,140 --> 00:24:36,040
+ال epsilon neighborhood ل x فهذا
+
+253
+00:24:36,040 --> 00:24:44,530
+is epsilon neighborhood of x صح؟وبالتالي حسب بي
+
+254
+00:24:44,530 --> 00:24:50,890
+لأي إبسلون neighborhood لهذا يوجد capital N إذا
+
+255
+00:24:50,890 --> 00:24:56,350
+يوجد capital N by
+
+256
+00:24:56,350 --> 00:25:02,930
+بي يوجد capital N يعتمد على الإبسلون neighborhood
+
+257
+00:25:02,930 --> 00:25:09,630
+وبالتالي يعتمد على إبسلون هذا عدد طبيعي بحيث
+
+258
+00:25:13,530 --> 00:25:19,590
+بحيث انه لو كان n أكبر من أو ساوي n of epsilon
+
+259
+00:25:19,590 --> 00:25:28,030
+فهذا بيقدي ان xn ينتمي ل v epsilon ل x اللي هو x
+
+260
+00:25:28,030 --> 00:25:35,630
+سالب epsilon وx موجة بepsilon طب وهذا معناه ان ال
+
+261
+00:25:35,630 --> 00:25:44,930
+xn أكبر من x سالب epsilon أصغر من x زاد epsilonهذا
+
+262
+00:25:44,930 --> 00:25:50,630
+الـ xn ينتمي للفترة المفتوحة هذه معناته هذا الكلام
+
+263
+00:25:50,630 --> 00:25:56,670
+صح هذا معناه xn minus x أصغر من epsilon أكبر من
+
+264
+00:25:56,670 --> 00:26:01,950
+سالب epsilon هذا معناه absolute xn minus x أصغر من
+
+265
+00:26:01,950 --> 00:26:10,800
+epsilon إذن هين أثبتنا إن لو كان b صحيحفلأي يبسلون
+
+266
+00:26:10,800 --> 00:26:18,300
+أكبر من السفر يوجد capital N يعتمد على يبسلون بحيث
+
+267
+00:26:18,300 --> 00:26:23,160
+لكل N أكبر من أو ساوي capital N طلع absolute xn
+
+268
+00:26:23,160 --> 00:26:29,920
+minus x أصغر من يبسلون وبالتالي العبارة C صحيحة
+
+269
+00:26:29,920 --> 00:26:38,500
+متحققة okay تمام؟ الآن بقى نثبت أن العبارة
+
+270
+00:26:38,500 --> 00:26:59,280
+Cبتقدي إلى العبارة A فأفرضي
+
+271
+00:26:59,280 --> 00:27:08,370
+أن العبارة C متحققة suppose C holdsبعدين، بدنا
+
+272
+00:27:08,370 --> 00:27:12,250
+نثبت أن x in converge ل x أو ال neighborhood
+
+273
+00:27:12,250 --> 00:27:17,730
+definition ل x بتحقق فبناخد أي let v be any
+
+274
+00:27:17,730 --> 00:27:24,590
+neighborhood of x فمن تعريف ال neighborhoodلأي
+
+275
+00:27:24,590 --> 00:27:28,910
+neighborhood كل neighborhood v ل x يحتوي داخله
+
+276
+00:27:28,910 --> 00:27:32,030
+epsilon neighborhood ل x هذا ما قلناه قبل هيك
+
+277
+00:27:32,030 --> 00:27:37,430
+وبالتالي يوجد epsilon عدد موجب بحيث ان ال epsilon
+
+278
+00:27:37,430 --> 00:27:44,890
+neighborhood هذه الفترة عبارة عن x in .. هذه
+
+279
+00:27:44,890 --> 00:27:51,090
+المفروضة تكون عفوا هذه المفروضة تكون x مش x in
+
+280
+00:27:51,090 --> 00:28:01,600
+وهذه x سلب epsilonهذا عبارة عن v epsilon ل x هذا
+
+281
+00:28:01,600 --> 00:28:08,880
+المفروض تكون x مش xm إذا لو كان v epsilon
+
+282
+00:28:08,880 --> 00:28:15,740
+neighborhood ففي عندي بقدر ألاقي جواته epsilon
+
+283
+00:28:15,740 --> 00:28:20,520
+neighborhood لل x اللي هو v epsilon ل x الآن من
+
+284
+00:28:20,520 --> 00:28:21,400
+الجزء c
+
+285
+00:28:25,470 --> 00:28:29,650
+لأي أبسلون من الجزء C لأي أبسلون لأ بما أن هذا
+
+286
+00:28:29,650 --> 00:28:33,170
+أبسلون أكبر من السفر إذا بنقدر نلاقي capital N
+
+287
+00:28:33,170 --> 00:28:36,310
+يعتمد على أبسلون بحيث لكل N أكبر من أو ساوية
+
+288
+00:28:36,310 --> 00:28:40,230
+capital N ال absolute value هذه أصغر من أبسلون هذا
+
+289
+00:28:40,230 --> 00:28:45,660
+من الجزء Cطب ما هذا معناه ال implication هذه
+
+290
+00:28:45,660 --> 00:28:50,920
+معناها لكل n أكبر من أو ساوي capital N لو فكيت
+
+291
+00:28:50,920 --> 00:28:58,800
+المتباينة هذه معناها xn ينتمي هذا عبارة عن x ينتمي
+
+292
+00:28:58,800 --> 00:29:06,480
+لفترة مفتوحة x minus y و x z epsilon اللي هو ال
+
+293
+00:29:06,480 --> 00:29:09,720
+epsilon neighborhood ل x اللي هو subset من V
+
+294
+00:29:11,670 --> 00:29:19,650
+وبالتالي هيك بنكون أثبتنا أن ال XIN ينتمي إلى ال
+
+295
+00:29:19,650 --> 00:29:24,530
+neighborhood V كمان
+
+296
+00:29:24,530 --> 00:29:30,830
+مرة أنا بدي أثبت أن العبارة C بتأدي ليه، افرض أن
+
+297
+00:29:30,830 --> 00:29:36,610
+العبارة C صحيحةالان لإثبات a اللى هى x in converge
+
+298
+00:29:36,610 --> 00:29:40,790
+ل x بتثبت أنه ال neighborhood definition لل
+
+299
+00:29:40,790 --> 00:29:45,750
+convergence بتحقق يعنى x عبارة عن limit لل
+
+300
+00:29:45,750 --> 00:29:48,650
+sequence x in فنرجع لتعريف ال neighborhood
+
+301
+00:29:48,650 --> 00:29:53,190
+definition of convergence نبدأ ب neighborhood ل x
+
+302
+00:29:53,190 --> 00:29:57,910
+ونستخدم الحقيقة أن كل neighborhood ل x يحتوي
+
+303
+00:29:57,910 --> 00:30:04,160
+epsilon neighborhoodالان من C .. C بيقول لي إذا في
+
+304
+00:30:04,160 --> 00:30:08,400
+عندك إبسلون موجبة تقدر تلاقي capital N يعتمد عليها
+
+305
+00:30:08,400 --> 00:30:12,940
+بحيث أنه لكل N أكبر من ما يساوي capital N المسافة
+
+306
+00:30:12,940 --> 00:30:17,660
+هذه أصغر من إبسلونطب هذه ال implication الأخيرة هي
+
+307
+00:30:17,660 --> 00:30:22,380
+N أكبر من أو ساوي capital N بتقدي في حل المتباين
+
+308
+00:30:22,380 --> 00:30:28,640
+هذه في Xn فبطلع Xn ينتمي إلى X سالب Y و X فاللي هو
+
+309
+00:30:28,640 --> 00:30:33,320
+هذا ال epsilon neighborhood اللي هوداخل V وبالتالي
+
+310
+00:30:33,320 --> 00:30:37,660
+لكل N أكبر من لو ساوي capital N طلع Xn ينتمي لل
+
+311
+00:30:37,660 --> 00:30:42,300
+neighborhood V هذا من التعريف معناه Xn converge ل
+
+312
+00:30:42,300 --> 00:30:48,820
+X وبالتالي اللي عبارة أيه صحيحة تمام؟ إذا هيك
+
+313
+00:30:48,820 --> 00:30:53,940
+بنكون أثبتنا النظرية أن التلت تعريفات هذه كلها
+
+314
+00:30:53,940 --> 00:30:54,840
+متكافئة
+
+315
+00:31:02,750 --> 00:31:06,990
+في تعريف الـ tail of a sequence او الـ M tail of a
+
+316
+00:31:06,990 --> 00:31:11,070
+sequence احنا عارفين ان لو في اندز اي .. لأي
+
+317
+00:31:11,070 --> 00:31:18,570
+sequence XN لو خدت M عدد طبيعي اي عدد طبيعي
+
+318
+00:31:18,570 --> 00:31:24,210
+natural number و XN اي sequence of real numbers
+
+319
+00:31:24,210 --> 00:31:31,330
+فالـ XN هذه ممكن انفرفتها نكتب حدودها X1 X2 و هكذا
+
+320
+00:31:32,450 --> 00:31:41,130
+الى x رقم m الان الحد اللي بعد xm عبارة عن xm زاد
+
+321
+00:31:41,130 --> 00:31:50,010
+واحد و اللي بعده xm زاد اتنين و هكذا اذا
+
+322
+00:31:50,010 --> 00:31:53,130
+ال sequence هذه ممكن اكتبها على الصورة هذه حيث م
+
+323
+00:31:53,130 --> 00:31:57,770
+هنا عدد طبيعي ما ثابت
+
+324
+00:31:59,680 --> 00:32:10,460
+الان لو انا ركزت على الجزء هذا من ال sequence و
+
+325
+00:32:10,460 --> 00:32:20,440
+الجزء هذا هو اول m من حدود ال sequence حذفتها فاذا
+
+326
+00:32:20,440 --> 00:32:22,400
+هذا بنسميه m tail
+
+327
+00:32:28,870 --> 00:32:37,630
+متل لسيكوينس xn الدنب م دنب م مش هذا دنب يعني تصور
+
+328
+00:32:37,630 --> 00:32:42,110
+إنها دي أفع هي الرأس تبعها أول م من الحدود ده هي
+
+329
+00:32:42,110 --> 00:32:47,570
+الرأس جاطعة الرأس تبعها فبقى الدنب مش هيك بيقولوا
+
+330
+00:32:47,570 --> 00:32:50,870
+الدنب
+
+331
+00:32:50,870 --> 00:32:56,090
+هذا طويلبنبدأ يعني في عدد لانها من الحدود الراس
+
+332
+00:32:56,090 --> 00:33:02,470
+محدود هي عدد منتهي من الحدود اذا ال sequence لو
+
+333
+00:33:02,470 --> 00:33:08,250
+انا حدفت اول M من حدودها فباقي الجزء المتبقى من ال
+
+334
+00:33:08,250 --> 00:33:16,070
+sequence بنسميه M tail واضح طيب اذا الان في نظرية
+
+335
+00:33:16,070 --> 00:33:18,250
+اتنين تلاتة او نظرية تالتة
+
+336
+00:33:20,720 --> 00:33:23,800
+ما هي هذه النظرية اللي بتقول؟ بتقول لو أنا في اندي
+
+337
+00:33:23,800 --> 00:33:29,500
+إذا هاي ال m tail هذا ال m tail ممكن كتابته على
+
+338
+00:33:29,500 --> 00:33:35,820
+صورة sequence هاي x المؤشر الحد العام تبع ال m
+
+339
+00:33:35,820 --> 00:33:40,660
+tail m زاد n حيث و اين العداد الطبيعي m ثابت و n
+
+340
+00:33:40,660 --> 00:33:43,980
+العداد الطبيعي وبالتالي هنا لو كانت n بالساوية
+
+341
+00:33:43,980 --> 00:33:50,800
+واحد اول حد xm زاد واحد و هكذا طيبالان النظرية
+
+342
+00:33:50,800 --> 00:33:57,980
+التالية بتقولني انه لو كان ال M tail convergent
+
+343
+00:34:02,380 --> 00:34:07,760
+فال sequence نفسها ال M بتكون convergent و العكس
+
+344
+00:34:07,760 --> 00:34:12,020
+لو كانت ال sequence convergent فأي M tail منها
+
+345
+00:34:12,020 --> 00:34:15,940
+هيكون convergent و اتنين لهم نفس ال limit اتنين
+
+346
+00:34:15,940 --> 00:34:20,020
+لهم نفس ال limit اذا مرة تانية لو كان في عندك
+
+347
+00:34:20,020 --> 00:34:27,500
+sequence XN M fixed natural number فال M tail اللي
+
+348
+00:34:27,500 --> 00:34:32,350
+هو ال sequence هذهconverges if and only if
+
+349
+00:34:32,350 --> 00:34:39,210
+الsequence نفسها converges وهي البرهان هذا ال part
+
+350
+00:34:39,210 --> 00:34:43,750
+f افرضي
+
+351
+00:34:43,750 --> 00:34:48,290
+ان x in convergent نثبت ان ال m ت ال convergent
+
+352
+00:34:48,290 --> 00:34:54,540
+ماشي الحال طيب اذا كانت x in convergent ل xيعني ال
+
+353
+00:34:54,540 --> 00:34:57,620
+limit تبعتها إذا كانت convergent فلازم يكون لها
+
+354
+00:34:57,620 --> 00:35:02,020
+limit فأفرض إن ال limit تبعتها xالأن حسب epsilon
+
+355
+00:35:02,020 --> 00:35:06,080
+capital N definition لل limit أو لل convergence
+
+356
+00:35:06,080 --> 00:35:11,140
+إذا لأي epsilon أكبر من 0 نقدر نلاقي N يعتمد على
+
+357
+00:35:11,140 --> 00:35:15,860
+epsilon عدد طبيعي كبير و ممكن ناخده يكون أكبر من
+
+358
+00:35:15,860 --> 00:35:22,040
+العدد الثابت العدد الطبيعي ثابت M بحيث أنه لكل N
+
+359
+00:35:22,040 --> 00:35:25,900
+أكبر من أو ساوي capital N المسافة بين X و N هو X
+
+360
+00:35:25,900 --> 00:35:31,410
+أصغر من Yهذا من تعريف الـ epsilon capital N
+
+361
+00:35:31,410 --> 00:35:37,590
+definition لل convergence طيب اللي انا بقدر اعرف
+
+362
+00:35:37,590 --> 00:35:43,930
+capital N prime على انه capital N مطروح منها
+
+363
+00:35:43,930 --> 00:35:50,060
+capital Mطبعا هنا capital N احنا اختارناها اكبر من
+
+364
+00:35:50,060 --> 00:35:54,220
+M فالفرق هذا موجب وهذا عدد طبيعي وهذا عدد طبيعي
+
+365
+00:35:54,220 --> 00:35:59,500
+اذا الفرق عدد صحيح موجب يعني عدد طبيعي هذا عدد
+
+366
+00:35:59,500 --> 00:36:03,220
+ثابت وهذا يعتمد على epsilon اذا N prime الفرق
+
+367
+00:36:03,220 --> 00:36:09,000
+بينهم يعتمد على epsilon تمام؟إذا هنا عرفنا N' عدد
+
+368
+00:36:09,000 --> 00:36:14,320
+طبيعي ويعتمد على epsilon الان لو أخدت اي M عدد
+
+369
+00:36:14,320 --> 00:36:16,960
+طبيعي أكبر من أو ساوي N'
+
+370
+00:36:20,020 --> 00:36:25,520
+فنجمع capital M للطرفين فبطلع capital M زاد small
+
+371
+00:36:25,520 --> 00:36:29,980
+m أكبر من أو ساوي N prime زاد capital M طب N prime
+
+372
+00:36:29,980 --> 00:36:34,540
+زاد capital M بساوي N إبسلون وبالتالي هذا أكبر من
+
+373
+00:36:34,540 --> 00:36:40,860
+أو ساوي N لإبسلون إذا حسب ال implication واحدالـ
+
+374
+00:36:40,860 --> 00:36:45,260
+implication واحد بتقوللي لأي عدد طبيعي .. لأي عدد
+
+375
+00:36:45,260 --> 00:36:50,560
+طبيعي أكبر من أو ساوي capital N لازم يطلع ال
+
+376
+00:36:50,560 --> 00:36:56,900
+absolute value ل X sub العدد الطبيعي اللي هو M زاد
+
+377
+00:36:56,900 --> 00:36:59,320
+M minus X أصغر من epsilon
+
+378
+00:37:03,110 --> 00:37:08,470
+وهذا بيدّي أن ال tail .. ال tail of the sequence
+
+379
+00:37:08,470 --> 00:37:13,110
+converge ل X حسب التعريف ما معناه أن ال tail هذا
+
+380
+00:37:13,110 --> 00:37:18,470
+convergent؟ معناه أن لأي epsilon أكبر من الصفر ..
+
+381
+00:37:18,470 --> 00:37:25,050
+لأي epsilon أكبر من الصفر هيوجد N prime .. هيوجد N
+
+382
+00:37:25,050 --> 00:37:29,130
+prime عدد طبيعي يعتمد على epsilon
+
+383
+00:37:31,850 --> 00:37:38,290
+يوجد عدد طبيعي N' يعتمد على إبسلون بحيث لكل M أكبر
+
+384
+00:37:38,290 --> 00:37:44,350
+من أو يساوي N' طلع المسافة بين الحد رقم capital M
+
+385
+00:37:44,350 --> 00:37:47,690
+زاد small m minus X أصغر من إبسلون هذا بالضبط
+
+386
+00:37:47,690 --> 00:37:53,310
+معناه إن ال sequence هذه converge ل X as M tends
+
+387
+00:37:53,310 --> 00:37:59,580
+to infinityإذاً هيك بنكون أثبتنا إنه لو كانت ال
+
+388
+00:37:59,580 --> 00:38:03,240
+sequence x in converge ل x فالتالت تبعها converge
+
+389
+00:38:03,240 --> 00:38:10,720
+ل x okay تمام العكس العكس يعني ضايق ممكن يعني
+
+390
+00:38:10,720 --> 00:38:20,220
+نبرهن العكس في دقيقة او دقيقتين العكس
+
+391
+00:38:20,220 --> 00:38:26,390
+يعني هذا العكس اللي هو ال only if partنفرض المرة
+
+392
+00:38:26,390 --> 00:38:30,450
+هذه أن الـ sequence الـ tail of a sequence الـ
+
+393
+00:38:30,450 --> 00:38:34,770
+tail of the sequence converged ل X وبينما نثبت أن
+
+394
+00:38:34,770 --> 00:38:40,170
+الـ sequence نفسها convergent ل X برضه فنستخدم
+
+395
+00:38:40,170 --> 00:38:42,930
+تعريف epsilon capital N definition للconvergence
+
+396
+00:38:42,930 --> 00:38:48,710
+اللي هو الجزء C من نظرية 2 2 فناخد given epsilon
+
+397
+00:38:48,710 --> 00:38:53,080
+أو let epsilon أكبر من الصفر بيه givenبما أن الـ
+
+398
+00:38:53,080 --> 00:38:56,560
+sequence هذه converge ل X إذا يوجد capital N يعتمد
+
+399
+00:38:56,560 --> 00:39:00,740
+على إبسلون بحيث لكل N أكبر من أو ساوي capital N
+
+400
+00:39:00,740 --> 00:39:04,560
+المسافة بين الحد العام للـ sequence هذه و X أصغر
+
+401
+00:39:04,560 --> 00:39:12,790
+من إبسلونالان بنعرف capital K على انه العدد
+
+402
+00:39:12,790 --> 00:39:18,250
+الطبيعي الثابت M زاد العدد الطبيعي capital N فطبعا
+
+403
+00:39:18,250 --> 00:39:22,490
+مجموعة دين الطبيعيين عدد طبيعي capital N يعتمد على
+
+404
+00:39:22,490 --> 00:39:26,670
+epsilon اذا المجموعة تبعهم بيطلع يعتمد على epsilon
+
+405
+00:39:26,670 --> 00:39:32,330
+اذا هنا انا وجدت او جدت او عرفت عدد طبيعي capital
+
+406
+00:39:32,330 --> 00:39:37,610
+K يعتمد على epsilonالان لو أخدت اي N أكبر من أو
+
+407
+00:39:37,610 --> 00:39:43,170
+ساوي ال capital A فاترحي .. اترحي N من هنا و اترحي
+
+408
+00:39:43,170 --> 00:39:50,350
+N من هنا M عفوا Mلو طرحنا M من الطرفين المتباينة
+
+409
+00:39:50,350 --> 00:39:55,330
+هذه فبطلع N negative capital M أكبر من أو ساوي K
+
+410
+00:39:55,330 --> 00:40:01,170
+minus M طب هاي K اطرحي منها M بساوي N وبالتالي
+
+411
+00:40:01,170 --> 00:40:05,950
+بطلع N سالب M أكبر من أو ساوي N الآن من ال
+
+412
+00:40:05,950 --> 00:40:11,550
+implication اتنين ال implication اتنين بتقول لأي N
+
+413
+00:40:11,550 --> 00:40:15,650
+أكبر من أو ساوي capital اي عدد طبيعيلو كان العدد
+
+414
+00:40:15,650 --> 00:40:20,950
+الطبيعي هذا أكبر من أو ساوي capital N فالمسافة بين
+
+415
+00:40:20,950 --> 00:40:27,390
+X للعدد الطبيعي واضيف عليه M إذا بدي أضيف على هذا
+
+416
+00:40:27,390 --> 00:40:32,230
+M المسافة بين X اللي المؤشر تبعها العدد الطبيعي
+
+417
+00:40:32,230 --> 00:40:37,770
+هذا زائد M اللي هو بيطلع N والمسافة بينه بين X
+
+418
+00:40:37,770 --> 00:40:42,770
+بيطلع أصغر من Epsilonإذاً هيك احنا أثبتنا أنه لأي
+
+419
+00:40:42,770 --> 00:40:46,970
+إبسلون أكبر من الصفر يوجد capital N يعتمد على
+
+420
+00:40:46,970 --> 00:40:53,790
+إبسلون بحيث أنه أو يوجد capital K لأي إبسلون أكبر
+
+421
+00:40:53,790 --> 00:40:57,570
+من الصفر يوجد عدد طبيعي capital K يعتمد على إبسلون
+
+422
+00:40:57,570 --> 00:41:06,250
+بحيث أنه لكل N أكبر من أو يساوي capital Kلكل n
+
+423
+00:41:06,250 --> 00:41:10,590
+أكبر من أو ساوي كابتل K طلع المسافة بين xn و x
+
+424
+00:41:10,590 --> 00:41:15,370
+أصغر من إبسل إذن هذا بالضبط معناه أن ال sequence
+
+425
+00:41:15,370 --> 00:41:22,590
+xn converge ل x زي ما هو مطلوب وهذا بكمل برهان
+
+426
+00:41:22,590 --> 00:41:26,410
+النظرية okay تمام واضح
+
+427
+00:41:31,150 --> 00:41:37,130
+طيب احنا بنكتفي بهذا القدر و ان شاء الله في
+
+428
+00:41:37,130 --> 00:41:42,010
+المحاضرة القادمة هناخد برضه بعض النظريات و ناخد
+
+429
+00:41:42,010 --> 00:41:46,350
+أمثلة كيف نثبت ان ال limit ل sequence ل
+
+430
+00:41:46,350 --> 00:41:51,090
+convergence sequence بالساوي عدد معين و هكذا طبعا
+
+431
+00:41:51,090 --> 00:41:54,130
+كل الأجزاء هذه موجودة عندكم ممكن تقرؤوها و تحضروها
+
+432
+00:41:54,130 --> 00:41:56,010
+للمحاضرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..380eff15f12992cf9c4a57ed94f1c51129709a8c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_raw.json
@@ -0,0 +1 @@
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{"start": 172.05, "end": 172.75, "word": " بدلالة", "probability": 0.96435546875}, {"start": 172.75, "end": 172.91, "word": " ال", "probability": 0.791015625}, {"start": 172.91, "end": 173.13, "word": " range", "probability": 0.92431640625}, {"start": 173.13, "end": 173.69, "word": " تبعها", "probability": 0.9471435546875}], "temperature": 1.0}, {"id": 8, "seek": 20434, "start": 174.84, "end": 204.34, "text": "يعني بدل ما اقول ال sequence هي function جرّت العادة ان احنا نحذف رمز ال function و نستبدله بال range تبع ال function اللي هو y ال range تبع ال function كل ال x n حيث n عدد طبيعي ببدأ من واحد من ت أنما إلى نهاية اذا ال sequence بدل ما نكتبها على صورة function هنكتبها على الصورة هذه او الصورة هذه او الصورة هذه او الصورة هذه okay", "tokens": [40228, 22653, 47525, 1211, 19446, 1975, 39648, 2423, 8310, 39896, 2445, 10874, 2288, 11703, 2655, 18863, 18513, 3660, 16472, 1975, 5016, 8315, 8717, 5016, 8848, 5172, 12602, 2304, 11622, 2423, 2445, 4032, 8717, 14851, 44510, 1211, 3224, 20666, 3613, 6055, 3555, 3615, 2423, 2445, 13672, 1829, 31439, 288, 2423, 3613, 6055, 3555, 3615, 2423, 2445, 28242, 2423, 2031, 297, 11331, 1829, 12984, 297, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 44510, 10721, 9154, 36764, 24401, 9154, 6055, 14739, 15042, 30731, 8717, 11296, 10632, 1975, 15730, 2423, 8310, 47525, 1211, 19446, 8717, 4117, 2655, 3555, 11296, 15844, 20328, 13063, 3660, 2445, 8032, 1863, 4117, 2655, 3555, 11296, 15844, 31767, 13063, 3660, 29538, 1975, 2407, 31767, 13063, 3660, 29538, 1975, 2407, 31767, 13063, 3660, 29538, 1975, 2407, 31767, 13063, 3660, 29538, 1392], "avg_logprob": -0.1847774675398162, "compression_ratio": 2.119341563786008, "no_speech_prob": 0.0, "words": [{"start": 174.84, "end": 175.14, "word": "يعني", "probability": 0.870361328125}, {"start": 175.14, "end": 175.48, "word": " بدل", "probability": 0.9853515625}, {"start": 175.48, "end": 175.62, "word": " ما", "probability": 0.8955078125}, {"start": 175.62, "end": 175.96, "word": " اقول", "probability": 0.689697265625}, {"start": 175.96, "end": 176.14, "word": " ال", "probability": 0.8154296875}, {"start": 176.14, "end": 176.4, "word": " sequence", "probability": 0.5400390625}, {"start": 176.4, "end": 176.72, "word": " هي", "probability": 0.95751953125}, {"start": 176.72, "end": 177.36, "word": " function", "probability": 0.931640625}, {"start": 177.36, "end": 178.38, "word": " جرّت", "probability": 0.54156494140625}, {"start": 178.38, "end": 178.76, "word": " العادة", "probability": 0.9371744791666666}, {"start": 178.76, "end": 178.88, "word": " ان", "probability": 0.50830078125}, {"start": 178.88, "end": 179.16, "word": " احنا", "probability": 0.9441731770833334}, {"start": 179.16, "end": 180.68, "word": " نحذف", "probability": 0.968994140625}, {"start": 180.68, "end": 181.16, "word": " رمز", "probability": 0.994140625}, {"start": 181.16, "end": 181.32, "word": " ال", "probability": 0.9892578125}, {"start": 181.32, "end": 181.8, "word": " function", "probability": 0.9423828125}, {"start": 181.8, "end": 182.4, "word": " و", "probability": 0.5751953125}, {"start": 182.4, "end": 183.0, "word": " نستبدله", "probability": 0.85634765625}, {"start": 183.0, "end": 183.16, "word": " بال", "probability": 0.9794921875}, {"start": 183.16, "end": 183.48, "word": " range", "probability": 0.92626953125}, {"start": 183.48, "end": 183.76, "word": " تبع", "probability": 0.9169921875}, {"start": 183.76, "end": 183.9, "word": " ال", "probability": 0.9716796875}, {"start": 183.9, "end": 184.38, "word": " function", "probability": 0.96484375}, {"start": 184.38, "end": 185.08, "word": " اللي", "probability": 0.8818359375}, {"start": 185.08, "end": 185.3, "word": " هو", "probability": 0.9482421875}, {"start": 185.3, "end": 185.58, "word": " y", "probability": 0.5078125}, {"start": 185.58, "end": 185.98, "word": " ال", "probability": 0.64208984375}, {"start": 185.98, "end": 186.24, "word": " range", "probability": 0.95556640625}, {"start": 186.24, "end": 186.44, "word": " تبع", "probability": 0.9816080729166666}, {"start": 186.44, "end": 186.58, "word": " ال", "probability": 0.9755859375}, {"start": 186.58, "end": 186.9, "word": " function", "probability": 0.9677734375}, {"start": 186.9, "end": 187.24, "word": " كل", "probability": 0.986328125}, {"start": 187.24, "end": 187.4, "word": " ال", "probability": 0.92626953125}, {"start": 187.4, "end": 187.62, "word": " x", "probability": 0.9013671875}, {"start": 187.62, "end": 187.96, "word": " n", "probability": 0.1793212890625}, {"start": 187.96, "end": 188.82, "word": " حيث", "probability": 0.9855143229166666}, {"start": 188.82, "end": 189.06, "word": " n", "probability": 0.349609375}, {"start": 189.06, "end": 189.42, "word": " عدد", "probability": 0.9825846354166666}, {"start": 189.42, "end": 189.96, "word": " طبيعي", "probability": 0.9697265625}, {"start": 189.96, "end": 190.4, "word": " ببدأ", "probability": 0.740234375}, {"start": 190.4, "end": 190.54, "word": " من", "probability": 0.990234375}, {"start": 190.54, "end": 190.9, "word": " واحد", "probability": 0.908935546875}, {"start": 190.9, "end": 191.18, "word": " من", "probability": 0.546875}, {"start": 191.18, "end": 191.38, "word": " ت", "probability": 0.325927734375}, {"start": 191.38, "end": 191.82, "word": " أنما", "probability": 0.24200439453125}, {"start": 191.82, "end": 191.96, "word": " إلى", "probability": 0.435791015625}, {"start": 191.96, "end": 193.06, "word": " نهاية", "probability": 0.92529296875}, {"start": 193.06, "end": 193.3, "word": " اذا", "probability": 0.606689453125}, {"start": 193.3, "end": 193.5, "word": " ال", "probability": 0.94189453125}, {"start": 193.5, "end": 193.98, "word": " sequence", "probability": 0.99267578125}, {"start": 193.98, "end": 194.58, "word": " بدل", "probability": 0.983154296875}, {"start": 194.58, "end": 194.72, "word": " ما", "probability": 0.7353515625}, {"start": 194.72, "end": 195.88, "word": " نكتبها", "probability": 0.9783203125}, {"start": 195.88, "end": 196.08, "word": " على", "probability": 0.921875}, {"start": 196.08, "end": 196.36, "word": " صورة", "probability": 0.9851888020833334}, {"start": 196.36, "end": 196.9, "word": " function", "probability": 0.97119140625}, {"start": 196.9, "end": 197.98, "word": " هنكتبها", "probability": 0.9261881510416666}, {"start": 197.98, "end": 198.2, "word": " على", "probability": 0.8603515625}, {"start": 198.2, "end": 198.6, "word": " الصورة", "probability": 0.9874674479166666}, {"start": 198.6, "end": 198.96, "word": " هذه", "probability": 0.83056640625}, {"start": 198.96, "end": 199.68, "word": " او", "probability": 0.826904296875}, {"start": 199.68, "end": 200.18, "word": " الصورة", "probability": 0.9860026041666666}, {"start": 200.18, "end": 200.58, "word": " هذه", "probability": 0.9384765625}, {"start": 200.58, "end": 201.3, "word": " او", "probability": 0.946533203125}, {"start": 201.3, "end": 201.8, "word": " الصورة", "probability": 0.9890950520833334}, {"start": 201.8, "end": 202.2, "word": " هذه", "probability": 0.95068359375}, {"start": 202.2, "end": 202.56, "word": " او", "probability": 0.89794921875}, {"start": 202.56, "end": 203.0, "word": " الصورة", "probability": 0.9850260416666666}, {"start": 203.0, "end": 203.28, "word": " هذه", "probability": 0.955078125}, {"start": 203.28, "end": 204.34, "word": " okay", "probability": 0.67822265625}], "temperature": 1.0}, {"id": 9, "seek": 21735, "start": 206.55, "end": 217.35, "text": "و طبعا ال sequence هذه يعني أسرها هذه أو أي واحدة منهم ممكن نكتبها برضه على الصورة x1, x2, x3 و هكذا", "tokens": [2407, 23032, 3555, 3615, 995, 2423, 8310, 29538, 37495, 22653, 5551, 3794, 2288, 11296, 29538, 34051, 36632, 36764, 24401, 3660, 9154, 16095, 3714, 43020, 8717, 4117, 2655, 3555, 11296, 4724, 43042, 3224, 15844, 31767, 13063, 3660, 2031, 16, 11, 2031, 17, 11, 2031, 18, 4032, 8032, 4117, 15730], "avg_logprob": -0.1257174793554812, "compression_ratio": 1.2348484848484849, "no_speech_prob": 0.0, "words": [{"start": 206.55, "end": 206.81, "word": "و", "probability": 0.91064453125}, {"start": 206.81, "end": 207.19, "word": " طبعا", "probability": 0.8994140625}, {"start": 207.19, "end": 207.31, "word": " ال", "probability": 0.91357421875}, {"start": 207.31, "end": 207.75, "word": " sequence", "probability": 0.94873046875}, {"start": 207.75, "end": 208.19, "word": " هذه", "probability": 0.66796875}, {"start": 208.19, "end": 208.39, "word": " يعني", "probability": 0.818359375}, {"start": 208.39, "end": 208.95, "word": " أسرها", "probability": 0.8748779296875}, {"start": 208.95, "end": 209.35, "word": " هذه", "probability": 0.5166015625}, {"start": 209.35, "end": 209.57, "word": " أو", "probability": 0.84228515625}, {"start": 209.57, "end": 209.75, "word": " أي", "probability": 0.66845703125}, {"start": 209.75, "end": 210.07, "word": " واحدة", "probability": 0.9851888020833334}, {"start": 210.07, "end": 210.49, "word": " منهم", "probability": 0.991943359375}, {"start": 210.49, "end": 211.47, "word": " ممكن", "probability": 0.971435546875}, {"start": 211.47, "end": 211.89, "word": " نكتبها", "probability": 0.99453125}, {"start": 211.89, "end": 212.19, "word": " برضه", "probability": 0.91064453125}, {"start": 212.19, "end": 212.29, "word": " على", "probability": 0.8046875}, {"start": 212.29, "end": 212.73, "word": " الصورة", "probability": 0.9680989583333334}, {"start": 212.73, "end": 213.29, "word": " x1,", "probability": 0.67626953125}, {"start": 213.53, "end": 214.33, "word": " x2,", "probability": 0.93017578125}, {"start": 214.65, "end": 216.29, "word": " x3", "probability": 0.954345703125}, {"start": 216.29, "end": 216.79, "word": " و", "probability": 0.736328125}, {"start": 216.79, "end": 217.35, "word": " هكذا", "probability": 0.9305013020833334}], "temperature": 1.0}, {"id": 10, "seek": 24950, "start": 220.84, "end": 249.5, "text": "فكل الرموز هذه ترمز إلى ال sequence هذه اللي هي ال function f اللي هي ال function f okay إذن أهم شيء في تعريفنا أن ال sequence هي function دلنا وبالتالي لها مجال مجالها العداد الطبيعي المجال المقابل هي المجموعة اللي عناصر ال sequence تنتمي لها ال sequences ممكن أعرفهم بطريقتين", "tokens": [5172, 28820, 34892, 2304, 2407, 11622, 29538, 6055, 2288, 2304, 11622, 30731, 2423, 8310, 29538, 13672, 1829, 39896, 2423, 2445, 283, 13672, 1829, 39896, 2423, 2445, 283, 1392, 11933, 8848, 1863, 5551, 16095, 44049, 38207, 8978, 37279, 16572, 5172, 8315, 14739, 2423, 8310, 39896, 2445, 11778, 1211, 8315, 46599, 6027, 2655, 6027, 1829, 5296, 11296, 3714, 7435, 6027, 3714, 7435, 6027, 11296, 18863, 3215, 18513, 41950, 21292, 3615, 1829, 9673, 7435, 6027, 9673, 4587, 16758, 1211, 39896, 9673, 7435, 2304, 2407, 27884, 13672, 1829, 18871, 33546, 2288, 2423, 8310, 6055, 29399, 2304, 1829, 5296, 11296, 2423, 22978, 3714, 43020, 5551, 3615, 28480, 16095, 4724, 9566, 16572, 38149, 9957], "avg_logprob": -0.19495412297205095, "compression_ratio": 1.9775784753363228, "no_speech_prob": 0.0, "words": [{"start": 220.84, "end": 221.38, "word": "فكل", "probability": 0.82568359375}, 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229.86, "word": " اللي", "probability": 0.652099609375}, {"start": 229.86, "end": 230.0, "word": " هي", "probability": 0.94140625}, {"start": 230.0, "end": 230.1, "word": " ال", "probability": 0.98291015625}, {"start": 230.1, "end": 230.38, "word": " function", "probability": 0.923828125}, {"start": 230.38, "end": 230.8, "word": " f", "probability": 0.8349609375}, {"start": 230.8, "end": 231.82, "word": " okay", "probability": 0.438232421875}, {"start": 231.82, "end": 232.66, "word": " إذن", "probability": 0.6170247395833334}, {"start": 232.66, "end": 233.18, "word": " أهم", "probability": 0.83154296875}, {"start": 233.18, "end": 233.4, "word": " شيء", "probability": 0.5264892578125}, {"start": 233.4, "end": 233.48, "word": " في", "probability": 0.86962890625}, {"start": 233.48, "end": 234.0, "word": " تعريفنا", "probability": 0.921875}, {"start": 234.0, "end": 234.18, "word": " أن", "probability": 0.442138671875}, {"start": 234.18, "end": 234.34, "word": " ال", "probability": 0.88671875}, {"start": 234.34, "end": 234.66, "word": " sequence", "probability": 0.98095703125}, {"start": 234.66, "end": 234.96, "word": " هي", "probability": 0.96875}, {"start": 234.96, "end": 235.38, "word": " function", "probability": 0.8955078125}, {"start": 235.38, "end": 236.48, "word": " دلنا", "probability": 0.545654296875}, {"start": 236.48, "end": 237.2, "word": " وبالتالي", "probability": 0.93779296875}, {"start": 237.2, "end": 237.44, "word": " لها", "probability": 0.677734375}, {"start": 237.44, "end": 237.86, "word": " مجال", "probability": 0.9977213541666666}, {"start": 237.86, "end": 238.36, "word": " مجالها", "probability": 0.80743408203125}, {"start": 238.36, "end": 238.74, "word": " العداد", "probability": 0.8994140625}, {"start": 238.74, "end": 239.4, "word": " الطبيعي", "probability": 0.895751953125}, {"start": 239.4, "end": 240.4, "word": " المجال", "probability": 0.8460286458333334}, {"start": 240.4, "end": 241.04, "word": " المقابل", "probability": 0.9923095703125}, {"start": 241.04, "end": 241.26, "word": " هي", "probability": 0.79150390625}, {"start": 241.26, "end": 241.86, "word": " المجموعة", "probability": 0.94990234375}, {"start": 241.86, "end": 242.14, "word": " اللي", "probability": 0.9658203125}, {"start": 242.14, "end": 242.66, "word": " عناصر", "probability": 0.9747721354166666}, {"start": 242.66, "end": 242.78, "word": " ال", "probability": 0.966796875}, {"start": 242.78, "end": 243.28, "word": " sequence", "probability": 0.9931640625}, {"start": 243.28, "end": 244.42, "word": " تنتمي", "probability": 0.9681396484375}, {"start": 244.42, "end": 244.84, "word": " لها", "probability": 0.86376953125}, {"start": 244.84, "end": 246.96, "word": " ال", "probability": 0.9638671875}, {"start": 246.96, "end": 247.5, "word": " sequences", "probability": 0.9541015625}, {"start": 247.5, "end": 247.96, "word": " ممكن", "probability": 0.979736328125}, {"start": 247.96, "end": 248.66, "word": " أعرفهم", "probability": 0.73187255859375}, {"start": 248.66, "end": 249.5, "word": " بطريقتين", "probability": 0.9767578125}], "temperature": 1.0}, {"id": 11, "seek": 26161, "start": 250.49, "end": 261.61, "text": "إذا في الملاحظة هذه sequences can be defined explicitly هذه أحد الطرق ممكن يعرف ال sequence بطريقة صريحة بطريقة بقانون", "tokens": [28814, 15730, 8978, 9673, 15040, 5016, 19913, 3660, 29538, 22978, 393, 312, 7642, 20803, 29538, 5551, 24401, 41950, 2288, 4587, 3714, 43020, 37495, 28480, 2423, 8310, 4724, 9566, 16572, 28671, 20328, 16572, 5016, 3660, 4724, 9566, 16572, 28671, 4724, 4587, 7649, 11536], "avg_logprob": -0.18768168327420257, "compression_ratio": 1.2992700729927007, "no_speech_prob": 0.0, "words": [{"start": 250.49, "end": 250.77, "word": "إذا", "probability": 0.4859619140625}, {"start": 250.77, "end": 250.95, "word": " في", "probability": 0.33544921875}, {"start": 250.95, "end": 251.43, "word": " الملاحظة", "probability": 0.99326171875}, {"start": 251.43, "end": 251.75, "word": " هذه", "probability": 0.84375}, {"start": 251.75, "end": 252.83, "word": " sequences", "probability": 0.4990234375}, {"start": 252.83, "end": 253.33, "word": " can", "probability": 0.93310546875}, {"start": 253.33, "end": 253.59, "word": " be", "probability": 0.9306640625}, {"start": 253.59, "end": 254.29, "word": " defined", "probability": 0.91015625}, {"start": 254.29, "end": 255.97, "word": " explicitly", "probability": 0.75439453125}, {"start": 255.97, "end": 256.51, "word": " هذه", "probability": 0.4599609375}, {"start": 256.51, "end": 256.77, "word": " أحد", "probability": 0.865966796875}, {"start": 256.77, "end": 257.29, "word": " الطرق", "probability": 0.8759765625}, {"start": 257.29, "end": 257.77, "word": " ممكن", "probability": 0.892578125}, {"start": 257.77, "end": 258.21, "word": " يعرف", "probability": 0.73095703125}, {"start": 258.21, "end": 258.35, "word": " ال", "probability": 0.912109375}, {"start": 258.35, "end": 258.75, "word": " sequence", "probability": 0.9541015625}, {"start": 258.75, "end": 259.35, "word": " بطريقة", "probability": 0.9853515625}, {"start": 259.35, "end": 259.91, "word": " صريحة", "probability": 0.972412109375}, {"start": 259.91, "end": 260.61, "word": " بطريقة", "probability": 0.936279296875}, {"start": 260.61, "end": 261.61, "word": " بقانون", "probability": 0.81591796875}], "temperature": 1.0}, {"id": 12, "seek": 28314, "start": 263.47, "end": 283.15, "text": "فمثلا ال sequence if بالساوية عناصرها اتنين اربعة ستة تمانية الاخرى هذه عبارة عن sequence وهي معرفة بطريقة صريحة فهذه عبارة عن sequence of even natural members العداد الطبيعية الزوجية", "tokens": [5172, 2304, 12984, 15040, 2423, 8310, 498, 20666, 3794, 995, 2407, 10632, 18871, 33546, 2288, 11296, 1975, 2655, 1863, 9957, 1975, 25513, 27884, 8608, 2655, 3660, 46811, 7649, 10632, 2423, 47283, 2288, 7578, 29538, 6225, 3555, 9640, 3660, 18871, 8310, 37037, 1829, 20449, 28480, 3660, 4724, 9566, 16572, 28671, 20328, 16572, 5016, 3660, 6156, 3224, 24192, 6225, 3555, 9640, 3660, 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نهائية إذا هنا أنا بعرف ال sequence برص حدودها أول تلات حدود إلى و هكذا أو بكتب قاعدة لحد العام xn و طبعا n أدى الطبيعي فمقدر من القاعدة هذه أجيب كل الحدود إذا هذا explicit definition of a sequence هذا تعريف صريح لل sequence", "tokens": [10721, 2407, 3714, 43020, 8717, 4117, 2655, 3555, 2423, 8310, 15844, 20328, 13063, 3660, 568, 77, 9154, 297, 20666, 3794, 995, 27884, 36764, 24401, 30731, 3714, 15040, 8717, 3224, 16373, 10632, 11933, 15730, 34105, 41850, 45030, 28480, 2423, 8310, 4724, 2288, 9381, 11331, 3215, 23328, 11296, 5551, 12610, 6055, 1211, 9307, 11331, 3215, 23328, 30731, 4032, 8032, 4117, 15730, 34051, 4724, 4117, 2655, 3555, 12174, 995, 22488, 3660, 5296, 24401, 18863, 10943, 2031, 77, 4032, 23032, 3555, 3615, 995, 297, 5551, 3215, 7578, 41950, 21292, 3615, 1829, 6156, 2304, 28543, 2288, 9154, 25062, 995, 22488, 3660, 29538, 5551, 7435, 1829, 3555, 28242, 21542, 3215, 23328, 11933, 15730, 23758, 13691, 7123, 295, 257, 8310, 23758, 37279, 16572, 5172, 20328, 16572, 5016, 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"في طريقة تانية لتعريف ال sequence وهي الطريقة الاستقرائية، إذا ال sequences can be defined inductively أو recursively", "tokens": [41185, 23032, 16572, 28671, 6055, 7649, 10632, 5296, 2655, 3615, 16572, 5172, 2423, 8310, 37037, 1829, 41950, 16572, 28671, 42963, 14851, 4587, 2288, 16373, 10632, 12399, 11933, 15730, 2423, 22978, 393, 312, 7642, 31612, 3413, 34051, 20560, 3413], "avg_logprob": -0.10997596306678577, "compression_ratio": 1.296875, "no_speech_prob": 0.0, "words": [{"start": 338.51, "end": 338.71, "word": "في", "probability": 0.87255859375}, {"start": 338.71, "end": 339.15, "word": " طريقة", "probability": 0.9915364583333334}, {"start": 339.15, "end": 339.67, "word": " تانية", "probability": 0.9957682291666666}, {"start": 339.67, "end": 340.73, "word": " لتعريف", "probability": 0.98291015625}, {"start": 340.73, "end": 340.89, "word": " ال", "probability": 0.9306640625}, {"start": 340.89, "end": 341.49, "word": " sequence", "probability": 0.5810546875}, {"start": 341.49, 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{"start": 552.61, "end": 553.57, "word": " الطبيعية", "probability": 0.8914794921875}, {"start": 553.57, "end": 554.41, "word": " بحيث", "probability": 0.9383544921875}, {"start": 554.41, "end": 554.83, "word": " أنه", "probability": 0.677001953125}, {"start": 554.83, "end": 555.33, "word": " لكل", "probability": 0.98583984375}, {"start": 555.33, "end": 555.75, "word": " small", "probability": 0.87158203125}, {"start": 555.75, "end": 556.07, "word": " n", "probability": 0.77197265625}, {"start": 556.07, "end": 556.47, "word": " أكبر", "probability": 0.9658203125}, {"start": 556.47, "end": 556.61, "word": " من", "probability": 0.9189453125}, {"start": 556.61, "end": 556.73, "word": " أو", "probability": 0.91943359375}, {"start": 556.73, "end": 557.03, "word": " سوى", "probability": 0.7462565104166666}, {"start": 557.03, "end": 557.41, "word": " capital", "probability": 0.7451171875}, {"start": 557.41, "end": 557.83, "word": " N،", "probability": 0.702392578125}, {"start": 557.83, "end": 558.29, "word": " Xn", "probability": 0.635498046875}, {"start": 558.29, "end": 558.85, "word": " ينتمي", "probability": 0.9710693359375}, {"start": 558.85, "end": 559.01, "word": " إلى", "probability": 0.9228515625}, {"start": 559.01, "end": 559.27, "word": " V", "probability": 0.97216796875}, {"start": 559.27, "end": 560.13, "word": " يعني", "probability": 0.796630859375}, {"start": 560.13, "end": 560.69, "word": " الجوار", "probability": 0.9677734375}, {"start": 560.69, "end": 560.91, "word": " V", "probability": 0.96630859375}, {"start": 560.91, "end": 561.23, "word": " هذا", "probability": 0.65869140625}, {"start": 561.23, "end": 561.77, "word": " يحتوي", "probability": 0.9827473958333334}, {"start": 561.77, "end": 562.09, "word": " كل", "probability": 0.9697265625}, {"start": 562.09, "end": 562.61, "word": " عناصر", "probability": 0.9918619791666666}, {"start": 562.61, "end": 562.75, "word": " ال", "probability": 0.92724609375}, {"start": 562.75, "end": 563.23, "word": " sequence", "probability": 0.90087890625}, {"start": 563.23, "end": 563.53, "word": " من", "probability": 0.990234375}, {"start": 563.53, "end": 564.03, "word": " capital", "probability": 0.77685546875}, {"start": 564.03, "end": 564.59, "word": " N", "probability": 0.9892578125}, {"start": 564.59, "end": 565.55, "word": " وانت", "probability": 0.9384765625}, {"start": 565.55, "end": 565.97, "word": " طالع", "probability": 0.8572591145833334}], "temperature": 1.0}, {"id": 25, "seek": 59314, "start": 568.5, "end": 593.14, "text": "فلو هذا الشرط اتحقق فبنقول ان الـ sequence converge و ال limit تبعتها هي العدد X في الحالة هذه بنقول ان X is the limit of sequence X in و بنكتب limit X in بالساوية X او نكتب X in tends to X as N tends to infinity", "tokens": [5172, 1211, 2407, 23758, 25124, 2288, 9566, 1975, 2655, 5016, 4587, 4587, 6156, 3555, 1863, 39648, 16472, 2423, 39184, 8310, 41881, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 39896, 18863, 3215, 3215, 1783, 8978, 21542, 6027, 3660, 29538, 44945, 39648, 16472, 1783, 307, 264, 4948, 295, 8310, 1783, 294, 4032, 44945, 4117, 2655, 3555, 4948, 1783, 294, 20666, 3794, 995, 2407, 10632, 1783, 1975, 2407, 8717, 4117, 2655, 3555, 1783, 294, 12258, 281, 1783, 382, 426, 12258, 281, 13202], "avg_logprob": -0.23496093675494195, "compression_ratio": 1.575268817204301, "no_speech_prob": 0.0, "words": [{"start": 568.5, "end": 568.9, "word": "فلو", "probability": 0.7333984375}, {"start": 568.9, "end": 569.1, "word": " هذا", "probability": 0.61328125}, {"start": 569.1, "end": 569.46, "word": " الشرط", "probability": 0.9488932291666666}, {"start": 569.46, "end": 570.18, "word": " اتحقق", "probability": 0.92529296875}, {"start": 570.18, "end": 571.32, "word": " فبنقول", "probability": 0.836181640625}, {"start": 571.32, "end": 571.58, "word": " ان", "probability": 0.6923828125}, {"start": 571.58, "end": 571.76, "word": " الـ", "probability": 0.57275390625}, {"start": 571.76, "end": 572.08, "word": " sequence", "probability": 0.64013671875}, {"start": 572.08, "end": 572.98, "word": " converge", "probability": 0.80517578125}, {"start": 572.98, "end": 573.88, "word": " و", "probability": 0.7568359375}, {"start": 573.88, "end": 574.02, "word": " ال", "probability": 0.37548828125}, {"start": 574.02, "end": 574.3, "word": " limit", "probability": 0.7841796875}, {"start": 574.3, "end": 575.0, "word": " تبعتها", "probability": 0.9110107421875}, {"start": 575.0, "end": 575.28, "word": " هي", "probability": 0.91845703125}, {"start": 575.28, "end": 575.72, "word": " العدد", "probability": 0.9637044270833334}, {"start": 575.72, "end": 576.1, "word": " X", "probability": 0.552734375}, {"start": 576.1, "end": 576.54, "word": " في", "probability": 0.4111328125}, {"start": 576.54, "end": 576.82, "word": " الحالة", "probability": 0.974609375}, {"start": 576.82, "end": 577.0, "word": " هذه", "probability": 0.47265625}, {"start": 577.0, "end": 577.36, "word": " بنقول", "probability": 0.660888671875}, {"start": 577.36, "end": 577.58, "word": " ان", "probability": 0.8046875}, {"start": 577.58, "end": 578.04, "word": " X", "probability": 0.87890625}, {"start": 578.04, "end": 579.38, "word": " is", "probability": 0.740234375}, {"start": 579.38, "end": 579.66, "word": " the", "probability": 0.8193359375}, {"start": 579.66, "end": 580.08, "word": " limit", "probability": 0.974609375}, {"start": 580.08, "end": 580.68, "word": " of", "probability": 0.97119140625}, {"start": 580.68, "end": 581.48, "word": " sequence", "probability": 0.56689453125}, {"start": 581.48, "end": 581.82, "word": " X", "probability": 0.8359375}, {"start": 581.82, "end": 582.14, "word": " in", "probability": 0.252197265625}, {"start": 582.14, "end": 586.08, "word": " و", "probability": 0.61328125}, {"start": 586.08, "end": 586.56, "word": " بنكتب", "probability": 0.9549560546875}, {"start": 586.56, "end": 586.86, "word": " limit", "probability": 0.9775390625}, {"start": 586.86, "end": 587.18, "word": " X", "probability": 0.9462890625}, {"start": 587.18, "end": 587.44, "word": " in", "probability": 0.88623046875}, {"start": 587.44, "end": 588.04, "word": " بالساوية", "probability": 0.70771484375}, {"start": 588.04, "end": 588.38, "word": " X", "probability": 0.97607421875}, {"start": 588.38, "end": 588.82, "word": " او", "probability": 0.7197265625}, {"start": 588.82, "end": 589.88, "word": " نكتب", "probability": 0.8629150390625}, {"start": 589.88, "end": 590.14, "word": " X", "probability": 0.951171875}, {"start": 590.14, "end": 590.5, "word": " in", "probability": 0.9091796875}, {"start": 590.5, "end": 590.94, "word": " tends", "probability": 0.72607421875}, {"start": 590.94, "end": 591.18, "word": " to", "probability": 0.97900390625}, {"start": 591.18, "end": 591.5, "word": " X", "probability": 0.98876953125}, {"start": 591.5, "end": 591.9, "word": " as", "probability": 0.947265625}, {"start": 591.9, "end": 592.2, "word": " N", "probability": 0.6455078125}, {"start": 592.2, "end": 592.48, "word": " tends", "probability": 0.85302734375}, {"start": 592.48, "end": 592.66, "word": " to", "probability": 0.95751953125}, {"start": 592.66, "end": 593.14, "word": " infinity", "probability": 0.7373046875}], "temperature": 1.0}, {"id": 26, "seek": 61821, "start": 594.69, "end": 618.21, "text": "هذا التعريف بنسميه ال neighborhood neighborhood definition neighborhood definition of convergence تعريف الجوار للتقارب", "tokens": [3224, 15730, 16712, 3615, 16572, 5172, 44945, 38251, 1829, 3224, 2423, 7630, 7630, 7123, 7630, 7123, 295, 32181, 37279, 16572, 5172, 25724, 2407, 9640, 24976, 2655, 4587, 9640, 3555], "avg_logprob": -0.1076171875, "compression_ratio": 1.4259259259259258, "no_speech_prob": 0.0, "words": [{"start": 594.69, "end": 595.19, "word": "هذا", "probability": 0.97900390625}, {"start": 595.19, "end": 595.83, "word": " التعريف", "probability": 0.9932861328125}, {"start": 595.83, "end": 597.55, "word": " بنسميه", "probability": 0.9410400390625}, {"start": 597.55, "end": 597.75, "word": " ال", "probability": 0.389404296875}, {"start": 597.75, "end": 598.19, "word": " neighborhood", "probability": 0.61376953125}, {"start": 598.19, "end": 600.95, "word": " neighborhood", "probability": 0.484130859375}, {"start": 600.95, "end": 602.05, "word": " definition", "probability": 0.94482421875}, {"start": 602.05, "end": 605.17, "word": " neighborhood", "probability": 0.705078125}, {"start": 605.17, "end": 605.91, "word": " definition", "probability": 0.93310546875}, {"start": 605.91, "end": 606.59, "word": " of", "probability": 0.978515625}, {"start": 606.59, "end": 609.05, "word": " convergence", "probability": 0.97216796875}, {"start": 609.05, "end": 616.71, "word": " تعريف", "probability": 0.95068359375}, {"start": 616.71, "end": 617.35, "word": " الجوار", "probability": 0.9617513020833334}, {"start": 617.35, "end": 618.21, "word": " للتقارب", "probability": 0.97197265625}], "temperature": 1.0}, {"id": 27, "seek": 65155, "start": 622.96, "end": 651.56, "text": "طيب لو ال sequence ماكانش لها limit يعني مافيش لا يوجد x ينتمي ل r بحقق الشرط هذا فبنقول ان ال sequence ليست not convergent او divergent اذا لو ال sequence مالهاش has no limit فبنسميها divergent اذا مثلا بتكون ال sequence convergent اذا كان في لها limit", "tokens": [9566, 1829, 3555, 45164, 2423, 8310, 19446, 41361, 8592, 5296, 11296, 4948, 37495, 22653, 19446, 41185, 8592, 20193, 7251, 29245, 3215, 2031, 7251, 29399, 2304, 1829, 5296, 367, 4724, 5016, 4587, 4587, 25124, 2288, 9566, 23758, 6156, 3555, 1863, 39648, 16472, 2423, 8310, 32239, 14851, 406, 9652, 6930, 1975, 2407, 18558, 6930, 1975, 15730, 45164, 2423, 8310, 3714, 6027, 3224, 33599, 575, 572, 4948, 6156, 3555, 1863, 38251, 1829, 11296, 18558, 6930, 1975, 15730, 50113, 15040, 39894, 30544, 2423, 8310, 9652, 6930, 1975, 15730, 25961, 8978, 5296, 11296, 4948], "avg_logprob": -0.20538195073604584, "compression_ratio": 1.8860103626943006, "no_speech_prob": 0.0, "words": [{"start": 622.96, "end": 623.32, "word": "طيب", "probability": 0.684814453125}, {"start": 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"probability": 0.91455078125}, {"start": 630.14, "end": 630.38, "word": " r", "probability": 0.315673828125}, {"start": 630.38, "end": 630.92, "word": " بحقق", "probability": 0.9442138671875}, {"start": 630.92, "end": 631.36, "word": " الشرط", "probability": 0.93310546875}, {"start": 631.36, "end": 631.72, "word": " هذا", "probability": 0.89892578125}, {"start": 631.72, "end": 634.26, "word": " فبنقول", "probability": 0.9351806640625}, {"start": 634.26, "end": 634.42, "word": " ان", "probability": 0.496337890625}, {"start": 634.42, "end": 634.56, "word": " ال", "probability": 0.95849609375}, {"start": 634.56, "end": 635.12, "word": " sequence", "probability": 0.994140625}, {"start": 635.12, "end": 636.3, "word": " ليست", "probability": 0.9326171875}, {"start": 636.3, "end": 636.58, "word": " not", "probability": 0.9111328125}, {"start": 636.58, "end": 637.32, "word": " convergent", "probability": 0.934814453125}, {"start": 637.32, "end": 637.58, "word": " او", "probability": 0.7437744140625}, {"start": 637.58, "end": 638.4, "word": " divergent", "probability": 0.6552734375}, {"start": 638.4, "end": 639.62, "word": " اذا", "probability": 0.6934814453125}, {"start": 639.62, "end": 640.06, "word": " لو", "probability": 0.81494140625}, {"start": 640.06, "end": 640.18, "word": " ال", "probability": 0.92724609375}, {"start": 640.18, "end": 640.56, "word": " sequence", "probability": 0.9951171875}, {"start": 640.56, "end": 641.22, "word": " مالهاش", "probability": 0.81884765625}, {"start": 641.22, "end": 641.6, "word": " has", "probability": 0.61328125}, {"start": 641.6, "end": 642.02, "word": " no", "probability": 0.9345703125}, {"start": 642.02, "end": 642.5, "word": " limit", "probability": 0.974609375}, {"start": 642.5, "end": 644.42, "word": " فبنسميها", "probability": 0.9790852864583334}, {"start": 644.42, "end": 645.22, "word": " divergent", "probability": 0.90576171875}, {"start": 645.22, "end": 647.14, "word": " اذا", "probability": 0.836669921875}, {"start": 647.14, "end": 647.54, "word": " مثلا", "probability": 0.758544921875}, {"start": 647.54, "end": 647.98, "word": " بتكون", "probability": 0.798828125}, {"start": 647.98, "end": 648.1, "word": " ال", "probability": 0.9736328125}, {"start": 648.1, "end": 648.5, "word": " sequence", "probability": 0.99462890625}, {"start": 648.5, "end": 649.34, "word": " convergent", "probability": 0.944091796875}, {"start": 649.34, "end": 650.3, "word": " اذا", "probability": 0.85400390625}, {"start": 650.3, "end": 650.62, "word": " كان", "probability": 0.96728515625}, {"start": 650.62, "end": 650.82, "word": " في", "probability": 0.92822265625}, {"start": 650.82, "end": 651.14, "word": " لها", "probability": 0.7406005859375}, {"start": 651.14, "end": 651.56, "word": " limit", "probability": 0.98291015625}], "temperature": 1.0}, {"id": 28, "seek": 67678, "start": 652.52, "end": 676.78, "text": "طب ما معناه ان ال sequence يكون لها limit؟ معناه ان يوجد عدد حقيقي X بحيث لكل جوار V ل X في عدد طبيعي capital N يعتمد على الجوار بحيث ان كل حدود ال sequence تنتمي للجوار هذا والمؤشر تبعها ببدأ من capital N وانت طالع", "tokens": [9566, 3555, 19446, 20449, 8315, 3224, 16472, 2423, 8310, 7251, 30544, 5296, 11296, 4948, 22807, 20449, 8315, 3224, 16472, 7251, 29245, 3215, 6225, 3215, 3215, 11331, 38436, 38436, 1783, 4724, 5016, 1829, 12984, 5296, 28820, 10874, 2407, 9640, 691, 5296, 1783, 8978, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4238, 426, 7251, 34268, 2304, 3215, 15844, 25724, 2407, 9640, 4724, 5016, 1829, 12984, 16472, 28242, 11331, 3215, 23328, 2423, 8310, 6055, 29399, 2304, 1829, 24976, 7435, 2407, 9640, 23758, 16070, 2304, 33604, 46309, 6055, 3555, 3615, 11296, 4724, 44510, 10721, 9154, 4238, 426, 4032, 7649, 2655, 23032, 6027, 3615], "avg_logprob": -0.16593750402331353, "compression_ratio": 1.6995073891625616, "no_speech_prob": 0.0, "words": [{"start": 652.52, "end": 652.86, "word": "طب", "probability": 0.511199951171875}, {"start": 652.86, "end": 652.96, "word": " ما", 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جوار v2 ل y و الجوارين هدول منفصلين بعتقد حلنا السؤال هذا اه فقولنا خدي epsilon بساوي نص المسافة بين x و y", "tokens": [39237, 10943, 22807, 45164, 25961, 8978, 18871, 16254, 6225, 3215, 3215, 9957, 11331, 4587, 4587, 9957, 2031, 20193, 7251, 3794, 995, 45865, 288, 4724, 28543, 2288, 5551, 15040, 38436, 10874, 2407, 9640, 371, 16, 5296, 2031, 4032, 10874, 2407, 9640, 371, 17, 5296, 288, 4032, 25724, 2407, 9640, 9957, 8032, 3215, 12610, 9154, 5172, 36520, 9957, 4724, 34268, 28543, 11331, 1211, 8315, 21136, 33604, 6027, 23758, 1975, 3224, 6156, 39648, 8315, 16490, 16254, 17889, 4724, 3794, 995, 45865, 8717, 9381, 9673, 3794, 31845, 3660, 49374, 2031, 4032, 288], "avg_logprob": -0.18872893124483944, "compression_ratio": 1.6054054054054054, "no_speech_prob": 0.0, "words": [{"start": 859.26, "end": 860.06, "word": "تمام؟", "probability": 0.6532389322916666}, {"start": 860.06, "end": 860.36, "word": " لو", "probability": 0.8212890625}, {"start": 860.36, "end": 860.64, "word": " كان", 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1146.02, "word": " convergence", "probability": 0.64697265625}, {"start": 1146.02, "end": 1148.04, "word": " للسيكوينس", "probability": 0.6595633370535714}, {"start": 1148.04, "end": 1149.74, "word": " فلو", "probability": 0.8429361979166666}, {"start": 1149.74, "end": 1149.86, "word": " في", "probability": 0.7939453125}, {"start": 1149.86, "end": 1150.1, "word": " عندي", "probability": 0.598388671875}, {"start": 1150.1, "end": 1150.58, "word": " سيكوينس", "probability": 0.91748046875}, {"start": 1150.58, "end": 1150.72, "word": " of", "probability": 0.74462890625}, {"start": 1150.72, "end": 1150.92, "word": " real", "probability": 0.9482421875}, {"start": 1150.92, "end": 1151.52, "word": " numbers", "probability": 0.69384765625}, {"start": 1151.52, "end": 1152.2, "word": " وعندي", "probability": 0.63397216796875}, {"start": 1152.2, "end": 1152.48, "word": " real", "probability": 0.84716796875}, {"start": 1152.48, "end": 1152.84, "word": " number", "probability": 0.9580078125}, {"start": 1152.84, "end": 1153.22, "word": " x", "probability": 0.49951171875}, {"start": 1153.22, "end": 1153.98, "word": " the", "probability": 0.1658935546875}, {"start": 1153.98, "end": 1154.36, "word": " following", "probability": 0.88232421875}, {"start": 1154.36, "end": 1154.72, "word": " are", "probability": 0.93896484375}, {"start": 1154.72, "end": 1155.24, "word": " equivalent", "probability": 0.94384765625}], "temperature": 1.0}, {"id": 48, "seek": 118481, "start": 1156.19, "end": 1184.81, "text": "هذا اختصار الكلمات the following are equivalent الاعبارات التالية متكافئة اول عبارة x in converge ل x هذا معناه حسب تعريف ال convergence ال neighborhood definition ان for every neighborhood V of X of X there exists capital N يعتمد على V", "tokens": [3224, 15730, 1975, 46456, 9381, 9640, 33251, 19528, 9307, 264, 3480, 366, 10344, 42963, 3615, 3555, 9640, 9307, 16712, 6027, 10632, 44650, 4117, 31845, 19986, 3660, 1975, 12610, 6225, 3555, 9640, 3660, 2031, 294, 41881, 5296, 2031, 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{"start": 1161.15, "end": 1161.53, "word": " التالية", "probability": 0.98291015625}, {"start": 1161.53, "end": 1162.45, "word": " متكافئة", "probability": 0.95615234375}, {"start": 1162.45, "end": 1163.21, "word": " اول", "probability": 0.799560546875}, {"start": 1163.21, "end": 1163.75, "word": " عبارة", "probability": 0.97509765625}, {"start": 1163.75, "end": 1164.01, "word": " x", "probability": 0.53076171875}, {"start": 1164.01, "end": 1164.27, "word": " in", "probability": 0.394775390625}, {"start": 1164.27, "end": 1164.91, "word": " converge", "probability": 0.95361328125}, {"start": 1164.91, "end": 1165.09, "word": " ل", "probability": 0.8955078125}, {"start": 1165.09, "end": 1165.47, "word": " x", "probability": 0.662109375}, {"start": 1165.47, "end": 1165.85, "word": " هذا", "probability": 0.7587890625}, {"start": 1165.85, "end": 1167.17, "word": " معناه", "probability": 0.9801432291666666}, {"start": 1167.17, "end": 1167.67, "word": " حسب", "probability": 0.970703125}, 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0.91650390625}, {"start": 1176.25, "end": 1176.89, "word": " X", "probability": 0.57373046875}, {"start": 1176.89, "end": 1178.43, "word": " of", "probability": 0.7900390625}, {"start": 1178.43, "end": 1179.09, "word": " X", "probability": 0.9189453125}, {"start": 1179.09, "end": 1181.63, "word": " there", "probability": 0.857421875}, {"start": 1181.63, "end": 1182.15, "word": " exists", "probability": 0.58154296875}, {"start": 1182.15, "end": 1182.67, "word": " capital", "probability": 0.83203125}, {"start": 1182.67, "end": 1183.09, "word": " N", "probability": 0.92578125}, {"start": 1183.09, "end": 1183.85, "word": " يعتمد", "probability": 0.964599609375}, {"start": 1183.85, "end": 1184.21, "word": " على", "probability": 0.880859375}, {"start": 1184.21, "end": 1184.81, "word": " V", "probability": 0.9619140625}], "temperature": 1.0}, {"id": 49, "seek": 121307, "start": 1185.99, "end": 1213.07, "text": "عدد طبيعي بحيث أنه لو كان n أكبر من أو ساوي capital N هذا بيقدر ان xn ينتمي إلى b هاي معناه xn converge ل x الان هذا ال neighborhood definition لل convergence بيكافئ العبارة بي وهذا بنسميها ال epsilon neighborhood definition لل convergence", "tokens": [3615, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 14739, 3224, 45164, 25961, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 23758, 4724, 1829, 4587, 3215, 2288, 16472, 2031, 77, 7251, 29399, 2304, 1829, 30731, 272, 8032, 47302, 20449, 8315, 3224, 2031, 77, 41881, 5296, 2031, 2423, 7649, 23758, 2423, 7630, 7123, 24976, 32181, 4724, 1829, 4117, 31845, 19986, 18863, 3555, 9640, 3660, 4724, 1829, 37037, 15730, 44945, 38251, 1829, 11296, 2423, 17889, 7630, 7123, 24976, 32181], "avg_logprob": -0.25150601978761605, "compression_ratio": 1.6298076923076923, "no_speech_prob": 0.0, "words": [{"start": 1185.99, "end": 1186.57, "word": "عدد", "probability": 0.791015625}, {"start": 1186.57, "end": 1187.49, "word": " طبيعي", "probability": 0.9276123046875}, {"start": 1187.49, "end": 1189.13, 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استعماله شائع اكتر من العبارات السابقة", "tokens": [7649, 995, 8978, 43242, 7578, 1975, 8315, 27188, 2288, 4587, 316, 23758, 6225, 3555, 9640, 3660, 18871, 17889, 6225, 3555, 9640, 3660, 18871, 7630, 7123, 295, 32181, 27188, 2288, 4587, 363, 44945, 38251, 1829, 3224, 2423, 17889, 7630, 7123, 24976, 32181, 27188, 2288, 4587, 383, 44945, 38251, 1829, 3224, 17889, 4238, 426, 7123, 295, 32181, 23758, 39896, 30544, 44713, 25957, 6027, 3224, 13412, 16373, 3615, 1975, 4117, 2655, 2288, 9154, 18863, 3555, 9640, 9307, 21136, 16758, 28671], "avg_logprob": -0.1896033642383722, "compression_ratio": 1.876923076923077, "no_speech_prob": 0.0, "words": [{"start": 1290.35, "end": 1290.69, "word": "انا", "probability": 0.25128173828125}, {"start": 1290.69, "end": 1290.85, "word": " في", "probability": 0.5400390625}, {"start": 1290.85, "end": 1291.21, "word": " عندى", "probability": 0.7734375}, {"start": 1291.21, "end": 1291.51, "word": " انا", "probability": 0.822998046875}, {"start": 1291.51, "end": 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"temperature": 1.0}, {"id": 60, "seek": 148220, "start": 1453.9, "end": 1482.2, "text": "لو أخدت أي epsilon أكبر من السفر for any epsilon أكبر من السفر take v epsilon of x اللي هو عبارة عن ال epsilon neighborhood ل x فهذا is epsilon neighborhood of x صح؟", "tokens": [1211, 2407, 5551, 9778, 3215, 2655, 36632, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 337, 604, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 747, 371, 17889, 295, 2031, 13672, 1829, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 17889, 7630, 5296, 2031, 6156, 3224, 15730, 307, 17889, 7630, 295, 2031, 20328, 5016, 22807], "avg_logprob": -0.21889204437082463, "compression_ratio": 1.5683453237410072, "no_speech_prob": 0.0, "words": [{"start": 1453.9, "end": 1454.24, "word": "لو", "probability": 0.893798828125}, {"start": 1454.24, "end": 1454.74, "word": " أخدت", "probability": 0.95849609375}, {"start": 1454.74, "end": 1455.0, "word": " أي", "probability": 0.88818359375}, {"start": 1455.0, "end": 1455.46, "word": " 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"word": " epsilon", "probability": 0.658203125}, {"start": 1466.68, "end": 1467.12, "word": " of", "probability": 0.93603515625}, {"start": 1467.12, "end": 1467.68, "word": " x", "probability": 0.91064453125}, {"start": 1467.68, "end": 1469.42, "word": " اللي", "probability": 0.868896484375}, {"start": 1469.42, "end": 1469.62, "word": " هو", "probability": 0.88134765625}, {"start": 1469.62, "end": 1469.98, "word": " عبارة", "probability": 0.9937744140625}, {"start": 1469.98, "end": 1470.14, "word": " عن", "probability": 0.9951171875}, {"start": 1470.14, "end": 1470.26, "word": " ال", "probability": 0.70947265625}, {"start": 1470.26, "end": 1470.64, "word": " epsilon", "probability": 0.40283203125}, {"start": 1470.64, "end": 1471.24, "word": " neighborhood", "probability": 0.56005859375}, {"start": 1471.24, "end": 1471.56, "word": " ل", "probability": 0.88818359375}, {"start": 1471.56, "end": 1471.94, "word": " x", "probability": 0.47998046875}, {"start": 1471.94, "end": 1476.04, 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"probability": 0.936279296875}, {"start": 1487.35, "end": 1488.07, "word": " يوجد", "probability": 0.9832356770833334}, {"start": 1488.07, "end": 1488.53, "word": " capital", "probability": 0.56591796875}, {"start": 1488.53, "end": 1488.95, "word": " N", "probability": 0.75341796875}, {"start": 1488.95, "end": 1490.89, "word": " إذا", "probability": 0.4295654296875}, {"start": 1490.89, "end": 1491.45, "word": " يوجد", "probability": 0.9763997395833334}, {"start": 1491.45, "end": 1491.81, "word": " capital", "probability": 0.712890625}, {"start": 1491.81, "end": 1492.35, "word": " N", "probability": 0.95068359375}, {"start": 1492.35, "end": 1496.35, "word": " by", "probability": 0.41650390625}, {"start": 1496.35, "end": 1496.93, "word": " بي", "probability": 0.907470703125}, {"start": 1496.93, "end": 1497.81, "word": " يوجد", "probability": 0.9781901041666666}, {"start": 1497.81, "end": 1498.29, "word": " capital", "probability": 0.7158203125}, {"start": 1498.29, "end": 1498.67, "word": " N", "probability": 0.9501953125}, {"start": 1498.67, "end": 1500.33, "word": " يعتمد", "probability": 0.938232421875}, {"start": 1500.33, "end": 1501.29, "word": " على", "probability": 0.890625}, {"start": 1501.29, "end": 1502.15, "word": " الإبسلون", "probability": 0.9021484375}, {"start": 1502.15, "end": 1502.93, "word": " neighborhood", "probability": 0.88818359375}, {"start": 1502.93, "end": 1503.95, "word": " وبالتالي", "probability": 0.94189453125}, {"start": 1503.95, "end": 1504.63, "word": " يعتمد", "probability": 0.969482421875}, {"start": 1504.63, "end": 1504.85, "word": " على", "probability": 0.9248046875}, {"start": 1504.85, "end": 1505.53, "word": " إبسلون", "probability": 0.99052734375}, {"start": 1505.53, "end": 1506.55, "word": " هذا", "probability": 0.80517578125}, {"start": 1506.55, "end": 1506.97, "word": " عدد", "probability": 0.98046875}, {"start": 1506.97, "end": 1508.39, "word": " طبيعي", "probability": 0.988037109375}, {"start": 1508.39, "end": 1509.63, "word": " بحيث", "probability": 0.9677734375}], "temperature": 1.0}, {"id": 62, "seek": 154259, "start": 1513.53, "end": 1542.59, "text": "بحيث انه لو كان n أكبر من أو ساوي n of epsilon فهذا بيقدي ان xn ينتمي ل v epsilon ل x اللي هو x سالب epsilon وx موجة بepsilon طب وهذا معناه ان ال xn أكبر من x سالب epsilon أصغر من x زاد epsilon", "tokens": [49628, 1829, 12984, 16472, 3224, 45164, 25961, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 297, 295, 17889, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 16472, 2031, 77, 7251, 29399, 2304, 1829, 5296, 371, 17889, 5296, 2031, 13672, 1829, 31439, 2031, 8608, 6027, 3555, 17889, 4032, 87, 3714, 29245, 3660, 4724, 10653, 15754, 23032, 3555, 37037, 15730, 20449, 8315, 3224, 16472, 2423, 2031, 77, 5551, 4117, 26890, 9154, 2031, 8608, 6027, 3555, 17889, 5551, 9381, 17082, 2288, 9154, 2031, 30767, 18513, 17889], "avg_logprob": -0.17857142928100767, "compression_ratio": 1.6964285714285714, "no_speech_prob": 0.0, "words": [{"start": 1513.53, 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"probability": 0.9405924479166666}, {"start": 1539.37, "end": 1539.61, "word": " من", "probability": 0.994140625}, {"start": 1539.61, "end": 1539.97, "word": " x", "probability": 0.978515625}, {"start": 1539.97, "end": 1540.41, "word": " سالب", "probability": 0.9720052083333334}, {"start": 1540.41, "end": 1540.85, "word": " epsilon", "probability": 0.97314453125}, {"start": 1540.85, "end": 1541.37, "word": " أصغر", "probability": 0.9442138671875}, {"start": 1541.37, "end": 1541.55, "word": " من", "probability": 0.9951171875}, {"start": 1541.55, "end": 1541.81, "word": " x", "probability": 0.95263671875}, {"start": 1541.81, "end": 1542.19, "word": " زاد", "probability": 0.92578125}, {"start": 1542.19, "end": 1542.59, "word": " epsilon", "probability": 0.9736328125}], "temperature": 1.0}, {"id": 63, "seek": 156783, "start": 1544.55, "end": 1567.83, "text": "هذا الـ xn ينتمي للفترة المفتوحة هذه معناته هذا الكلام صح هذا معناه xn minus x أصغر من epsilon أكبر من سالب epsilon هذا معناه absolute xn minus x أصغر من epsilon إذن هين أثبتنا إن لو كان b صحيح", "tokens": [3224, 15730, 2423, 39184, 2031, 77, 7251, 29399, 2304, 1829, 24976, 5172, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 29538, 20449, 8315, 47395, 23758, 2423, 28820, 10943, 20328, 5016, 23758, 20449, 8315, 3224, 2031, 77, 3175, 2031, 5551, 9381, 17082, 2288, 9154, 17889, 5551, 4117, 26890, 9154, 8608, 6027, 3555, 17889, 23758, 20449, 8315, 3224, 8236, 2031, 77, 3175, 2031, 5551, 9381, 17082, 2288, 9154, 17889, 11933, 8848, 1863, 8032, 9957, 5551, 12984, 3555, 2655, 8315, 36145, 45164, 25961, 272, 20328, 5016, 1829, 5016], "avg_logprob": -0.1646075648970382, "compression_ratio": 1.7383720930232558, "no_speech_prob": 0.0, "words": [{"start": 1544.55, "end": 1544.93, "word": "هذا", "probability": 0.75927734375}, {"start": 1544.93, "end": 1545.07, "word": " الـ", "probability": 0.671630859375}, {"start": 1545.07, "end": 1545.67, "word": " xn", "probability": 0.5205078125}, {"start": 1545.67, "end": 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{"start": 1553.35, "end": 1553.69, "word": " x", "probability": 0.98291015625}, {"start": 1553.69, "end": 1554.29, "word": " أصغر", "probability": 0.9776611328125}, {"start": 1554.29, "end": 1554.49, "word": " من", "probability": 0.99267578125}, {"start": 1554.49, "end": 1554.95, "word": " epsilon", "probability": 0.331787109375}, {"start": 1554.95, "end": 1555.61, "word": " أكبر", "probability": 0.9749348958333334}, {"start": 1555.61, "end": 1556.67, "word": " من", "probability": 0.9921875}, {"start": 1556.67, "end": 1557.19, "word": " سالب", "probability": 0.8743489583333334}, {"start": 1557.19, "end": 1557.75, "word": " epsilon", "probability": 0.76904296875}, {"start": 1557.75, "end": 1558.71, "word": " هذا", "probability": 0.8564453125}, {"start": 1558.71, "end": 1559.09, "word": " معناه", "probability": 0.98046875}, {"start": 1559.09, "end": 1559.51, "word": " absolute", "probability": 0.912109375}, {"start": 1559.51, "end": 1560.29, "word": " xn", "probability": 0.9560546875}, 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"temperature": 1.0}, {"id": 66, "seek": 164291, "start": 1627.27, "end": 1642.91, "text": "بعدين، بدنا نثبت أن x in converge ل x أو ال neighborhood definition ل x بتحقق فبناخد أي let v be any neighborhood of x فمن تعريف ال neighborhood", "tokens": [3555, 22488, 9957, 12399, 47525, 8315, 8717, 12984, 3555, 2655, 14739, 2031, 294, 41881, 5296, 2031, 34051, 2423, 7630, 7123, 5296, 2031, 39894, 5016, 4587, 4587, 6156, 3555, 1863, 47283, 3215, 36632, 718, 371, 312, 604, 7630, 295, 2031, 6156, 27842, 37279, 16572, 5172, 2423, 7630], "avg_logprob": -0.2937167591237007, "compression_ratio": 1.3309859154929577, "no_speech_prob": 0.0, "words": [{"start": 1627.27, "end": 1628.03, "word": "بعدين،", "probability": 0.5352935791015625}, {"start": 1628.03, "end": 1628.37, "word": " بدنا", "probability": 0.590087890625}, {"start": 1628.37, "end": 1628.81, "word": " نثبت", "probability": 0.988525390625}, {"start": 1628.81, "end": 1629.05, "word": " أن", "probability": 0.30712890625}, {"start": 1629.05, 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عن limit لل sequence x in فنرجع لتعريف ال neighborhood definition of convergence نبدأ ب neighborhood ل x ونستخدم الحقيقة أن كل neighborhood ل x يحتوي epsilon neighborhood", "tokens": [6027, 7649, 5296, 28814, 12984, 3555, 9307, 257, 13672, 7578, 8032, 7578, 2031, 294, 41881, 5296, 2031, 39894, 12984, 3555, 2655, 14739, 3224, 2423, 7630, 7123, 24976, 32181, 39894, 5016, 4587, 4587, 37495, 1863, 7578, 2031, 6225, 3555, 9640, 3660, 18871, 4948, 24976, 8310, 2031, 294, 6156, 1863, 47341, 3615, 5296, 2655, 3615, 16572, 5172, 2423, 7630, 7123, 295, 32181, 8717, 44510, 10721, 4724, 7630, 5296, 2031, 4032, 1863, 14851, 9778, 40448, 21542, 38436, 28671, 14739, 28242, 7630, 5296, 2031, 7251, 33753, 45865, 17889, 7630], "avg_logprob": -0.18731831291387246, "compression_ratio": 1.75, "no_speech_prob": 0.0, "words": [{"start": 1773.11, "end": 1773.73, "word": "الان", "probability": 0.745849609375}, {"start": 1773.73, "end": 1774.57, "word": " لإثبات", "probability": 0.769482421875}, {"start": 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"end": 1814.1, "text": "الان من C .. C بيقول لي إذا في عندك إبسلون موجبة تقدر تلاقي capital N يعتمد عليها بحيث أنه لكل N أكبر من ما يساوي capital N المسافة هذه أصغر من إبسلون", "tokens": [6027, 7649, 9154, 383, 4386, 383, 4724, 1829, 39648, 32239, 11933, 15730, 8978, 43242, 4117, 11933, 3555, 3794, 1211, 11536, 3714, 29245, 49401, 6055, 28543, 2288, 6055, 15040, 38436, 4238, 426, 7251, 34268, 2304, 3215, 25894, 11296, 4724, 5016, 1829, 12984, 14739, 3224, 5296, 28820, 426, 5551, 4117, 26890, 9154, 19446, 7251, 3794, 995, 45865, 4238, 426, 9673, 3794, 31845, 3660, 29538, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536], "avg_logprob": -0.20772688846065573, "compression_ratio": 1.496969696969697, "no_speech_prob": 0.0, "words": [{"start": 1800.98, "end": 1801.52, "word": "الان", "probability": 0.818603515625}, {"start": 1801.52, "end": 1801.84, "word": " من", "probability": 0.88720703125}, {"start": 1801.84, "end": 1802.14, "word": " C", "probability": 0.50830078125}, {"start": 1802.14, "end": 1802.96, "word": 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N", "probability": 0.98046875}, {"start": 1812.28, "end": 1812.94, "word": " المسافة", "probability": 0.9263916015625}, {"start": 1812.94, "end": 1813.16, "word": " هذه", "probability": 0.72021484375}, {"start": 1813.16, "end": 1813.52, "word": " أصغر", "probability": 0.986083984375}, {"start": 1813.52, "end": 1813.64, "word": " من", "probability": 0.994140625}, {"start": 1813.64, "end": 1814.1, "word": " إبسلون", "probability": 0.9736328125}], "temperature": 1.0}, {"id": 74, "seek": 183058, "start": 1815.58, "end": 1830.58, "text": "طب هذه ال implication الأخيرة هي N أكبر من أو ساوي capital N بتقدي في حل المتباين هذه في Xn فبطلع Xn ينتمي إلى X سالب Y و X فاللي هو هذا ال epsilon neighborhood اللي هو", "tokens": [9566, 3555, 29538, 2423, 37814, 16247, 9778, 48923, 39896, 426, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 39894, 4587, 16254, 8978, 11331, 1211, 9673, 2655, 3555, 995, 9957, 29538, 8978, 1783, 77, 6156, 3555, 9566, 1211, 3615, 1783, 77, 7251, 29399, 2304, 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sequence XN لو خدت M عدد طبيعي اي عدد طبيعي natural number و XN اي sequence of real numbers فالـ XN هذه ممكن انفرفتها نكتب حدودها X1 X2 و هكذا", "tokens": [41185, 37279, 16572, 5172, 2423, 39184, 6838, 295, 257, 8310, 1975, 2407, 2423, 39184, 376, 6838, 295, 257, 8310, 1975, 5016, 8315, 6225, 9640, 5172, 9957, 16472, 45164, 8978, 16472, 3215, 11622, 1975, 1829, 4386, 5296, 10721, 1829, 8310, 1783, 45, 45164, 16490, 3215, 2655, 376, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 1975, 1829, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 3303, 1230, 4032, 1783, 45, 1975, 1829, 8310, 295, 957, 3547, 6156, 6027, 39184, 1783, 45, 29538, 3714, 43020, 16472, 5172, 28480, 2655, 11296, 8717, 4117, 2655, 3555, 11331, 3215, 23328, 11296, 1783, 16, 1783, 17, 4032, 8032, 4117, 15730], "avg_logprob": -0.21859830039218792, "compression_ratio": 1.6407766990291262, "no_speech_prob": 0.0, "words": [{"start": 1862.75, "end": 1863.11, "word": "في", "probability": 0.53955078125}, {"start": 1863.11, "end": 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"end": 1993.61, "word": " طيب", "probability": 0.83251953125}, {"start": 1993.61, "end": 1994.43, "word": " اذا", "probability": 0.9267578125}, {"start": 1994.43, "end": 1995.37, "word": " الان", "probability": 0.78759765625}, {"start": 1995.37, "end": 1995.59, "word": " في", "probability": 0.8134765625}, {"start": 1995.59, "end": 1996.07, "word": " نظرية", "probability": 0.98095703125}, {"start": 1996.07, "end": 1996.39, "word": " اتنين", "probability": 0.76873779296875}, {"start": 1996.39, "end": 1996.99, "word": " تلاتة", "probability": 0.911376953125}, {"start": 1996.99, "end": 1997.21, "word": " او", "probability": 0.978271484375}, {"start": 1997.21, "end": 1997.61, "word": " نظرية", "probability": 0.9822998046875}, {"start": 1997.61, "end": 1998.25, "word": " تالتة", "probability": 0.9644775390625}], "temperature": 1.0}, {"id": 81, "seek": 202768, "start": 2000.72, "end": 2027.68, "text": "ما هي هذه النظرية اللي بتقول؟ بتقول لو أنا في اندي إذا هاي ال m tail هذا ال m tail ممكن كتابته على صورة sequence هاي x المؤشر الحد العام تبع ال m tail m زاد n حيث و اين العداد الطبيعي m ثابت و n العداد الطبيعي وبالتالي هنا لو كانت n بالساوية واحد اول حد xm زاد واحد و هكذا طيب", "tokens": [15042, 39896, 29538, 28239, 19913, 2288, 10632, 13672, 1829, 39894, 39648, 22807, 39894, 39648, 45164, 41850, 8978, 16472, 16254, 11933, 15730, 8032, 47302, 2423, 275, 6838, 23758, 2423, 275, 6838, 3714, 43020, 9122, 2655, 16758, 47395, 15844, 20328, 13063, 3660, 8310, 8032, 47302, 2031, 9673, 33604, 46309, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 2423, 275, 6838, 275, 30767, 18513, 297, 11331, 1829, 12984, 4032, 1975, 9957, 18863, 3215, 18513, 41950, 21292, 3615, 1829, 275, 38637, 16758, 2655, 4032, 297, 18863, 3215, 18513, 41950, 21292, 3615, 1829, 46599, 6027, 2655, 6027, 1829, 34105, 45164, 25961, 2655, 297, 20666, 3794, 995, 2407, 10632, 36764, 24401, 1975, 12610, 11331, 3215, 2031, 76, 30767, 18513, 36764, 24401, 4032, 8032, 4117, 15730, 23032, 1829, 3555], "avg_logprob": 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2022.0, "word": " الطبيعي", "probability": 0.9786376953125}, {"start": 2022.0, "end": 2022.58, "word": " وبالتالي", "probability": 0.9078125}, {"start": 2022.58, "end": 2022.78, "word": " هنا", "probability": 0.94580078125}, {"start": 2022.78, "end": 2023.02, "word": " لو", "probability": 0.87255859375}, {"start": 2023.02, "end": 2023.38, "word": " كانت", "probability": 0.927734375}, {"start": 2023.38, "end": 2023.52, "word": " n", "probability": 0.77783203125}, {"start": 2023.52, "end": 2023.98, "word": " بالساوية", "probability": 0.743212890625}, {"start": 2023.98, "end": 2024.4, "word": " واحد", "probability": 0.91064453125}, {"start": 2024.4, "end": 2024.7, "word": " اول", "probability": 0.712646484375}, {"start": 2024.7, "end": 2024.96, "word": " حد", "probability": 0.9931640625}, {"start": 2024.96, "end": 2025.4, "word": " xm", "probability": 0.7314453125}, {"start": 2025.4, "end": 2025.72, "word": " زاد", "probability": 0.820556640625}, {"start": 2025.72, "end": 2026.1, "word": " واحد", "probability": 0.943115234375}, {"start": 2026.1, "end": 2026.22, "word": " و", "probability": 0.8564453125}, {"start": 2026.22, "end": 2026.64, "word": " هكذا", "probability": 0.8629557291666666}, {"start": 2026.64, "end": 2027.68, "word": " طيب", "probability": 0.86865234375}], "temperature": 1.0}, {"id": 82, "seek": 203798, "start": 2029.22, "end": 2037.98, "text": "الان النظرية التالية بتقولني انه لو كان ال M tail convergent", "tokens": [6027, 7649, 28239, 19913, 2288, 10632, 16712, 6027, 10632, 39894, 39648, 22653, 16472, 3224, 45164, 25961, 2423, 376, 6838, 9652, 6930], "avg_logprob": -0.2585227367552844, "compression_ratio": 1.1585365853658536, "no_speech_prob": 0.0, "words": [{"start": 2029.22, "end": 2030.16, "word": "الان", "probability": 0.802734375}, {"start": 2030.16, "end": 2030.8, "word": " النظرية", "probability": 0.8887939453125}, {"start": 2030.8, "end": 2031.94, "word": " التالية", "probability": 0.9889322916666666}, {"start": 2031.94, "end": 2032.74, "word": " بتقولني", "probability": 0.8580729166666666}, {"start": 2032.74, "end": 2034.06, "word": " انه", "probability": 0.738037109375}, {"start": 2034.06, "end": 2036.12, "word": " لو", "probability": 0.841796875}, {"start": 2036.12, "end": 2036.46, "word": " كان", "probability": 0.99267578125}, {"start": 2036.46, "end": 2036.58, "word": " ال", "probability": 0.9541015625}, {"start": 2036.58, "end": 2036.8, "word": " M", "probability": 0.304443359375}, {"start": 2036.8, "end": 2037.1, "word": " tail", "probability": 0.291259765625}, {"start": 2037.1, "end": 2037.98, "word": " convergent", "probability": 0.7978515625}], "temperature": 1.0}, {"id": 83, "seek": 206869, "start": 2042.38, "end": 2068.7, "text": "فال sequence نفسها ال M بتكون convergent و العكس لو كانت ال sequence convergent فأي M tail منها هيكون convergent و اتنين لهم نفس ال limit اتنين لهم نفس ال limit اذا مرة تانية لو كان في عندك sequence XN M fixed natural number فال M tail اللي هو ال sequence هذه", "tokens": [5172, 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"end": 2092.85, "text": "converges if and only if الsequence نفسها converges وهي البرهان هذا ال part f افرضي ان x in convergent نثبت ان ال m ت ال convergent ماشي الحال طيب اذا كانت x in convergent ل x", "tokens": [1671, 331, 2880, 498, 293, 787, 498, 2423, 11834, 655, 8717, 36178, 11296, 9652, 2880, 37037, 1829, 2423, 26890, 3224, 7649, 23758, 2423, 644, 283, 1975, 5172, 43042, 1829, 16472, 2031, 294, 9652, 6930, 8717, 12984, 3555, 2655, 16472, 2423, 275, 6055, 2423, 9652, 6930, 3714, 33599, 1829, 21542, 6027, 23032, 1829, 3555, 1975, 15730, 25961, 2655, 2031, 294, 9652, 6930, 5296, 2031], "avg_logprob": -0.4287109449505806, "compression_ratio": 1.5666666666666667, "no_speech_prob": 0.0, "words": [{"start": 2069.93, "end": 2070.79, "word": "converges", "probability": 0.7164713541666666}, {"start": 2070.79, "end": 2071.17, "word": " if", "probability": 0.93310546875}, {"start": 2071.17, "end": 2071.57, "word": " and", "probability": 0.8828125}, {"start": 2071.57, "end": 2071.89, "word": 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{"start": 2479.39, "end": 2481.27, "word": " وهذا", "probability": 0.887939453125}, {"start": 2481.27, "end": 2481.89, "word": " بكمل", "probability": 0.9353841145833334}, {"start": 2481.89, "end": 2482.59, "word": " برهان", "probability": 0.9134521484375}, {"start": 2482.59, "end": 2483.67, "word": " النظرية", "probability": 0.9769287109375}, {"start": 2483.67, "end": 2485.37, "word": " okay", "probability": 0.484619140625}, {"start": 2485.37, "end": 2485.91, "word": " تمام", "probability": 0.89306640625}, {"start": 2485.91, "end": 2486.41, "word": " واضح", "probability": 0.9485677083333334}], "temperature": 1.0}, {"id": 103, "seek": 251601, "start": 2491.15, "end": 2516.01, "text": "طيب احنا بنكتفي بهذا القدر و ان شاء الله في المحاضرة القادمة هناخد برضه بعض النظريات و ناخد أمثلة كيف نثبت ان ال limit ل sequence ل convergence sequence بالساوي عدد معين و هكذا طبعا كل الأجزاء هذه موجودة عندكم ممكن تقرؤوها و تحضروها للمحاضرة الجاية", "tokens": [9566, 1829, 3555, 1975, 5016, 8315, 44945, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288, 4032, 16472, 13412, 16606, 21984, 8978, 9673, 5016, 46958, 25720, 25062, 18513, 46007, 8032, 1863, 47283, 3215, 4724, 43042, 3224, 45030, 11242, 28239, 19913, 16572, 9307, 4032, 8717, 47283, 3215, 5551, 2304, 12984, 37977, 9122, 33911, 8717, 12984, 3555, 2655, 16472, 2423, 4948, 5296, 8310, 5296, 32181, 8310, 20666, 3794, 995, 45865, 6225, 3215, 3215, 20449, 9957, 4032, 8032, 4117, 15730, 23032, 3555, 3615, 995, 28242, 16247, 7435, 11622, 16606, 29538, 3714, 29245, 23328, 3660, 43242, 24793, 3714, 43020, 6055, 4587, 2288, 33604, 2407, 11296, 4032, 6055, 5016, 11242, 32887, 11296, 5296, 19528, 5016, 46958, 25720, 25724, 995, 10632], "avg_logprob": -0.11039402160955512, "compression_ratio": 1.6827309236947792, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2491.15, "end": 2491.65, "word": "طيب", "probability": 0.9739583333333334}, {"start": 2491.65, "end": 2492.05, "word": " احنا", "probability": 0.8797200520833334}, {"start": 2492.05, "end": 2492.67, "word": " بنكتفي", "probability": 0.871337890625}, {"start": 2492.67, "end": 2493.07, "word": " بهذا", "probability": 0.944091796875}, {"start": 2493.07, "end": 2493.87, "word": " القدر", "probability": 0.9962565104166666}, {"start": 2493.87, "end": 2495.89, "word": " و", "probability": 0.92431640625}, {"start": 2495.89, "end": 2496.19, "word": " ان", "probability": 0.4091796875}, {"start": 2496.19, "end": 2496.49, "word": " شاء", "probability": 0.987548828125}, {"start": 2496.49, "end": 2496.67, "word": " الله", "probability": 0.94384765625}, {"start": 2496.67, "end": 2497.13, "word": " في", "probability": 0.626953125}, {"start": 2497.13, "end": 2497.61, "word": " المحاضرة", "probability": 0.98583984375}, {"start": 2497.61, "end": 2498.19, "word": " القادمة", "probability": 0.9934895833333334}, {"start": 2498.19, "end": 2499.89, "word": " هناخد", "probability": 0.9576416015625}, {"start": 2499.89, "end": 2500.33, "word": " برضه", "probability": 0.9402669270833334}, {"start": 2500.33, "end": 2500.71, "word": " بعض", "probability": 0.977294921875}, {"start": 2500.71, "end": 2501.49, "word": " النظريات", "probability": 0.966552734375}, {"start": 2501.49, "end": 2501.63, "word": " و", "probability": 0.98828125}, {"start": 2501.63, "end": 2502.01, "word": " ناخد", "probability": 0.93798828125}, {"start": 2502.01, "end": 2502.57, "word": " أمثلة", "probability": 0.8922119140625}, {"start": 2502.57, "end": 2503.61, "word": " كيف", "probability": 0.974365234375}, {"start": 2503.61, "end": 2504.19, "word": " نثبت", "probability": 0.9945068359375}, {"start": 2504.19, "end": 2504.39, "word": " ان", "probability": 0.51220703125}, {"start": 2504.39, "end": 2504.65, "word": " ال", "probability": 0.7265625}, {"start": 2504.65, "end": 2504.95, "word": " limit", "probability": 0.9228515625}, {"start": 2504.95, "end": 2505.09, "word": " ل", "probability": 0.82958984375}, {"start": 2505.09, "end": 2505.71, "word": " sequence", "probability": 0.94091796875}, {"start": 2505.71, "end": 2506.35, "word": " ل", "probability": 0.385009765625}, {"start": 2506.35, "end": 2506.83, "word": " convergence", "probability": 0.6201171875}, {"start": 2506.83, "end": 2507.51, "word": " sequence", "probability": 0.96044921875}, {"start": 2507.51, "end": 2507.99, "word": " بالساوي", "probability": 0.7930908203125}, {"start": 2507.99, "end": 2508.37, "word": " عدد", "probability": 0.9786783854166666}, {"start": 2508.37, "end": 2508.97, "word": " معين", "probability": 0.98486328125}, {"start": 2508.97, "end": 2509.49, "word": " و", "probability": 0.94580078125}, {"start": 2509.49, "end": 2509.95, "word": " هكذا", "probability": 0.9615885416666666}, {"start": 2509.95, "end": 2511.09, "word": " طبعا", "probability": 0.976806640625}, {"start": 2511.09, "end": 2511.33, "word": " كل", "probability": 0.97998046875}, {"start": 2511.33, "end": 2511.67, "word": " الأجزاء", "probability": 0.904541015625}, {"start": 2511.67, "end": 2511.89, "word": " هذه", "probability": 0.6337890625}, {"start": 2511.89, "end": 2512.41, "word": " موجودة", "probability": 0.9866943359375}, {"start": 2512.41, "end": 2512.79, "word": " عندكم", "probability": 0.828857421875}, {"start": 2512.79, "end": 2513.03, "word": " ممكن", "probability": 0.72314453125}, {"start": 2513.03, "end": 2513.51, "word": " تقرؤوها", "probability": 0.8933919270833334}, {"start": 2513.51, "end": 2513.61, "word": " و", "probability": 0.798828125}, {"start": 2513.61, "end": 2514.13, "word": " تحضروها", "probability": 0.96669921875}, {"start": 2514.13, "end": 2514.93, "word": " للمحاضرة", "probability": 0.971484375}, {"start": 2514.93, "end": 2516.01, "word": " الجاية", "probability": 0.787109375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2520.71175, "duration_after_vad": 2317.237499999993}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CINg1xNQafM_raw.srt
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@@ -0,0 +1,1728 @@
+1
+00:00:21,320 --> 00:00:25,400
+هنبدأ ان شاء الله اليوم chapter جديد و هو ال
+
+2
+00:00:25,400 --> 00:00:30,060
+chapter التاني عنوان ال chapter sequences and
+
+3
+00:00:30,060 --> 00:00:35,960
+series المتتاليات و المتسلسلات طبعا الموضوع هذا
+
+4
+00:00:35,960 --> 00:00:43,220
+مار معاكم في تفاضل ألف .. تفاضل با عفوا و درسنا
+
+5
+00:00:43,220 --> 00:00:46,860
+خواص ال sequences بطريقة مختصرة و ال series
+
+6
+00:00:46,860 --> 00:00:53,710
+اتوسعنا فيهاالمرة هذه هنتوسع في ال sequences و
+
+7
+00:00:53,710 --> 00:00:58,750
+هنختصر في ال series العكس يعني و هنتناول دراسة كل
+
+8
+00:00:58,750 --> 00:01:06,130
+منهم بطريقة تحليلية و طريقة موضعية أكتر يعني من
+
+9
+00:01:06,130 --> 00:01:07,270
+وجه اتناظر رياضية
+
+10
+00:01:10,330 --> 00:01:13,590
+فأول section في هذا ال chapter هيكون عنوانه
+
+11
+00:01:13,590 --> 00:01:17,610
+sequences and their limits المتتاليات و نهاياتهم
+
+12
+00:01:22,470 --> 00:01:28,630
+فنشوف تعريف ال sequence ال sequence in X ما معنى
+
+13
+00:01:28,630 --> 00:01:33,110
+sequence in X، X مجموعة، أي مجموعة ممكن طبعا هناخد
+
+14
+00:01:33,110 --> 00:01:37,470
+هنا X مجموعة الأعداد الحقيقية، هذه المجموعة اللي
+
+15
+00:01:37,470 --> 00:01:42,450
+احنا بنهتم فيها في ال course هذا ف sequence in X
+
+16
+00:01:42,450 --> 00:01:47,410
+يعني ال sequence على سرها تنتمي للمجموعة Xفلو أخدت
+
+17
+00:01:47,410 --> 00:01:52,610
+أي مجموعة x فعشان أعرف sequence عناصرها في x فما
+
+18
+00:01:52,610 --> 00:01:55,470
+هي ال sequence في المجموعة x؟ هي عبارة مجرد
+
+19
+00:01:55,470 --> 00:02:00,970
+function دالة المجال تبعها الأعداد الطبيعية أو أي
+
+20
+00:02:00,970 --> 00:02:04,970
+مجموعة جزئية منها والمجال المقابل تبعها هي
+
+21
+00:02:04,970 --> 00:02:09,820
+المجموعة x اللي ال sequence تنتمي إليهاو في الحالة
+
+22
+00:02:09,820 --> 00:02:13,360
+هذه إذا ال sequence هي function دالة بس دالة من
+
+23
+00:02:13,360 --> 00:02:19,320
+نوع خاص مجالها مجموعة الأعداد الحقيقية و عادة احنا
+
+24
+00:02:19,320 --> 00:02:23,320
+بنهتم بال sequences of real numbers او المتتاليات
+
+25
+00:02:23,320 --> 00:02:27,280
+اللي عناصرها أعداد حقيقية وبالتالي X هذه هتكون
+
+26
+00:02:27,280 --> 00:02:31,460
+اللي هو مجموعة الأعداد الحقيقية طيب هذه ال
+
+27
+00:02:31,460 --> 00:02:35,410
+sequence functionمجالها العداد الطبيعي وبالتالي
+
+28
+00:02:35,410 --> 00:02:40,350
+ممكن نعرفها F هي عند أي عدد طبيعي N هي عبارة عن XN
+
+29
+00:02:40,350 --> 00:02:47,030
+XN طبعا هذا ينتمي للمجموعة X وبالتالي ال .. ال ..
+
+30
+00:02:47,030 --> 00:02:52,910
+ال sequence FN هذه احنا بنحاول نعرفها بدلالة ال
+
+31
+00:02:52,910 --> 00:02:56,720
+range تبعهايعني بدل ما اقول ال sequence هي
+
+32
+00:02:56,720 --> 00:03:01,800
+function جرّت العادة ان احنا نحذف رمز ال function
+
+33
+00:03:01,800 --> 00:03:05,980
+و نستبدله بال range تبع ال function اللي هو y ال
+
+34
+00:03:05,980 --> 00:03:09,960
+range تبع ال function كل ال x n حيث n عدد طبيعي
+
+35
+00:03:09,960 --> 00:03:13,980
+ببدأ من واحد من ت أنما إلى نهاية اذا ال sequence
+
+36
+00:03:13,980 --> 00:03:18,600
+بدل ما نكتبها على صورة function هنكتبها على الصورة
+
+37
+00:03:18,600 --> 00:03:24,340
+هذه او الصورة هذه او الصورة هذه او الصورة هذه okay
+
+38
+00:03:26,550 --> 00:03:30,070
+و طبعا ال sequence هذه يعني أسرها هذه أو أي واحدة
+
+39
+00:03:30,070 --> 00:03:37,350
+منهم ممكن نكتبها برضه على الصورة x1, x2, x3 و هكذا
+
+40
+00:03:40,840 --> 00:03:45,180
+فكل الرموز هذه ترمز إلى ال sequence هذه اللي هي ال
+
+41
+00:03:45,180 --> 00:03:53,400
+function f اللي هي ال function f okay إذن أهم شيء
+
+42
+00:03:53,400 --> 00:03:56,480
+في تعريفنا أن ال sequence هي function دلنا
+
+43
+00:03:56,480 --> 00:04:00,400
+وبالتالي لها مجال مجالها العداد الطبيعي المجال
+
+44
+00:04:00,400 --> 00:04:04,420
+المقابل هي المجموعة اللي عناصر ال sequence تنتمي
+
+45
+00:04:04,420 --> 00:04:10,950
+لها ال sequences ممكن أعرفهم بطريقتينإذا في
+
+46
+00:04:10,950 --> 00:04:15,970
+الملاحظة هذه sequences can be defined explicitly
+
+47
+00:04:15,970 --> 00:04:19,910
+هذه أحد الطرق ممكن يعرف ال sequence بطريقة صريحة
+
+48
+00:04:19,910 --> 00:04:27,890
+بطريقة بقانونفمثلا ال sequence if بالساوية عناصرها
+
+49
+00:04:27,890 --> 00:04:31,670
+اتنين اربعة ستة تمانية الاخرى هذه عبارة عن
+
+50
+00:04:31,670 --> 00:04:38,130
+sequence وهي معرفة بطريقة صريحة فهذه عبارة عن
+
+51
+00:04:38,130 --> 00:04:42,630
+sequence of even natural members العداد الطبيعية
+
+52
+00:04:42,630 --> 00:04:47,790
+الزوجيةممكن نكتب الحد العام الانف هذا بنسميه الانف
+
+53
+00:04:47,790 --> 00:04:53,710
+term اكس ان هذا هنا بنسميه الانف term الحد النوني
+
+54
+00:04:53,710 --> 00:04:59,190
+الحد النوني او الحد العام فال انف term هنا هو
+
+55
+00:04:59,190 --> 00:05:08,180
+اتنين ان اكس ان بساوي اتنين ان حيث ان عدد طبيعيأو
+
+56
+00:05:08,180 --> 00:05:12,620
+ممكن نكتب ال sequence على صورة 2n من n بالساعة
+
+57
+00:05:12,620 --> 00:05:16,740
+واحد إلى ملا نهائية إذا هنا أنا بعرف ال sequence
+
+58
+00:05:16,740 --> 00:05:22,960
+برص حدودها أول تلات حدود إلى و هكذا أو بكتب قاعدة
+
+59
+00:05:22,960 --> 00:05:27,880
+لحد العام xn و طبعا n أدى الطبيعي فمقدر من القاعدة
+
+60
+00:05:27,880 --> 00:05:32,740
+هذه أجيب كل الحدود إذا هذا explicit definition of
+
+61
+00:05:32,740 --> 00:05:39,150
+a sequence هذا تعريف صريح لل sequenceفي طريقة
+
+62
+00:05:39,150 --> 00:05:44,870
+تانية لتعريف ال sequence وهي الطريقة الاستقرائية،
+
+63
+00:05:44,870 --> 00:05:49,330
+إذا ال sequences can be defined inductively أو
+
+64
+00:05:49,330 --> 00:05:55,970
+recursivelyبطريقة استقرائية او بطريقة تكرارية كيف
+
+65
+00:05:55,970 --> 00:06:02,290
+هذه الطريقة باجي لل sequence و باخد اول حد فيها زي
+
+66
+00:06:02,290 --> 00:06:07,250
+X1 او اول حدين او اول تلات حدود و بعطيهم قيم
+
+67
+00:06:07,250 --> 00:06:16,010
+بحددهم قيم محددة بعطيهم قيم محددة بعدين باجيبباجي
+
+68
+00:06:16,010 --> 00:06:21,990
+بعبّر عن الحد xn زايد واحد او xn بدلالة الحدود
+
+69
+00:06:21,990 --> 00:06:27,850
+اللي جابله وبستخدم طبعا لهذا formula بنسميها
+
+70
+00:06:27,850 --> 00:06:32,070
+recursive formula او inductive formula كما في
+
+71
+00:06:32,070 --> 00:06:39,550
+المثال التالي يعني انا عند ال sequence 2n هذه انا
+
+72
+00:06:39,550 --> 00:06:48,000
+عند ال sequence xn بساوة 2nهذه ممكن أعرفها بطريقة
+
+73
+00:06:48,000 --> 00:06:57,140
+استقرائية كيف باخد بعطي أول حد فيه x1 بعطيله قيمة
+
+74
+00:06:57,140 --> 00:07:01,220
+محددة وهي 2 طبعا أول حد في ال sequence هذه هو 2
+
+75
+00:07:01,220 --> 00:07:06,760
+صح؟ لأن هنا أخدت x1 وعطيته قيمة محددة ممكن في بعض
+
+76
+00:07:06,760 --> 00:07:12,140
+الأمثلة أعطي قيمة قيمة محددة ل x1 و x2 و x3بعدين
+
+77
+00:07:12,140 --> 00:07:19,100
+باجي إلى الحد رقم n زياد واحد و بعبر عنه ب
+
+78
+00:07:19,100 --> 00:07:23,000
+recursive formula بعبر عنه بدلالة الحد اللي جابله
+
+79
+00:07:23,000 --> 00:07:26,760
+او الحد اللي جابله مباشرة و الجاب اللي جابله و
+
+80
+00:07:26,760 --> 00:07:32,510
+هكذافهذه بنسميها recursive أو inductive formula
+
+81
+00:07:32,510 --> 00:07:37,150
+تعطيني لحد رقم n زاد واحد بدالة الحد اللي جابله xn
+
+82
+00:07:37,150 --> 00:07:43,870
+فمثلا لو بده أحسب x2 فباخد n بساوي واحد هنا صح
+
+83
+00:07:43,870 --> 00:07:50,110
+فبطل عند x2 بساوي x1 زاد اتنين x1 بساوي اتنين زاد
+
+84
+00:07:50,110 --> 00:07:56,400
+اتنين بطلع أربعةX3 برضه عشان اجيب X3 بستخدم ال
+
+85
+00:07:56,400 --> 00:08:00,480
+recursive formula و باخد N بساوي 2 فبطلع عند X3
+
+86
+00:08:00,480 --> 00:08:06,600
+بساوي X2 زائد 2 X2 أربعة و اتنين بطلع ستة و هكذا
+
+87
+00:08:06,600 --> 00:08:13,340
+اذا هيك بحصل على ال sequence 2N اللي حدودها 2 4 6
+
+88
+00:08:13,340 --> 00:08:20,460
+8 و هكذا اه okay تمام ال
+
+89
+00:08:20,460 --> 00:08:30,520
+..طيب الان بدي اعرف ما معناه ان ال sequence تكون
+
+90
+00:08:30,520 --> 00:08:36,500
+convergent او لها limit لو في عندى sequence من
+
+91
+00:08:36,500 --> 00:08:37,720
+العداد الحقيقية
+
+92
+00:08:41,200 --> 00:08:45,480
+فبقول إن ال sequence converge
+
+93
+00:08:45,480 --> 00:08:51,860
+ال sequence of real numbers بتكون converge أو
+
+94
+00:08:51,860 --> 00:08:59,940
+convergent إذا قدرت ألاقي X ينتمي ل R بحيث إنه لكل
+
+95
+00:08:59,940 --> 00:09:06,200
+neighborhood V ل X لكل جوار V ل X بقدر أو جد أو
+
+96
+00:09:06,200 --> 00:09:12,250
+ألاقيعدد طبيعي capital N يعتمد على الجوار V ينتمي
+
+97
+00:09:12,250 --> 00:09:17,030
+لعداد الطبيعية بحيث أنه لكل small n أكبر من أو سوى
+
+98
+00:09:17,030 --> 00:09:21,770
+capital N، Xn ينتمي إلى V يعني الجوار V هذا يحتوي
+
+99
+00:09:21,770 --> 00:09:29,100
+كل عناصر ال sequence من capital N وانت طالعفلو هذا
+
+100
+00:09:29,100 --> 00:09:34,020
+الشرط اتحقق فبنقول ان الـ sequence converge و ال
+
+101
+00:09:34,020 --> 00:09:38,040
+limit تبعتها هي العدد X في الحالة هذه بنقول ان X
+
+102
+00:09:38,040 --> 00:09:46,080
+is the limit of sequence X in و
+
+103
+00:09:46,080 --> 00:09:51,180
+بنكتب limit X in بالساوية X او نكتب X in tends to
+
+104
+00:09:51,180 --> 00:09:57,750
+X as N tends to infinityهذا التعريف بنسميه ال
+
+105
+00:09:57,750 --> 00:10:05,170
+neighborhood neighborhood definition neighborhood
+
+106
+00:10:05,170 --> 00:10:16,710
+definition of convergence تعريف
+
+107
+00:10:16,710 --> 00:10:18,210
+الجوار للتقارب
+
+108
+00:10:22,960 --> 00:10:28,200
+طيب لو ال sequence ماكانش لها limit يعني مافيش لا
+
+109
+00:10:28,200 --> 00:10:34,560
+يوجد x ينتمي ل r بحقق الشرط هذا فبنقول ان ال
+
+110
+00:10:34,560 --> 00:10:40,060
+sequence ليست not convergent او divergent اذا لو
+
+111
+00:10:40,060 --> 00:10:45,220
+ال sequence مالهاش has no limit فبنسميها divergent
+
+112
+00:10:45,220 --> 00:10:50,820
+اذا مثلا بتكون ال sequence convergent اذا كان في
+
+113
+00:10:50,820 --> 00:10:54,560
+لها limitطب ما معناه ان ال sequence يكون لها
+
+114
+00:10:54,560 --> 00:11:01,680
+limit؟ معناه ان يوجد عدد حقيقي X بحيث لكل جوار V ل
+
+115
+00:11:01,680 --> 00:11:08,260
+X في عدد طبيعي capital N يعتمد على الجوار بحيث ان
+
+116
+00:11:08,260 --> 00:11:14,120
+كل حدود ال sequence تنتمي للجوار هذا والمؤشر تبعها
+
+117
+00:11:14,120 --> 00:11:20,130
+ببدأ من capital N وانت طالعيعني معنى الكلام هذا ..
+
+118
+00:11:20,130 --> 00:11:28,290
+هذا الكلام معناه ان X capital N و X capital N زائد
+
+119
+00:11:28,290 --> 00:11:35,990
+واحد و X capital N زائد اتنين و هكذا كل هدول
+
+120
+00:11:35,990 --> 00:11:38,630
+بينتموا الى الجوار دي
+
+121
+00:11:44,830 --> 00:11:48,590
+لو ال sequence مالهاش limit فبنسميها divergent
+
+122
+00:11:48,590 --> 00:11:56,190
+okay طبعا؟ V جوار .. جوار يعني .. مجموعة .. اه
+
+123
+00:11:56,190 --> 00:12:01,410
+جوار ل X يعني مجموعة تحتوي ال X و الجوار عشان V
+
+124
+00:12:01,410 --> 00:12:05,710
+يكون جوار لازم يكون داخله .. لازم نلاقي داخله
+
+125
+00:12:05,710 --> 00:12:10,010
+epsilon نبرهون كل جوار لازم يحتوي epsilon نبرهون
+
+126
+00:12:15,360 --> 00:12:23,300
+يعني مش اي مجموعة طيب
+
+127
+00:12:23,300 --> 00:12:27,780
+ال .. ان لو
+
+128
+00:12:27,780 --> 00:12:32,800
+في اندي سيكوانس و السيكوانس هاد convergent ف ال
+
+129
+00:12:32,800 --> 00:12:34,600
+limit تبعتها بتطلع unique
+
+130
+00:12:41,740 --> 00:12:45,620
+النظرية الأولى بتقول لو كانت x in sequence of real
+
+131
+00:12:45,620 --> 00:12:51,320
+numbers و converge ل x و converge ل y يعني لها two
+
+132
+00:12:51,320 --> 00:12:55,740
+limits فلازم ال limits يكونوا متساويتين يعني ممنوع
+
+133
+00:12:55,740 --> 00:12:59,940
+ال convergence sequence يكون لها أكتر من limit
+
+134
+00:12:59,940 --> 00:13:05,400
+يعني معناه بعبارة أخرى a convergent sequence has a
+
+135
+00:13:05,400 --> 00:13:06,140
+unique limit
+
+136
+00:13:09,340 --> 00:13:13,560
+خلّينا نبرهن الكلام هذا، افرض إنه في عندي sequence
+
+137
+00:13:13,560 --> 00:13:20,440
+x in converge ل x و أيضا converge ل y المطلوب
+
+138
+00:13:20,440 --> 00:13:25,540
+إثبات إن x بساوي y لبرهان ذلك نعمل برهان بالتناقض
+
+139
+00:13:25,540 --> 00:13:30,680
+assume on contrary إن x لا تساوي y اللي هو نفي
+
+140
+00:13:30,680 --> 00:13:36,600
+النتيجة و بينصل لتناقض في exercise 15 في section 2
+
+141
+00:13:36,600 --> 00:13:41,810
+.2أخذناها في ال chapter السابق بقول لو في عندي أي
+
+142
+00:13:41,810 --> 00:13:49,130
+عددين حقيقيين x و y فبقدر
+
+143
+00:13:49,130 --> 00:13:57,250
+ألاقي v1 جوار ل x و
+
+144
+00:13:57,250 --> 00:14:05,390
+بقدر ألاقي v2 v2
+
+145
+00:14:05,390 --> 00:14:06,610
+جوار ل y
+
+146
+00:14:09,920 --> 00:14:17,120
+بحيث ان تقاطعهم بساوي five يعني اثنين disjoint
+
+147
+00:14:19,260 --> 00:14:24,660
+تمام؟ لو كان في عندي عددين حققين x لا يساوي y بقدر
+
+148
+00:14:24,660 --> 00:14:31,280
+ألاقي جوار v1 ل x و جوار v2 ل y و الجوارين هدول
+
+149
+00:14:31,280 --> 00:14:36,660
+منفصلين بعتقد حلنا السؤال هذا اه فقولنا خدي
+
+150
+00:14:36,660 --> 00:14:45,290
+epsilon بساوي نص المسافة بين x و yو هد خلّي x زاد
+
+151
+00:14:45,290 --> 00:14:50,410
+y و النقطة هد x سالب y هد عبارة عن y neighborhood
+
+152
+00:14:50,410 --> 00:14:55,570
+ل x وبالتالي neighborhood ل x و خدي هنا برضه هد
+
+153
+00:14:55,570 --> 00:15:01,030
+عبارة عن y سالب y و النقطة هد y زاد y
+
+154
+00:15:03,680 --> 00:15:09,460
+فال .. واضح أن الجوارين هدول متقاطعوش لأن أنا أخدت
+
+155
+00:15:09,460 --> 00:15:13,180
+epsilon نص المسافة هذه و هذه فترة مفتوعة و هذه
+
+156
+00:15:13,180 --> 00:15:18,560
+مفتوعة فمافيش بينهم نقاط مشتركة okay إذا هذا
+
+157
+00:15:18,560 --> 00:15:23,620
+الكلام موجود إذا هذا صحيح exercise 15 بيقول لي إذا
+
+158
+00:15:23,620 --> 00:15:30,310
+كان x لا يساوي yفطبعا ممكن نفرض ان x أصغر من y أو
+
+159
+00:15:30,310 --> 00:15:35,170
+y أصغر من x وبالتالي بقدر ألاقي this joint this
+
+160
+00:15:35,170 --> 00:15:43,630
+joint neighborhoods v1 ل x وv2 ل y على التوالي و 2
+
+161
+00:15:43,630 --> 00:15:50,910
+منفصلين الان احنا فرضين ان x in converge ل xحسب
+
+162
+00:15:50,910 --> 00:15:54,790
+الـ Neighborhood Definition لـ Convergence لما أن
+
+163
+00:15:54,790 --> 00:16:00,550
+المتتالي Xn converge ل X و V1 جوار ل X إذا يوجد
+
+164
+00:16:00,550 --> 00:16:07,710
+عدد طبيعي N1 يعتمد على الجوار V1 بحيث أن Xn تنتمي
+
+165
+00:16:07,710 --> 00:16:13,260
+للجوار V1 لكل N أكبر من أو ساوى N1كذلك احنا فرضين
+
+166
+00:16:13,260 --> 00:16:18,320
+في النظرية ان sequence xn converge ل y و الان v2
+
+167
+00:16:18,320 --> 00:16:23,660
+neighborhood ل y، اذا حسب تعريف ال convergence بما
+
+168
+00:16:23,660 --> 00:16:27,680
+ان xn converge ل y و v2 neighborhood ل y، اذا
+
+169
+00:16:27,680 --> 00:16:32,440
+بنقدر نلاقي عدد طبيعي n2 يعتمد على v2، بحيث ان xn
+
+170
+00:16:32,440 --> 00:16:38,840
+ينتمي لv2 لكل n أكبر من أو ساوي n2الان لو عرفت
+
+171
+00:16:38,840 --> 00:16:42,320
+capital N على Nها ال maximum الاكبر بين N واحد و N
+
+172
+00:16:42,320 --> 00:16:47,360
+اتنين هذا معناه ان capital N عدد طبيعي لان الاكبر
+
+173
+00:16:47,360 --> 00:16:52,320
+بين هدول هيكون واحد منهم فهو عدد طبيعي و capital N
+
+174
+00:16:52,320 --> 00:16:55,640
+اكبر من او ساوي N واحد و اكبر من او ساوي N اتنين
+
+175
+00:16:55,640 --> 00:16:59,820
+لان الكبير فيهم الان
+
+176
+00:16:59,820 --> 00:17:04,120
+لو اخدت small n اكبر من او ساوي capital N فمن
+
+177
+00:17:04,120 --> 00:17:09,540
+تعريف capital Nهذا بيقدي ان capital N أكبر من أو
+
+178
+00:17:09,540 --> 00:17:14,760
+ساوي N واحد اذا الان انا عندي small n أكبر من أو
+
+179
+00:17:14,760 --> 00:17:23,820
+ساوي N واحد وبالتالي اذا Xn تنتمي ل D واحد كذلك
+
+180
+00:17:23,820 --> 00:17:29,560
+انا عندي من تعريف capital N capital N أكبر من أو
+
+181
+00:17:29,560 --> 00:17:34,950
+ساوي N اتنينوبالتالي small n أكبر من أو ساوي
+
+182
+00:17:34,950 --> 00:17:38,970
+capital N اتنين لما تكون small n أكبر من أو ساوي
+
+183
+00:17:38,970 --> 00:17:45,450
+capital N اتنين فبطلع xn ينتمي إلى v2 إذا الأن أنا
+
+184
+00:17:45,450 --> 00:17:49,110
+أثبتت أنه لو كانت small n أكبر من أو ساوي capital
+
+185
+00:17:49,110 --> 00:17:57,090
+N فبطلع xn ينتمي إلىV1 و الى V2 وبالتالي تنتمي
+
+186
+00:17:57,090 --> 00:18:01,290
+لتقاطعهم إذا المعنى أن التقاطع هذا لا يساوي فيه
+
+187
+00:18:01,290 --> 00:18:05,810
+وهذا بيديني contradiction لأنه exercise 15 بيقول
+
+188
+00:18:05,810 --> 00:18:10,450
+لي أن V1 و V2 هدول disjoint فكيف طلع مش disjoint
+
+189
+00:18:10,450 --> 00:18:16,070
+تناقض تناقض هذا بيقول لي أن ال assumption تبعيإن X
+
+190
+00:18:16,070 --> 00:18:20,390
+لا تساوي Y كان خطأ إذن الصح إن X بالساوي Y
+
+191
+00:18:20,390 --> 00:18:25,430
+وبالتالي ال limit لل sequence لازم تكون واحدة
+
+192
+00:18:25,430 --> 00:18:33,990
+unique تمام؟ واضح البرهان؟ في أي استفسار؟
+
+193
+00:18:33,990 --> 00:18:37,510
+في أي سؤال؟
+
+194
+00:18:50,080 --> 00:19:02,120
+النظرية التانية تعطيني
+
+195
+00:19:02,120 --> 00:19:09,740
+شروط متكافئة لتعريف ال convergence للسيكوينس فلو
+
+196
+00:19:09,740 --> 00:19:12,840
+في عندي سيكوينس of real numbers وعندي real number
+
+197
+00:19:12,840 --> 00:19:17,630
+x the following are equivalentهذا اختصار الكلمات
+
+198
+00:19:17,630 --> 00:19:21,530
+the following are equivalent الاعبارات التالية
+
+199
+00:19:21,530 --> 00:19:27,670
+متكافئة اول عبارة x in converge ل x هذا معناه حسب
+
+200
+00:19:27,670 --> 00:19:31,070
+تعريف ال convergence ال neighborhood definition ان
+
+201
+00:19:31,070 --> 00:19:42,150
+for every neighborhood V of X of X there exists
+
+202
+00:19:42,150 --> 00:19:50,590
+capital N يعتمد على Vعدد طبيعي بحيث أنه لو كان n
+
+203
+00:19:50,590 --> 00:19:56,150
+أكبر من أو ساوي capital N هذا بيقدر ان xn ينتمي
+
+204
+00:19:56,150 --> 00:20:03,390
+إلى b هاي معناه xn converge ل x الان هذا ال
+
+205
+00:20:03,390 --> 00:20:06,990
+neighborhood definition لل convergence بيكافئ
+
+206
+00:20:06,990 --> 00:20:11,770
+العبارة بي وهذا بنسميها ال epsilon neighborhood
+
+207
+00:20:11,770 --> 00:20:16,150
+definition لل convergenceهذا بقى بنسميه epsilon
+
+208
+00:20:16,150 --> 00:20:20,210
+neighborhood definition of convergence ليه؟
+
+209
+00:20:20,210 --> 00:20:22,850
+العبارة دي بتقول لكل for every epsilon
+
+210
+00:20:22,850 --> 00:20:27,930
+neighborhood V epsilon ل X يعني بدل لكل
+
+211
+00:20:27,930 --> 00:20:32,550
+neighborhood بدلناها لكل epsilon neighborhood ل X
+
+212
+00:20:32,550 --> 00:20:35,630
+يوجد capital N يعتمد على ال epsilon neighborhood
+
+213
+00:20:35,630 --> 00:20:42,160
+وبالتالي يعتمد على ال epsilon عدد طبيعيبحيث أنه
+
+214
+00:20:42,160 --> 00:20:46,200
+لكل N أكبر من أوسعه capital N بطلع XN ينتمي لبي
+
+215
+00:20:46,200 --> 00:20:52,820
+نفس العادلالعبارة التالتة بتقول لكل إبسلون لأي عدد
+
+216
+00:20:52,820 --> 00:20:56,260
+إبسلون موجة بنقدر نلاقي عدد طبيعي يعتمد على إبسلون
+
+217
+00:20:56,260 --> 00:21:01,500
+بحيث لو كان n أكبر من أو ساوي capital N فالمسافة
+
+218
+00:21:01,500 --> 00:21:07,800
+بين x and x تطلع أصغر من إبسلون هذا بنسميه الجزء C
+
+219
+00:21:07,800 --> 00:21:13,180
+وهذا الجزء الأكتر جزء هنستخدمه في إثبات ال
+
+220
+00:21:13,180 --> 00:21:18,080
+convergence لsequences معينةهذا بيسميه epsilon
+
+221
+00:21:18,080 --> 00:21:25,600
+capital N definition of
+
+222
+00:21:25,600 --> 00:21:26,500
+convergence
+
+223
+00:21:30,350 --> 00:21:34,970
+انا في عندى انا الفرق A هذا عبارة عن epsilon عبارة
+
+224
+00:21:34,970 --> 00:21:38,530
+عن neighborhood definition of convergence الفرق B
+
+225
+00:21:38,530 --> 00:21:42,230
+بنسميه ال epsilon neighborhood definition لل
+
+226
+00:21:42,230 --> 00:21:46,210
+convergence الفرق C بنسميه epsilon capital N
+
+227
+00:21:46,210 --> 00:21:49,770
+definition of convergence هذا هيكون استعماله شائع
+
+228
+00:21:49,770 --> 00:21:57,370
+اكتر من العبارات السابقةالبرهان ان هذا ال تلاتة
+
+229
+00:21:57,370 --> 00:22:02,490
+إبراهيم بتكافئ بعض هنثبت ان a implies b و b
+
+230
+00:22:02,490 --> 00:22:10,610
+implies c و بعد هيك هنثبت ان c implies a وبالتالي
+
+231
+00:22:10,610 --> 00:22:14,370
+هيك بيطلع التلاتة متكافئة حسب قوانين ال logic
+
+232
+00:22:14,370 --> 00:22:21,830
+مظبوط صح؟طيب نشوف الأول a implies b افرض ان x in
+
+233
+00:22:21,830 --> 00:22:28,010
+converge ل x يعني هذا الكلام صحيح حسب تعريف ال
+
+234
+00:22:28,010 --> 00:22:34,510
+neighborhood definition لل convergence طيب .. طيب
+
+235
+00:22:34,510 --> 00:22:39,150
+احنا عارفين ان كل epsilon .. طيب لإثبات ان b صحيح
+
+236
+00:22:39,150 --> 00:22:45,130
+ناخد أي epsilon neighborhood ل xطب احنا لما درسنا
+
+237
+00:22:45,130 --> 00:22:48,990
+ال neighborhoods قلنا ان كل epsilon neighborhood
+
+238
+00:22:48,990 --> 00:22:52,130
+.. every epsilon neighborhood على الصورة هذه ل X
+
+239
+00:22:52,130 --> 00:22:57,490
+هو ايضا neighborhood ل X صح؟ هذه حقيقة معروفة ..
+
+240
+00:22:57,490 --> 00:23:02,570
+كل epsilon neighborhood ل X is also a neighborhood
+
+241
+00:23:02,570 --> 00:23:09,280
+of Xوبالتالي إذا هنا لو أخدت أي إبسلون
+
+242
+00:23:09,280 --> 00:23:13,140
+neighborhood ل X فهذا neighborhood ل X وبالتالي
+
+243
+00:23:13,140 --> 00:23:15,820
+يوجد capital N يعتمد على الإبسلون neighborhood
+
+244
+00:23:15,820 --> 00:23:24,080
+وهذا الكلام صح وبالتالي A بيؤدي ل B نشوف
+
+245
+00:23:24,080 --> 00:23:27,460
+الآن بيؤدي العبارة بيؤدي إلى C
+
+246
+00:23:42,950 --> 00:23:55,970
+طيب العبارة P هذا هي لو كان P صحيح فبنثبت
+
+247
+00:23:55,970 --> 00:24:05,490
+ان C صحيح فخلينا ناخد خلينا
+
+248
+00:24:05,490 --> 00:24:09,250
+ناخد أبسلون أكبر من السفر ناخد أبسلون أكبر من
+
+249
+00:24:09,250 --> 00:24:09,730
+السفر
+
+250
+00:24:13,900 --> 00:24:22,140
+لو أخدت أي epsilon أكبر من السفر for any epsilon
+
+251
+00:24:22,140 --> 00:24:30,140
+أكبر من السفر take v epsilon of x اللي هو عبارة عن
+
+252
+00:24:30,140 --> 00:24:36,040
+ال epsilon neighborhood ل x فهذا
+
+253
+00:24:36,040 --> 00:24:44,530
+is epsilon neighborhood of x صح؟وبالتالي حسب بي
+
+254
+00:24:44,530 --> 00:24:50,890
+لأي إبسلون neighborhood لهذا يوجد capital N إذا
+
+255
+00:24:50,890 --> 00:24:56,350
+يوجد capital N by
+
+256
+00:24:56,350 --> 00:25:02,930
+بي يوجد capital N يعتمد على الإبسلون neighborhood
+
+257
+00:25:02,930 --> 00:25:09,630
+وبالتالي يعتمد على إبسلون هذا عدد طبيعي بحيث
+
+258
+00:25:13,530 --> 00:25:19,590
+بحيث انه لو كان n أكبر من أو ساوي n of epsilon
+
+259
+00:25:19,590 --> 00:25:28,030
+فهذا بيقدي ان xn ينتمي ل v epsilon ل x اللي هو x
+
+260
+00:25:28,030 --> 00:25:35,630
+سالب epsilon وx موجة بepsilon طب وهذا معناه ان ال
+
+261
+00:25:35,630 --> 00:25:44,930
+xn أكبر من x سالب epsilon أصغر من x زاد epsilonهذا
+
+262
+00:25:44,930 --> 00:25:50,630
+الـ xn ينتمي للفترة المفتوحة هذه معناته هذا الكلام
+
+263
+00:25:50,630 --> 00:25:56,670
+صح هذا معناه xn minus x أصغر من epsilon أكبر من
+
+264
+00:25:56,670 --> 00:26:01,950
+سالب epsilon هذا معناه absolute xn minus x أصغر من
+
+265
+00:26:01,950 --> 00:26:10,800
+epsilon إذن هين أثبتنا إن لو كان b صحيحفلأي يبسلون
+
+266
+00:26:10,800 --> 00:26:18,300
+أكبر من السفر يوجد capital N يعتمد على يبسلون بحيث
+
+267
+00:26:18,300 --> 00:26:23,160
+لكل N أكبر من أو ساوي capital N طلع absolute xn
+
+268
+00:26:23,160 --> 00:26:29,920
+minus x أصغر من يبسلون وبالتالي العبارة C صحيحة
+
+269
+00:26:29,920 --> 00:26:38,500
+متحققة okay تمام؟ الآن بقى نثبت أن العبارة
+
+270
+00:26:38,500 --> 00:26:59,280
+Cبتقدي إلى العبارة A فأفرضي
+
+271
+00:26:59,280 --> 00:27:08,370
+أن العبارة C متحققة suppose C holdsبعدين، بدنا
+
+272
+00:27:08,370 --> 00:27:12,250
+نثبت أن x in converge ل x أو ال neighborhood
+
+273
+00:27:12,250 --> 00:27:17,730
+definition ل x بتحقق فبناخد أي let v be any
+
+274
+00:27:17,730 --> 00:27:24,590
+neighborhood of x فمن تعريف ال neighborhoodلأي
+
+275
+00:27:24,590 --> 00:27:28,910
+neighborhood كل neighborhood v ل x يحتوي داخله
+
+276
+00:27:28,910 --> 00:27:32,030
+epsilon neighborhood ل x هذا ما قلناه قبل هيك
+
+277
+00:27:32,030 --> 00:27:37,430
+وبالتالي يوجد epsilon عدد موجب بحيث ان ال epsilon
+
+278
+00:27:37,430 --> 00:27:44,890
+neighborhood هذه الفترة عبارة عن x in .. هذه
+
+279
+00:27:44,890 --> 00:27:51,090
+المفروضة تكون عفوا هذه المفروضة تكون x مش x in
+
+280
+00:27:51,090 --> 00:28:01,600
+وهذه x سلب epsilonهذا عبارة عن v epsilon ل x هذا
+
+281
+00:28:01,600 --> 00:28:08,880
+المفروض تكون x مش xm إذا لو كان v epsilon
+
+282
+00:28:08,880 --> 00:28:15,740
+neighborhood ففي عندي بقدر ألاقي جواته epsilon
+
+283
+00:28:15,740 --> 00:28:20,520
+neighborhood لل x اللي هو v epsilon ل x الآن من
+
+284
+00:28:20,520 --> 00:28:21,400
+الجزء c
+
+285
+00:28:25,470 --> 00:28:29,650
+لأي أبسلون من الجزء C لأي أبسلون لأ بما أن هذا
+
+286
+00:28:29,650 --> 00:28:33,170
+أبسلون أكبر من السفر إذا بنقدر نلاقي capital N
+
+287
+00:28:33,170 --> 00:28:36,310
+يعتمد على أبسلون بحيث لكل N أكبر من أو ساوية
+
+288
+00:28:36,310 --> 00:28:40,230
+capital N ال absolute value هذه أصغر من أبسلون هذا
+
+289
+00:28:40,230 --> 00:28:45,660
+من الجزء Cطب ما هذا معناه ال implication هذه
+
+290
+00:28:45,660 --> 00:28:50,920
+معناها لكل n أكبر من أو ساوي capital N لو فكيت
+
+291
+00:28:50,920 --> 00:28:58,800
+المتباينة هذه معناها xn ينتمي هذا عبارة عن x ينتمي
+
+292
+00:28:58,800 --> 00:29:06,480
+لفترة مفتوحة x minus y و x z epsilon اللي هو ال
+
+293
+00:29:06,480 --> 00:29:09,720
+epsilon neighborhood ل x اللي هو subset من V
+
+294
+00:29:11,670 --> 00:29:19,650
+وبالتالي هيك بنكون أثبتنا أن ال XIN ينتمي إلى ال
+
+295
+00:29:19,650 --> 00:29:24,530
+neighborhood V كمان
+
+296
+00:29:24,530 --> 00:29:30,830
+مرة أنا بدي أثبت أن العبارة C بتأدي ليه، افرض أن
+
+297
+00:29:30,830 --> 00:29:36,610
+العبارة C صحيحةالان لإثبات a اللى هى x in converge
+
+298
+00:29:36,610 --> 00:29:40,790
+ل x بتثبت أنه ال neighborhood definition لل
+
+299
+00:29:40,790 --> 00:29:45,750
+convergence بتحقق يعنى x عبارة عن limit لل
+
+300
+00:29:45,750 --> 00:29:48,650
+sequence x in فنرجع لتعريف ال neighborhood
+
+301
+00:29:48,650 --> 00:29:53,190
+definition of convergence نبدأ ب neighborhood ل x
+
+302
+00:29:53,190 --> 00:29:57,910
+ونستخدم الحقيقة أن كل neighborhood ل x يحتوي
+
+303
+00:29:57,910 --> 00:30:04,160
+epsilon neighborhoodالان من C .. C بيقول لي إذا في
+
+304
+00:30:04,160 --> 00:30:08,400
+عندك إبسلون موجبة تقدر تلاقي capital N يعتمد عليها
+
+305
+00:30:08,400 --> 00:30:12,940
+بحيث أنه لكل N أكبر من ما يساوي capital N المسافة
+
+306
+00:30:12,940 --> 00:30:17,660
+هذه أصغر من إبسلونطب هذه ال implication الأخيرة هي
+
+307
+00:30:17,660 --> 00:30:22,380
+N أكبر من أو ساوي capital N بتقدي في حل المتباين
+
+308
+00:30:22,380 --> 00:30:28,640
+هذه في Xn فبطلع Xn ينتمي إلى X سالب Y و X فاللي هو
+
+309
+00:30:28,640 --> 00:30:33,320
+هذا ال epsilon neighborhood اللي هوداخل V وبالتالي
+
+310
+00:30:33,320 --> 00:30:37,660
+لكل N أكبر من لو ساوي capital N طلع Xn ينتمي لل
+
+311
+00:30:37,660 --> 00:30:42,300
+neighborhood V هذا من التعريف معناه Xn converge ل
+
+312
+00:30:42,300 --> 00:30:48,820
+X وبالتالي اللي عبارة أيه صحيحة تمام؟ إذا هيك
+
+313
+00:30:48,820 --> 00:30:53,940
+بنكون أثبتنا النظرية أن التلت تعريفات هذه كلها
+
+314
+00:30:53,940 --> 00:30:54,840
+متكافئة
+
+315
+00:31:02,750 --> 00:31:06,990
+في تعريف الـ tail of a sequence او الـ M tail of a
+
+316
+00:31:06,990 --> 00:31:11,070
+sequence احنا عارفين ان لو في اندز اي .. لأي
+
+317
+00:31:11,070 --> 00:31:18,570
+sequence XN لو خدت M عدد طبيعي اي عدد طبيعي
+
+318
+00:31:18,570 --> 00:31:24,210
+natural number و XN اي sequence of real numbers
+
+319
+00:31:24,210 --> 00:31:31,330
+فالـ XN هذه ممكن انفرفتها نكتب حدودها X1 X2 و هكذا
+
+320
+00:31:32,450 --> 00:31:41,130
+الى x رقم m الان الحد اللي بعد xm عبارة عن xm زاد
+
+321
+00:31:41,130 --> 00:31:50,010
+واحد و اللي بعده xm زاد اتنين و هكذا اذا
+
+322
+00:31:50,010 --> 00:31:53,130
+ال sequence هذه ممكن اكتبها على الصورة هذه حيث م
+
+323
+00:31:53,130 --> 00:31:57,770
+هنا عدد طبيعي ما ثابت
+
+324
+00:31:59,680 --> 00:32:10,460
+الان لو انا ركزت على الجزء هذا من ال sequence و
+
+325
+00:32:10,460 --> 00:32:20,440
+الجزء هذا هو اول m من حدود ال sequence حذفتها فاذا
+
+326
+00:32:20,440 --> 00:32:22,400
+هذا بنسميه m tail
+
+327
+00:32:28,870 --> 00:32:37,630
+متل لسيكوينس xn الدنب م دنب م مش هذا دنب يعني تصور
+
+328
+00:32:37,630 --> 00:32:42,110
+إنها دي أفع هي الرأس تبعها أول م من الحدود ده هي
+
+329
+00:32:42,110 --> 00:32:47,570
+الرأس جاطعة الرأس تبعها فبقى الدنب مش هيك بيقولوا
+
+330
+00:32:47,570 --> 00:32:50,870
+الدنب
+
+331
+00:32:50,870 --> 00:32:56,090
+هذا طويلبنبدأ يعني في عدد لانها من الحدود الراس
+
+332
+00:32:56,090 --> 00:33:02,470
+محدود هي عدد منتهي من الحدود اذا ال sequence لو
+
+333
+00:33:02,470 --> 00:33:08,250
+انا حدفت اول M من حدودها فباقي الجزء المتبقى من ال
+
+334
+00:33:08,250 --> 00:33:16,070
+sequence بنسميه M tail واضح طيب اذا الان في نظرية
+
+335
+00:33:16,070 --> 00:33:18,250
+اتنين تلاتة او نظرية تالتة
+
+336
+00:33:20,720 --> 00:33:23,800
+ما هي هذه النظرية اللي بتقول؟ بتقول لو أنا في اندي
+
+337
+00:33:23,800 --> 00:33:29,500
+إذا هاي ال m tail هذا ال m tail ممكن كتابته على
+
+338
+00:33:29,500 --> 00:33:35,820
+صورة sequence هاي x المؤشر الحد العام تبع ال m
+
+339
+00:33:35,820 --> 00:33:40,660
+tail m زاد n حيث و اين العداد الطبيعي m ثابت و n
+
+340
+00:33:40,660 --> 00:33:43,980
+العداد الطبيعي وبالتالي هنا لو كانت n بالساوية
+
+341
+00:33:43,980 --> 00:33:50,800
+واحد اول حد xm زاد واحد و هكذا طيبالان النظرية
+
+342
+00:33:50,800 --> 00:33:57,980
+التالية بتقولني انه لو كان ال M tail convergent
+
+343
+00:34:02,380 --> 00:34:07,760
+فال sequence نفسها ال M بتكون convergent و العكس
+
+344
+00:34:07,760 --> 00:34:12,020
+لو كانت ال sequence convergent فأي M tail منها
+
+345
+00:34:12,020 --> 00:34:15,940
+هيكون convergent و اتنين لهم نفس ال limit اتنين
+
+346
+00:34:15,940 --> 00:34:20,020
+لهم نفس ال limit اذا مرة تانية لو كان في عندك
+
+347
+00:34:20,020 --> 00:34:27,500
+sequence XN M fixed natural number فال M tail اللي
+
+348
+00:34:27,500 --> 00:34:32,350
+هو ال sequence هذهconverges if and only if
+
+349
+00:34:32,350 --> 00:34:39,210
+الsequence نفسها converges وهي البرهان هذا ال part
+
+350
+00:34:39,210 --> 00:34:43,750
+f افرضي
+
+351
+00:34:43,750 --> 00:34:48,290
+ان x in convergent نثبت ان ال m ت ال convergent
+
+352
+00:34:48,290 --> 00:34:54,540
+ماشي الحال طيب اذا كانت x in convergent ل xيعني ال
+
+353
+00:34:54,540 --> 00:34:57,620
+limit تبعتها إذا كانت convergent فلازم يكون لها
+
+354
+00:34:57,620 --> 00:35:02,020
+limit فأفرض إن ال limit تبعتها xالأن حسب epsilon
+
+355
+00:35:02,020 --> 00:35:06,080
+capital N definition لل limit أو لل convergence
+
+356
+00:35:06,080 --> 00:35:11,140
+إذا لأي epsilon أكبر من 0 نقدر نلاقي N يعتمد على
+
+357
+00:35:11,140 --> 00:35:15,860
+epsilon عدد طبيعي كبير و ممكن ناخده يكون أكبر من
+
+358
+00:35:15,860 --> 00:35:22,040
+العدد الثابت العدد الطبيعي ثابت M بحيث أنه لكل N
+
+359
+00:35:22,040 --> 00:35:25,900
+أكبر من أو ساوي capital N المسافة بين X و N هو X
+
+360
+00:35:25,900 --> 00:35:31,410
+أصغر من Yهذا من تعريف الـ epsilon capital N
+
+361
+00:35:31,410 --> 00:35:37,590
+definition لل convergence طيب اللي انا بقدر اعرف
+
+362
+00:35:37,590 --> 00:35:43,930
+capital N prime على انه capital N مطروح منها
+
+363
+00:35:43,930 --> 00:35:50,060
+capital Mطبعا هنا capital N احنا اختارناها اكبر من
+
+364
+00:35:50,060 --> 00:35:54,220
+M فالفرق هذا موجب وهذا عدد طبيعي وهذا عدد طبيعي
+
+365
+00:35:54,220 --> 00:35:59,500
+اذا الفرق عدد صحيح موجب يعني عدد طبيعي هذا عدد
+
+366
+00:35:59,500 --> 00:36:03,220
+ثابت وهذا يعتمد على epsilon اذا N prime الفرق
+
+367
+00:36:03,220 --> 00:36:09,000
+بينهم يعتمد على epsilon تمام؟إذا هنا عرفنا N' عدد
+
+368
+00:36:09,000 --> 00:36:14,320
+طبيعي ويعتمد على epsilon الان لو أخدت اي M عدد
+
+369
+00:36:14,320 --> 00:36:16,960
+طبيعي أكبر من أو ساوي N'
+
+370
+00:36:20,020 --> 00:36:25,520
+فنجمع capital M للطرفين فبطلع capital M زاد small
+
+371
+00:36:25,520 --> 00:36:29,980
+m أكبر من أو ساوي N prime زاد capital M طب N prime
+
+372
+00:36:29,980 --> 00:36:34,540
+زاد capital M بساوي N إبسلون وبالتالي هذا أكبر من
+
+373
+00:36:34,540 --> 00:36:40,860
+أو ساوي N لإبسلون إذا حسب ال implication واحدالـ
+
+374
+00:36:40,860 --> 00:36:45,260
+implication واحد بتقوللي لأي عدد طبيعي .. لأي عدد
+
+375
+00:36:45,260 --> 00:36:50,560
+طبيعي أكبر من أو ساوي capital N لازم يطلع ال
+
+376
+00:36:50,560 --> 00:36:56,900
+absolute value ل X sub العدد الطبيعي اللي هو M زاد
+
+377
+00:36:56,900 --> 00:36:59,320
+M minus X أصغر من epsilon
+
+378
+00:37:03,110 --> 00:37:08,470
+وهذا بيدّي أن ال tail .. ال tail of the sequence
+
+379
+00:37:08,470 --> 00:37:13,110
+converge ل X حسب التعريف ما معناه أن ال tail هذا
+
+380
+00:37:13,110 --> 00:37:18,470
+convergent؟ معناه أن لأي epsilon أكبر من الصفر ..
+
+381
+00:37:18,470 --> 00:37:25,050
+لأي epsilon أكبر من الصفر هيوجد N prime .. هيوجد N
+
+382
+00:37:25,050 --> 00:37:29,130
+prime عدد طبيعي يعتمد على epsilon
+
+383
+00:37:31,850 --> 00:37:38,290
+يوجد عدد طبيعي N' يعتمد على إبسلون بحيث لكل M أكبر
+
+384
+00:37:38,290 --> 00:37:44,350
+من أو يساوي N' طلع المسافة بين الحد رقم capital M
+
+385
+00:37:44,350 --> 00:37:47,690
+زاد small m minus X أصغر من إبسلون هذا بالضبط
+
+386
+00:37:47,690 --> 00:37:53,310
+معناه إن ال sequence هذه converge ل X as M tends
+
+387
+00:37:53,310 --> 00:37:59,580
+to infinityإذاً هيك بنكون أثبتنا إنه لو كانت ال
+
+388
+00:37:59,580 --> 00:38:03,240
+sequence x in converge ل x فالتالت تبعها converge
+
+389
+00:38:03,240 --> 00:38:10,720
+ل x okay تمام العكس العكس يعني ضايق ممكن يعني
+
+390
+00:38:10,720 --> 00:38:20,220
+نبرهن العكس في دقيقة او دقيقتين العكس
+
+391
+00:38:20,220 --> 00:38:26,390
+يعني هذا العكس اللي هو ال only if partنفرض المرة
+
+392
+00:38:26,390 --> 00:38:30,450
+هذه أن الـ sequence الـ tail of a sequence الـ
+
+393
+00:38:30,450 --> 00:38:34,770
+tail of the sequence converged ل X وبينما نثبت أن
+
+394
+00:38:34,770 --> 00:38:40,170
+الـ sequence نفسها convergent ل X برضه فنستخدم
+
+395
+00:38:40,170 --> 00:38:42,930
+تعريف epsilon capital N definition للconvergence
+
+396
+00:38:42,930 --> 00:38:48,710
+اللي هو الجزء C من نظرية 2 2 فناخد given epsilon
+
+397
+00:38:48,710 --> 00:38:53,080
+أو let epsilon أكبر من الصفر بيه givenبما أن الـ
+
+398
+00:38:53,080 --> 00:38:56,560
+sequence هذه converge ل X إذا يوجد capital N يعتمد
+
+399
+00:38:56,560 --> 00:39:00,740
+على إبسلون بحيث لكل N أكبر من أو ساوي capital N
+
+400
+00:39:00,740 --> 00:39:04,560
+المسافة بين الحد العام للـ sequence هذه و X أصغر
+
+401
+00:39:04,560 --> 00:39:12,790
+من إبسلونالان بنعرف capital K على انه العدد
+
+402
+00:39:12,790 --> 00:39:18,250
+الطبيعي الثابت M زاد العدد الطبيعي capital N فطبعا
+
+403
+00:39:18,250 --> 00:39:22,490
+مجموعة دين الطبيعيين عدد طبيعي capital N يعتمد على
+
+404
+00:39:22,490 --> 00:39:26,670
+epsilon اذا المجموعة تبعهم بيطلع يعتمد على epsilon
+
+405
+00:39:26,670 --> 00:39:32,330
+اذا هنا انا وجدت او جدت او عرفت عدد طبيعي capital
+
+406
+00:39:32,330 --> 00:39:37,610
+K يعتمد على epsilonالان لو أخدت اي N أكبر من أو
+
+407
+00:39:37,610 --> 00:39:43,170
+ساوي ال capital A فاترحي .. اترحي N من هنا و اترحي
+
+408
+00:39:43,170 --> 00:39:50,350
+N من هنا M عفوا Mلو طرحنا M من الطرفين المتباينة
+
+409
+00:39:50,350 --> 00:39:55,330
+هذه فبطلع N negative capital M أكبر من أو ساوي K
+
+410
+00:39:55,330 --> 00:40:01,170
+minus M طب هاي K اطرحي منها M بساوي N وبالتالي
+
+411
+00:40:01,170 --> 00:40:05,950
+بطلع N سالب M أكبر من أو ساوي N الآن من ال
+
+412
+00:40:05,950 --> 00:40:11,550
+implication اتنين ال implication اتنين بتقول لأي N
+
+413
+00:40:11,550 --> 00:40:15,650
+أكبر من أو ساوي capital اي عدد طبيعيلو كان العدد
+
+414
+00:40:15,650 --> 00:40:20,950
+الطبيعي هذا أكبر من أو ساوي capital N فالمسافة بين
+
+415
+00:40:20,950 --> 00:40:27,390
+X للعدد الطبيعي واضيف عليه M إذا بدي أضيف على هذا
+
+416
+00:40:27,390 --> 00:40:32,230
+M المسافة بين X اللي المؤشر تبعها العدد الطبيعي
+
+417
+00:40:32,230 --> 00:40:37,770
+هذا زائد M اللي هو بيطلع N والمسافة بينه بين X
+
+418
+00:40:37,770 --> 00:40:42,770
+بيطلع أصغر من Epsilonإذاً هيك احنا أثبتنا أنه لأي
+
+419
+00:40:42,770 --> 00:40:46,970
+إبسلون أكبر من الصفر يوجد capital N يعتمد على
+
+420
+00:40:46,970 --> 00:40:53,790
+إبسلون بحيث أنه أو يوجد capital K لأي إبسلون أكبر
+
+421
+00:40:53,790 --> 00:40:57,570
+من الصفر يوجد عدد طبيعي capital K يعتمد على إبسلون
+
+422
+00:40:57,570 --> 00:41:06,250
+بحيث أنه لكل N أكبر من أو يساوي capital Kلكل n
+
+423
+00:41:06,250 --> 00:41:10,590
+أكبر من أو ساوي كابتل K طلع المسافة بين xn و x
+
+424
+00:41:10,590 --> 00:41:15,370
+أصغر من إبسل إذن هذا بالضبط معناه أن ال sequence
+
+425
+00:41:15,370 --> 00:41:22,590
+xn converge ل x زي ما هو مطلوب وهذا بكمل برهان
+
+426
+00:41:22,590 --> 00:41:26,410
+النظرية okay تمام واضح
+
+427
+00:41:31,150 --> 00:41:37,130
+طيب احنا بنكتفي بهذا القدر و ان شاء الله في
+
+428
+00:41:37,130 --> 00:41:42,010
+المحاضرة القادمة هناخد برضه بعض النظريات و ناخد
+
+429
+00:41:42,010 --> 00:41:46,350
+أمثلة كيف نثبت ان ال limit ل sequence ل
+
+430
+00:41:46,350 --> 00:41:51,090
+convergence sequence بالساوي عدد معين و هكذا طبعا
+
+431
+00:41:51,090 --> 00:41:54,130
+كل الأجزاء هذه موجودة عندكم ممكن تقرؤوها و تحضروها
+
+432
+00:41:54,130 --> 00:41:56,010
+للمحاضرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CRzAwh3Ypto.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CRzAwh3Ypto.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d8a186e596504b72834e803cc16d6604dc73d499
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CRzAwh3Ypto.srt
@@ -0,0 +1,1703 @@
+1
+00:00:21,580 --> 00:00:26,600
+بسم الله الرحمن الرحيم إن شاء الله اليوم هناخد 
+
+2
+00:00:26,600 --> 00:00:31,760
+section خمسة اتنين اللي عنوانه combination of
+
+3
+00:00:31,760 --> 00:00:38,560
+continuous functions قبل ما ناخد أول نظرية عن الـ
+
+4
+00:00:38,560 --> 00:00:41,860
+combination of continuous functions نستذكر أو 
+
+5
+00:00:41,860 --> 00:00:45,300
+نسترجع مع بعض تعريف الـ continuous  الـ continuity
+
+6
+00:00:45,300 --> 00:00:49,300
+عند نقطة ف a function f from a to r is continuous
+
+7
+00:00:49,300 --> 00:00:55,620
+at c نقطة c تنتمي لـ a f and only f لكل إبسلون في
+
+8
+00:00:55,620 --> 00:00:59,740
+دلتا تعتمد على إبسلون عدد موجب بهات لكل x في a
+
+9
+00:01:00,390 --> 00:01:03,710
+المسافة بينها وبين الـC أصغر من دلتا لازم هذا
+
+10
+00:01:03,710 --> 00:01:08,610
+يتضمن أن absolute F of X minus F of C أصغر من
+
+11
+00:01:08,610 --> 00:01:13,270
+إبسلون طبعا شوفنا أن هذا التعريف بيكافئ التعريف
+
+12
+00:01:13,270 --> 00:01:18,970
+اللي أخدناه في calculus A هو الشرط
+
+13
+00:01:18,970 --> 00:01:23,980
+اللي هو بيتألف من تلت شروط وهو أن limit f عن c تكون
+
+14
+00:01:23,980 --> 00:01:30,900
+موجودة و f عن c موجودة و الاثنين بسوء نفس القيمة
+
+15
+00:01:30,900 --> 00:01:43,420
+الآن لو في عندي تلت دوال f و g و h بيه functions
+
+16
+00:01:43,420 --> 00:01:48,700
+from a to r بيه 
+
+17
+00:01:48,700 --> 00:01:49,460
+functions
+
+18
+00:01:54,460 --> 00:02:06,860
+و c تنتمي إلى a و b real number الـ
+
+19
+00:02:06,860 --> 00:02:17,440
+functions
+
+20
+00:02:17,440 --> 00:02:23,820
+are continuous at c
+
+21
+00:02:28,450 --> 00:02:34,350
+إذا الدوال الثلاث F وG وH كلهم متصلين عند النقطة
+
+22
+00:02:34,350 --> 00:02:44,150
+C اللي بتنتمي إليها النتيجة F plus أو minus G F
+
+23
+00:02:44,150 --> 00:02:53,630
+ضرب G B ضرب F are continuous at C
+
+24
+00:02:55,230 --> 00:03:11,750
+B إذا كان H H of X لا تساوي صفر لكل X في A then F
+
+25
+00:03:11,750 --> 00:03:19,710
+على H الدالة F على H is continuous is continuous
+
+26
+00:03:19,710 --> 00:03:20,950
+at C
+
+27
+00:03:25,450 --> 00:03:38,190
+وهي البرهان proof to
+
+28
+00:03:38,190 --> 00:03:48,530
+show مثلا الـ function fg is continuous at c
+
+29
+00:03:51,910 --> 00:04:02,370
+We have لدينا التالي بتثبت 
+
+30
+00:04:02,370 --> 00:04:09,010
+أن الـ F حاصل ضرب الدالتين F و G متصل حاصل ضرب متصل
+
+31
+00:04:09,010 --> 00:04:14,990
+and C فالاثبات دالك بتثبت ان الشرط هذا تبع الاتصال
+
+32
+00:04:14,990 --> 00:04:23,830
+على النقطة بتحقق فتعالى نشوف high limit F ضرب G عند
+
+33
+00:04:23,830 --> 00:04:33,190
+X لما X تقول لـC بنثبت أن هذا بيساوي FG عند C فهذا 
+
+34
+00:04:33,190 --> 00:04:42,190
+بيساوي limit F of X ضرب G of X as X tends to C هذا 
+
+35
+00:04:42,190 --> 00:04:48,610
+من تعريف حاصل ضرب اختراعين وهذا بيساوي أنا عندي
+
+36
+00:04:48,610 --> 00:04:56,150
+limit F of X لما X تقول لـC existو limit الـ
+
+37
+00:04:56,150 --> 00:05:02,250
+function g of x لما x تقول ل c exist لأن الـ
+
+38
+00:05:02,250 --> 00:05:05,110
+function f continuous عند الـ c و الـ function g 
+
+39
+00:05:05,110 --> 00:05:11,250
+احنا فرضينها continuous عند c مش هيكو بس ومن اتصال
+
+40
+00:05:11,250 --> 00:05:17,410
+ده ل F عن C الـ limit هذه بيساوي قيمة F عن C وكذلك 
+
+41
+00:05:17,410 --> 00:05:20,810
+من اتصال الـ function G عن C الـ limit هذه بتطلع 
+
+42
+00:05:20,810 --> 00:05:30,610
+بيساوي G عن C وهذا بيساوي F ضرب G of C إذن هاي
+
+43
+00:05:30,610 --> 00:05:36,480
+الشرط تبع الاتصال عن نقطة متحقق للـ function f ضارب
+
+44
+00:05:36,480 --> 00:05:42,720
+g وبالتالي therefore by definition الـ function f g
+
+45
+00:05:42,720 --> 00:05:59,940
+is continuous at c تمام الـ proof الـ proof of the
+
+46
+00:05:59,940 --> 00:06:00,580
+other
+
+47
+00:06:05,540 --> 00:06:14,200
+parts is similar مشابه للبرهان اللي احنا لسه 
+
+48
+00:06:14,200 --> 00:06:19,180
+ماخدينه يعني لإثبات أن مثلا مجموعة دالتين
+
+49
+00:06:19,180 --> 00:06:24,660
+continuous برضه ممكن إثبات أن limit f زائد g لما x
+
+50
+00:06:24,660 --> 00:06:31,220
+تقول ل c بساوي f زائد g and c لو بدنا نثبت ان limit 
+
+51
+00:06:31,220 --> 00:06:39,480
+f على g او f على h continuous عن c فبناخد limit f
+
+52
+00:06:39,480 --> 00:06:47,420
+على h عن c وهذا بيطلع بساوي limit f of x على h of
+
+53
+00:06:47,420 --> 00:06:53,810
+x ومع أن limit المقامه لا يساوي صفر لأن H ب X لا
+
+54
+00:06:53,810 --> 00:07:00,210
+يساوي صفر لكل X في A فممكن نوزع الـ limit نقول
+
+55
+00:07:00,210 --> 00:07:02,910
+limit البسط يساوي limit البسط على limit 
+
+56
+00:07:02,910 --> 00:07:06,770
+المقام و limit البسط بيساوي F عن C لأن F
+
+57
+00:07:06,770 --> 00:07:13,070
+continuous عن C و limit المقام عن C اللي هو H عن C 
+
+58
+00:07:13,070 --> 00:07:15,950
+بيساوي قيمة الدالة H عن C لإن أنا متصل عن C
+
+59
+00:07:16,620 --> 00:07:21,880
+وبالتالي بيطلع limit f على h لما x تقول ل c بس هو
+
+60
+00:07:21,880 --> 00:07:27,660
+قيمة الدالة f على h and c okay إذا البرهين
+
+61
+00:07:27,660 --> 00:07:34,860
+المتبقية ممكن يعني أعطاها بنفس الطريقة okay تمام
+
+62
+00:07:34,860 --> 00:07:38,720
+النظرية 
+
+63
+00:07:38,720 --> 00:07:40,640
+هذه ممكن تعميمها 
+
+64
+00:07:43,880 --> 00:07:51,460
+يعني بدل لو كانت الدالة F و G و H متصلين are
+
+65
+00:07:51,460 --> 00:07:56,640
+continuous are
+
+66
+00:07:56,640 --> 00:08:08,380
+continuous على كل المجموعة A على كل المجال على
+
+67
+00:08:08,380 --> 00:08:15,140
+كل المجال A الـ F والـ G والـ H المجال المشترك
+
+68
+00:08:15,140 --> 00:08:18,800
+تبعهم المجموعة A فلو كانت الدوال الثلاث كلهم
+
+69
+00:08:18,800 --> 00:08:30,280
+continuous على كل المجموعة A ف .. then فبتطلع
+
+70
+00:08:30,280 --> 00:08:36,520
+كل الدوال هذه متصلة على كل المجموعة A على كل
+
+71
+00:08:36,520 --> 00:08:51,320
+المجموعة A يعني هذا بيصير on A وهذه on .. on A فلو 
+
+72
+00:08:51,320 --> 00:08:52,380
+بدي أبرهن
+
+73
+00:08:58,870 --> 00:09:03,330
+أي واحدة من الدوال هذه continuous على كل ال A
+
+74
+00:09:03,330 --> 00:09:15,770
+فإيش بعمل بقول fix C تنتمي إلى A and 
+
+75
+00:09:15,770 --> 00:09:21,870
+then by
+
+76
+00:09:21,870 --> 00:09:23,090
+above theorem
+
+77
+00:09:28,740 --> 00:09:35,220
+by above theorem أنا 
+
+78
+00:09:35,220 --> 00:09:40,520
+الآن عندي كل واحدة من الدوال هدول continuous على
+
+79
+00:09:40,520 --> 00:09:45,240
+المجموعة a وبالتالي
+
+80
+00:09:45,240 --> 00:09:47,040
+then
+
+81
+00:09:48,850 --> 00:09:52,490
+بما أنه F و G و H continuous على كل المجموعة A فهي
+
+82
+00:09:52,490 --> 00:09:55,870
+continuous عند أي نقطة مش هيك تعرف الاتصال على
+
+83
+00:09:55,870 --> 00:10:08,510
+مجموعة اذا F و G و H are continuous at C وبالتالي 
+
+84
+00:10:08,510 --> 00:10:10,130
+حسب النظرية السابقة
+
+85
+00:10:19,920 --> 00:10:26,160
+So by above theorem
+
+86
+00:10:26,160 --> 00:10:32,600
+all functions in
+
+87
+00:10:32,600 --> 00:10:36,660
+parts A
+
+88
+00:10:36,660 --> 00:10:45,920
+and B are continuous at C مش هي كثبتنا احنا في 
+
+89
+00:10:45,920 --> 00:10:48,120
+النظرية السابقة هذه اللي جاب ال head اللي انا
+
+90
+00:10:48,120 --> 00:10:52,520
+عدلتهالو كان في عندي تلت دوال و كلهم متصلين عن
+
+91
+00:10:52,520 --> 00:10:57,040
+النقطة فكل الدول الموجودة في الفرق a و الدول
+
+92
+00:10:57,040 --> 00:11:01,300
+الموجودة في الفرق b كلهم بيطلعوا continuous عن نفس
+
+93
+00:11:01,300 --> 00:11:09,660
+النقطة الان بما أن النقطة c was arbitrary since c
+
+94
+00:11:09,660 --> 00:11:17,880
+belonged to a was arbitrary the above
+
+95
+00:11:25,240 --> 00:11:32,060
+All functions in A
+
+96
+00:11:32,060 --> 00:11:37,260
+and B are
+
+97
+00:11:37,260 --> 00:11:39,580
+continuous
+
+98
+00:11:41,100 --> 00:11:46,400
+على كل المجموعة A لأن كل واحدة منهم continuous على 
+
+99
+00:11:46,400 --> 00:11:51,060
+أي و كل نقطة C في A وبالتالي هذا يكون برنامج 
+
+100
+00:11:51,060 --> 00:11:56,540
+النظرية إذا النظرية هذه تنتج مباشرة من نظرية
+
+101
+00:11:56,540 --> 00:12:04,020
+السابقتها وذلك بتثبيت C عنصر في A وطبعا النظرية
+
+102
+00:12:04,020 --> 00:12:08,300
+السابقة بتقول عند أي عنصرC بما أن الثلاث دوال 
+
+103
+00:12:08,300 --> 00:12:12,440
+متصلة إذا كل ال combinations هدولة بطلعوا متصلين 
+
+104
+00:12:12,440 --> 00:12:17,220
+عن نفس النقطة هذا صحيح لأي نقطة ل C وبالتالي كلهم
+
+105
+00:12:17,220 --> 00:12:21,840
+متصلين على كل المجال تبعهم اللي هو المجموعة A
+
+106
+00:12:21,840 --> 00:12:28,040
+تمام ناخد 
+
+107
+00:12:28,040 --> 00:12:29,100
+بعض الأمثلة
+
+108
+00:12:40,050 --> 00:12:46,710
+every polynomial .. every polynomial function على
+
+109
+00:12:46,710 --> 00:12:56,190
+الصورة P of X بيساوي A N في X to N plus A N minus
+
+110
+00:12:56,190 --> 00:13:03,290
+one في X to N minus one زائد .. زائد A one في X
+
+111
+00:13:03,290 --> 00:13:08,330
+زائد A zero is continuous
+
+112
+00:13:10,930 --> 00:13:15,470
+on R proof
+
+113
+00:13:15,470 --> 00:13:20,310
+fix
+
+114
+00:13:20,310 --> 00:13:23,750
+fix
+
+115
+00:13:23,750 --> 00:13:29,150
+C ينتمي لـ R و بدي أثبت أن الـ polynomial function P
+
+116
+00:13:29,150 --> 00:13:36,470
+هذه متصلة عند النقطة C طيب we should أثبتنا في 
+
+117
+00:13:36,470 --> 00:13:45,400
+chapter 4 we should in chapter in chapter four that
+
+118
+00:13:45,400 --> 00:13:48,960
+لو
+
+119
+00:13:48,960 --> 00:13:53,420
+في عندي polynomial P
+
+120
+00:13:53,420 --> 00:13:57,660
+polynomial في X فأثبتنا أن الـ limit للـ polynomial
+
+121
+00:13:57,660 --> 00:14:03,280
+P عند أي real number
+
+122
+00:14:03,280 --> 00:14:11,430
+C بسوء قيمتها عن C therefore حسب تعريف تبع الاتصال
+
+123
+00:14:11,430 --> 00:14:22,830
+النقطة إذا P is continuous at C بما أن الـ C was 
+
+124
+00:14:22,830 --> 00:14:28,510
+arbitrary element
+
+125
+00:14:28,510 --> 00:14:35,610
+إذا P continuous عند كل الـ C في R وبالتالي P is 
+
+126
+00:14:35,610 --> 00:14:43,190
+continuous على كل المجموعة R هنا ال A اللي هي R
+
+127
+00:14:43,190 --> 00:14:49,190
+تمام مثال 
+
+128
+00:14:49,190 --> 00:15:04,390
+ثاني if R بتساوي P على Q P على Q where P 
+
+129
+00:15:04,390 --> 00:15:05,930
+و Q R
+
+130
+00:15:08,300 --> 00:15:19,440
+Polynomials are كثيرات حدود then R is continuous
+
+131
+00:15:19,440 --> 00:15:29,720
+on الست اللي هي R كل الأعداد الحقيقية معدّى أسفار 
+
+132
+00:15:29,720 --> 00:15:36,720
+المقام كل ال X حيث Q of X بتساوي صفر 
+
+133
+00:15:50,720 --> 00:15:56,680
+Proof برضه Fix C 
+
+134
+00:15:56,680 --> 00:16:08,860
+تنتمي الى R معدّى كل ال X حيث Q of X بتساوي صفر 
+
+135
+00:16:08,860 --> 00:16:18,260
+معدّى أسفار الـ function Q إذن Q and C لا يساوي صفر 
+
+136
+00:16:20,990 --> 00:16:30,050
+So by chapter .. By chapter four احنا أثبتنا انه
+
+137
+00:16:30,050 --> 00:16:37,310
+في الحالة هذه الـ limit ل R of X as X tends to C
+
+138
+00:16:37,310 --> 00:16:48,030
+بساوي R of C وبالتالي therefore R is continuous
+
+139
+00:16:50,660 --> 00:16:58,640
+at C ولما كانت الـ C موجودة في R minus أسفار
+
+140
+00:16:58,640 --> 00:17:04,520
+المقام was arbitrarily إذن الـ R continuous على كل
+
+141
+00:17:04,520 --> 00:17:18,080
+الـ sign هذه okay دي الأبارع بنكتبها طيب
+
+142
+00:17:18,080 --> 00:17:19,900
+في الدوال المثلثية
+
+143
+00:17:25,880 --> 00:17:41,480
+في الدوال المثلثية زي الدالة مثلا sin مثال
+
+144
+00:17:41,480 --> 00:17:52,660
+رقم تلاتة f of x بساوي sin x is continuous
+
+145
+00:17:56,130 --> 00:18:07,970
+on R متصلة على جميع الأعداد الحقيقية proof we
+
+146
+00:18:07,970 --> 00:18:08,650
+use
+
+147
+00:18:13,350 --> 00:18:21,010
+هنستخدم الحقائق التالية |sin z| أصغر من أو 
+
+148
+00:18:21,010 --> 00:18:30,290
+ساوي 1 لكل z في R هذا معروف من الرسمة بتاعت ال
+
+149
+00:18:30,290 --> 00:18:33,690
+sin function ال sin function أكبر قيمة لها 
+
+150
+00:18:33,690 --> 00:18:38,190
+maximum value 1 وال absolute minimum -1
+
+151
+00:18:38,190 --> 00:18:43,220
+إذاً قيمها محصورة بينهما، إذن هذه واضحة من الرسم أو من
+
+152
+00:18:43,220 --> 00:18:50,960
+تعريف ال function كذلك في هندسة كمان |
+
+153
+00:18:50,960 --> 00:18:59,040
+sin z| أصغر من أو ساوي |z| for all z في R
+
+154
+00:18:59,040 --> 00:19:02,240
+إذن
+
+155
+00:19:02,240 --> 00:19:08,260
+هذه موجود برهانها في chapter chapter
+
+156
+00:19:08,260 --> 00:19:16,030
+8 الناس اللي هياخدوا تحليل حقيقي 2 هيشوفوا البرهان
+
+157
+00:19:16,030 --> 00:19:20,890
+والناس اللي مش هياخدوا تحليل حقيقي 2 ممكن يقرؤوا
+
+158
+00:19:20,890 --> 00:19:27,890
+البرهان من chapter 8 حتى تعرفوا يعني إيه تتحققوا 
+
+159
+00:19:27,890 --> 00:19:35,870
+أن هذه فعلاً المتباينة الصحيحة كذلك من حساب المثلثات
+
+160
+00:19:35,870 --> 00:19:39,970
+من ال trigonometry اللي درسناها في calculus A أو
+
+161
+00:19:39,970 --> 00:19:45,030
+ما حتى في الثانوية العامة كان في متطابقات مثلثية و
+
+162
+00:19:45,030 --> 00:19:54,690
+من المتطابقات هذه ممكن نستنتج أن sin x - sin
+
+163
+00:19:54,690 --> 00:20:11,220
+c = 2 في sin (½ (x - c)) × cos (½
+
+164
+00:20:11,220 --> 00:20:23,100
+(x + c))
+
+165
+00:20:23,100 --> 00:20:26,200
+في
+
+166
+00:20:26,200 --> 00:20:27,680
+x + c
+
+167
+00:20:37,480 --> 00:20:46,140
+إذن هذه المتطابقة ممكن أثبتها كيف نثبتها sin
+
+168
+00:20:46,140 --> 00:20:51,900
+الفرق x/2 - c/2 sin الفرق = sin
+
+169
+00:20:51,900 --> 00:21:00,860
+cos - cos sin و cos المجموعة =
+
+170
+00:21:00,860 --> 00:21:04,420
+cos cos - sin sin وبعدين نجمعهم و
+
+171
+00:21:04,420 --> 00:21:09,160
+نضربهم وفي 2 فهيطلع في الآخر بتتصف عليه okay
+
+172
+00:21:12,120 --> 00:21:16,040
+بالمناسبة في برضه كمان هندسة مش |sin z|
+
+173
+00:21:16,040 --> 00:21:22,100
+أصغر من أو ساوي 1 وكذلك في هندسة |
+
+174
+00:21:22,100 --> 00:21:28,820
+cos z| برضه أصغر من أو ساوي 1 لكل z في R لأنه
+
+175
+00:21:28,820 --> 00:21:32,260
+برضه ال cos ال | مجزمة منها 1 وال
+
+176
+00:21:32,260 --> 00:21:35,600
+absolute minimum -1 وبالتالي قيمة محصورة
+
+177
+00:21:35,600 --> 00:21:40,020
+بين -1 و 1 الآن خلينا ناخد ال ..
+
+178
+00:21:42,890 --> 00:21:46,090
+من المعادلة الأخيرة
+
+179
+00:21:56,720 --> 00:21:59,960
+من المعادلة الأخيرة بيطلع عندي لو أخدت ال |
+
+180
+00:21:59,960 --> 00:22:05,840
+value للطرفين فبيطلع عندي |sin x - sin
+
+181
+00:22:05,840 --> 00:22:12,700
+c| طبعاً هذا الكلام كله صحيح لكل x و c أعداد حقيقية
+
+182
+00:22:14,590 --> 00:22:20,190
+فهذا بيطلع = أو < أو ≤ 2 في
+
+183
+00:22:20,190 --> 00:22:28,230
+|sin (½(x-c))| |sin (½(x-c))| ≤
+
+184
+00:22:28,230 --> 00:22:35,070
+|½(x-c)| اللي هو ½ في |x - c| ×
+
+185
+00:22:35,070 --> 00:22:41,650
+|cos (½(x+c))| ≤ 1 ≤
+
+186
+00:22:41,650 --> 00:22:52,580
+أو ≤ 1 تمام؟ وهذا صحيح لكل x و c في R طبعاً
+
+187
+00:22:52,580 --> 00:23:00,660
+هذا = |x - c| و
+
+188
+00:23:00,660 --> 00:23:06,260
+من المتباينة هذه بينتج أن ده ل sin متصل عن c okay؟
+
+189
+00:23:06,260 --> 00:23:20,770
+إذاً to show fix c ∈ R to show أن f of x
+
+190
+00:23:20,770 --> 00:23:32,290
+= sin x is continuous at c let ε >
+
+191
+00:23:32,290 --> 00:23:37,050
+الصفر be given it shows
+
+192
+00:23:40,310 --> 00:23:44,950
+δ = ε > الصفر إذاً يوجد δ
+
+193
+00:23:44,950 --> 00:23:51,430
+تعتمد على ε عدد موجب فلهذه ال δ لو كان x
+
+194
+00:23:51,430 --> 00:23:56,950
+∈ R اللي هو مجال الدالة A و |x
+
+195
+00:23:56,950 --> 00:24:04,070
+- c| < δ فهذا بتضمن أنه |f of
+
+196
+00:24:04,070 --> 00:24:15,190
+x - f of c| اللي هو |sin x - sin c| شوفنا
+
+197
+00:24:15,190 --> 00:24:21,870
+هذا ≤ أو < |x - c| من هنا الآن
+
+198
+00:24:21,870 --> 00:24:25,530
+ال x هذه ماخدها أنا بحيث المسافة بينها وبين ال c
+
+199
+00:24:25,530 --> 00:24:30,410
+أصغر من δ وأنا اخترت ال δ = ε
+
+200
+00:24:30,410 --> 00:24:34,810
+عشان يطلع | الفرق بين f of x وf of c| ≤
+
+201
+00:24:34,810 --> 00:24:39,370
+من ε إذاً هاي شرط ε δ لتعريف ال
+
+202
+00:24:39,370 --> 00:24:44,910
+continuity والنقطة المتحققة بما أن ε was
+
+203
+00:24:44,910 --> 00:24:51,090
+arbitrary since ε > الصفر was arbitrary
+
+204
+00:24:51,090 --> 00:24:56,550
+إذاً حسب تعريف ε δ للاتصال بيطلع عندي ال
+
+205
+00:24:56,550 --> 00:25:05,710
+function f of x = sin x is continuous at c
+
+206
+00:25:05,710 --> 00:25:11,130
+وبما أن ال c was arbitrary since
+
+207
+00:25:14,280 --> 00:25:22,700
+c ∈ R since c ∈ R was
+
+208
+00:25:22,700 --> 00:25:29,940
+arbitrary وهنا أثبتنا أن ال f continuous at c ف f
+
+209
+00:25:29,940 --> 00:25:36,980
+is continuous على كل ال R وهو المطلوب
+
+210
+00:25:40,210 --> 00:25:43,290
+أن ال sin function continuous على كل ال R
+
+211
+00:25:43,290 --> 00:25:52,970
+بالمثل ممكن إثبات أن ال function g of x =
+
+212
+00:25:52,970 --> 00:26:01,630
+cos x أيضاً continuous on R هنستخدم ال ..
+
+213
+00:26:01,630 --> 00:26:10,410
+هنستخدم يعني الحاجات هذه أو 2 منهم و .. بدل ال
+
+214
+00:26:10,410 --> 00:26:16,710
+sin هنستخدم معادلة أو متطابقة زي هذه بس نبدل ال
+
+215
+00:26:16,710 --> 00:26:27,010
+sin ب cos فهنا
+
+216
+00:26:27,010 --> 00:26:34,800
+هيصير في عندي اختلاف هذا هيصير -2 بدل 2
+
+217
+00:26:34,800 --> 00:26:43,640
+وهيكون عند هنا sin (½(x+c)) sin (½(x+c)) × sin
+
+218
+00:26:43,640 --> 00:26:48,820
+(½(x-c)) تمام؟
+
+219
+00:26:48,820 --> 00:26:54,040
+وطبعاً هناخد ال | value للطرفين
+
+220
+00:26:56,180 --> 00:26:59,400
+فهذا = ال | value للطرف الثاني
+
+221
+00:26:59,400 --> 00:27:06,700
+وباستخدام المتطابقات هذه فهذا هيطلع أصغر من
+
+222
+00:27:06,700 --> 00:27:11,380
+|-2| بيطلع 2 وهذا أصغر من
+
+223
+00:27:11,380 --> 00:27:17,000
+|sin z| أصغر من أو ساوي 1 و
+
+224
+00:27:17,000 --> 00:27:18,960
+|cos z|
+
+225
+00:27:30,570 --> 00:27:37,920
+لأ هذه مش cos هذه sin هذه ال sin فهي sin ال
+
+226
+00:27:37,920 --> 00:27:40,800
+z ال | value لها أصغر من أو يساوي 1
+
+227
+00:27:40,800 --> 00:27:47,800
+وهي كمان sin أو | value ل sin ال z أصغر
+
+228
+00:27:47,800 --> 00:27:53,360
+من أو يساوي | ال z ال z هنا اللي هو ½ في x
+
+229
+00:27:53,360 --> 00:28:00,620
+- c فبيطلع ½ في | في |x - c|
+
+230
+00:28:00,620 --> 00:28:06,150
+بيطلع هذا = |x - c| وباقي البرهان زي
+
+231
+00:28:06,150 --> 00:28:10,110
+ما عملنا هنا okay تمام لأي ε > الصفر
+
+232
+00:28:10,110 --> 00:28:15,130
+choose δ = ε ف this δ will work
+
+233
+00:28:15,130 --> 00:28:22,370
+تمام إذا باقي البرهان كما عملنا في حالة ال sin
+
+234
+00:28:22,370 --> 00:28:29,210
+إذاً هذا المثال الرابع شوفنا فيه كيف نثبت أن ال
+
+235
+00:28:29,210 --> 00:28:33,870
+cos function is continuous تمام واضح
+
+236
+00:28:37,340 --> 00:28:48,220
+الآن ممكن إثبات بعد هيك أن ال tangent function
+
+237
+00:28:48,220 --> 00:28:58,040
+tan x اللي هي = sin x / cos x is
+
+238
+00:28:58,040 --> 00:28:58,800
+continuous
+
+239
+00:29:01,890 --> 00:29:06,770
+ال sin مستمر على ال R وال cos مستمر على ال R هذه
+
+240
+00:29:06,770 --> 00:29:10,670
+rational function rational function مستمر
+
+241
+00:29:10,670 --> 00:29:14,370
+على ال R ما عدا عند أسفار المقام ما هي أسفار
+
+242
+00:29:14,370 --> 00:29:19,910
+ال cos المضاعفات
+
+243
+00:29:19,910 --> 00:29:27,970
+ال فردية ل π/2 مستمر على ال R ما عدا
+
+244
+00:29:31,580 --> 00:29:42,960
+2n + 1 في π/2 حيث أن n عدد صحيح
+
+245
+00:29:42,960 --> 00:29:46,040
+صح؟
+
+246
+00:29:46,040 --> 00:29:49,100
+هيك
+
+247
+00:29:49,100 --> 00:29:57,940
+بمضاعفات الفردية ل π/2 وكذلك cot x
+
+248
+00:29:57,940 --> 00:30:06,200
+= cos x / sin x is continuous على R ما عدا
+
+249
+00:30:06,200 --> 00:30:14,260
+أسفار المقام اللي هي مضاعفات ال π مضاعفات ال π
+
+250
+00:30:14,260 --> 00:30:21,160
+ما عدا n π حيث أن n عدد صحيح
+
+251
+00:30:27,460 --> 00:30:32,160
+وكذلك بالمثل
+
+252
+00:30:32,160 --> 00:30:39,460
+ال .. ال .. ال secant .. لأ ال cosecant x اللي
+
+253
+00:30:39,460 --> 00:30:45,240
+= 1 / sin x متصل على R ما عدا عند
+
+254
+00:30:45,240 --> 00:30:52,570
+أسفار المقام، إذاً زيها زي ال cotangent وال secant
+
+255
+00:30:52,570 --> 00:30:58,430
+x اللي هي 1 / cos برضه متصلة زيها زي ال
+
+256
+00:30:58,430 --> 00:31:02,690
+tangent على R ما عدا المضاعفات الفردية ل π/2
+
+257
+00:31:02,690 --> 00:31:10,190
+okay تمام طيب
+
+258
+00:31:10,190 --> 00:31:10,790
+ناخد
+
+259
+00:31:28,820 --> 00:31:39,340
+ناخد النظرية التالية let f
+
+260
+00:31:39,340 --> 00:31:43,440
+be a function from A to R
+
+261
+00:31:56,070 --> 00:32:09,810
+وإذاً if |f| is continuous if |f| is continuous at c
+
+262
+00:32:09,810 --> 00:32:14,370
+∈ A then |f|
+
+263
+00:32:17,670 --> 00:32:27,990
+is continuous at c then if |f| is continuous on A
+
+264
+00:32:27,990 --> 00:32:41,190
+then |f| is continuous on A proof
+
+265
+00:32:41,190 --> 00:32:44,230
+we
+
+266
+00:32:44,230 --> 00:32:44,850
+use
+
+267
+00:32:47,240 --> 00:32:51,480
+we use exercise
+
+268
+00:32:51,480 --> 00:32:54,760
+exercise
+
+269
+00:32:54,760 --> 00:33:00,600
+رقم 13
+
+270
+00:33:00,600 --> 00:33:09,220
+في section 4.2 نرجع لل exercise هذا و
+
+271
+00:33:09,220 --> 00:33:09,900
+نكتبه
+
+272
+00:33:16,290 --> 00:33:29,470
+ال exercise هذا بيقول if
+
+273
+00:33:29,470 --> 00:33:38,790
+lim f of x عندما x → c
+
+274
+00:33:38,790 --> 00:33:41,470
+exists
+
+275
+00:33:47,480 --> 00:33:57,760
+then lim |f of x| عندما x → c
+
+276
+00:33:57,760 --> 00:34:04,600
+exists
+
+277
+00:34:04,600 --> 00:34:11,180
+and equals |lim| |
+
+278
+00:34:11,180 --> 00:34:16,900
+lim f of x عندما x → c
+
+279
+00:34:22,160 --> 00:34:26,760
+طبعاً وهنا c is cluster point ال c هنا cluster
+
+280
+00:34:26,760 --> 00:34:30,700
+point cluster 
+
+281
+00:34:30,700 --> 00:34:41,220
+point of A و طبعاً F function من A إلى R فهذا
+
+282
+00:34:41,220 --> 00:34:46,480
+التمرين موجود في section 4-2 لو كانت ال function F
+
+283
+00:34:46,480 --> 00:34:54,090
+ال limit تبعتها عن C موجودة ف limit absolute f of c
+
+284
+00:34:54,090 --> 00:34:58,170
+برضه بتكون موجودة و بساوي قيمتها ال absolute
+
+285
+00:34:58,170 --> 00:35:02,350
+value ل limit f of x when x tends to c يعني مقدر نبدل ال
+
+286
+00:35:02,350 --> 00:35:06,170
+absolute value مع ال limit الآن باستخدام هذا ال
+
+287
+00:35:06,170 --> 00:35:18,290
+exercise ممكن نبره هنا النظرية السابقة إذا 
+
+288
+00:35:18,290 --> 00:35:18,690
+هنا
+
+289
+00:35:23,870 --> 00:35:30,210
+لبرهان الجزء الأول to
+
+290
+00:35:30,210 --> 00:35:36,410
+show that
+
+291
+00:35:36,410 --> 00:35:44,890
+if f is continuous at c to show absolute if is
+
+292
+00:35:44,890 --> 00:35:51,710
+continuous at c تنتمي ل a
+
+293
+00:36:03,810 --> 00:36:09,350
+لدينا اتصالين اتصال
+
+294
+00:36:09,350 --> 00:36:16,650
+اتصال اتصال اتصال اتصال اتصال اتصال اتصال اتصال
+
+295
+00:36:23,200 --> 00:36:26,500
+فشوفنا ان لو كانت الـ C ماهياش cluster point
+
+296
+00:36:26,500 --> 00:36:31,780
+فالاتصال عندها بيطلع متحقق اوتوماتيكي شوفنا في
+
+297
+00:36:31,780 --> 00:36:40,600
+التعريف then the continuity of
+
+298
+00:36:40,600 --> 00:36:47,700
+absolute f at C is automatic
+
+299
+00:36:47,700 --> 00:36:49,560
+اوتوماتيكي
+
+300
+00:36:50,790 --> 00:36:56,590
+إذا احنا بنهتم بالحالة التانية انه C is a cluster
+
+301
+00:36:56,590 --> 00:37:12,550
+point of A ففي الحالة هذه by exercise 13
+
+302
+00:37:12,550 --> 00:37:19,170
+of section أربعة
+
+303
+00:37:19,170 --> 00:37:19,930
+اتنين
+
+304
+00:37:28,440 --> 00:37:37,660
+بما أنه limit ل f of x as x tends to c بيساوي c
+
+305
+00:37:37,660 --> 00:37:46,940
+احنا فرضين ان f continuous by continuity of f at c
+
+306
+00:37:48,200 --> 00:37:51,740
+بما ان f continuous at c احنا فرضين ان f is
+
+307
+00:37:51,740 --> 00:37:55,920
+continuous at c فبالتالي
+
+308
+00:37:55,920 --> 00:38:01,020
+limit f of x لما x تقول ل c بيساوي f of c اذا هاي
+
+309
+00:38:01,020 --> 00:38:05,320
+في عندي limit f of x لما x تقول ل c exist و بيساوي
+
+310
+00:38:05,320 --> 00:38:13,860
+f of c اذا by exercise 13 بطلع عندي limit
+
+311
+00:38:16,480 --> 00:38:25,460
+absolute f of x as x tends to c موجودة وبساوي
+
+312
+00:38:25,460 --> 00:38:37,480
+absolute limit ل f of x لما x تقول ل c اللي هي
+
+313
+00:38:37,480 --> 00:38:45,580
+بتطلع بساوي absolute f of cاللي هي عبارة عن
+
+314
+00:38:45,580 --> 00:38:50,780
+absolute f محسوب عن c إذا هي شرط الاتصال لل
+
+315
+00:38:50,780 --> 00:38:55,980
+function absolute f عند النقطة c متحقق وبالتالي
+
+316
+00:38:55,980 --> 00:39:04,620
+therefore absolute f is continuous at c إذا هذا
+
+317
+00:39:04,620 --> 00:39:09,020
+بثبت الجزء الأول الجزء التاني corollary على الجزء
+
+318
+00:39:09,020 --> 00:39:14,920
+الأول نتيجة الجزء الأول لأن إذا كانت الدالة F
+
+319
+00:39:14,920 --> 00:39:20,640
+continuous على كل الـ A معناته F continuous عند كل
+
+320
+00:39:20,640 --> 00:39:26,600
+C في A وبالتالي بيطلع absolute F متصل عند كل C في
+
+321
+00:39:26,600 --> 00:39:34,680
+A صح؟ إذن هذا إيه برهن النظرية إذن التاني نتيجة
+
+322
+00:39:34,680 --> 00:39:40,900
+على الجزء الأول في كمان نظرية أخرى مشابهة زي هذه
+
+323
+00:39:43,790 --> 00:39:50,770
+لكن بدل absolute f ففي عندى هنا let f be a function
+
+324
+00:39:50,770 --> 00:39:57,510
+from a to r such that f of x أكبر من أو يساوي صفر
+
+325
+00:39:57,510 --> 00:40:05,170
+لكل x في a يعني هنا ال function قيمها غير سالبة فلو
+
+326
+00:40:05,170 --> 00:40:14,660
+كانت f continuous at c فال square root ل f بطلع
+
+327
+00:40:14,660 --> 00:40:21,580
+continuous at C كذلك لو كانت F continuous on A ف
+
+328
+00:40:21,580 --> 00:40:29,760
+ال square root ل F is continuous على كل ال A و
+
+329
+00:40:29,760 --> 00:40:34,500
+المرة هذه البرهان بستخدم exercise ثاني في section
+
+330
+00:40:34,500 --> 00:40:40,090
+4-2 اللي هو exercise 14الـ exercise هذا بيقول إذا
+
+331
+00:40:40,090 --> 00:40:44,510
+كانت ال limit للدالة هذه، يعني عند C موجودة، then ال
+
+332
+00:40:44,510 --> 00:40:49,030
+limit للـ square root .. لل function اللي هي square
+
+333
+00:40:49,030 --> 00:40:56,970
+root of F عند الـ C موجودة وبتساوي ال square root
+
+334
+00:40:56,970 --> 00:41:04,110
+وبتساوي جذر التربيع ليه؟ ال limit لل square root
+
+335
+00:41:05,350 --> 00:41:09,530
+يعني بمعنى آخر أنا ممكن ابدل ال limit مع ال square
+
+336
+00:41:09,530 --> 00:41:15,750
+root و البرهان زي برهان النظرية السابقة
+
+337
+00:41:34,960 --> 00:41:37,360
+الحالة التانية اللي هي المهمة لو كانت C cluster
+
+338
+00:41:37,360 --> 00:41:44,180
+point ل A فحسب exercise 14من section أربعة - اثنين
+
+339
+00:41:44,180 --> 00:41:49,120
+اللي هو كتبناه هناك بما أنه ال limit بما أنه ال
+
+340
+00:41:49,120 --> 00:41:54,160
+function if continuous at c إذا ال limit f of x من
+
+341
+00:41:54,160 --> 00:41:58,900
+x تقوى ل c exist و بساوي f of c الآن من exercise
+
+342
+00:41:58,900 --> 00:42:03,400
+أربعة عشر إذا
+
+343
+00:42:03,400 --> 00:42:07,680
+ال limit هي عند ال limit ل f of x من x تقوى ل c
+
+344
+00:42:07,680 --> 00:42:10,440
+exist إذا by exercise
+
+345
+00:42:14,160 --> 00:42:19,740
+أربعتاش limit ال square root لل function f لما X
+
+346
+00:42:19,740 --> 00:42:27,200
+تقول ل C exist و بساوي ال square root لل limit of
+
+347
+00:42:27,200 --> 00:42:31,460
+the function f when x tends to c وهذا بساوي
+
+348
+00:42:33,950 --> 00:42:37,990
+الـ square root أنا عندي limit f of x عند c exist
+
+349
+00:42:37,990 --> 00:42:44,870
+و بتساوي f of c إذن هذا بيطلع بساوي ال square root
+
+350
+00:42:44,870 --> 00:42:50,870
+ل f هذه ك function محسوبة عن c إذن أنا في عند ال
+
+351
+00:42:50,870 --> 00:42:57,510
+function جذر ال f بالمناسبة جذر f and x كيف
+
+352
+00:42:57,510 --> 00:43:02,430
+بنعرفها؟ بيه عبارة عن الجذر التربيعي ل f of x
+
+353
+00:43:05,740 --> 00:43:11,800
+فإذا أنا عندي الدالة تبعتي جذر F هي دي function ال
+
+354
+00:43:11,800 --> 00:43:16,920
+function هي حسبنا ال limit اللي عند C طلعت موجودة
+
+355
+00:43:16,920 --> 00:43:24,140
+و بتساوي قيمتها عند C إذا ال square root ل F ك
+
+356
+00:43:24,140 --> 00:43:29,560
+function is continuous at C تمام؟ إذا هذا بثبت
+
+357
+00:43:29,560 --> 00:43:33,980
+الجزء الأول من النظرية هذه الآن الجزء التاني
+
+358
+00:43:33,980 --> 00:43:41,050
+Corollary to the first part نتيجة على الجزء الأول
+
+359
+00:43:41,050 --> 00:43:45,510
+لأنه إذا كانت إذا
+
+360
+00:43:45,510 --> 00:43:52,210
+كانت ال F continuous على كل ال A فهي continuous
+
+361
+00:43:52,210 --> 00:43:56,370
+عند كل C في A وبالتالي ال square root من الجزء
+
+362
+00:43:56,370 --> 00:44:01,250
+الأول إلها continuous عند ال C وهذا ال C هذا طبعا
+
+363
+00:44:01,250 --> 00:44:04,170
+ال C was arbitrary إذا ال square root continuous
+
+364
+00:44:04,170 --> 00:44:15,650
+على كل ال A تمام؟ إذن هذه الحاجات .. هذا هو برهانها
+
+365
+00:44:15,650 --> 00:44:24,030
+ال exercise 13 و 14 هدول نظريات فالمفروض أن احنا
+
+366
+00:44:24,030 --> 00:44:31,910
+يعني إيه .. ان .. نبرهنهم فلو
+
+367
+00:44:31,910 --> 00:44:52,750
+بدنا نبرهن مثلا الجزء الأخير هذا فممكن
+
+368
+00:44:52,750 --> 00:45:02,030
+نستخدم ال sequential criterion يعني
+
+369
+00:45:02,030 --> 00:45:03,070
+مثلا ال proof
+
+370
+00:45:06,120 --> 00:45:25,180
+of exercise 14 section أربعة اتنين we
+
+371
+00:45:25,180 --> 00:45:28,920
+use sequential
+
+372
+00:45:28,920 --> 00:45:29,920
+criterion
+
+373
+00:45:32,750 --> 00:45:37,670
+أنا بتثبت أن عندي limit f of x عن c exist و بتثبت
+
+374
+00:45:37,670 --> 00:45:42,450
+limit الجذر ال f عن c exist و بساوي الجذر التربيعي ال
+
+375
+00:45:42,450 --> 00:45:55,150
+limit ف let x n be a sequence طبعا
+
+376
+00:45:55,150 --> 00:45:56,530
+في مجال الدالة
+
+377
+00:46:01,100 --> 00:46:10,900
+be a sequence in a such that limit x n بساوي c تمام
+
+378
+00:46:10,900 --> 00:46:18,060
+then x n
+
+379
+00:46:18,060 --> 00:46:24,120
+أكبر من أو يساوي صفر لأي قيمة للدالة
+
+380
+00:46:43,880 --> 00:46:53,240
+طيب اذا ال function عندي f of x إذا
+
+381
+00:46:53,240 --> 00:47:01,820
+since limit f of x as x tends to c exist هذا
+
+382
+00:47:01,820 --> 00:47:09,850
+بيقدّي انه ال limitالـ f of x n as n tends to
+
+383
+00:47:09,850 --> 00:47:14,530
+infinity موجودة
+
+384
+00:47:14,530 --> 00:47:21,010
+exist و
+
+385
+00:47:21,010 --> 00:47:29,270
+بتساوي and مثلا equals عدد L كويس هذا by
+
+386
+00:47:29,270 --> 00:47:32,810
+sequential criterion
+
+387
+00:47:35,150 --> 00:47:39,110
+الـ function لها limit عن c إذا كان لكل sequence x
+
+388
+00:47:39,110 --> 00:47:46,570
+in تتقارب ل c نهاية صورتها موجودة وبتساوي عدد معين
+
+389
+00:47:46,570 --> 00:47:55,910
+الآن أنا عندي since f of x n أكبر من أو يساوي 0
+
+390
+00:47:55,910 --> 00:48:01,350
+لكل n لأن الدالة قيمها موجبة الدالة هذه قيمها
+
+391
+00:48:01,350 --> 00:48:10,190
+موجبةفالـ limit فالـ L اللي هي limit f
+
+392
+00:48:10,190 --> 00:48:14,510
+of x n تطلع موجب ايضا اكبر من أو يساوي صفر
+
+393
+00:48:14,510 --> 00:48:21,610
+وبالتالي
+
+394
+00:48:21,610 --> 00:48:26,410
+ال limit وفي
+
+395
+00:48:26,410 --> 00:48:30,310
+عندي أنا الآن ال sequence هذه by
+
+396
+00:48:32,240 --> 00:48:41,100
+في مثال أخدناه سابقا او نظرية by theorem 3
+
+397
+00:48:41,100 --> 00:48:46,260
+و12 في الكتاب بتقول لو في عندي sequence زي
+
+398
+00:48:46,260 --> 00:48:55,330
+هذه حدودها غير سالبة فال limitللـ square root ل F
+
+399
+00:48:55,330 --> 00:49:05,390
+of X N as N tends to infinity تطلع موجودة
+
+400
+00:49:05,390 --> 00:49:11,610
+و
+
+401
+00:49:11,610 --> 00:49:16,970
+بتساوي جذر ال Lحسب النظرية هذه إذا كان في end
+
+402
+00:49:16,970 --> 00:49:22,170
+sequence حدودها غير سالبة ومتقاربة إذا ال limit
+
+403
+00:49:22,170 --> 00:49:25,630
+square root لحدودها بساوي square root ل limit
+
+404
+00:49:25,630 --> 00:49:29,330
+تبعتها طبما ال square root ل L هي عبارة عن ال
+
+405
+00:49:29,330 --> 00:49:37,150
+square root ل limit f of x n
+
+406
+00:49:41,810 --> 00:49:47,030
+من هنا الـ square root لإيه اللي بيساوي ال square
+
+407
+00:49:47,030 --> 00:49:56,990
+root لlimit f of x n لما n طول لإنفينتيز إذا
+
+408
+00:49:56,990 --> 00:50:04,550
+انا هيطلع عندي ال limit وهذه عبارة عن limit
+
+409
+00:50:07,530 --> 00:50:15,030
+للـ square root ل F of XN لما N تقول infinity اذا
+
+410
+00:50:15,030 --> 00:50:19,650
+انا بدأت ب XN sequence contained in A ونهايتها C
+
+411
+00:50:19,650 --> 00:50:25,330
+فطلع نهايت نهايت
+
+412
+00:50:25,330 --> 00:50:30,250
+صورتها صورة ال sequence موجودة وبساوي ال square
+
+413
+00:50:30,250 --> 00:50:35,010
+root ل L موجودة وبالتالي therefore by sequential
+
+414
+00:50:39,060 --> 00:50:47,080
+criterion ال limit لل square root ل F of X لما X
+
+415
+00:50:47,080 --> 00:50:55,780
+تقول إلى C بساوي exist و بساوي ال square root ل F
+
+416
+00:50:55,780 --> 00:51:00,980
+when x tends to c أو
+
+417
+00:51:00,980 --> 00:51:03,820
+اللي هو اللي بساوي .. لأ بساوي اللي هو
+
+418
+00:51:09,800 --> 00:51:20,620
+السكوير روت ال L اللي هو برضه اللي هو
+
+419
+00:51:20,620 --> 00:51:23,100
+نعم نعم
+
+420
+00:51:31,480 --> 00:51:37,500
+يعني هاد ممكن هاد يسميها L من الأول فإذا بطلع عندي
+
+421
+00:51:37,500 --> 00:51:40,940
+the square root function لها limit، limit عن سي
+
+422
+00:51:40,940 --> 00:51:46,260
+موجودة بساوي square root لـ L إذا هاد بكمل البرهن
+
+423
+00:51:46,260 --> 00:51:52,320
+بالمثل ممكن نبرهن exercise اللي جابله 13
+
+424
+00:51:57,010 --> 00:52:01,530
+فحاولوا يكونوا تبرهنوا exercise 13 بنفس الطريقة،
+
+425
+00:52:01,530 --> 00:52:07,570
+في أي سؤال أو استفسار؟ okay إذا المرة الجاية بال
+
+426
+00:52:07,570 --> 00:52:08,070
+Campbell 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CRzAwh3Ypto_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CRzAwh3Ypto_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..498a73dcfab7727eed31eceb464ac765b266afdc
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/CRzAwh3Ypto_raw.srt
@@ -0,0 +1,1704 @@
+1
+00:00:21,580 --> 00:00:26,600
+بسم الله الرحمن الرحيم ان شاء الله اليوم هناخد
+
+2
+00:00:26,600 --> 00:00:31,760
+section خمسة اتنين اللي عنوانه combination of
+
+3
+00:00:31,760 --> 00:00:38,560
+continuous functions قبل ما ناخد اول نظريةعن الـ
+
+4
+00:00:38,560 --> 00:00:41,860
+combination of continuous functions نستذكر أو
+
+5
+00:00:41,860 --> 00:00:45,300
+نسترجع مع بعض تعريف ال continuous ال continuity
+
+6
+00:00:45,300 --> 00:00:49,300
+عند نقطة ف a function f from a to r is continuous
+
+7
+00:00:49,300 --> 00:00:55,620
+at c نقطة c تنتمي ل a f and only f لكل إبسلون في
+
+8
+00:00:55,620 --> 00:00:59,740
+دلتا تعتمد على إبسلون عدد موجة بهات لكل x في a
+
+9
+00:01:00,390 --> 00:01:03,710
+المسافة بينها وبين الـC أصغر من دلتا لازم هذا
+
+10
+00:01:03,710 --> 00:01:08,610
+يتضمن أن absolute F of X minus F of C أصغر من
+
+11
+00:01:08,610 --> 00:01:13,270
+إبسلون طبعا شوفنا أن هذا التعريف بيكافئ التعريف
+
+12
+00:01:13,270 --> 00:01:18,970
+اللي أخدناه في calculus A هو الشرط
+
+13
+00:01:18,970 --> 00:01:23,980
+اللي هو بتاوي تلت شروطوهو ان limit f عن c تكون
+
+14
+00:01:23,980 --> 00:01:30,900
+موجودة و f عن c موجودة و اتنين بسوء نفس القيمة
+
+15
+00:01:30,900 --> 00:01:43,420
+الان لو فى عندي تلت دولة f و g و h بيه functions
+
+16
+00:01:43,420 --> 00:01:48,700
+from a to r بيه
+
+17
+00:01:48,700 --> 00:01:49,460
+functions
+
+18
+00:01:54,460 --> 00:02:06,860
+و c تنتمي إلى a و b real number ال
+
+19
+00:02:06,860 --> 00:02:17,440
+functions
+
+20
+00:02:17,440 --> 00:02:23,820
+are continuous at c
+
+21
+00:02:28,450 --> 00:02:34,350
+إذا الدوالة التلاتة F وG وH كلهم متصلين عند النقطة
+
+22
+00:02:34,350 --> 00:02:44,150
+C اللي بتنتمي إليها النتيجة F plus أو minus G F
+
+23
+00:02:44,150 --> 00:02:53,630
+ضرب G B ضرب F are continuous at C
+
+24
+00:02:55,230 --> 00:03:11,750
+B إذا كان H H of X لا تساوي سفر لكل X في A then F
+
+25
+00:03:11,750 --> 00:03:19,710
+على H الدالة F على H is continuous is continuous
+
+26
+00:03:19,710 --> 00:03:20,950
+at C
+
+27
+00:03:25,450 --> 00:03:38,190
+وهي البرهان proof to
+
+28
+00:03:38,190 --> 00:03:48,530
+show مثلا ال function fg is continuous at c
+
+29
+00:03:51,910 --> 00:04:02,370
+We have لدينا التالي بتثبت
+
+30
+00:04:02,370 --> 00:04:09,010
+ان ال F حصل ضرب الدالتين F و G متصل اخترار متصل
+
+31
+00:04:09,010 --> 00:04:14,990
+and C فالاثبات دالك بتثبت ان الشرط هذا تبع الاتصال
+
+32
+00:04:14,990 --> 00:04:23,830
+على النقطة بتحقق فتعالى نشوف high limit F ضرب Gعند
+
+33
+00:04:23,830 --> 00:04:33,190
+X لما X تقول لـC بنثبت أن هذا بيساوي FG عند C فهذا
+
+34
+00:04:33,190 --> 00:04:42,190
+بيساوي limit F of X ضرب G of X as X tends to C هذا
+
+35
+00:04:42,190 --> 00:04:48,610
+من تعريف حصل ضرب اخترانين وهذا بيساوي أنا عندي
+
+36
+00:04:48,610 --> 00:04:56,150
+limit F of X لما X تقول لـC existو limit ال
+
+37
+00:04:56,150 --> 00:05:02,250
+function g of x لما x تقول ل c exist لأن ال
+
+38
+00:05:02,250 --> 00:05:05,110
+function f continuous عند ال c و ال function g
+
+39
+00:05:05,110 --> 00:05:11,250
+احنا فرضينها continuous عند cمش هيكو بس ومن اتصال
+
+40
+00:05:11,250 --> 00:05:17,410
+ده ل F عن C ال limit هذه بيساوي قيمة F عن C وكذلك
+
+41
+00:05:17,410 --> 00:05:20,810
+من اتصال ال function G عن C ال limit هذه بتطلع
+
+42
+00:05:20,810 --> 00:05:30,610
+بيساوي G عن C وهذا بيساوي F ضرب G of C إذن هاي
+
+43
+00:05:30,610 --> 00:05:36,480
+الشرطتبع الاتصال عن نقطة متحقق لل function f ضارب
+
+44
+00:05:36,480 --> 00:05:42,720
+g وبالتالي therefore by definition ال function f g
+
+45
+00:05:42,720 --> 00:05:59,940
+is continuous at c تمام ال proof ال proof of the
+
+46
+00:05:59,940 --> 00:06:00,580
+other
+
+47
+00:06:05,540 --> 00:06:14,200
+parts is similar مشابه للبرهان اللي احنا لسه
+
+48
+00:06:14,200 --> 00:06:19,180
+ماخدينه يعني لإثبات ان مثلا مجموعة دلتين
+
+49
+00:06:19,180 --> 00:06:24,660
+continuous برضه ممكن اثبات ان limit f زائد g لما x
+
+50
+00:06:24,660 --> 00:06:31,220
+تقول ل c بساوي f زائد g and cلو بدنا نثبت ان limit
+
+51
+00:06:31,220 --> 00:06:39,480
+f على g او f على h continuous عن c فبناخد limit f
+
+52
+00:06:39,480 --> 00:06:47,420
+على h عن c وهذا بيطلع بساوي limit f of x على h of
+
+53
+00:06:47,420 --> 00:06:53,810
+x ومع ان limit المقاملأ يساوي سفر لأن H ب X لأ
+
+54
+00:06:53,810 --> 00:07:00,210
+يساوي سفر لكل X في A فممكن نوزع ال limit نقول
+
+55
+00:07:00,210 --> 00:07:02,910
+limit خارج كسمها بيساوي limit ال bus علي limit
+
+56
+00:07:02,910 --> 00:07:06,770
+المقام و limit ال bus بيساوي F عن C لأن F
+
+57
+00:07:06,770 --> 00:07:13,070
+continuous عن C و limit المقام عن C اللي هو H عن C
+
+58
+00:07:13,070 --> 00:07:15,950
+بيساوي قيمة الدالة H عن C لإن أنا متصل عن C
+
+59
+00:07:16,620 --> 00:07:21,880
+وبالتالي بيطلع limit f على h لما x تقول ل c بس هو
+
+60
+00:07:21,880 --> 00:07:27,660
+قيمة الدالة f على h and c okay إذا البرهين
+
+61
+00:07:27,660 --> 00:07:34,860
+المتبقية ممكن يعني أعطاها بنفس الطريقة okay تمام
+
+62
+00:07:34,860 --> 00:07:38,720
+النظرية
+
+63
+00:07:38,720 --> 00:07:40,640
+هذه ممكن تعميمها
+
+64
+00:07:43,880 --> 00:07:51,460
+يعني بدل لو كانت الدالة F و G و H متصلين are
+
+65
+00:07:51,460 --> 00:07:56,640
+continuous are
+
+66
+00:07:56,640 --> 00:08:08,380
+continuous على كل المجموعة A على كل المجال على
+
+67
+00:08:08,380 --> 00:08:15,140
+كل المجال Aالـ F والـ G والـ H المجال المشترك
+
+68
+00:08:15,140 --> 00:08:18,800
+تبعهم المجموعة A فلو كانت الدوالة اتا كلهم
+
+69
+00:08:18,800 --> 00:08:30,280
+continuous على كل المجموعة A ف .. then فبتطلع
+
+70
+00:08:30,280 --> 00:08:36,520
+كل الدوالة هذه متصلة على كل المجموعة Aعلى كل
+
+71
+00:08:36,520 --> 00:08:51,320
+المجموع A يعني هذا بيصير on A وهذه on .. on A فلو
+
+72
+00:08:51,320 --> 00:08:52,380
+بدي أبرهن
+
+73
+00:08:58,870 --> 00:09:03,330
+أي واحدة من الدوايا الهادئة continuous على كل ال A
+
+74
+00:09:03,330 --> 00:09:15,770
+فإيش بعمل بقول fix C تنتمي إلى A and
+
+75
+00:09:15,770 --> 00:09:21,870
+then by
+
+76
+00:09:21,870 --> 00:09:23,090
+above theorem
+
+77
+00:09:28,740 --> 00:09:35,220
+by above theorem أنا
+
+78
+00:09:35,220 --> 00:09:40,520
+الأن عندي كل واحدة من الدوال هدولة continuous على
+
+79
+00:09:40,520 --> 00:09:45,240
+المجموعة a وبالتالي
+
+80
+00:09:45,240 --> 00:09:47,040
+then
+
+81
+00:09:48,850 --> 00:09:52,490
+بما أنه F و G و H continuous على كل المجموعة A فهي
+
+82
+00:09:52,490 --> 00:09:55,870
+continuous عند أي نقطة مش هيك تعرف الاتصال على
+
+83
+00:09:55,870 --> 00:10:08,510
+مجموعة اذا F و G و H are continuous at C وبالتالي
+
+84
+00:10:08,510 --> 00:10:10,130
+حسب النظرية السابقة
+
+85
+00:10:19,920 --> 00:10:26,160
+So by above theorem
+
+86
+00:10:26,160 --> 00:10:32,600
+all functions in
+
+87
+00:10:32,600 --> 00:10:36,660
+parts A
+
+88
+00:10:36,660 --> 00:10:45,920
+and B are continuous at C مش هي كثبتنا احنا في
+
+89
+00:10:45,920 --> 00:10:48,120
+النظرية السابقة هذه اللي جاب ال head اللي انا
+
+90
+00:10:48,120 --> 00:10:52,520
+عدلتهالو كان في عندي تلت دوال و كلهم متصلين عن
+
+91
+00:10:52,520 --> 00:10:57,040
+النقطة فكل الدول الموجودة في الفرق a و الدول
+
+92
+00:10:57,040 --> 00:11:01,300
+الموجودة في الفرق b كلهم بيطلعوا continuous عن نفس
+
+93
+00:11:01,300 --> 00:11:09,660
+النقطة الان بما أن النقطة c was arbitrary since c
+
+94
+00:11:09,660 --> 00:11:17,880
+belonged to a was arbitrary the above
+
+95
+00:11:25,240 --> 00:11:32,060
+All functions in A
+
+96
+00:11:32,060 --> 00:11:37,260
+and B are
+
+97
+00:11:37,260 --> 00:11:39,580
+continuous
+
+98
+00:11:41,100 --> 00:11:46,400
+على كل المجموعة A لأن كل واحدة منهم continuous على
+
+99
+00:11:46,400 --> 00:11:51,060
+أي و كل نقطة C في A وبالتالي هذا يكون برنامج
+
+100
+00:11:51,060 --> 00:11:56,540
+النظرية إذا النظرية هذه تنتج مباشرة من نظرية
+
+101
+00:11:56,540 --> 00:12:04,020
+السابقتها وذلك بتثبيت C عنصر في A وطبعا النظرية
+
+102
+00:12:04,020 --> 00:12:08,300
+السابقة بتقول عند أي عنصرC بما أن التلات دوال
+
+103
+00:12:08,300 --> 00:12:12,440
+متصلة إذا كل ال combinations هدولة بطلعوا متصلين
+
+104
+00:12:12,440 --> 00:12:17,220
+عن نفس النقطة هذا صحيح لأي نقطة ل C وبالتالي كلهم
+
+105
+00:12:17,220 --> 00:12:21,840
+متصلين على كل المجال تباعهم اللي هو المجموعة A
+
+106
+00:12:21,840 --> 00:12:28,040
+تمام ناخد
+
+107
+00:12:28,040 --> 00:12:29,100
+بعض الأمثلة
+
+108
+00:12:40,050 --> 00:12:46,710
+every polynomial .. every polynomial function على
+
+109
+00:12:46,710 --> 00:12:56,190
+الصورة P of X بيساوي A N في X to N plus A N minus
+
+110
+00:12:56,190 --> 00:13:03,290
+one في X to N minus one زائد .. زائد A one في X
+
+111
+00:13:03,290 --> 00:13:08,330
+زائد A zero is continuous
+
+112
+00:13:10,930 --> 00:13:15,470
+on R proof
+
+113
+00:13:15,470 --> 00:13:20,310
+fix
+
+114
+00:13:20,310 --> 00:13:23,750
+fix
+
+115
+00:13:23,750 --> 00:13:29,150
+C ينتمي ل R و بد أثبت أن ال polynomial function P
+
+116
+00:13:29,150 --> 00:13:36,470
+هذه متصلة عند النقطة C طيب we should أثبتنا في
+
+117
+00:13:36,470 --> 00:13:45,400
+chapter 4 we shouldin chapter in chapter four that
+
+118
+00:13:45,400 --> 00:13:48,960
+لو
+
+119
+00:13:48,960 --> 00:13:53,420
+في عندي polynomial P
+
+120
+00:13:53,420 --> 00:13:57,660
+polynomial في X فأثبتنا أن ال limit لل polynomial
+
+121
+00:13:57,660 --> 00:14:03,280
+P عند أي real number
+
+122
+00:14:03,280 --> 00:14:11,430
+C بسوء قيمتها عن C thereforeحسب تعريف تبع الاتصال
+
+123
+00:14:11,430 --> 00:14:22,830
+النقطة إذا P is continuous at C بما أن الـ C was
+
+124
+00:14:22,830 --> 00:14:28,510
+arbitrary element
+
+125
+00:14:28,510 --> 00:14:35,610
+إذا P continuous عند كل الـ C في R وبالتالي P is
+
+126
+00:14:35,610 --> 00:14:43,190
+continuousعلى كل المجموعة R هنا ال A اللي هي R
+
+127
+00:14:43,190 --> 00:14:49,190
+تمام مثال
+
+128
+00:14:49,190 --> 00:15:04,390
+تاني if R بتساوي P على Q P على Q where P
+
+129
+00:15:04,390 --> 00:15:05,930
+و Q R
+
+130
+00:15:08,300 --> 00:15:19,440
+Polynomials are كثيرات حدود then R is continuous
+
+131
+00:15:19,440 --> 00:15:29,720
+on الست اللي هي R كل الأعداد الحقيقية معدى أسفار
+
+132
+00:15:29,720 --> 00:15:36,720
+المقام كل ال X حيث Q of X بتساوي سفر
+
+133
+00:15:50,720 --> 00:15:56,680
+Proof برضه Fix C
+
+134
+00:15:56,680 --> 00:16:08,860
+تنتمي الى R معدى كل ال X حيث Q of X بتساوي سفر
+
+135
+00:16:08,860 --> 00:16:18,260
+معدى أسفار ال function Q إذن Q and C لا يساوي سفر
+
+136
+00:16:20,990 --> 00:16:30,050
+So by chapter .. By chapter four احنا أثبتنا انه
+
+137
+00:16:30,050 --> 00:16:37,310
+في الحالة هذه ال limit ل R of X as X tends to C
+
+138
+00:16:37,310 --> 00:16:48,030
+بساوي R of C وبالتالي therefore R is continuous
+
+139
+00:16:50,660 --> 00:16:58,640
+at C ولمّا كانت الـ C موجودة في R minus أسفار
+
+140
+00:16:58,640 --> 00:17:04,520
+المقام was arbitrarily إذن الـ R continuous على كل
+
+141
+00:17:04,520 --> 00:17:18,080
+الـ A الست هذه okay دي الأبارع بنكتبها طيب
+
+142
+00:17:18,080 --> 00:17:19,900
+في الدول المثلثية
+
+143
+00:17:25,880 --> 00:17:41,480
+في الدوان المثلثية زي الدالة مثلا sign مثال
+
+144
+00:17:41,480 --> 00:17:52,660
+رقم تلاتة f of x بساوي sign x is continuous
+
+145
+00:17:56,130 --> 00:18:07,970
+on R مبتصل على جميع الأعداد الحقيقية proof we
+
+146
+00:18:07,970 --> 00:18:08,650
+use
+
+147
+00:18:13,350 --> 00:18:21,010
+هنستخدم الحقائق التالية absolute sign z أصغر من أو
+
+148
+00:18:21,010 --> 00:18:30,290
+ساوى واحد لكل z في R هذا معروف من الرسمة تبعت ال
+
+149
+00:18:30,290 --> 00:18:33,690
+sign function ال sign function أكبر قيمة إلها ال
+
+150
+00:18:33,690 --> 00:18:38,190
+maximum value واحد وال absolute minimum سالب واحد
+
+151
+00:18:38,190 --> 00:18:43,220
+إذا قيمها محصورة بينهمإذن هذه واضحة من الرسم أو من
+
+152
+00:18:43,220 --> 00:18:50,960
+تعريف ال function كذلك في اندي كمان absolute
+
+153
+00:18:50,960 --> 00:18:59,040
+sin z أصغر من أو ساوي absolute z for all z في R
+
+154
+00:18:59,040 --> 00:19:02,240
+إذن
+
+155
+00:19:02,240 --> 00:19:08,260
+هذه موجود برهانها في chapter chapter
+
+156
+00:19:08,260 --> 00:19:16,030
+8الناس اللي هياخدوا تحليل حقيقة 2 هيشوفوا البرهان
+
+157
+00:19:16,030 --> 00:19:20,890
+والناس اللي مش هياخدوا تحليل حقيقة 2 ممكن يقرؤوا
+
+158
+00:19:20,890 --> 00:19:27,890
+البرهان من chapter 8 حتى تعرفوا يعني ايه تتحققوا
+
+159
+00:19:27,890 --> 00:19:35,870
+ان هذه فعلا المتباينة الصحيحة كذلكمن حساب المثلثات
+
+160
+00:19:35,870 --> 00:19:39,970
+من الـ trigonometry اللي درسناها في calculus A أو
+
+161
+00:19:39,970 --> 00:19:45,030
+ما حتى في الثانوية العامة كان في متطابقات مثلثية و
+
+162
+00:19:45,030 --> 00:19:54,690
+من المتطابقات هذه ممكن نستنتج ان sign x minus sign
+
+163
+00:19:54,690 --> 00:20:11,220
+c بساوي اتنين في signنص في x minus c ضرب
+
+164
+00:20:11,220 --> 00:20:23,100
+cosine نص
+
+165
+00:20:23,100 --> 00:20:26,200
+في
+
+166
+00:20:26,200 --> 00:20:27,680
+x زاد c
+
+167
+00:20:37,480 --> 00:20:46,140
+إذن هذه المتطابقة ممكن أثبتها كيف نثبتها sign
+
+168
+00:20:46,140 --> 00:20:51,900
+الفرق x ع 2 سالب c ع 2 sign الفرق بيسوي sign
+
+169
+00:20:51,900 --> 00:21:00,860
+cosine سالب cosine sign و cosine المجموعة بيسوي
+
+170
+00:21:00,860 --> 00:21:04,420
+cosine cosine سالب sine sine و بعدين نجمعهم و
+
+171
+00:21:04,420 --> 00:21:09,160
+نضربهموفي اتنين فهيطلع في الآخر بتتصف عليه okay
+
+172
+00:21:12,120 --> 00:21:16,040
+بالمناسبة في برضه كمان هندي مش absolute sine z
+
+173
+00:21:16,040 --> 00:21:22,100
+أصغر من أو ساوي الواحد وكذلك في هندي absolute
+
+174
+00:21:22,100 --> 00:21:28,820
+cosine z برضه أصغر من أو ساوي واحد لكل z في R لأنه
+
+175
+00:21:28,820 --> 00:21:32,260
+برضه ال cosine ال absolute مجزمة منها واحد وال
+
+176
+00:21:32,260 --> 00:21:35,600
+absolute minimum سالب واحد وبالتالي قيمة محصورة
+
+177
+00:21:35,600 --> 00:21:40,020
+بين سالب واحد واحد الآن خلينا ناخد ال ..
+
+178
+00:21:42,890 --> 00:21:46,090
+من المعادلة الأخيرة
+
+179
+00:21:56,720 --> 00:21:59,960
+من المعادلة الأخيرة بطلع عندي لو أخدت ال absolute
+
+180
+00:21:59,960 --> 00:22:05,840
+value للطرفين فبطلع عندي absolute sin x minus sin
+
+181
+00:22:05,840 --> 00:22:12,700
+c طبعا هذا الكلام كله صحيح لكل x و c أعداد حقيقية
+
+182
+00:22:14,590 --> 00:22:20,190
+فهذا بيطلع بساوي او اصغر من او ساوي اتنين في
+
+183
+00:22:20,190 --> 00:22:28,230
+absolute sin z absolute sin z اصغر من او ساوي
+
+184
+00:22:28,230 --> 00:22:35,070
+absolute z اللي هو نص في absolute x minus z ضرب
+
+185
+00:22:35,070 --> 00:22:41,650
+absolute cosine z اصغر من او ساوي الواحد اصغر من
+
+186
+00:22:41,650 --> 00:22:52,580
+او ساوي الواحدتمام؟ و هذا صحيح لكل x و c في R طبعا
+
+187
+00:22:52,580 --> 00:23:00,660
+هذا بيساوي absolute x minus c و
+
+188
+00:23:00,660 --> 00:23:06,260
+من المتباين هذي بينتج ان ده ل sign متصل عن c okay؟
+
+189
+00:23:06,260 --> 00:23:20,770
+اذا to show fix c belong to Rto show أن f of x
+
+190
+00:23:20,770 --> 00:23:32,290
+بساوي sin x is continuous at c let epsilon أكبر من
+
+191
+00:23:32,290 --> 00:23:37,050
+السفر be given it shows
+
+192
+00:23:40,310 --> 00:23:44,950
+دلتا بساوي إبسلون أكبر من الصفر إذا هيوجد دلتا
+
+193
+00:23:44,950 --> 00:23:51,430
+تعتمد على إبسلون عدد موجب فلهذه الدلتا لو كان X
+
+194
+00:23:51,430 --> 00:23:56,950
+بينتمي إلى R اللي هو مجال الدالة A و absolute X
+
+195
+00:23:56,950 --> 00:24:04,070
+minus C أصغر من دلتا فهذا بتضمن أنه absolute F of
+
+196
+00:24:04,070 --> 00:24:15,190
+X-f of c اللي هو absolute sin x minus sin c شوفنا
+
+197
+00:24:15,190 --> 00:24:21,870
+هذا أصغر من أو ساوي absolute x minus c من هنا الآن
+
+198
+00:24:21,870 --> 00:24:25,530
+ال x هذه ماخدها أنا بحيث المسافة بينها وبين ال c
+
+199
+00:24:25,530 --> 00:24:30,410
+أصغرمن delta وانا اختار ال delta تساوي epsilon
+
+200
+00:24:30,410 --> 00:24:34,810
+عشان يطلع absolute الفرق بين f of x وf of z أصغر
+
+201
+00:24:34,810 --> 00:24:39,370
+من epsilon إذا هاي شرط epsilon delta لتعريف ال
+
+202
+00:24:39,370 --> 00:24:44,910
+continuity و النقطة المتحققةبما ان ابسلون was
+
+203
+00:24:44,910 --> 00:24:51,090
+arbitrary since ابسلون اكبر من السفر was arbitrary
+
+204
+00:24:51,090 --> 00:24:56,550
+اذا حسب تعريف ابسلون دلتا للاتصال بيطلع عندي ال
+
+205
+00:24:56,550 --> 00:25:05,710
+function f of x بتساوي sin x is continuous at c
+
+206
+00:25:05,710 --> 00:25:11,130
+وبما ان ال c was arbitrary since
+
+207
+00:25:14,280 --> 00:25:22,700
+C belonged to R since C belonged to R was
+
+208
+00:25:22,700 --> 00:25:29,940
+arbitrary وهين أثبتنا أن ال F continuous at C فF
+
+209
+00:25:29,940 --> 00:25:36,980
+is continuous على كل ال R وهو المطلوب
+
+210
+00:25:40,210 --> 00:25:43,290
+ان الـ sine function continuous على كل الـ R
+
+211
+00:25:43,290 --> 00:25:52,970
+بالمثل ممكن اثبات ان ال function g of x بساوي
+
+212
+00:25:52,970 --> 00:26:01,630
+cosine x ايضا continuous on R هنستخدم ال ..
+
+213
+00:26:01,630 --> 00:26:10,410
+هنستخدم يعني الحاجات هذه او اتنين منهم و ..بدل ال
+
+214
+00:26:10,410 --> 00:26:16,710
+sign هنستخدم معادلة أو متطابقة زي هذه بس نبدل ال
+
+215
+00:26:16,710 --> 00:26:27,010
+sign ب cosine فهنا
+
+216
+00:26:27,010 --> 00:26:34,800
+هيصير في عندي اختلاف هذا هصير سالب اتنينبدل اتنين
+
+217
+00:26:34,800 --> 00:26:43,640
+و هيكون عند هنا sign نص sign نص المجموعة ضرب sign
+
+218
+00:26:43,640 --> 00:26:48,820
+نص الفرق تمام؟
+
+219
+00:26:48,820 --> 00:26:54,040
+و طبعا هناخد ال absolute value للطرفين
+
+220
+00:26:56,180 --> 00:26:59,400
+فهذا بيساوي ال absolute value للطرف التاني
+
+221
+00:26:59,400 --> 00:27:06,700
+وباستخدام المتطابقات هذه فهذا هيطلع أصغر من
+
+222
+00:27:06,700 --> 00:27:11,380
+absolute سالب اتنين بيطلع اتنين وهذا أصغر من
+
+223
+00:27:11,380 --> 00:27:17,000
+absolute sine of z أصغر من أو ساوي الواحد و
+
+224
+00:27:17,000 --> 00:27:18,960
+absolute cosine of z
+
+225
+00:27:30,570 --> 00:27:37,920
+لأ هذه مش cosine هذه sinهذه الـ sine فهي sine الـ
+
+226
+00:27:37,920 --> 00:27:40,800
+z ال absolute value لها أصغر من أو يساوي الواحد
+
+227
+00:27:40,800 --> 00:27:47,800
+وهي كمان sine أو absolute value ل sine ال z أصغر
+
+228
+00:27:47,800 --> 00:27:53,360
+من أو يساوي absolute ال z ال z هنا اللي هو نص في x
+
+229
+00:27:53,360 --> 00:28:00,620
+minus z فبطلع نص في absolute في absolute x minus z
+
+230
+00:28:00,620 --> 00:28:06,150
+بطلع هذا بساوي absolute x minus zو باقي البرهان زي
+
+231
+00:28:06,150 --> 00:28:10,110
+ما عملنا هنا okay تمام لأي epsilon أكبر من السفر
+
+232
+00:28:10,110 --> 00:28:15,130
+choose delta بساوي epsilon ف this delta will work
+
+233
+00:28:15,130 --> 00:28:22,370
+تمام إذا باقي البرهان كما عملنا في حالة ال sine
+
+234
+00:28:22,370 --> 00:28:29,210
+إذا هذا المثال الرابع شوفنا فيه كيف نثبت إن ال
+
+235
+00:28:29,210 --> 00:28:33,870
+cosine function is continuous تمام واضح
+
+236
+00:28:37,340 --> 00:28:48,220
+الان ممكن اثبات بعد هيك انه ال tangent function
+
+237
+00:28:48,220 --> 00:28:58,040
+tangent x اللي هي بساوي sin x على cos x is
+
+238
+00:28:58,040 --> 00:28:58,800
+continuous
+
+239
+00:29:01,890 --> 00:29:06,770
+الصين مستمر على الار والكوسين مستمر على الار هذه
+
+240
+00:29:06,770 --> 00:29:10,670
+راشيونال فانتشار فانتشار راشيونال فانتشار مستمر
+
+241
+00:29:10,670 --> 00:29:14,370
+على الار ما عدا عند أسفار المخام ما هي أسفار
+
+242
+00:29:14,370 --> 00:29:19,910
+الكوسين المضاعفات
+
+243
+00:29:19,910 --> 00:29:27,970
+الفردية لا πاية اتنين مستمر على الار ما عدا
+
+244
+00:29:31,580 --> 00:29:42,960
+تنين n زياد واحد في πاي على اتنين حيث ان عدد صحيح
+
+245
+00:29:42,960 --> 00:29:46,040
+صح؟
+
+246
+00:29:46,040 --> 00:29:49,100
+هيك
+
+247
+00:29:49,100 --> 00:29:57,940
+بقطين المضاعفات الفردية لπاي على اتنينو كذلك cot x
+
+248
+00:29:57,940 --> 00:30:06,200
+بيساوي cosine x على sin x is continuous على r مادة
+
+249
+00:30:06,200 --> 00:30:14,260
+أسفار المقام اللي هي مضاعفات الـ pi مضاعفات الـ pi
+
+250
+00:30:14,260 --> 00:30:21,160
+مادة n pi حيث ان عدد صحيح
+
+251
+00:30:27,460 --> 00:30:32,160
+و كذلك بالمثل
+
+252
+00:30:32,160 --> 00:30:39,460
+ال .. ال .. ال secant .. لأ ال cosecant x اللي
+
+253
+00:30:39,460 --> 00:30:45,240
+بيساوي واحد على sign ال x متصل على R معدى عند
+
+254
+00:30:45,240 --> 00:30:52,570
+أسفار المقان، اذا زيها زيالـ cotangent و ال secant
+
+255
+00:30:52,570 --> 00:30:58,430
+x اللي هي واحد على cos برضه متصلة زيها زي ال
+
+256
+00:30:58,430 --> 00:31:02,690
+tangent على R بعد المضاعفات الفردية ل πاية اتنين
+
+257
+00:31:02,690 --> 00:31:10,190
+okay تمام طيب
+
+258
+00:31:10,190 --> 00:31:10,790
+ناخد
+
+259
+00:31:28,820 --> 00:31:39,340
+ناخد النظرية التالية let f
+
+260
+00:31:39,340 --> 00:31:43,440
+be a function from A to R
+
+261
+00:31:56,070 --> 00:32:09,810
+وحد if if is continuous if if is continuous at c
+
+262
+00:32:09,810 --> 00:32:14,370
+تنتمي إلى a then absolute if
+
+263
+00:32:17,670 --> 00:32:27,990
+is continuous at c then if if is continuous on a
+
+264
+00:32:27,990 --> 00:32:41,190
+then absolute if is continuous on a proof
+
+265
+00:32:41,190 --> 00:32:44,230
+we
+
+266
+00:32:44,230 --> 00:32:44,850
+use
+
+267
+00:32:47,240 --> 00:32:51,480
+we use exercise
+
+268
+00:32:51,480 --> 00:32:54,760
+exercise
+
+269
+00:32:54,760 --> 00:33:00,600
+رقم تلتاش
+
+270
+00:33:00,600 --> 00:33:09,220
+في section أربعة اتنين نرجع لل exercise هذا و
+
+271
+00:33:09,220 --> 00:33:09,900
+نكتبه
+
+272
+00:33:16,290 --> 00:33:29,470
+الـ exercise هذا بيقول if
+
+273
+00:33:29,470 --> 00:33:38,790
+ال limit ل ال function f of x لما x تقول إلى c
+
+274
+00:33:38,790 --> 00:33:41,470
+exists
+
+275
+00:33:47,480 --> 00:33:57,760
+then ال limit ل absolute f of x لما x تقول إلى c
+
+276
+00:33:57,760 --> 00:34:04,600
+exist
+
+277
+00:34:04,600 --> 00:34:11,180
+and equals absolute limit absolute
+
+278
+00:34:11,180 --> 00:34:16,900
+limit f of x لما x تقول إلى c
+
+279
+00:34:22,160 --> 00:34:26,760
+طبعاً و هنا C is cluster point الـ C هنا cluster
+
+280
+00:34:26,760 --> 00:34:30,700
+point cluster
+
+281
+00:34:30,700 --> 00:34:41,220
+point of A و طبعاً F function من A إلى R فهذا
+
+282
+00:34:41,220 --> 00:34:46,480
+التمرين موجود في section 4-2 لو كانت ال function F
+
+283
+00:34:46,480 --> 00:34:54,090
+ال limit تبعتها عن C موجودةف limit absolute f and
+
+284
+00:34:54,090 --> 00:34:58,170
+c برضه بتكون موجودة و بساوي قيمتها ال absolute
+
+285
+00:34:58,170 --> 00:35:02,350
+value ل limit f of x and z يعني مقدر نبدل ال
+
+286
+00:35:02,350 --> 00:35:06,170
+absolute value مع ال limit الآن باستخدام هذا ال
+
+287
+00:35:06,170 --> 00:35:18,290
+exercise ممكن نبره هنا النظرية السابقة إذا
+
+288
+00:35:18,290 --> 00:35:18,690
+هنا
+
+289
+00:35:23,870 --> 00:35:30,210
+لبرهان الجزء الأول to
+
+290
+00:35:30,210 --> 00:35:36,410
+show if
+
+291
+00:35:36,410 --> 00:35:44,890
+is .. to show absolute if is
+
+292
+00:35:44,890 --> 00:35:51,710
+continuous at c تنتمي ل a
+
+293
+00:36:03,810 --> 00:36:09,350
+لدينا اتصالين اتصال
+
+294
+00:36:09,350 --> 00:36:16,650
+اتصال اتصال اتصال اتصال اتصال اتصال اتصال اتصال
+
+295
+00:36:23,200 --> 00:36:26,500
+فشوفنا ان لو كانت الـ C ماهياش cluster point
+
+296
+00:36:26,500 --> 00:36:31,780
+فالاتصال عندها بيطلع متحقق اوتوماتيكي شوفنا في
+
+297
+00:36:31,780 --> 00:36:40,600
+التعريف then the continuity of
+
+298
+00:36:40,600 --> 00:36:47,700
+absolute if at C is automatic
+
+299
+00:36:47,700 --> 00:36:49,560
+اوتوماتيكي
+
+300
+00:36:50,790 --> 00:36:56,590
+إذا احنا بنهتم بالحالة التانية انه C is a cluster
+
+301
+00:36:56,590 --> 00:37:12,550
+point of A ففي الحالة هذه by exercise تلتاش
+
+302
+00:37:12,550 --> 00:37:19,170
+of section اربعة
+
+303
+00:37:19,170 --> 00:37:19,930
+اتنين
+
+304
+00:37:28,440 --> 00:37:37,660
+بما أنه limit ل f of x as x tends to c بيساوي c
+
+305
+00:37:37,660 --> 00:37:46,940
+احنا فرضين ان f continuous by continuity of f at c
+
+306
+00:37:48,200 --> 00:37:51,740
+بما ان f continuous at c احنا فرضين ان f is
+
+307
+00:37:51,740 --> 00:37:55,920
+continuous at c فبالتالي
+
+308
+00:37:55,920 --> 00:38:01,020
+limit f of x لما x تقول ل c بيساوي f of c اذا هاي
+
+309
+00:38:01,020 --> 00:38:05,320
+في عندي limit f of x لما x تقول ل c exist و بيساوي
+
+310
+00:38:05,320 --> 00:38:13,860
+f of c اذا by exercise 13 بطلع عندي limit
+
+311
+00:38:16,480 --> 00:38:25,460
+absolute f of x as x tends to c موجودة وبساوي
+
+312
+00:38:25,460 --> 00:38:37,480
+absolute limit ل f of x لما x تقول ل c اللي هي
+
+313
+00:38:37,480 --> 00:38:45,580
+بتطلع بساوي absolute f of cاللي هي عبارة عن
+
+314
+00:38:45,580 --> 00:38:50,780
+absolute f محسوب عن c إذا هي شرط الاتصال لل
+
+315
+00:38:50,780 --> 00:38:55,980
+function absolute f عند النقطة c متحقق وبالتالي
+
+316
+00:38:55,980 --> 00:39:04,620
+therefore absolute f is continuous at c إذا هذا
+
+317
+00:39:04,620 --> 00:39:09,020
+بثبت الجزء الأول الجزء التاني corollary على الجزء
+
+318
+00:39:09,020 --> 00:39:14,920
+الأول نتيجة الجزء الأوللأن إذا كانت الدالة F
+
+319
+00:39:14,920 --> 00:39:20,640
+continuous على كل الـ A معناته F continuous عند كل
+
+320
+00:39:20,640 --> 00:39:26,600
+C في A وبالتالي بيطلع absolute F متصل عند كل C في
+
+321
+00:39:26,600 --> 00:39:34,680
+A صح؟ إذن هذا إيه برهن النظرية إذن التاني نتيجة
+
+322
+00:39:34,680 --> 00:39:40,900
+على الجزء الأول في كمان نظرية أخرى مشابهة زي هذه
+
+323
+00:39:43,790 --> 00:39:50,770
+لكن بدل absolute f ففي عندى هنا let f be function
+
+324
+00:39:50,770 --> 00:39:57,510
+from a to r such that f of x أكبر من أو يساوي سفر
+
+325
+00:39:57,510 --> 00:40:05,170
+لكل x في a يعني هنا ال dialer قيمها غير سالبة فلو
+
+326
+00:40:05,170 --> 00:40:14,660
+كانت f continuous at c فال square root ل fبطلع
+
+327
+00:40:14,660 --> 00:40:21,580
+continuous at C كذلك لو كانت F continuous on A ف
+
+328
+00:40:21,580 --> 00:40:29,760
+ال square root ل F is continuous على كل ال A و
+
+329
+00:40:29,760 --> 00:40:34,500
+المرة هذه البرهان بستخدم exercise ثاني في section
+
+330
+00:40:34,500 --> 00:40:40,090
+42 اللي هو exercise 14الـ exercise هذا بيقول إذا
+
+331
+00:40:40,090 --> 00:40:44,510
+كانت ال limit للدالة هذه، يعني C موجودة، then ال
+
+332
+00:40:44,510 --> 00:40:49,030
+limit للـ square .. لل function اللي هي square
+
+333
+00:40:49,030 --> 00:40:56,970
+root of F عند الـ C موجودة وبتساوي ال square root
+
+334
+00:40:56,970 --> 00:41:04,110
+وبتساوي جذر التربيع أيه؟ ال limit لل square root
+
+335
+00:41:05,350 --> 00:41:09,530
+يعني بمعنى اخر انا ممكن ابدل ال limit مع ال square
+
+336
+00:41:09,530 --> 00:41:15,750
+root و البرهان زي برهان النظرية السابقة
+
+337
+00:41:34,960 --> 00:41:37,360
+الحالة التانية اللي هي المهمة لو كانت C cluster
+
+338
+00:41:37,360 --> 00:41:44,180
+point ل A فحسب exercise 14من سكتشن أربعة اتنين
+
+339
+00:41:44,180 --> 00:41:49,120
+اللي هو كتبناه هناك بما أنه ال limit بما أنه ال
+
+340
+00:41:49,120 --> 00:41:54,160
+function if continuous at c إذا ال limit f of x من
+
+341
+00:41:54,160 --> 00:41:58,900
+x تقوى ل c exist و بساوي f of c الآن من exercise
+
+342
+00:41:58,900 --> 00:42:03,400
+أربعة عشر إذا
+
+343
+00:42:03,400 --> 00:42:07,680
+ال limit هي عند ال limit ل f of x من x تقوى ل c
+
+344
+00:42:07,680 --> 00:42:10,440
+exist إذا by exercise
+
+345
+00:42:14,160 --> 00:42:19,740
+أربعتاش limit ال square root لل function f لما X
+
+346
+00:42:19,740 --> 00:42:27,200
+تقول ل C exist و بساوي ال square root لل limit of
+
+347
+00:42:27,200 --> 00:42:31,460
+the function يعني C وهذا بساوي
+
+348
+00:42:33,950 --> 00:42:37,990
+الـ square root أنا عندي limit f of x عند c exist
+
+349
+00:42:37,990 --> 00:42:44,870
+و بتساوي f of c إذن هذا بيطلع بساوي ال square root
+
+350
+00:42:44,870 --> 00:42:50,870
+ل f هذه ك function محسوبة عن c إذن أنا في عند ال
+
+351
+00:42:50,870 --> 00:42:57,510
+function جدر ال f بالمناسبة جدر f and x كيف
+
+352
+00:42:57,510 --> 00:43:02,430
+بنعرفها؟ بيه عبارة عن الجدر التربيهي ل f of x
+
+353
+00:43:05,740 --> 00:43:11,800
+فإذا أنا عندي الدالة تبعتي جذر F هي دي function ال
+
+354
+00:43:11,800 --> 00:43:16,920
+function هي حسبنا ال limit اللي عند C طلعت موجودة
+
+355
+00:43:16,920 --> 00:43:24,140
+و بتساوي قيمتها عند C إذا ال square root ل F ك
+
+356
+00:43:24,140 --> 00:43:29,560
+function is continuous at C تمام؟ إذا هذا بثبت
+
+357
+00:43:29,560 --> 00:43:33,980
+الجزء الأول من النظرية هذه الآن الجزء التاني
+
+358
+00:43:33,980 --> 00:43:41,050
+Corollaryto the first part نتيجة على الجزء الأول
+
+359
+00:43:41,050 --> 00:43:45,510
+لأنه إذا كانت إذا
+
+360
+00:43:45,510 --> 00:43:52,210
+كانت ال F continuous على كل ال A فهي continuous
+
+361
+00:43:52,210 --> 00:43:56,370
+عند كل C في A وبالتالي ال square root من الجزء
+
+362
+00:43:56,370 --> 00:44:01,250
+الأول إلها continuous عند ال C وهذا ال C هذا طبعا
+
+363
+00:44:01,250 --> 00:44:04,170
+ال C was arbitrary إذا ال square root continuous
+
+364
+00:44:04,170 --> 00:44:15,650
+على كل ال Aتمام؟ إذن هذه الحاجات .. هذا هو برهانها
+
+365
+00:44:15,650 --> 00:44:24,030
+ال exercise 13 و 14 هدول نظريات فالمفروض أن احنا
+
+366
+00:44:24,030 --> 00:44:31,910
+يعني إيه .. ان .. نبرهنهم فلو
+
+367
+00:44:31,910 --> 00:44:52,750
+بدنا نبرهن مثلاالجزء الأخير هذا فممكن
+
+368
+00:44:52,750 --> 00:45:02,030
+نستخدم ال sequential criterion يعني
+
+369
+00:45:02,030 --> 00:45:03,070
+مثلا ال proof
+
+370
+00:45:06,120 --> 00:45:25,180
+of exercise أربعة طعش section أربعة اتنين we
+
+371
+00:45:25,180 --> 00:45:28,920
+use sequential
+
+372
+00:45:28,920 --> 00:45:29,920
+criterion
+
+373
+00:45:32,750 --> 00:45:37,670
+أنا بتثبت أن عندي limit f of x عن c exist و بتثبت
+
+374
+00:45:37,670 --> 00:45:42,450
+limit الجذر ال f عن c exist و بساوي الجذر تربية ال
+
+375
+00:45:42,450 --> 00:45:55,150
+unlimited ف let x in be sequence طبعا
+
+376
+00:45:55,150 --> 00:45:56,530
+في مجال الدالة
+
+377
+00:46:01,100 --> 00:46:10,900
+b sequence in a such that limit xn بساوي c تمام
+
+378
+00:46:10,900 --> 00:46:18,060
+then xn
+
+379
+00:46:18,060 --> 00:46:24,120
+أكبر من أو يساوي سفر لأ قيمة الدالة
+
+380
+00:46:43,880 --> 00:46:53,240
+طيب اذا ال function عندي f of x اذا
+
+381
+00:46:53,240 --> 00:47:01,820
+since limit f of x as x tends to c exist هذا
+
+382
+00:47:01,820 --> 00:47:09,850
+بيقدّي انه ال limitالـ f of x in as n tends to
+
+383
+00:47:09,850 --> 00:47:14,530
+infinity موجودة
+
+384
+00:47:14,530 --> 00:47:21,010
+exist و
+
+385
+00:47:21,010 --> 00:47:29,270
+بتساوي and مثلا equals عدد L كويس هذا by
+
+386
+00:47:29,270 --> 00:47:32,810
+sequential criterion
+
+387
+00:47:35,150 --> 00:47:39,110
+الـ function لها limit عن c إذا كان لكل sequence x
+
+388
+00:47:39,110 --> 00:47:46,570
+in تتقارب ل c نهاية صورتها موجودة وبتساوي عدد معين
+
+389
+00:47:46,570 --> 00:47:55,910
+الأن أنا عندي sense f of x in أكبر من أو ساوى 0
+
+390
+00:47:55,910 --> 00:48:01,350
+لكل in لأن الدالة قيمها موجبة الدالة هذه قيمها
+
+391
+00:48:01,350 --> 00:48:10,190
+موجبةفالـ limit فالـ L اللي هي limit f
+
+392
+00:48:10,190 --> 00:48:14,510
+of x in تطلع موجب ايضا اكبر من أو ساوي سفر
+
+393
+00:48:14,510 --> 00:48:21,610
+وبالتالي
+
+394
+00:48:21,610 --> 00:48:26,410
+ال limit وفي
+
+395
+00:48:26,410 --> 00:48:30,310
+عندي انا الآن ال sequence هذه by
+
+396
+00:48:32,240 --> 00:48:41,100
+في مثال أخدناه سابقا او نظرية by theorem تلاتة
+
+397
+00:48:41,100 --> 00:48:46,260
+اتنين عشرة في الكتاب بتقول لو في عندي sequence زي
+
+398
+00:48:46,260 --> 00:48:55,330
+هذه حدودها غير سالبة فال limitللـ square root ل F
+
+399
+00:48:55,330 --> 00:49:05,390
+of X N as N tends to infinity تطلع موجودة
+
+400
+00:49:05,390 --> 00:49:11,610
+و
+
+401
+00:49:11,610 --> 00:49:16,970
+بالساوي جذر ال Lحسب النظرية هذه إذا كان في end
+
+402
+00:49:16,970 --> 00:49:22,170
+sequence حدودها غير سالبة ومتقاربة إذا ال limit
+
+403
+00:49:22,170 --> 00:49:25,630
+square root لحدودها بساوي square root ل limit
+
+404
+00:49:25,630 --> 00:49:29,330
+تبعتها طبما ال square root ل L هي عبارة عن ال
+
+405
+00:49:29,330 --> 00:49:37,150
+square root ل limit f of x in
+
+406
+00:49:41,810 --> 00:49:47,030
+من هنا الـ square root لإيه اللي بيساوي ال square
+
+407
+00:49:47,030 --> 00:49:56,990
+root لlimit f of x n لما n طول لإنفينتيز اذا
+
+408
+00:49:56,990 --> 00:50:04,550
+انا هيطلع عندي ال limit وهذه عبارة عن limit
+
+409
+00:50:07,530 --> 00:50:15,030
+للـ square root ل F of XN لما N تقول infinity اذا
+
+410
+00:50:15,030 --> 00:50:19,650
+انا بدأت ب XN sequence contained in A ونهايتها C
+
+411
+00:50:19,650 --> 00:50:25,330
+فطلع نهايت نهايت
+
+412
+00:50:25,330 --> 00:50:30,250
+صورتها صورة ال sequence موجودة وبساوي ال square
+
+413
+00:50:30,250 --> 00:50:35,010
+root ل L موجودة وبالتالي therefore by sequential
+
+414
+00:50:39,060 --> 00:50:47,080
+criterion ال limit لل square root ل F of X لما X
+
+415
+00:50:47,080 --> 00:50:55,780
+تقول إلى C بساوي exist و بساوي ال square root ل F
+
+416
+00:50:55,780 --> 00:51:00,980
+and C أو
+
+417
+00:51:00,980 --> 00:51:03,820
+اللي هو اللي بساوي .. لأ بساوي اللي هو
+
+418
+00:51:09,800 --> 00:51:20,620
+السكوير روت ال L اللي هو برضه اللي هو
+
+419
+00:51:20,620 --> 00:51:23,100
+نعم نعم
+
+420
+00:51:31,480 --> 00:51:37,500
+يعني هاد ممكن هاد يسميها L من الأول فإذا بطلع عندي
+
+421
+00:51:37,500 --> 00:51:40,940
+ال square root function لها limit، limit عن سي
+
+422
+00:51:40,940 --> 00:51:46,260
+موجودة بساوي square root ل L إذا هاد بكمل البرهن
+
+423
+00:51:46,260 --> 00:51:52,320
+بالمثل ممكن نبرهن exercise اللي جابله 13
+
+424
+00:51:57,010 --> 00:52:01,530
+فحاولوا يكونوا تبرهنوا exercise 13 بنفس الطريقة،
+
+425
+00:52:01,530 --> 00:52:07,570
+في أي سؤال أو استفسار؟ okay إذا المرة الجاية بال
+
+426
+00:52:07,570 --> 00:52:08,070
+campbell
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..01bc4bdff2ebe09d17ddfc5ba51c05f92b5d3e6a
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ehj01gka7EU_raw.srt
@@ -0,0 +1,1740 @@
+1
+00:00:19,740 --> 00:00:27,020
+بسم الله الرحمن الرحيم هنواصل اليوم تغطية section
+
+2
+00:00:27,020 --> 00:00:34,550
+5-3اللي بتعلق ب .. موضوع ال continuous functions
+
+3
+00:00:34,550 --> 00:00:40,590
+على ال intervals على الفترات احنا بدينا ال ..
+
+4
+00:00:40,590 --> 00:00:46,690
+بدينا الموضوع هذا المحاضرة السابقة و كان اخر نظرية
+
+5
+00:00:46,690 --> 00:00:50,890
+أخدناها اللي هي ال maximum .. maximum minimum
+
+6
+00:00:50,890 --> 00:00:56,870
+theorem نعود نستذكر بس نظرية الأخيرة هذه ال
+
+7
+00:00:56,870 --> 00:00:57,670
+maximum
+
+8
+00:01:12,090 --> 00:01:21,410
+ال maximum minimum minimum
+
+9
+00:01:21,410 --> 00:01:28,050
+theorem يقول
+
+10
+00:01:28,050 --> 00:01:36,050
+إذا كانت if I is a closed and bounded interval is
+
+11
+00:01:36,050 --> 00:01:40,290
+closed and bounded
+
+12
+00:01:46,350 --> 00:01:56,730
+وإذا كانت العملية من I إلى R مستمرة
+
+13
+00:01:56,730 --> 00:02:01,270
+على
+
+14
+00:02:01,270 --> 00:02:02,470
+الفترة I
+
+15
+00:02:07,590 --> 00:02:20,690
+there exist x lower star و x upper star عناصر في I
+
+16
+00:02:20,690 --> 00:02:22,070
+بحيث انه
+
+17
+00:02:24,550 --> 00:02:32,550
+f of x lower star بساوي ال minimum ل range ال
+
+18
+00:02:32,550 --> 00:02:41,450
+function f and f of x super star بساوي ال supremum
+
+19
+00:02:41,450 --> 00:02:49,410
+ل range ال function f وبالتالي هذه بسميها ال
+
+20
+00:02:49,410 --> 00:02:52,930
+absolute maximum
+
+21
+00:02:54,540 --> 00:03:03,820
+value و القيمة هتبسميها ال absolute minimum
+
+22
+00:03:03,820 --> 00:03:06,900
+value
+
+23
+00:03:06,900 --> 00:03:15,920
+لل function f على الفترة I طبعا okay okay في اليوم
+
+24
+00:03:15,920 --> 00:03:22,580
+هناخد نظريات برضه خاصة باتصال الدوال على الفترات
+
+25
+00:03:23,810 --> 00:03:30,090
+فأول نظرية هتكون location location
+
+26
+00:03:30,090 --> 00:03:36,970
+of roots theorem
+
+27
+00:03:36,970 --> 00:03:45,570
+نظرية تحديد ال roots فنفس
+
+28
+00:03:45,570 --> 00:03:46,790
+الحاجة let
+
+29
+00:03:49,750 --> 00:03:57,890
+I be closed and bounded interval على الصورة AB and
+
+30
+00:03:57,890 --> 00:04:06,190
+let f be function from I to R be continuous
+
+31
+00:04:06,190 --> 00:04:09,790
+function
+
+32
+00:04:09,790 --> 00:04:15,390
+على الفترة المغلقة والمحدودة I if
+
+33
+00:04:17,630 --> 00:04:29,370
+لو كان f of a أصغر من صفر أصغر من f of b او f of b
+
+34
+00:04:29,370 --> 00:04:38,610
+أصغر من صفر أصغر من f of a then
+
+35
+00:04:38,610 --> 00:04:48,980
+there exist c ينتميللفترة المفتوحة من a إلى b بحيث
+
+36
+00:04:48,980 --> 00:04:57,540
+أن f of c بيساوي سفر فالنظرية
+
+37
+00:04:57,540 --> 00:05:08,100
+هذه ممكن أنلخصها بالرسمة التالية محاور
+
+38
+00:05:08,100 --> 00:05:13,280
+الأحداثيات وممكن يكون في ending حاجة زي هذه
+
+39
+00:05:22,650 --> 00:05:28,930
+فهي function هذه عبارة عن ال graph y بساوي f of x
+
+40
+00:05:28,930 --> 00:05:37,750
+ال function هذه متصلة على الفترة المغلطة من a ل d
+
+41
+00:05:37,750 --> 00:05:42,890
+وهي
+
+42
+00:05:42,890 --> 00:05:51,450
+عندي f of a أصغر من سفر وهي عندي
+
+43
+00:05:59,190 --> 00:06:02,510
+النظرية بتقول لو كان في اندرالا متصلة زي هذه على
+
+44
+00:06:02,510 --> 00:06:07,830
+فترة مغلقة من a لb وكان f of a أصغر من الصفر و
+
+45
+00:06:07,830 --> 00:06:16,370
+الصفر أصغر من f of bلابد ان نجد نقطة C بين A وB
+
+46
+00:06:16,370 --> 00:06:21,030
+بحيث ان قيمة الـ function عندها بالساوي سفر و واضح
+
+47
+00:06:21,030 --> 00:06:26,270
+ان نقطة C هي قيمة الـ function عندها بالساوي سفر
+
+48
+00:06:26,270 --> 00:06:30,830
+ممكن برضه يكون العكس يعني الملحانة هذا يكون شكله
+
+49
+00:06:30,830 --> 00:06:31,430
+زي هيك
+
+50
+00:06:35,680 --> 00:06:41,700
+فيكون يعني عندي هنا ال F of B هي السالة بقى وهي
+
+51
+00:06:41,700 --> 00:06:46,580
+عند ال A فال F of B هي الموجة بقى برضه نفس النتيجة
+
+52
+00:06:46,580 --> 00:06:47,620
+okay تمام؟
+
+53
+00:06:55,410 --> 00:06:59,790
+البرهان النظرية هذه يعني it's زي ما بيقولوا it's
+
+54
+00:06:59,790 --> 00:07:06,630
+quite technical يعني فيه شوية تفاصيل تقنية زاد انه
+
+55
+00:07:06,630 --> 00:07:13,090
+طويل شوية فاحنا عشان بصدر نهاية الفصل مابناش ناخد
+
+56
+00:07:13,090 --> 00:07:16,330
+.. ناخد .. ناخد في البرهين الطويلة فحسيبكم تقراوا
+
+57
+00:07:16,330 --> 00:07:19,530
+البرهان اذا see the textbook
+
+58
+00:07:25,030 --> 00:07:32,130
+إذا الممكن بدؤوكم يمكن تقرؤوا البرهان من الكتاب و
+
+59
+00:07:32,130 --> 00:07:36,770
+تحاولوا تفهموه طبعا البرهان طويل مابنجوبش طبعا
+
+60
+00:07:36,770 --> 00:07:41,990
+البرهين طويلة جه هدف الامتحانات okay فهذا بالنسبة
+
+61
+00:07:41,990 --> 00:07:46,930
+للبرهان الآن هاي مثال مثلا مثال example
+
+62
+00:07:54,050 --> 00:07:58,270
+Show that the
+
+63
+00:07:58,270 --> 00:08:03,470
+equation f
+
+64
+00:08:03,470 --> 00:08:11,510
+of x بتساوي x في e أُس x سالب اتنين بتساوي سفر has
+
+65
+00:08:11,510 --> 00:08:14,210
+a root
+
+66
+00:08:20,420 --> 00:08:29,980
+in الـ interval من سفر لواحد لنثبت
+
+67
+00:08:29,980 --> 00:08:35,200
+ان المعادلة f of x بالساوي سفر عشان f of x بالساوي
+
+68
+00:08:35,200 --> 00:08:43,100
+الدالة هذه لها جدر يعني بنقدر اللاجم اي
+
+69
+00:08:43,100 --> 00:08:54,460
+هذا يعني showان يوجد C ينتمي للفترة المغلقة من سفر
+
+70
+00:08:54,460 --> 00:09:01,900
+لواحد بحيث انه اخه C مساره سفر ففي الحالة اللي
+
+71
+00:09:01,900 --> 00:09:07,360
+بنقول انه C root جدر للمعادلة او C zero لل
+
+72
+00:09:07,360 --> 00:09:14,900
+function F فبنرفبت الكلام هذا فحسب النظرية هذه
+
+73
+00:09:27,630 --> 00:09:35,370
+F of X بساوي X في E to X سالب اتنين is continuous
+
+74
+00:09:35,370 --> 00:09:40,650
+متصلة على الفترة المغلقة من سفر لواحد
+
+75
+00:09:47,890 --> 00:09:51,510
+لأن X في E to X هي دالة متصلة اتراحى منها ثابت
+
+76
+00:09:51,510 --> 00:09:56,750
+دالة متصلة على R كذلك
+
+77
+00:09:56,750 --> 00:10:06,460
+انا عندي F of سفر بساوي سالب اتنين اصغر من سفرو F
+
+78
+00:10:06,460 --> 00:10:15,300
+of واحد بالساوي E ثاند اتنين وال E معروف انه عدد
+
+79
+00:10:15,300 --> 00:10:21,100
+اكبر من اتنين فهذا اكبر من ساكنة اذا هاي شروط ال
+
+80
+00:10:21,100 --> 00:10:28,500
+location of roots ال theorem كلها متحققة hence by
+
+81
+00:10:28,500 --> 00:10:33,700
+location of roots theorem
+
+82
+00:10:36,360 --> 00:10:42,300
+يوجد C أنتمي للفترة المغلقة من ستة إلى واحد بحيث
+
+83
+00:10:42,300 --> 00:10:55,640
+انه F of C بساوي سفر اذا هنا اثبتنا ان C is a root
+
+84
+00:10:55,640 --> 00:11:01,400
+of equation F of X
+
+85
+00:11:04,080 --> 00:11:09,640
+بساوي X في E أس X minus اتنين بساوي سفر وهو
+
+86
+00:11:09,640 --> 00:11:16,360
+المطلوب اذا هنا اثبتنا ان فعلا المعادلة هذه لها
+
+87
+00:11:16,360 --> 00:11:22,720
+جذر في الفترة هذا الجذر يقع هو عدد C يقع في الفترة
+
+88
+00:11:22,720 --> 00:11:28,180
+من سفر لوحده عدد من سفر لوحده طبعا ممكن هذا العدد
+
+89
+00:11:28,180 --> 00:11:35,450
+C نعمله تقريب إلى أقرب يعنيبحيث يكون النسبة الخطأ
+
+90
+00:11:35,450 --> 00:11:41,470
+من القيمة الحقيقية تبقى تكون أقل من واحد على ألف
+
+91
+00:11:41,470 --> 00:11:46,210
+أو واحد على مية أو واحد على عشر ألف فالكتاب الموضح
+
+92
+00:11:46,210 --> 00:11:51,610
+لكم هي هنا في المثال كيف نجيب تقريب نحصل العدد
+
+93
+00:11:51,610 --> 00:11:55,730
+سيرة بحيث نطلع تقريبا قريب من القيمة الحقيقية
+
+94
+00:11:55,730 --> 00:11:59,030
+والفرق بينها ومن القيمة الحقيقية واللي هنجيبها في
+
+95
+00:11:59,030 --> 00:12:05,240
+المثال تكون أقل من واحد على ألف أوش زيهافممكن تقرأ
+
+96
+00:12:05,240 --> 00:12:09,960
+و تشوف الكلام هذا في الكتاب لكن احنا اللي بهمنا ان
+
+97
+00:12:09,960 --> 00:12:15,460
+ال equation هذه ضمننا انه في لها root في الفترة
+
+98
+00:12:15,460 --> 00:12:18,740
+هذه حسب ال location of roots في الفترة الباقية
+
+99
+00:12:18,740 --> 00:12:24,520
+كانت تخلي ال root هذا يعني تجيبله قيمة قريبة جدا
+
+100
+00:12:24,520 --> 00:12:30,020
+من القيمة الحقيقية هذه مجرد يعني تفاصيل حسابية
+
+101
+00:12:30,020 --> 00:12:36,320
+okay فحاولوا تقراوها من الكتاب لو سمحتالان هذه
+
+102
+00:12:36,320 --> 00:12:47,540
+النظرية بتقود الى نظرية تانية وهي
+
+103
+00:12:47,540 --> 00:12:55,380
+Bolzano's
+
+104
+00:12:55,380 --> 00:12:57,140
+intermediate
+
+105
+00:13:04,990 --> 00:13:25,730
+value theorem let
+
+106
+00:13:25,730 --> 00:13:29,210
+I be any interval
+
+107
+00:13:36,870 --> 00:13:50,310
+and if from I to R be continuous على
+
+108
+00:13:50,310 --> 00:14:00,830
+الفترة I إذا كان A و B أعداد في الفترة I and
+
+109
+00:14:03,830 --> 00:14:16,170
+K عدد حقيقي such that F of A أصغر من K أصغر من F
+
+110
+00:14:16,170 --> 00:14:20,150
+of B then
+
+111
+00:14:20,150 --> 00:14:31,610
+النتيجة أنه يوجد C ينتمي للفترة I وهذا العدد C يقع
+
+112
+00:14:31,610 --> 00:14:32,010
+بين
+
+113
+00:14:38,340 --> 00:14:48,280
+between a and b such that بحيث ان f and c تطلع
+
+114
+00:14:48,280 --> 00:14:56,720
+بالساوية قيمة k لنعمل
+
+115
+00:14:56,720 --> 00:14:58,740
+رسمة قبل أن أظهر المظهر
+
+116
+00:15:17,560 --> 00:15:37,280
+فممكن يكون في عندي function زي هذه مثلا فهي
+
+117
+00:15:37,280 --> 00:15:43,320
+في عندي فترة I ال dialer معرفة و متصل عليها
+
+118
+00:15:45,810 --> 00:15:52,110
+يعني هذه الفترة من هنا إلى هنا I وممكن يكون في
+
+119
+00:15:52,110 --> 00:15:59,510
+عندي أعداد A وB فممكن يكون مثلا هذه ال A وهذه ال B
+
+120
+00:15:59,510 --> 00:16:04,630
+فهذه
+
+121
+00:16:04,630 --> 00:16:10,450
+F of A فهذه
+
+122
+00:16:10,450 --> 00:16:11,430
+F of A
+
+123
+00:16:16,610 --> 00:16:22,070
+وهي F of B فلو
+
+124
+00:16:22,070 --> 00:16:25,290
+كان
+
+125
+00:16:25,290 --> 00:16:38,180
+K عدد بين F of A و F of B فهي F of B وهي F of Aف K
+
+126
+00:16:38,180 --> 00:16:45,220
+عدد بين F of A و F of B فلهذا العدد نقدر نلاقي C
+
+127
+00:16:45,220 --> 00:16:49,420
+عدد C عدد
+
+128
+00:16:49,420 --> 00:16:53,960
+C بين A و B وبالتالي ينتمي للفترة I
+
+129
+00:16:57,560 --> 00:17:06,760
+إذا C بين A وB وينتمي للفترة I بحيث إن صورة C
+
+130
+00:17:06,760 --> 00:17:12,380
+هي صورة الـ C بساوي العدد P هذا هو بولزانو
+
+131
+00:17:12,380 --> 00:17:16,560
+intermediate value theorem نظرية القيمة الوسيطية
+
+132
+00:17:16,560 --> 00:17:22,740
+نظرية القيمة الوسيطية لبولزانو مرهانة نظرية هذه مش
+
+133
+00:17:22,740 --> 00:17:23,880
+صعبة سهل
+
+134
+00:17:43,340 --> 00:17:48,440
+Proof البرهان بعتمد على ال maximum minimum theorem
+
+135
+00:17:48,440 --> 00:17:55,160
+وعلى اللي هو location of roads theorem ففي عندي
+
+136
+00:17:55,160 --> 00:17:58,740
+هنا حلتين لاحظوا أن a و b أعداد دراوي
+
+137
+00:18:19,180 --> 00:18:25,800
+النتيجة بتكون واضحة لو كان a بساوي b ف f of a
+
+138
+00:18:25,800 --> 00:18:31,470
+بتطلع بساوي f of bوبالتالي اي k بين f of a وf of b
+
+139
+00:18:31,470 --> 00:18:35,690
+هيساوي واحدة منهم وبالتالي ال k بيساوي f of a خد
+
+140
+00:18:35,690 --> 00:18:40,790
+ال c بيساوي a او b فالنتيجة ايه واضح بدهية يعني
+
+141
+00:18:40,790 --> 00:18:49,030
+متحققات القائمة so assume ان
+
+142
+00:18:49,030 --> 00:18:52,630
+a لايساوي b then
+
+143
+00:18:54,390 --> 00:18:58,750
+by tricotomy property إذا كان في عددين بيسويش بعض
+
+144
+00:18:58,750 --> 00:19:06,610
+فبطلع a أصغر من b or b أصغر من a فناخد الحالة
+
+145
+00:19:06,610 --> 00:19:14,850
+الأولى case one لو كان a أصغر من b ففي الحالة هذه
+
+146
+00:19:21,390 --> 00:19:29,810
+لو كان ال a أصغر من b فبدي أعرف define
+
+147
+00:19:29,810 --> 00:19:39,130
+في الحالة هذه define g of x علي أنها الدالة
+
+148
+00:19:39,130 --> 00:19:43,990
+اللي هي بالساوي f
+
+149
+00:19:43,990 --> 00:19:47,990
+of x minus
+
+150
+00:19:47,990 --> 00:19:48,470
+k
+
+151
+00:19:51,460 --> 00:19:56,340
+فطبعا الـ function g الـ function f متصل على
+
+152
+00:19:56,340 --> 00:20:01,680
+الفترة I هو متصل على الفترة المغلقة من a إلى b
+
+153
+00:20:01,680 --> 00:20:07,080
+اللي هي جزء من الفترة I فالـ function g اللي
+
+154
+00:20:07,080 --> 00:20:11,760
+بتساوي f ثالث ثابت مثلها متصل على نفس الفترة اذا g
+
+155
+00:20:11,760 --> 00:20:18,450
+is continuous على الفترة المغلقة من a إلى bاللي هي
+
+156
+00:20:18,450 --> 00:20:22,130
+بالمناسبة مجموعة جزئية من I لأن ال A و ال B
+
+157
+00:20:22,130 --> 00:20:26,530
+موجودين في I و
+
+158
+00:20:26,530 --> 00:20:35,210
+كذلك لاحظوا أن G of A بساوي F of A minus K وهذا من
+
+159
+00:20:35,210 --> 00:20:44,570
+هنا من الفرض هذا بيطلع أصغر من سفروهذا أصغر من F
+
+160
+00:20:44,570 --> 00:20:52,070
+of B minus K F of B minus K بيطلع عموجة اللي هو
+
+161
+00:20:52,070 --> 00:20:58,110
+بساوي G of B اذا هاي شروط ال location of roots ال
+
+162
+00:20:58,110 --> 00:21:01,990
+theorem كلها متحققة هي اندي فانش جي متصلة على فترة
+
+163
+00:21:01,990 --> 00:21:06,560
+مغلقة ومحدودةوقيمة الـ G عند الـ left endpoint
+
+164
+00:21:06,560 --> 00:21:11,980
+سالبة وقيمة الـ G عند ال right endpoint موجبة and
+
+165
+00:21:11,980 --> 00:21:16,220
+then by then
+
+166
+00:21:16,220 --> 00:21:28,020
+by location of roots theorem يوجد
+
+167
+00:21:28,020 --> 00:21:37,570
+C ينتميللفترة I يعني ينتمي يوجد C ينتمي للفترة
+
+168
+00:21:37,570 --> 00:21:46,150
+المطوحة من A وB اللي هي subset من I بحيث انه صورة
+
+169
+00:21:46,150 --> 00:21:54,170
+الـ C عندها بساوي سفر لكن انا عندي G of C من تعريف
+
+170
+00:21:54,170 --> 00:22:02,490
+ال function GG of C بساوي F of C negative K حل
+
+171
+00:22:02,490 --> 00:22:09,850
+المعادلة هذه في F of C فبطلع F of C بساوي K كما هو
+
+172
+00:22:09,850 --> 00:22:15,270
+مطلوب زي ما هو مطلوب ان هيك بتكون برهانة نظرية بس
+
+173
+00:22:15,270 --> 00:22:20,540
+A في الحالة اللي فيها بتكون A أصغر من Bيبقى ندرين
+
+174
+00:22:20,540 --> 00:22:25,300
+النظرية في الحالة التالية case 2 اللي فيها ال b
+
+175
+00:22:25,300 --> 00:22:29,920
+أصغر من a ففي
+
+176
+00:22:29,920 --> 00:22:37,080
+الحالة هذه خلّيني أعرف المرة هذه function h على
+
+177
+00:22:37,080 --> 00:22:45,820
+أنها بتساوي k minus f of x فواضح clearly
+
+178
+00:22:48,290 --> 00:22:57,210
+واضح ان الـ H زيها زي ال F متصلة is continuous على
+
+179
+00:22:57,210 --> 00:23:06,690
+الفترة المغلقة والمحدودة من A ل B and H
+
+180
+00:23:06,690 --> 00:23:15,910
+of A بيساوي K minus K minus F of A بيطلع سالب K
+
+181
+00:23:15,910 --> 00:23:23,460
+minus F of Aومن الفرب هذا بيطلع سالب وهذا أصغر من
+
+182
+00:23:23,460 --> 00:23:36,300
+k minus f of b اللي هو بيطلع h of b كذلك كي لو
+
+183
+00:23:36,300 --> 00:23:41,600
+طرحت من ال k f of b فبيطلع سالب فرق إذا الأن في
+
+184
+00:23:41,600 --> 00:23:45,200
+اندي function h continuous على فترة مغلقة ومحدودة
+
+185
+00:23:45,970 --> 00:23:49,610
+وقيمتها عند الـ left endpoint سالبة وعند ال right
+
+186
+00:23:49,610 --> 00:23:58,170
+point موجبة اذا كل شروط ال location of roots في
+
+187
+00:23:58,170 --> 00:24:04,550
+المحققة so by
+
+188
+00:24:04,550 --> 00:24:12,790
+locationof roots theorem يوجد
+
+189
+00:24:12,790 --> 00:24:23,950
+C ينتمي الى الفترة مظبوط هيك؟ كده كده كده كده كده
+
+190
+00:24:23,950 --> 00:24:30,130
+كده كده كده كده
+
+191
+00:24:30,130 --> 00:24:31,150
+كده كده كده كده كده
+
+192
+00:24:36,660 --> 00:24:43,600
+هك صح K سالب F of B بطلع سالب و هنا هاد المفروض
+
+193
+00:24:43,600 --> 00:24:53,120
+تكون A و هاد A صحيح، بظبط، صح، اذا H of A اللي هي
+
+194
+00:24:53,120 --> 00:24:58,180
+K minus F of A هي K اطرح منها F of A بطلع موجة
+
+195
+00:24:58,940 --> 00:25:03,460
+بينما K سالم F of B بيطلع سالم، مظبوط هيك، إذا U
+
+196
+00:25:03,460 --> 00:25:10,920
+جان C بين B وA وهي طبعا فترة contained in R بحيث
+
+197
+00:25:10,920 --> 00:25:19,420
+انه H of C بيساوي سفر، لكن H of C من تعريفها هي
+
+198
+00:25:19,420 --> 00:25:24,400
+عبارة عن K minus F of C وبالتالي هذا بيقدر حل
+
+199
+00:25:24,400 --> 00:25:30,190
+المعادلة هذه في F of Cفبطلع F of C بساوي K وهو
+
+200
+00:25:30,190 --> 00:25:35,010
+المطلوب إذا في الحالتين أثبتنا أن يوجد C في الفترة
+
+201
+00:25:35,010 --> 00:25:43,510
+I بين A وB وقيمتها عند C بساوي LK إذا نليك بيكون
+
+202
+00:25:43,510 --> 00:25:47,930
+برهاننا Bolzano's Intermediate Value
+
+203
+00:25:55,030 --> 00:26:03,170
+الان هذه النظرية في عليها نتيجة مهمة
+
+204
+00:26:20,170 --> 00:26:26,910
+let I بساوي closed and bounded interval and if the
+
+205
+00:26:26,910 --> 00:26:37,070
+function from I to R be continuous واتصلة على
+
+206
+00:26:37,070 --> 00:26:42,590
+الفترة I تمام؟
+
+207
+00:26:42,590 --> 00:26:45,910
+لو كان
+
+208
+00:26:48,570 --> 00:27:01,130
+فك عدد حقيقي satisfies
+
+209
+00:27:01,130 --> 00:27:08,250
+بيحقق الشرط التالي انه ك .. العدد ك هذا أكبر من أو
+
+210
+00:27:08,250 --> 00:27:16,090
+ساوي ال infimum لست f of I اللي هو range ال Fاللي
+
+211
+00:27:16,090 --> 00:27:20,590
+هي القيمة الصغيرة المطلقة ل F على I وأصغر من أوسعه
+
+212
+00:27:20,590 --> 00:27:24,890
+ال supremum ل range ال F اللي هي ال absolute
+
+213
+00:27:24,890 --> 00:27:29,850
+maximum value ل ال function F على I ففي الحالة هذه
+
+214
+00:27:29,850 --> 00:27:42,260
+من نقدر نلاقي C there existC ينتمي للفترة I بحيث
+
+215
+00:27:42,260 --> 00:27:51,400
+ان F of C بيساوي العدد K وبرهان
+
+216
+00:27:51,400 --> 00:28:00,440
+النظرية هذه سهل By
+
+217
+00:28:00,440 --> 00:28:05,440
+maximum minimum theorem
+
+218
+00:28:11,040 --> 00:28:14,040
+الـ maximum minimum theorem بتقول لو كان في أندي
+
+219
+00:28:14,040 --> 00:28:18,640
+function مفتصلة على فترة مغلقة ومحدودة فال
+
+220
+00:28:18,640 --> 00:28:24,020
+function هذه بتأخذ قيمها العظمى المطلقة وقيمتها
+
+221
+00:28:24,020 --> 00:28:29,360
+العظمى المطلقة وقيمتها العظمى المطلقة على الفترة I
+
+222
+00:28:29,360 --> 00:28:34,920
+يعني في أعداد في الفترة I أندها ال function بتاخد
+
+223
+00:28:34,920 --> 00:28:37,760
+قيمتها العظمى المطلقة وقيمتها العظمى المطلقة
+
+224
+00:28:40,910 --> 00:28:51,330
+إذاً there exist x lower star و x super star عناصر
+
+225
+00:28:51,330 --> 00:29:00,870
+في I بحيث أن ال F of x lower star بساول infimum
+
+226
+00:29:01,750 --> 00:29:10,230
+لسيت f of i and f of x super star بيساوي ال
+
+227
+00:29:10,230 --> 00:29:15,050
+supremum لسيت
+
+228
+00:29:15,050 --> 00:29:26,270
+f of i تمام
+
+229
+00:29:26,270 --> 00:29:27,750
+hence
+
+230
+00:29:31,910 --> 00:29:41,670
+by حسب ال hypothesis ال hypothesis star من الفرض
+
+231
+00:29:41,670 --> 00:29:44,750
+ال star اللي هو احنا فرضين انه ال key عدد key هذا
+
+232
+00:29:44,750 --> 00:29:55,670
+بحق المتباينة يعني we have لديناالـ k أكبر من أو
+
+233
+00:29:55,670 --> 00:30:06,970
+ساوي f of x lower star أصغر من أو ساوي f
+
+234
+00:30:06,970 --> 00:30:15,370
+of upper star و
+
+235
+00:30:15,370 --> 00:30:21,630
+ال function and if is continuousعلى الفترة المغلقة
+
+236
+00:30:21,630 --> 00:30:33,690
+من x lower star ل x super star او
+
+237
+00:30:33,690 --> 00:30:41,650
+لعكس ممكن يكونوا متبادلة تانية او x super star x
+
+238
+00:30:41,650 --> 00:30:46,320
+lower starتعتمد على مين اللي أصغر من التانية إذا
+
+239
+00:30:46,320 --> 00:30:50,760
+كانت هذه أصغر من هذه فهذه تطلع فترة داخل I و F
+
+240
+00:30:50,760 --> 00:30:54,340
+continuous على I ايضا continuous على أي فترة جزئية
+
+241
+00:30:54,340 --> 00:30:58,060
+منها وإذا كان ال X Superstar أصغر من X Lower Star
+
+242
+00:30:58,060 --> 00:30:59,380
+فمناخد الفترة أيضا
+
+243
+00:31:03,410 --> 00:31:08,850
+شروط بولزانو فيروس تراسي فيرم هاي في عندي نقطتين A
+
+244
+00:31:08,850 --> 00:31:16,070
+و B بينتموا للفترة I و F continuous على I
+
+245
+00:31:28,770 --> 00:31:35,190
+وعندي a و b بينتموا للفترة I وعندي K أكبر من أو
+
+246
+00:31:35,190 --> 00:31:47,070
+ساوي F of A أصغر من أو ساوي F of B so by Bolzano's
+
+247
+00:31:57,180 --> 00:32:09,500
+يوجد C ينتمي للفترة I بين X
+
+248
+00:32:09,500 --> 00:32:21,000
+lower star و X super starبحيث ان f of c بساوي
+
+249
+00:32:21,000 --> 00:32:25,580
+العدد k وهذا
+
+250
+00:32:25,580 --> 00:32:30,820
+اللي بدنا نقيله يعني اثبتنا يوجد c ينتمي للفترة I
+
+251
+00:32:30,820 --> 00:32:38,800
+وصورة c بساوي العدد k وهو المطلوب اذا هذه النتيجة
+
+252
+00:32:38,800 --> 00:32:44,440
+على بلزانو intermediate valley theoremبرهنها بكل
+
+253
+00:32:44,440 --> 00:32:51,920
+بساطة وبكل أريحية واضح البرهان في أي استفسار أن
+
+254
+00:32:51,920 --> 00:32:55,420
+البرهان هنا تبع النظرية هذه بعتمد على maximum
+
+255
+00:32:55,420 --> 00:32:59,680
+minimum maximum minimum theorem نظرية القيام
+
+256
+00:32:59,680 --> 00:33:03,360
+القصوى نخدناها المحاضرة اللي فاتت وعلى Bolzano
+
+257
+00:33:03,360 --> 00:33:11,040
+intermediate value theorem okay
+
+258
+00:33:11,040 --> 00:33:11,480
+تمام
+
+259
+00:33:15,690 --> 00:33:23,230
+طيب ال ..
+
+260
+00:33:23,230 --> 00:33:29,290
+ناخد نظرية
+
+261
+00:33:29,290 --> 00:33:38,950
+يمكن
+
+262
+00:33:38,950 --> 00:33:43,350
+ما نحتاجش هدول نمسح
+
+263
+00:33:43,350 --> 00:33:44,010
+اللوح هذا
+
+264
+00:34:03,530 --> 00:34:11,490
+فيرم let I بساوي closed bounded interval be closed
+
+265
+00:34:11,490 --> 00:34:25,090
+and bounded closed and bounded interval and let f
+
+266
+00:34:25,090 --> 00:34:49,240
+from I to Rدي continuous متصلة على الفترة I then
+
+267
+00:34:49,240 --> 00:34:57,280
+النتيجة انه ال set او ال rangeالـ range للـ
+
+268
+00:34:57,280 --> 00:35:08,740
+function I is a closed and bounded closed and
+
+269
+00:35:08,740 --> 00:35:17,460
+bounded interval that
+
+270
+00:35:17,460 --> 00:35:18,920
+is هذا يعني
+
+271
+00:35:21,810 --> 00:35:26,930
+هذا يعني .. يعني النص او نتيجة النظرية دي من كلها
+
+272
+00:35:26,930 --> 00:35:33,850
+خصها في عبارة واحدة وهي انه a continuous function
+
+273
+00:35:33,850 --> 00:35:44,230
+a continuous function preserves .. preserves
+
+274
+00:35:44,230 --> 00:35:49,610
+بتحافظ closed
+
+275
+00:35:51,530 --> 00:35:56,510
+and bounded intervals
+
+276
+00:35:56,510 --> 00:36:00,870
+الدوال
+
+277
+00:36:00,870 --> 00:36:05,130
+المتصلة بتحافظ على ال closed و ال bounded interval
+
+278
+00:36:05,130 --> 00:36:10,350
+يعني ال function f بتاخد I اللي هي closed bounded
+
+279
+00:36:10,350 --> 00:36:13,610
+interval بتعطيني صلتها closed bounded interval
+
+280
+00:36:13,610 --> 00:36:19,410
+زيها من نفس الصنف من نفس النوع لبرهان ذلك
+
+281
+00:36:29,060 --> 00:36:44,440
+ف let M بساوي الالفمن ل range ال F و
+
+282
+00:36:44,440 --> 00:36:50,720
+capital M بساوي ال superman ل range ال F
+
+283
+00:36:56,430 --> 00:37:11,770
+بOTH M AND N EXIST IN R BY MAXIMUM
+
+284
+00:37:11,770 --> 00:37:15,290
+MINIMUM THEOREM
+
+285
+00:37:19,900 --> 00:37:25,120
+نظرية القيام القصوى بتقول إنه إذا كانت f function
+
+286
+00:37:25,120 --> 00:37:30,180
+متصة على closed bounded interval فال .. ال .. ال
+
+287
+00:37:30,180 --> 00:37:34,560
+function إلها قيمة صغيرة مطلقة و إلها قيمة أضمة
+
+288
+00:37:34,560 --> 00:37:39,300
+مطلقة سمها قيمة صغيرة المطلقة M و قيمة الأضمة
+
+289
+00:37:39,300 --> 00:37:44,760
+المطلقة capital M تمام؟
+
+290
+00:37:44,760 --> 00:37:47,580
+clearly
+
+291
+00:37:54,210 --> 00:38:02,370
+F of X أكبر من أو ساوي M أصغر من أو ساوي م لكل X
+
+292
+00:38:02,370 --> 00:38:10,630
+في I قيمة الدالة عند أي X في المجال تبعها أصغر من
+
+293
+00:38:10,630 --> 00:38:15,090
+أو ساوي قيمة العظمى المطلقة و أكبر في نفس المجال
+
+294
+00:38:15,090 --> 00:38:17,370
+أكبر من أو ساوي قيمة الصغر المطلقة
+
+295
+00:38:21,460 --> 00:38:26,040
+فهذا بيقدي which
+
+296
+00:38:26,040 --> 00:38:40,440
+implies هذا بيقدي انه ال .. انه f of I contained
+
+297
+00:38:40,440 --> 00:38:47,680
+في الفترة المغلقة من small m لcapital Mالمتبادلة
+
+298
+00:38:47,680 --> 00:38:53,400
+الأخيرة هذه تثبت أن ال set هذه subset من هذه لأنه
+
+299
+00:38:53,400 --> 00:38:59,880
+خدوا أي عنصر هنا فأي عنصر هنا عبارة عن f of x for
+
+300
+00:38:59,880 --> 00:39:07,100
+some x ينتمي ل I صح فأي f of x for some x ينتمي ل
+
+301
+00:39:07,100 --> 00:39:12,730
+I هيمحصور من small m وcapital M وبالتالي ينتمي
+
+302
+00:39:12,730 --> 00:39:16,610
+للفترة المغلقة هذه، لأن كل أنصر أنا هو أنصر في
+
+303
+00:39:16,610 --> 00:39:22,490
+الفترة المغلقة، لأن هذا الاحتواء صحيح، تمام؟ الان
+
+304
+00:39:22,490 --> 00:39:24,770
+احنا بنثبت المساواة
+
+305
+00:39:30,090 --> 00:39:36,190
+إن ال range لل function f بساوي كل الفترة المغلقة
+
+306
+00:39:36,190 --> 00:39:44,150
+من small m لcapital M فلإثبات
+
+307
+00:39:44,150 --> 00:39:55,350
+ذلك هي عندي أنا to prove this it
+
+308
+00:39:55,350 --> 00:39:56,090
+remains
+
+309
+00:39:59,390 --> 00:40:07,930
+it remains to show يبقى اثبات دا في اثبات ان احنا
+
+310
+00:40:07,930 --> 00:40:12,370
+لثبت الاحتواء المعاكس the reverse inclusion
+
+311
+00:40:18,550 --> 00:40:28,670
+إذا يبقى إثبات إن الفترة المغلقة من small m to
+
+312
+00:40:28,670 --> 00:40:35,970
+capital M contained in F of I فكيف نثبت إحنا set
+
+313
+00:40:35,970 --> 00:40:41,570
+subset من الأخرى نسميه
+
+314
+00:40:41,570 --> 00:40:45,950
+برهان بإيه بتتبع العناصر يعني بناخد أنصر في
+
+315
+00:40:45,950 --> 00:40:49,540
+المجموعة الأولىنثبت العناصر في المجموعة التانية
+
+316
+00:40:49,540 --> 00:40:53,080
+هذا بيسموه في رياضيات الـ chasing of elements
+
+317
+00:40:53,080 --> 00:41:03,880
+argument برهان بتطبع العناصر فقالت why تنتمي
+
+318
+00:41:03,880 --> 00:41:12,580
+للفترة المغلقة من small m لcapital M طيب هذا
+
+319
+00:41:12,580 --> 00:41:19,540
+بيقدّيالـ y أكبر من أو ساوي small m أصغر من أو
+
+320
+00:41:19,540 --> 00:41:31,720
+ساوي capital M وهذا عبارة عن الـ infimum لـ range
+
+321
+00:41:31,720 --> 00:41:39,460
+الـ function f وهذا بساوي الـ supremum لـ range
+
+322
+00:41:39,460 --> 00:41:40,500
+الـ function f
+
+323
+00:41:47,540 --> 00:41:57,540
+وعندي ال .. إذا حسب ال .. ال corollary تبع النظرية
+
+324
+00:41:57,540 --> 00:42:04,060
+هذه فإن عندي ال function if continuous على الفترة
+
+325
+00:42:04,060 --> 00:42:10,520
+المغلقة a,b فعندي if continuous على الفترة المغلقة
+
+326
+00:42:10,520 --> 00:42:18,800
+a,bوعندي k اللي هو y عدد محصور بين ال infimum ل f
+
+327
+00:42:18,800 --> 00:42:28,520
+of i و ال suprem ل f of i by
+
+328
+00:42:28,520 --> 00:42:36,090
+above corollaryالقرن اللي لـ Bolzano Intermediate
+
+329
+00:42:36,090 --> 00:42:43,390
+Value Theorem يقول إن يوجد C ينتمي للفترة I بحيث
+
+330
+00:42:43,390 --> 00:42:50,090
+أن F of C بساوي
+
+331
+00:42:50,090 --> 00:42:58,170
+العدد Y اللي هو قابل الـK في نص النظريةالـ C ينتمي
+
+332
+00:42:58,170 --> 00:43:05,170
+لـ I إذاً F of C تنتمي لـ F للست F of I إذا هاني
+
+333
+00:43:05,170 --> 00:43:09,410
+بدأت بـ Y ينتمي للفترة المغلقة طلع Y ينتمي لـ F of
+
+334
+00:43:09,410 --> 00:43:17,570
+I Therefore Hence هيك
+
+335
+00:43:17,570 --> 00:43:20,750
+منكون أثباتنا أن الفترة المغلقة من small m
+
+336
+00:43:20,750 --> 00:43:31,990
+لcapital M is containedفي ال set f of i هذا
+
+337
+00:43:31,990 --> 00:43:37,490
+ببرهن ال claim و النظرية لأن هيك بيكون برهننا ال
+
+338
+00:43:37,490 --> 00:43:42,010
+claim وبالتالي برهننا النظرية لأن هيك هي اثبتت ان
+
+339
+00:43:42,010 --> 00:43:45,950
+ال image ل ال closed bounded interval I طلعت
+
+340
+00:43:45,950 --> 00:43:49,970
+closed bounded interval صح و هو المطلوب
+
+341
+00:43:53,870 --> 00:44:00,730
+Okay واضح البرهان؟ في أي استفسار على البرهان؟
+
+342
+00:44:00,730 --> 00:44:08,030
+في هنا تحذير warning تحذير
+
+343
+00:44:08,030 --> 00:44:15,070
+in
+
+344
+00:44:15,070 --> 00:44:30,000
+the above theorem we hadF of I التي هي F للفترة
+
+345
+00:44:30,000 --> 00:44:35,320
+المغلقة من A لB طلعت
+
+346
+00:44:35,320 --> 00:44:39,900
+بالساوي الفترة المغلقة من small m لcapital M حيث
+
+347
+00:44:39,900 --> 00:44:43,980
+small m is the absolute minimum value وcapital M
+
+348
+00:44:43,980 --> 00:44:47,140
+is the absolute maximum value of the function F on
+
+349
+00:44:47,140 --> 00:44:54,980
+the interval Iو هذا ليس بالضرورة مش شرط هذه الفترة
+
+350
+00:44:54,980 --> 00:45:03,750
+تكون الفترة من F of A ل F of Bهذه الفترة ماحدش جال
+
+351
+00:45:03,750 --> 00:45:08,370
+او مقدر يزم انها الفترة المغلقة من F of A لF of B
+
+352
+00:45:08,370 --> 00:45:13,750
+هذا مش صحيح okay النظرية ما بتقولي الكلام هذا
+
+353
+00:45:13,750 --> 00:45:18,490
+بتقولي الكلام هذا فقط هذا غلط مش شرط ال image
+
+354
+00:45:18,490 --> 00:45:23,050
+للفترة I بالساوي الفترة المغلقة من F of A لF of B
+
+355
+00:45:23,050 --> 00:45:30,640
+فاخدوا بالكم من ايه من التحذير هذاOkay إذا هين
+
+356
+00:45:30,640 --> 00:45:34,720
+أثبتنا إن لو كانت ال function تبعتي متصلة على فترة
+
+357
+00:45:34,720 --> 00:45:39,420
+مغلقة أو محدودة فصورتها بتطلع مغلقة أو محدودة
+
+358
+00:45:39,420 --> 00:45:45,280
+وبالتالي ال function preserves ال .. ال .. ال
+
+359
+00:45:45,280 --> 00:45:50,640
+intervals طيب
+
+360
+00:45:50,640 --> 00:45:53,940
+ال .. النظرية دي إلها تعميم
+
+361
+00:46:03,890 --> 00:46:10,730
+preservation of intervals
+
+362
+00:46:10,730 --> 00:46:14,770
+theorem لو
+
+363
+00:46:14,770 --> 00:46:21,050
+كانت الفترة let I be any interval مش شرط تكون ..
+
+364
+00:46:21,050 --> 00:46:28,070
+مش شرط تكون close about it .. be any interval and
+
+365
+00:46:28,070 --> 00:46:43,200
+letإذا من I إلى R يكون مستمر على الفترة I ثم
+
+366
+00:46:43,200 --> 00:46:49,260
+ستة F من I هي عرفة
+
+367
+00:46:53,620 --> 00:46:57,220
+النظرية هذه بتقول لو كانت f دالة متصلة المجال
+
+368
+00:46:57,220 --> 00:47:03,960
+تبعها أي فترة مغلقة، محدودة، مش محدودة، half-open،
+
+369
+00:47:03,960 --> 00:47:06,580
+open-half-open interval اللي لقاش، أي لوحة من ال
+
+370
+00:47:06,580 --> 00:47:11,820
+intervals اللي شفناهم في جبتر واحد فصورتها أيضا
+
+371
+00:47:11,820 --> 00:47:15,500
+لازم تطلع interval وبالتالي ال continuous function
+
+372
+00:47:15,500 --> 00:47:19,880
+بتحافظ على الفترة، على الفترات يعني بتاكن فترة في
+
+373
+00:47:19,880 --> 00:47:24,780
+مجالها بتعطيل صورتها فترةالفترة هذه ما بنعرفش كيف
+
+374
+00:47:24,780 --> 00:47:29,620
+نوعها لكن اللي بنقدر نزّمه في النظرية السابقة أنه
+
+375
+00:47:29,620 --> 00:47:33,300
+لو كانت الفترة I هذه closed bounded فصورتها هتطلع
+
+376
+00:47:33,300 --> 00:47:36,960
+closed bounded أما لو كانت من نوع أخر فصورتها مش
+
+377
+00:47:36,960 --> 00:47:42,200
+شرط تكون من نفس النوع ماحدش جال الكلام هذا فلبرهان
+
+378
+00:47:42,200 --> 00:47:47,780
+ذلك لبرهان
+
+379
+00:47:47,780 --> 00:47:48,260
+ذلك
+
+380
+00:47:53,440 --> 00:48:01,040
+خلّينا ناخد let alpha و beta belong to except f of
+
+381
+00:48:01,040 --> 00:48:14,580
+I with alpha أصغر من beta خلّينا
+
+382
+00:48:14,580 --> 00:48:15,640
+نستذكر بس
+
+383
+00:48:22,460 --> 00:48:27,560
+في نظرية أخدناها قبل هيك ال theorem اتنين خمسة
+
+384
+00:48:27,560 --> 00:48:44,300
+واحد بتقول if S subset of R contains at least two
+
+385
+00:48:44,300 --> 00:48:48,500
+elements and satisfies
+
+386
+00:48:52,530 --> 00:48:58,810
+Satisfies الخاصية واحد إن لو كان X و Y تلوي ل S و
+
+387
+00:48:58,810 --> 00:49:04,850
+X أصغر من Y هذا بيقدي إن الفترة من X إلى Y
+
+388
+00:49:04,850 --> 00:49:10,670
+contained in S then
+
+389
+00:49:10,670 --> 00:49:13,790
+set S is an interval
+
+390
+00:49:17,430 --> 00:49:19,470
+إن ان هذه النظرية أخدناها في ال chapter .. في ال
+
+391
+00:49:19,470 --> 00:49:23,830
+chapter الأولاري بتقول لو كان في عندي set subset
+
+392
+00:49:23,830 --> 00:49:29,520
+من R فيها على الأقل أنصر Lوبتحقق ال set هذه بتحقق
+
+393
+00:49:29,520 --> 00:49:33,580
+الخاصية واحد property one انه لأي x و y في ال set
+
+394
+00:49:33,580 --> 00:49:39,300
+و x أصغر من y الفترة من x ل y بتكون موجودة داخل ال
+
+395
+00:49:39,300 --> 00:49:43,880
+set في الحالة هذه ال set نفسها S تطلع interval اذا
+
+396
+00:49:43,880 --> 00:49:50,820
+انا بدي اثبت to show طيب
+
+397
+00:49:50,820 --> 00:49:51,780
+انا عندي
+
+398
+00:49:54,280 --> 00:50:00,060
+هذه أخدت نقطتين في ال set هذه هي ال set ال set S
+
+399
+00:50:00,060 --> 00:50:04,600
+هذه أخدت نقطتين و Alpha أصغر من Beta و بتثبت أنها
+
+400
+00:50:04,600 --> 00:50:08,700
+بتحقق الخاصية واحد عشان أثبت أنها interval أنا
+
+401
+00:50:08,700 --> 00:50:13,160
+عندي Alpha و Beta تنتمي ل F of I لأن Alpha بتساوي
+
+402
+00:50:13,160 --> 00:50:22,570
+F of Afor some a تنتمي إلى I و Beta بساوي F of B
+
+403
+00:50:22,570 --> 00:50:30,330
+for some B تنتمي إلى I وبالتالي
+
+404
+00:50:30,330 --> 00:50:36,950
+..
+
+405
+00:50:36,950 --> 00:50:39,790
+بالتالي ..
+
+406
+00:50:47,200 --> 00:50:59,180
+انا عندي ال bolzanova طيب طيب to show to
+
+407
+00:50:59,180 --> 00:51:09,420
+show f of I is an interval we
+
+408
+00:51:09,420 --> 00:51:20,550
+need to showان الset f of i satisfies property
+
+409
+00:51:20,550 --> 00:51:23,830
+واحد
+
+410
+00:51:23,830 --> 00:51:33,430
+of theorem اتنين خمسة واحد فهي
+
+411
+00:51:33,430 --> 00:51:36,830
+عندي alpha و beta تنتمي ل f of i و alpha أصغر من
+
+412
+00:51:36,830 --> 00:51:40,690
+beta ف
+
+413
+00:51:40,690 --> 00:51:42,290
+to show
+
+414
+00:51:44,880 --> 00:51:56,020
+الفترة من Alpha إلى Beta content in F of I let K
+
+415
+00:51:56,020 --> 00:52:00,300
+ينتمي إلى الفترة المغلقة من Alpha إلى Beta
+
+416
+00:52:04,220 --> 00:52:11,680
+أكبر من أو يساوي alpha هي بيساوي f of a وأصغر من
+
+417
+00:52:11,680 --> 00:52:21,460
+أو يساوي beta هي بيساوي f of b وبالتالي so by
+
+418
+00:52:21,460 --> 00:52:26,440
+Bolzano's
+
+419
+00:52:26,440 --> 00:52:31,120
+intermediate
+
+420
+00:52:31,120 --> 00:52:32,940
+value theorem
+
+421
+00:52:35,620 --> 00:52:48,960
+يوجد K عفوا يوجد C ينتمي إلى I between Alpha
+
+422
+00:52:48,960 --> 00:52:59,400
+و Beta بحيث ان F of C بساوي K او
+
+423
+00:52:59,400 --> 00:53:08,640
+K بساوي F of Cطبما ال C تنتمي ل I إذا F of C تنتمي
+
+424
+00:53:08,640 --> 00:53:13,380
+ل F of I إذا
+
+425
+00:53:13,380 --> 00:53:18,200
+هاني أثبتت إنه كل K في الفترة المغلقة من Alpha إلى
+
+426
+00:53:18,200 --> 00:53:25,850
+Beta طلع ينتمي لفترة F of I وبالتالي إذابطلع عند
+
+427
+00:53:25,850 --> 00:53:31,270
+الفترة المغلقة من Alpha إلى Beta ال subset من F of
+
+428
+00:53:31,270 --> 00:53:38,830
+I وبالتالي إذا ال set F of I بتحقق ال property
+
+429
+00:53:38,830 --> 00:53:46,810
+واحد إذا by theorem .. by theorem اتنين خمسة واحد
+
+430
+00:53:46,810 --> 00:53:53,650
+ال set F of I بتطلع interval is an interval
+
+431
+00:53:56,670 --> 00:54:03,570
+و هذا بيكمل النظرية اذا هذا بيكمل البرهان هيك
+
+432
+00:54:03,570 --> 00:54:10,630
+بنكون خلصنا ال section خمسة تلاتة و باقي عننا
+
+433
+00:54:10,630 --> 00:54:16,190
+section خمسة أربعة هناخده في المحاضرة الجاية نحاول
+
+434
+00:54:16,190 --> 00:54:24,130
+نشوف زمنا نخلصه ولا لأفال .. شكرا لحصن إصراعكم و
+
+435
+00:54:24,130 --> 00:54:26,910
+يعطيكم العافية و نشوفكم ان شاء الله المرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..502ba7bda5cee06ac757e9bd638fac6d27a5e325
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_raw.json
@@ -0,0 +1 @@
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fails to be continuous at C we say ان F is discontinuous", "tokens": [1211, 2407, 25961, 2655, 2423, 479, 32239, 14851, 44650, 36520, 3660, 43242, 28239, 47432, 3660, 383, 6156, 3555, 1863, 39648, 498, 479, 18199, 281, 312, 10957, 412, 383, 321, 584, 16472, 479, 307, 31420, 12549], "avg_logprob": -0.2510850793785519, "compression_ratio": 1.1300813008130082, "no_speech_prob": 0.0, "words": [{"start": 306.75, "end": 307.09, "word": "لو", "probability": 0.854248046875}, {"start": 307.09, "end": 307.47, "word": " كانت", "probability": 0.8076171875}, {"start": 307.47, "end": 307.59, "word": " ال", "probability": 0.76318359375}, {"start": 307.59, "end": 307.75, "word": " F", "probability": 0.33984375}, {"start": 307.75, "end": 308.21, "word": " ليست", "probability": 0.893798828125}, {"start": 308.21, "end": 308.89, "word": " متصلة", "probability": 0.86181640625}, {"start": 308.89, "end": 309.09, "word": " عند", "probability": 0.97802734375}, {"start": 309.09, "end": 309.99, "word": " 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{"start": 666.76, "end": 667.22, "word": " neighborhood", "probability": 0.875}, {"start": 667.22, "end": 667.82, "word": " هذا", "probability": 0.64697265625}, {"start": 667.82, "end": 670.22, "word": " بـ", "probability": 0.3992919921875}, {"start": 670.22, "end": 670.62, "word": " Delta", "probability": 0.466796875}, {"start": 670.62, "end": 674.58, "word": " يعني", "probability": 0.7003173828125}, {"start": 674.58, "end": 675.96, "word": " أكوّن", "probability": 0.961328125}, {"start": 675.96, "end": 676.32, "word": " Delta", "probability": 0.64013671875}, {"start": 676.32, "end": 676.8, "word": " neighborhood", "probability": 0.9140625}, {"start": 676.8, "end": 677.38, "word": " هنا", "probability": 0.9384765625}, {"start": 677.38, "end": 679.08, "word": " C", "probability": 0.703125}, {"start": 679.08, "end": 679.58, "word": " minus", "probability": 0.44140625}, {"start": 679.58, "end": 680.04, "word": " Delta", "probability": 0.79541015625}, {"start": 680.04, "end": 680.48, 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هو مقاومة مقاومة A ثم F مستمر في X بساوي", "tokens": [28814, 15730, 383, 31439, 3714, 4587, 995, 20498, 3660, 3714, 4587, 995, 20498, 3660, 316, 38637, 2304, 479, 3714, 14851, 29973, 8978, 1783, 4724, 3794, 995, 45865], "avg_logprob": -0.6863839541162763, "compression_ratio": 1.1, "no_speech_prob": 0.0, "words": [{"start": 856.0, "end": 856.48, "word": "إذا", "probability": 0.659423828125}, {"start": 856.48, "end": 857.1, "word": " C", "probability": 0.461181640625}, {"start": 857.1, "end": 857.6, "word": " هو", "probability": 0.29443359375}, {"start": 857.6, "end": 858.4, "word": " مقاومة", "probability": 0.4960693359375}, {"start": 858.4, "end": 862.64, "word": " مقاومة", "probability": 0.624169921875}, {"start": 862.64, "end": 862.64, "word": " A", "probability": 0.1885986328125}, {"start": 862.64, "end": 870.18, "word": " ثم", "probability": 0.56298828125}, {"start": 870.18, "end": 872.64, "word": " F", "probability": 0.548828125}, {"start": 872.64, "end": 873.94, "word": " مستمر", 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"probability": 0.8878580729166666}, {"start": 968.48, "end": 968.84, "word": " الدالة", "probability": 0.9920247395833334}, {"start": 968.84, "end": 969.06, "word": " تكون", "probability": 0.980224609375}, {"start": 969.06, "end": 969.64, "word": " متصلة", "probability": 0.89404296875}, {"start": 969.64, "end": 969.98, "word": " عند", "probability": 0.923828125}, {"start": 969.98, "end": 970.58, "word": " النقطة", "probability": 0.9298502604166666}, {"start": 970.58, "end": 970.78, "word": " c", "probability": 0.74267578125}, {"start": 970.78, "end": 970.98, "word": " في", "probability": 0.95068359375}, {"start": 970.98, "end": 971.56, "word": " مجالها", "probability": 0.9959716796875}], "temperature": 1.0}, {"id": 34, "seek": 97642, "start": 973.54, "end": 976.42, "text": "و لو كانت الـ C هي cluster point طبعا", "tokens": [2407, 45164, 25961, 2655, 2423, 39184, 383, 39896, 13630, 935, 23032, 3555, 3615, 995], "avg_logprob": -0.540625023841858, "compression_ratio": 0.8833333333333333, "no_speech_prob": 0.0, "words": [{"start": 973.54, "end": 974.26, "word": "و", "probability": 0.458251953125}, {"start": 974.26, "end": 974.42, "word": " لو", "probability": 0.2626953125}, {"start": 974.42, "end": 974.8, "word": " كانت", "probability": 0.886962890625}, {"start": 974.8, "end": 974.94, "word": " الـ", "probability": 0.6380615234375}, {"start": 974.94, "end": 975.06, "word": " C", "probability": 0.494873046875}, {"start": 975.06, "end": 975.22, "word": " هي", "probability": 0.244140625}, {"start": 975.22, "end": 975.58, "word": " cluster", "probability": 0.66064453125}, {"start": 975.58, "end": 976.02, "word": " point", "probability": 0.90966796875}, {"start": 976.02, "end": 976.42, "word": " طبعا", "probability": 0.8941650390625}], "temperature": 1.0}, {"id": 35, "seek": 100497, "start": 978.09, "end": 1004.97, "text": "أو حتى لو ماكنتش cluster point فلازم التلاتة صوروطها تتحقق الدالة معرفة عن C طبعا هذا لأن C نقطة في مجال الدالة فلازم تكون معرفة عن ده لازم ال limit ل F عن C تكون موجودة وقيمة ال limit بساوي قيمة الدالة عند النقطة C لو أي واحد ماليش صوروط التلاتة هدول اختل فبنقول ان ال function مش متصلة عند النقطة C", "tokens": [10721, 2407, 11331, 49975, 45164, 19446, 19452, 2655, 8592, 13630, 935, 6156, 1211, 31377, 2304, 16712, 1211, 9307, 3660, 20328, 13063, 2407, 9566, 11296, 6055, 2655, 5016, 4587, 4587, 32748, 6027, 3660, 20449, 28480, 3660, 18871, 383, 23032, 3555, 3615, 995, 23758, 5296, 33456, 383, 8717, 47432, 3660, 8978, 3714, 7435, 6027, 32748, 6027, 3660, 6156, 1211, 31377, 2304, 6055, 30544, 20449, 28480, 3660, 18871, 11778, 3224, 5296, 31377, 2304, 2423, 4948, 5296, 479, 18871, 383, 6055, 30544, 3714, 29245, 23328, 3660, 4032, 4587, 32640, 3660, 2423, 4948, 4724, 3794, 995, 45865, 12174, 32640, 3660, 32748, 6027, 3660, 43242, 28239, 47432, 3660, 383, 45164, 36632, 36764, 24401, 19446, 20292, 8592, 20328, 13063, 2407, 9566, 16712, 1211, 9307, 3660, 8032, 3215, 12610, 1975, 46456, 1211, 6156, 3555, 1863, 39648, 16472, 2423, 2445, 37893, 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{"start": 1023.85, "end": 1024.01, "word": " C", "probability": 0.97900390625}, {"start": 1024.01, "end": 1024.53, "word": " تنتمي", "probability": 0.982177734375}, {"start": 1024.53, "end": 1024.65, "word": " ل", "probability": 0.5595703125}, {"start": 1024.65, "end": 1024.93, "word": " A", "probability": 0.6962890625}, {"start": 1024.93, "end": 1025.51, "word": " اه", "probability": 0.773681640625}, {"start": 1025.51, "end": 1025.89, "word": " طبعا", "probability": 0.98583984375}, {"start": 1025.89, "end": 1026.07, "word": " ال", "probability": 0.9794921875}, {"start": 1026.07, "end": 1026.23, "word": " C", "probability": 0.97705078125}, {"start": 1026.23, "end": 1026.85, "word": " تنتمي", "probability": 0.9820556640625}, {"start": 1026.85, "end": 1026.99, "word": " ل", "probability": 0.94921875}, {"start": 1026.99, "end": 1027.29, "word": " A", "probability": 0.92578125}, {"start": 1027.29, "end": 1029.31, "word": " ال", "probability": 0.246826171875}, {"start": 1029.31, "end": 1029.51, "word": " C", "probability": 0.97607421875}, {"start": 1029.51, "end": 1030.01, "word": " دائما", "probability": 0.86279296875}, {"start": 1030.01, "end": 1030.77, "word": " تنتمي", "probability": 0.990234375}, {"start": 1030.77, "end": 1030.91, "word": " ل", "probability": 0.96923828125}, {"start": 1030.91, "end": 1031.13, "word": " A", "probability": 0.90771484375}], "temperature": 1.0}, {"id": 37, "seek": 106409, "start": 1037.1, "end": 1064.1, "text": "طب لو ماكناش ال c cluster point الملاحظة التانية if c is not يعني لو كان ال c تنتمي طبعا دايما ال c تنتمي ل a is not a cluster point is not cluster point of a", "tokens": [9566, 3555, 45164, 19446, 19452, 33599, 2423, 269, 13630, 935, 9673, 15040, 5016, 19913, 3660, 16712, 7649, 10632, 498, 269, 307, 406, 37495, 22653, 45164, 25961, 2423, 269, 6055, 29399, 2304, 1829, 23032, 3555, 3615, 995, 11778, 47302, 15042, 2423, 269, 6055, 29399, 2304, 1829, 5296, 257, 307, 406, 257, 13630, 935, 307, 406, 13630, 935, 295, 257], 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1062.56, "end": 1063.76, "word": " of", "probability": 0.97705078125}, {"start": 1063.76, "end": 1064.1, "word": " a", "probability": 0.86962890625}], "temperature": 1.0}, {"id": 38, "seek": 108684, "start": 1068.95, "end": 1086.85, "text": "then من تعريف ال cluster point لازم نلاقي delta أكبر من السفر such that a تقاطع v delta of c بساوي singleton c", "tokens": [19096, 9154, 37279, 16572, 5172, 2423, 13630, 935, 5296, 31377, 2304, 8717, 15040, 38436, 8289, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 1270, 300, 257, 6055, 4587, 41193, 3615, 371, 8289, 295, 269, 4724, 3794, 995, 45865, 1522, 14806, 269], "avg_logprob": -0.18206936557118486, "compression_ratio": 1.1732283464566928, "no_speech_prob": 0.0, "words": [{"start": 1068.95, "end": 1069.51, "word": "then", "probability": 0.1217041015625}, {"start": 1069.51, "end": 1070.23, "word": " من", "probability": 0.9619140625}, {"start": 1070.23, "end": 1070.65, "word": " تعريف", "probability": 0.9689127604166666}, {"start": 1070.65, "end": 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{"start": 1105.02, "end": 1105.32, "word": " في", "probability": 0.92431640625}, {"start": 1105.32, "end": 1105.82, "word": " نقطة", "probability": 0.9847005208333334}, {"start": 1105.82, "end": 1106.46, "word": " مختلفة", "probability": 0.97998046875}, {"start": 1106.46, "end": 1106.6, "word": " عن", "probability": 0.96142578125}, {"start": 1106.6, "end": 1106.76, "word": " ال", "probability": 0.56005859375}, {"start": 1106.76, "end": 1107.0, "word": " C", "probability": 0.78076171875}, {"start": 1107.0, "end": 1107.28, "word": " على", "probability": 0.93505859375}, {"start": 1107.28, "end": 1107.78, "word": " الأقل", "probability": 0.9285481770833334}], "temperature": 1.0}, {"id": 40, "seek": 112294, "start": 1108.96, "end": 1122.94, "text": "معناه ان الـ C ما تكونش cluster point معناه ان يوجد delta neighborhood واحد يعني يوجد delta عدد موجب وبالتالي يوجد على الأقل delta neighborhood للـ C وهذا ال delta neighborhood", "tokens": [2304, 3615, 8315, 3224, 16472, 2423, 39184, 383, 19446, 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"probability": 0.67138671875}, {"start": 1307.11, "end": 1314.95, "word": " فكان", "probability": 0.90087890625}, {"start": 1314.95, "end": 1315.11, "word": " في", "probability": 0.93896484375}, {"start": 1315.11, "end": 1315.35, "word": " عندي", "probability": 0.782470703125}, {"start": 1315.35, "end": 1315.85, "word": " sequential", "probability": 0.9453125}, {"start": 1315.85, "end": 1316.71, "word": " criterion", "probability": 0.9521484375}, {"start": 1316.71, "end": 1317.03, "word": " for", "probability": 0.9521484375}, {"start": 1317.03, "end": 1317.59, "word": " limits", "probability": 0.9541015625}], "temperature": 1.0}, {"id": 49, "seek": 134511, "start": 1322.27, "end": 1345.11, "text": "بنفس الطريقة في هنا sequential criterion for continuity للاتصال إذا في عندي هنا sequential criterion for continuity", "tokens": [3555, 1863, 36178, 41950, 16572, 28671, 8978, 34105, 42881, 46691, 337, 23807, 24976, 9307, 9381, 6027, 11933, 15730, 8978, 18871, 16254, 34105, 42881, 46691, 337, 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0.970703125}, {"start": 1361.45, "end": 1361.99, "word": " a", "probability": 0.9501953125}, {"start": 1361.99, "end": 1364.43, "word": " then", "probability": 0.7236328125}, {"start": 1364.43, "end": 1366.43, "word": " the", "probability": 0.8544921875}, {"start": 1366.43, "end": 1367.19, "word": " following", "probability": 0.888671875}, {"start": 1367.19, "end": 1368.33, "word": " statements", "probability": 0.89306640625}, {"start": 1368.33, "end": 1370.99, "word": " are", "probability": 0.955078125}, {"start": 1370.99, "end": 1371.71, "word": " equivalent", "probability": 0.9501953125}, {"start": 1371.71, "end": 1376.17, "word": " واحد", "probability": 0.90966796875}, {"start": 1376.17, "end": 1377.95, "word": " if", "probability": 0.56494140625}, {"start": 1377.95, "end": 1378.67, "word": " is", "probability": 0.9306640625}, {"start": 1378.67, "end": 1379.45, "word": " continuous", "probability": 0.89453125}], "temperature": 1.0}, {"id": 51, "seek": 140967, "start": 1381.35, "end": 1409.67, "text": "if is continuous at c for every for every sequence x in contained in a with limit x in بساوي c", "tokens": [351, 307, 10957, 412, 269, 337, 633, 337, 633, 8310, 2031, 294, 16212, 294, 257, 365, 4948, 2031, 294, 4724, 3794, 995, 45865, 269], "avg_logprob": -0.37625000953674315, "compression_ratio": 1.1379310344827587, "no_speech_prob": 0.0, "words": [{"start": 1381.35, "end": 1381.75, "word": "if", "probability": 0.03656005859375}, {"start": 1381.75, "end": 1382.05, "word": " is", "probability": 0.6279296875}, {"start": 1382.05, "end": 1382.59, "word": " continuous", "probability": 0.880859375}, {"start": 1382.59, "end": 1383.27, "word": " at", "probability": 0.94921875}, {"start": 1383.27, "end": 1383.89, "word": " c", "probability": 0.60205078125}, {"start": 1383.89, "end": 1388.01, "word": " for", "probability": 0.338134765625}, {"start": 1388.01, "end": 1388.89, "word": " every", "probability": 0.78955078125}, {"start": 1388.89, "end": 1391.91, "word": " for", "probability": 0.53466796875}, {"start": 1391.91, "end": 1392.75, "word": " every", "probability": 0.80615234375}, {"start": 1392.75, "end": 1393.55, "word": " sequence", "probability": 0.97314453125}, {"start": 1393.55, "end": 1395.65, "word": " x", "probability": 0.89501953125}, {"start": 1395.65, "end": 1396.07, "word": " in", "probability": 0.5087890625}, {"start": 1396.07, "end": 1397.05, "word": " contained", "probability": 0.572265625}, {"start": 1397.05, "end": 1397.33, "word": " in", "probability": 0.94921875}, {"start": 1397.33, "end": 1397.81, "word": " a", "probability": 0.6796875}, {"start": 1397.81, "end": 1402.05, "word": " with", "probability": 0.8896484375}, {"start": 1402.05, "end": 1405.37, "word": " limit", "probability": 0.9638671875}, {"start": 1405.37, "end": 1407.29, "word": " x", "probability": 0.97705078125}, {"start": 1407.29, "end": 1407.95, "word": " in", "probability": 0.62353515625}, {"start": 1407.95, "end": 1409.13, "word": " بساوي", "probability": 0.840576171875}, {"start": 1409.13, "end": 1409.67, "word": " c", "probability": 0.8486328125}], "temperature": 1.0}, {"id": 52, "seek": 142579, "start": 1411.09, "end": 1425.79, "text": "نحن لدينا ان ال limit ل f of x n as n tensor infinity بسوي f of c", "tokens": [1863, 5016, 1863, 5296, 16254, 8315, 16472, 2423, 4948, 5296, 283, 295, 2031, 297, 382, 297, 2064, 539, 81, 13202, 4724, 3794, 2407, 1829, 283, 295, 269], "avg_logprob": -0.5764508779559817, "compression_ratio": 0.9879518072289156, "no_speech_prob": 0.0, "words": [{"start": 1411.09, "end": 1411.71, "word": "نحن", "probability": 0.4895426432291667}, {"start": 1411.71, "end": 1414.13, "word": " لدينا", "probability": 0.6939697265625}, {"start": 1414.13, "end": 1414.63, "word": " ان", "probability": 0.3720703125}, {"start": 1414.63, "end": 1414.81, "word": " ال", "probability": 0.2156982421875}, {"start": 1414.81, "end": 1415.27, "word": " limit", "probability": 0.2066650390625}, {"start": 1415.27, "end": 1417.25, "word": " ل", "probability": 0.84033203125}, {"start": 1417.25, "end": 1417.69, "word": " f", "probability": 0.322998046875}, {"start": 1417.69, "end": 1417.97, "word": " of", "probability": 0.314697265625}, {"start": 1417.97, "end": 1418.57, "word": " x", "probability": 0.9150390625}, {"start": 1418.57, "end": 1419.29, "word": " n", "probability": 0.1751708984375}, {"start": 1419.29, "end": 1420.87, "word": " as", "probability": 0.64111328125}, {"start": 1420.87, "end": 1421.27, "word": " n", "probability": 0.72607421875}, {"start": 1421.27, "end": 1421.71, "word": " tensor", "probability": 0.7649739583333334}, {"start": 1421.71, "end": 1422.33, "word": " infinity", "probability": 0.80517578125}, {"start": 1422.33, "end": 1423.57, "word": " بسوي", "probability": 0.524169921875}, {"start": 1423.57, "end": 1425.23, "word": " f", "probability": 0.81005859375}, {"start": 1425.23, "end": 1425.51, "word": " of", "probability": 0.7861328125}, {"start": 1425.51, "end": 1425.79, "word": " c", "probability": 0.74658203125}], "temperature": 1.0}, {"id": 53, "seek": 145990, "start": 1431.74, "end": 1459.9, "text": "الان الـ sequential criterion for continuity بتقول عشان اثبت ان الدالة F continuous عند نقطة يكفي ان انا اثبت ان لو اخدت اي sequence نهايتها اي sequence في مجال الدالة طبعا كنا في ال limits نشترط ان X in كل انصر في ال sequence مختلف عن ال C هنا لأ ممكن يساوي ال C مش مشكلة هاي الاختلاف بس بين ال sequential criterion for limits", "tokens": [6027, 7649, 2423, 39184, 42881, 46691, 337, 23807, 39894, 39648, 6225, 8592, 7649, 1975, 12984, 3555, 2655, 16472, 32748, 6027, 3660, 479, 10957, 43242, 8717, 47432, 3660, 7251, 4117, 41185, 16472, 1975, 8315, 1975, 12984, 3555, 2655, 16472, 45164, 1975, 9778, 3215, 2655, 1975, 1829, 8310, 8717, 11296, 36081, 11296, 1975, 1829, 8310, 8978, 3714, 7435, 6027, 32748, 6027, 3660, 23032, 3555, 3615, 995, 9122, 8315, 8978, 2423, 10406, 8717, 8592, 2655, 2288, 9566, 16472, 1783, 294, 28242, 16472, 9381, 2288, 8978, 2423, 8310, 3714, 46456, 46538, 18871, 2423, 383, 34105, 5296, 10721, 3714, 43020, 7251, 3794, 995, 45865, 2423, 383, 37893, 37893, 28820, 3660, 8032, 47302, 2423, 47283, 2655, 15040, 5172, 4724, 3794, 49374, 2423, 42881, 46691, 337, 10406], "avg_logprob": -0.2040289305458384, "compression_ratio": 1.8615384615384616, "no_speech_prob": 0.0, "words": [{"start": 1431.74, "end": 1432.14, "word": "الان", "probability": 0.153717041015625}, {"start": 1432.14, "end": 1432.38, "word": " الـ", "probability": 0.4952392578125}, {"start": 1432.38, "end": 1432.74, "word": " sequential", "probability": 0.70849609375}, {"start": 1432.74, "end": 1433.46, "word": " criterion", "probability": 0.91845703125}, {"start": 1433.46, "end": 1433.92, "word": " for", "probability": 0.93017578125}, {"start": 1433.92, "end": 1434.54, "word": " continuity", "probability": 0.80126953125}, {"start": 1434.54, "end": 1434.94, "word": " بتقول", "probability": 0.877685546875}, {"start": 1434.94, "end": 1435.48, "word": " عشان", "probability": 0.77783203125}, {"start": 1435.48, "end": 1436.52, "word": " اثبت", "probability": 0.8707275390625}, {"start": 1436.52, "end": 1436.68, "word": " ان", "probability": 0.81005859375}, {"start": 1436.68, "end": 1437.12, "word": " الدالة", "probability": 0.7762044270833334}, {"start": 1437.12, "end": 1437.32, "word": " F", "probability": 0.47607421875}, {"start": 1437.32, "end": 1437.86, "word": " continuous", "probability": 0.8056640625}, {"start": 1437.86, "end": 1438.26, "word": " عند", "probability": 0.896484375}, {"start": 1438.26, "end": 1438.8, "word": " نقطة", "probability": 0.7805989583333334}, {"start": 1438.8, "end": 1440.22, "word": " يكفي", "probability": 0.9192708333333334}, {"start": 1440.22, "end": 1440.38, "word": " ان", "probability": 0.826171875}, {"start": 1440.38, "end": 1440.52, "word": " انا", "probability": 0.76318359375}, {"start": 1440.52, "end": 1441.04, "word": " اثبت", "probability": 0.9725341796875}, {"start": 1441.04, "end": 1441.24, "word": " ان", "probability": 0.9248046875}, {"start": 1441.24, "end": 1441.42, "word": " لو", "probability": 0.71630859375}, {"start": 1441.42, "end": 1441.88, "word": " اخدت", "probability": 0.891845703125}, {"start": 1441.88, "end": 1442.12, "word": " اي", "probability": 0.83544921875}, {"start": 1442.12, "end": 1442.7, "word": " sequence", "probability": 0.9150390625}, {"start": 1442.7, "end": 1443.58, "word": " نهايتها", "probability": 0.9227294921875}, {"start": 1443.58, "end": 1444.9, "word": " اي", "probability": 0.901123046875}, {"start": 1444.9, "end": 1445.36, "word": " sequence", "probability": 0.9755859375}, {"start": 1445.36, "end": 1445.52, "word": " في", "probability": 0.90869140625}, {"start": 1445.52, "end": 1445.86, "word": " مجال", "probability": 0.9964192708333334}, {"start": 1445.86, "end": 1446.4, "word": " الدالة", "probability": 0.9915364583333334}, {"start": 1446.4, "end": 1446.72, "word": " طبعا", "probability": 0.9638671875}, {"start": 1446.72, "end": 1446.96, "word": " كنا", "probability": 0.9814453125}, {"start": 1446.96, "end": 1447.12, "word": " في", "probability": 0.845703125}, {"start": 1447.12, "end": 1447.24, "word": " ال", "probability": 0.900390625}, {"start": 1447.24, "end": 1447.66, "word": " limits", "probability": 0.96826171875}, {"start": 1447.66, "end": 1449.06, "word": " نشترط", "probability": 0.6869140625}, {"start": 1449.06, "end": 1449.22, "word": " ان", "probability": 0.916015625}, {"start": 1449.22, "end": 1449.52, "word": " X", "probability": 0.73388671875}, {"start": 1449.52, "end": 1449.84, "word": " in", "probability": 0.3447265625}, {"start": 1449.84, "end": 1450.34, "word": " كل", "probability": 0.96630859375}, {"start": 1450.34, "end": 1450.7, "word": " انصر", "probability": 0.6695149739583334}, {"start": 1450.7, "end": 1450.82, "word": " في", "probability": 0.9345703125}, {"start": 1450.82, "end": 1450.88, "word": " ال", "probability": 0.96044921875}, {"start": 1450.88, "end": 1451.28, "word": " sequence", "probability": 0.9833984375}, {"start": 1451.28, "end": 1451.94, "word": " مختلف", "probability": 0.986328125}, {"start": 1451.94, "end": 1452.66, "word": " عن", "probability": 0.9794921875}, {"start": 1452.66, "end": 1452.86, "word": " ال", "probability": 0.8125}, {"start": 1452.86, "end": 1453.02, "word": " C", "probability": 0.7490234375}, {"start": 1453.02, "end": 1453.28, "word": " هنا", "probability": 0.81591796875}, {"start": 1453.28, "end": 1453.52, "word": " لأ", "probability": 0.930419921875}, {"start": 1453.52, "end": 1453.82, "word": " ممكن", "probability": 0.96826171875}, {"start": 1453.82, "end": 1454.18, "word": " يساوي", "probability": 0.7977294921875}, {"start": 1454.18, "end": 1454.32, "word": " ال", "probability": 0.75830078125}, {"start": 1454.32, "end": 1454.42, "word": " C", "probability": 0.92236328125}, {"start": 1454.42, "end": 1454.62, "word": " مش", "probability": 0.94775390625}, {"start": 1454.62, "end": 1455.18, "word": " مشكلة", "probability": 0.9514973958333334}, {"start": 1455.18, "end": 1456.42, "word": " هاي", "probability": 0.698974609375}, {"start": 1456.42, "end": 1456.92, "word": " الاختلاف", "probability": 0.88662109375}, {"start": 1456.92, "end": 1457.2, "word": " بس", "probability": 0.98681640625}, {"start": 1457.2, "end": 1457.48, "word": " بين", "probability": 0.91845703125}, {"start": 1457.48, "end": 1457.8, "word": " ال", "probability": 0.97119140625}, {"start": 1457.8, "end": 1458.24, "word": " sequential", "probability": 0.94921875}, {"start": 1458.24, "end": 1458.9, "word": " criterion", "probability": 0.95703125}, {"start": 1458.9, "end": 1459.4, "word": " for", "probability": 0.9560546875}, {"start": 1459.4, "end": 1459.9, "word": " limits", "probability": 0.9482421875}], "temperature": 1.0}, {"id": 54, "seek": 147869, "start": 1460.65, "end": 1478.69, "text": "و Sequential criterion for continuity إنه لكل sequence x in في مجال الدالة و نهايتها بتساوي c لازم اطلع عندي نهاية ال image تبعت ال sequence x in بتساوي العدد f و c", "tokens": [2407, 46859, 2549, 46691, 337, 23807, 36145, 3224, 5296, 28820, 8310, 2031, 294, 8978, 3714, 7435, 6027, 32748, 6027, 3660, 4032, 8717, 11296, 36081, 11296, 39894, 3794, 995, 45865, 269, 5296, 31377, 2304, 1975, 9566, 1211, 3615, 18871, 16254, 8717, 11296, 10632, 2423, 3256, 6055, 3555, 34268, 2423, 8310, 2031, 294, 39894, 3794, 995, 45865, 18863, 3215, 3215, 283, 4032, 269], "avg_logprob": -0.23828125, "compression_ratio": 1.4444444444444444, "no_speech_prob": 0.0, "words": [{"start": 1460.65, "end": 1460.85, "word": "و", "probability": 0.69677734375}, {"start": 1460.85, "end": 1461.43, "word": " Sequential", "probability": 0.565185546875}, {"start": 1461.43, "end": 1461.97, "word": " criterion", "probability": 0.77734375}, {"start": 1461.97, "end": 1462.37, "word": " for", "probability": 0.94189453125}, {"start": 1462.37, "end": 1463.21, "word": " continuity", "probability": 0.80078125}, {"start": 1463.21, "end": 1464.47, "word": " إنه", "probability": 0.29693603515625}, {"start": 1464.47, "end": 1464.89, "word": " لكل", "probability": 0.755859375}, {"start": 1464.89, "end": 1465.39, "word": " sequence", "probability": 0.77685546875}, {"start": 1465.39, "end": 1465.65, "word": " x", "probability": 0.5986328125}, {"start": 1465.65, "end": 1465.85, "word": " in", "probability": 0.556640625}, {"start": 1465.85, "end": 1466.03, "word": " في", "probability": 0.890625}, {"start": 1466.03, "end": 1466.41, "word": " مجال", "probability": 0.9925130208333334}, {"start": 1466.41, "end": 1467.11, "word": " الدالة", "probability": 0.9724934895833334}, {"start": 1467.11, "end": 1467.61, "word": " و", "probability": 0.908203125}, {"start": 1467.61, "end": 1468.41, "word": " نهايتها", "probability": 0.8529052734375}, {"start": 1468.41, "end": 1469.79, "word": " بتساوي", "probability": 0.8492431640625}, {"start": 1469.79, "end": 1470.13, "word": " c", "probability": 0.349609375}, {"start": 1470.13, "end": 1471.83, "word": " لازم", "probability": 0.9332682291666666}, {"start": 1471.83, "end": 1472.19, "word": " اطلع", "probability": 0.82647705078125}, {"start": 1472.19, "end": 1472.55, "word": " عندي", "probability": 0.6900634765625}, {"start": 1472.55, "end": 1473.35, "word": " نهاية", "probability": 0.9646809895833334}, {"start": 1473.35, "end": 1474.25, "word": " ال", "probability": 0.92578125}, {"start": 1474.25, "end": 1474.55, "word": " image", "probability": 0.84912109375}, {"start": 1474.55, "end": 1474.95, "word": " تبعت", "probability": 0.8271484375}, {"start": 1474.95, "end": 1475.09, "word": " ال", "probability": 0.70263671875}, {"start": 1475.09, "end": 1475.47, "word": " sequence", "probability": 0.97509765625}, {"start": 1475.47, "end": 1475.83, "word": " x", "probability": 0.95166015625}, {"start": 1475.83, "end": 1476.17, "word": " in", "probability": 0.87548828125}, {"start": 1476.17, "end": 1477.35, "word": " بتساوي", "probability": 0.883056640625}, {"start": 1477.35, "end": 1477.99, "word": " العدد", "probability": 0.9348958333333334}, {"start": 1477.99, "end": 1478.27, "word": " f", "probability": 0.7490234375}, {"start": 1478.27, "end": 1478.47, "word": " و", "probability": 0.962890625}, {"start": 1478.47, "end": 1478.69, "word": " c", "probability": 0.77685546875}], "temperature": 1.0}, {"id": 55, "seek": 150774, "start": 1480.02, "end": 1507.74, "text": "وبرهان النظرية هذه زي برهان sequential criterion for limits مع تعديلات طفيفة مع التعديلات الطفيفة في التعريفين او في التعريف تبع الاتصال اذا ال proof similar to proof of", "tokens": [2407, 26890, 3224, 7649, 28239, 19913, 2288, 10632, 29538, 30767, 1829, 4724, 2288, 3224, 7649, 42881, 46691, 337, 10406, 20449, 37279, 16254, 1211, 9307, 23032, 5172, 33911, 3660, 20449, 16712, 3615, 16254, 1211, 9307, 41950, 5172, 33911, 3660, 8978, 16712, 3615, 16572, 5172, 9957, 1975, 2407, 8978, 16712, 3615, 16572, 5172, 6055, 3555, 3615, 2423, 9307, 9381, 6027, 1975, 15730, 2423, 8177, 2531, 281, 8177, 295], "avg_logprob": 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"probability": 0.409423828125}, {"start": 2013.83, "end": 2015.93, "word": " is", "probability": 0.87060546875}, {"start": 2015.93, "end": 2016.83, "word": " continuous", "probability": 0.90576171875}, {"start": 2016.83, "end": 2018.91, "word": " if", "probability": 0.53515625}, {"start": 2018.91, "end": 2019.35, "word": " is", "probability": 0.921875}, {"start": 2019.35, "end": 2020.25, "word": " continuous", "probability": 0.90234375}], "temperature": 1.0}, {"id": 73, "seek": 202429, "start": 2021.83, "end": 2024.29, "text": "على مجموعة الأعداد الحقيقية", "tokens": [3615, 23942, 3714, 7435, 2304, 2407, 27884, 16247, 22488, 18513, 21542, 38436, 4587, 10632], "avg_logprob": -0.068359375, "compression_ratio": 0.9444444444444444, "no_speech_prob": 0.0, "words": [{"start": 2021.83, "end": 2022.31, "word": "على", "probability": 0.876708984375}, {"start": 2022.31, "end": 2023.05, "word": " مجموعة", "probability": 0.973828125}, {"start": 2023.05, "end": 2023.53, "word": " الأعداد", "probability": 0.8546549479166666}, {"start": 2023.53, "end": 2024.29, "word": " الحقيقية", "probability": 0.98779296875}], "temperature": 1.0}, {"id": 74, "seek": 207561, "start": 2047.95, "end": 2075.61, "text": "فممكن أن نثبت C ينتمي إلى R و أثبتنا احنا في section أربعة واحد ان limit F of X لما X تقول C طلعت بساوي C صح؟ وهذا عبارة عن F of C فالـ F is continuous at C", "tokens": [5172, 2304, 43020, 14739, 8717, 12984, 3555, 2655, 383, 7251, 29399, 2304, 1829, 30731, 497, 4032, 5551, 12984, 3555, 2655, 8315, 1975, 5016, 8315, 8978, 3541, 5551, 25513, 27884, 36764, 24401, 16472, 4948, 479, 295, 1783, 5296, 15042, 1783, 6055, 39648, 383, 23032, 1211, 34268, 4724, 3794, 995, 45865, 383, 20328, 5016, 22807, 37037, 15730, 6225, 3555, 9640, 3660, 18871, 479, 295, 383, 6156, 6027, 39184, 479, 307, 10957, 412, 383], "avg_logprob": -0.2875434060891469, "compression_ratio": 1.3163841807909604, "no_speech_prob": 0.0, "words": [{"start": 2047.9499999999998, "end": 2048.87, "word": "فممكن", 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"probability": 0.6745198567708334}, {"start": 2058.71, "end": 2059.33, "word": " واحد", "probability": 0.963134765625}, {"start": 2059.33, "end": 2060.53, "word": " ان", "probability": 0.583984375}, {"start": 2060.53, "end": 2060.99, "word": " limit", "probability": 0.490478515625}, {"start": 2060.99, "end": 2061.47, "word": " F", "probability": 0.291748046875}, {"start": 2061.47, "end": 2061.73, "word": " of", "probability": 0.64599609375}, {"start": 2061.73, "end": 2062.13, "word": " X", "probability": 0.95458984375}, {"start": 2062.13, "end": 2063.19, "word": " لما", "probability": 0.79541015625}, {"start": 2063.19, "end": 2063.53, "word": " X", "probability": 0.95947265625}, {"start": 2063.53, "end": 2063.95, "word": " تقول", "probability": 0.510986328125}, {"start": 2063.95, "end": 2064.39, "word": " C", "probability": 0.51171875}, {"start": 2064.39, "end": 2064.85, "word": " طلعت", "probability": 0.8732096354166666}, {"start": 2064.85, "end": 2065.33, "word": " بساوي", 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0.84228515625}, {"start": 2095.18, "end": 2095.42, "word": " ال", "probability": 0.87744140625}, {"start": 2095.42, "end": 2095.72, "word": " R", "probability": 0.7451171875}, {"start": 2095.72, "end": 2096.26, "word": " وبالتالي", "probability": 0.91923828125}, {"start": 2096.26, "end": 2096.82, "word": " continuous", "probability": 0.89306640625}, {"start": 2096.82, "end": 2097.18, "word": " على", "probability": 0.91064453125}, {"start": 2097.18, "end": 2097.66, "word": " كل", "probability": 0.97314453125}, {"start": 2097.66, "end": 2098.62, "word": " ال", "probability": 0.75341796875}, {"start": 2098.62, "end": 2098.88, "word": " R", "probability": 0.96142578125}, {"start": 2098.88, "end": 2103.22, "word": " ممكن", "probability": 0.95947265625}, {"start": 2103.22, "end": 2103.74, "word": " برضه", "probability": 0.98876953125}], "temperature": 1.0}, {"id": 76, "seek": 212867, "start": 2105.0, "end": 2128.68, "text": "نستخدم تعريف epsilon دلتا مباشرة بلاش نقول ان احنا اثبتنا ان ال limit ل ال function f and c بالساوية c في section اربعة واحدة انا ممكن اثبت يعني استخدم تعريف epsilon دلتا مباشرة و اقول let if fix اول حاجة fix c تنتمي ل R to show", "tokens": [1863, 14851, 9778, 40448, 37279, 16572, 5172, 17889, 11778, 1211, 2655, 995, 3714, 3555, 33599, 25720, 4724, 1211, 33599, 8717, 39648, 16472, 1975, 5016, 8315, 1975, 12984, 3555, 2655, 8315, 16472, 2423, 4948, 5296, 2423, 2445, 283, 293, 269, 20666, 3794, 995, 2407, 10632, 269, 8978, 3541, 1975, 25513, 27884, 36764, 24401, 3660, 1975, 8315, 3714, 43020, 1975, 12984, 3555, 2655, 37495, 22653, 44713, 9778, 40448, 37279, 16572, 5172, 17889, 11778, 1211, 2655, 995, 3714, 3555, 33599, 25720, 4032, 1975, 39648, 718, 498, 3191, 1975, 12610, 11331, 26108, 3660, 3191, 269, 6055, 29399, 2304, 1829, 5296, 497, 281, 855], "avg_logprob": -0.28937500923871995, "compression_ratio": 1.7241379310344827, "no_speech_prob": 0.0, "words": [{"start": 2105.0, "end": 2105.82, "word": "نستخدم", "probability": 0.762481689453125}, 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السفر be given it shows .. زي ما عملنا في ال limits it shows delta بساوي epsilon لذن هي يوجد delta تعتمد على epsilon", "tokens": [351, 307, 10957, 412, 269, 718, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 312, 2212, 309, 3110, 4386, 30767, 1829, 19446, 6225, 42213, 8315, 8978, 2423, 10406, 309, 3110, 8289, 4724, 3794, 995, 45865, 17889, 5296, 8848, 1863, 39896, 7251, 29245, 3215, 8289, 6055, 34268, 2304, 3215, 15844, 17889], "avg_logprob": -0.3155637313337887, "compression_ratio": 1.3246753246753247, "no_speech_prob": 0.0, "words": [{"start": 2130.38, "end": 2130.88, "word": "if", "probability": 0.061187744140625}, {"start": 2130.88, "end": 2132.18, "word": " is", "probability": 0.75439453125}, {"start": 2132.18, "end": 2133.2, "word": " continuous", "probability": 0.8876953125}, {"start": 2133.2, "end": 2134.96, "word": " at", "probability": 0.833984375}, {"start": 2134.96, "end": 2135.4, "word": " c", "probability": 0.66796875}, {"start": 2135.4, "end": 2135.86, "word": " let", "probability": 0.465576171875}, {"start": 2135.86, "end": 2136.72, "word": " epsilon", "probability": 0.70751953125}, {"start": 2136.72, "end": 2138.26, "word": " أكبر", "probability": 0.9254557291666666}, {"start": 2138.26, "end": 2138.48, "word": " من", "probability": 0.9931640625}, {"start": 2138.48, "end": 2139.08, "word": " السفر", "probability": 0.7945963541666666}, {"start": 2139.08, "end": 2139.3, "word": " be", "probability": 0.9072265625}, {"start": 2139.3, "end": 2139.82, "word": " given", "probability": 0.89208984375}, {"start": 2139.82, "end": 2144.72, "word": " it", "probability": 0.39013671875}, {"start": 2144.72, "end": 2145.3, "word": " shows", "probability": 0.6650390625}, {"start": 2145.3, "end": 2145.68, "word": " ..", "probability": 0.13427734375}, {"start": 2145.68, "end": 2146.8, "word": " زي", "probability": 0.7626953125}, {"start": 2146.8, "end": 2146.96, "word": " ما", "probability": 0.96533203125}, {"start": 2146.96, "end": 2147.32, "word": " عملنا", 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since C تنتمي ل R was arbitrary اذا F is continuous على كل الأعداد الحقيقية تمام؟ اذا هذا ممكن استخدم تعريف Epsilon Delta مباشرة", "tokens": [267, 383, 1670, 383, 6055, 29399, 2304, 1829, 5296, 497, 390, 23211, 1975, 15730, 479, 307, 10957, 15844, 28242, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 46811, 10943, 22807, 1975, 15730, 23758, 3714, 43020, 44713, 9778, 40448, 37279, 16572, 5172, 462, 16592, 18183, 3714, 3555, 33599, 25720], "avg_logprob": -0.27376303014655906, "compression_ratio": 1.204968944099379, "no_speech_prob": 0.0, "words": [{"start": 2216.73, "end": 2217.63, "word": "at", "probability": 0.11419677734375}, {"start": 2217.63, "end": 2218.39, "word": " C", "probability": 0.53857421875}, {"start": 2218.39, "end": 2220.81, "word": " since", "probability": 0.5205078125}, {"start": 2220.81, "end": 2221.27, "word": " C", "probability": 0.92138671875}, {"start": 2221.27, "end": 2221.85, "word": " تنتمي", "probability": 0.9354248046875}, {"start": 2221.85, "end": 2221.93, 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83, "seek": 231927, "start": 2293.11, "end": 2319.27, "text": "نفس تعريف epsilon دلتا زي ما عملنا في اثبات ان ال limit لل function f of x and x بساوي c بساوي c تربيه اللي هو f of c وذلك بياخد اي epsilon اكبر من صفر و بنجيب دلتا زي ما عملنا في section اربعة واحد دلتا بساوي ال minimum لقمتين", "tokens": [1863, 36178, 37279, 16572, 5172, 17889, 11778, 1211, 2655, 995, 30767, 1829, 19446, 6225, 42213, 8315, 8978, 1975, 12984, 3555, 9307, 16472, 2423, 4948, 24976, 2445, 283, 295, 2031, 293, 2031, 4724, 3794, 995, 45865, 269, 4724, 3794, 995, 45865, 269, 6055, 2288, 21292, 3224, 13672, 1829, 31439, 283, 295, 269, 4032, 8848, 23275, 4724, 1829, 47283, 3215, 1975, 1829, 17889, 1975, 4117, 26890, 9154, 20328, 5172, 2288, 4032, 44945, 7435, 1829, 3555, 11778, 1211, 2655, 995, 30767, 1829, 19446, 6225, 42213, 8315, 8978, 3541, 1975, 25513, 27884, 36764, 24401, 11778, 1211, 2655, 995, 4724, 3794, 995, 45865, 2423, 7285, 5296, 4587, 2304, 2655, 9957], "avg_logprob": -0.21816037173541086, 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"probability": 0.7333984375}, {"start": 2313.31, "end": 2314.21, "word": " و", "probability": 0.875}, {"start": 2314.21, "end": 2314.97, "word": " بنجيب", "probability": 0.904541015625}, {"start": 2314.97, "end": 2315.39, "word": " دلتا", "probability": 0.843505859375}, {"start": 2315.39, "end": 2315.61, "word": " زي", "probability": 0.980224609375}, {"start": 2315.61, "end": 2315.71, "word": " ما", "probability": 0.9638671875}, {"start": 2315.71, "end": 2316.11, "word": " عملنا", "probability": 0.9903971354166666}, {"start": 2316.11, "end": 2316.23, "word": " في", "probability": 0.8603515625}, {"start": 2316.23, "end": 2316.49, "word": " section", "probability": 0.84521484375}, {"start": 2316.49, "end": 2316.89, "word": " اربعة", "probability": 0.6676432291666666}, {"start": 2316.89, "end": 2317.27, "word": " واحد", "probability": 0.966796875}, {"start": 2317.27, "end": 2317.65, "word": " دلتا", "probability": 0.942138671875}, {"start": 2317.65, "end": 2318.07, "word": " بساوي", 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"probability": 0.966796875}, {"start": 2704.73, "end": 2706.29, "word": " tends", "probability": 0.8212890625}, {"start": 2706.29, "end": 2706.47, "word": " to", "probability": 0.984375}, {"start": 2706.47, "end": 2706.97, "word": " zero", "probability": 0.82470703125}, {"start": 2706.97, "end": 2707.37, "word": " does", "probability": 0.85009765625}, {"start": 2707.37, "end": 2707.69, "word": " not", "probability": 0.96728515625}, {"start": 2707.69, "end": 2708.29, "word": " exist", "probability": 0.9580078125}, {"start": 2708.29, "end": 2710.11, "word": " أثبتنا", "probability": 0.9673828125}, {"start": 2710.11, "end": 2710.29, "word": " إن", "probability": 0.34912109375}, {"start": 2710.29, "end": 2710.67, "word": " الـ", "probability": 0.5570068359375}, {"start": 2710.67, "end": 2710.95, "word": " function", "probability": 0.90185546875}, {"start": 2710.95, "end": 2711.37, "word": " هذه", "probability": 0.51513671875}, {"start": 2711.37, "end": 2711.49, "word": " ما", "probability": 0.53076171875}, {"start": 2711.49, "end": 2711.79, "word": " لهاش", "probability": 0.653564453125}, {"start": 2711.79, "end": 2712.03, "word": " limit", "probability": 0.9306640625}, {"start": 2712.03, "end": 2712.35, "word": " عند", "probability": 0.98974609375}, {"start": 2712.35, "end": 2712.85, "word": " السفر", "probability": 0.8251953125}], "temperature": 1.0}, {"id": 98, "seek": 274255, "start": 2715.83, "end": 2742.55, "text": "فا استخدمنا ال divergence criterion ا شفنا ان هناك sequence اللى هى واحد عال ان converge للسفر but limit ال image لل sequence واحد على ان as n tends to infinity بساوي limit in بساوي infinity does not exist in R", "tokens": [5172, 995, 44713, 9778, 40448, 8315, 2423, 47387, 46691, 1975, 13412, 5172, 8315, 16472, 34105, 4117, 8310, 13672, 7578, 8032, 7578, 36764, 24401, 6225, 6027, 16472, 41881, 24976, 3794, 5172, 2288, 457, 4948, 2423, 3256, 24976, 8310, 36764, 24401, 15844, 16472, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 4948, 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0.430908203125}, {"start": 2737.81, "end": 2738.77, "word": " بساوي", "probability": 0.90185546875}, {"start": 2738.77, "end": 2739.39, "word": " infinity", "probability": 0.8642578125}, {"start": 2739.39, "end": 2739.85, "word": " does", "probability": 0.51123046875}, {"start": 2739.85, "end": 2740.17, "word": " not", "probability": 0.970703125}, {"start": 2740.17, "end": 2740.87, "word": " exist", "probability": 0.9482421875}, {"start": 2740.87, "end": 2742.17, "word": " in", "probability": 0.96728515625}, {"start": 2742.17, "end": 2742.55, "word": " R", "probability": 0.80712890625}], "temperature": 1.0}, {"id": 99, "seek": 277265, "start": 2744.21, "end": 2772.65, "text": "وبالتالي by divergence criterion ال function هذه مالهاش limit وبالتالي مش ممكن تكون continuous so if I can't be continuous at x بساوي سفر تمام؟ لأن واحد من الشروط التلاتة تبعت الاتصال عن نقطة غير متحققة تمام؟", "tokens": [37746, 6027, 2655, 6027, 1829, 538, 47387, 46691, 2423, 2445, 29538, 19446, 43761, 33599, 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section", "probability": 0.8525390625}, {"start": 2840.93, "end": 2841.39, "word": " أربعة", "probability": 0.6333821614583334}, {"start": 2841.39, "end": 2841.93, "word": " واحد", "probability": 0.96875}, {"start": 2841.93, "end": 2842.93, "word": " انه", "probability": 0.58984375}, {"start": 2842.93, "end": 2843.33, "word": " limit", "probability": 0.79833984375}, {"start": 2843.33, "end": 2843.55, "word": " ل", "probability": 0.89013671875}, {"start": 2843.55, "end": 2844.41, "word": " signum", "probability": 0.6656494140625}, {"start": 2844.41, "end": 2845.45, "word": " x", "probability": 0.7236328125}, {"start": 2845.45, "end": 2846.83, "word": " لما", "probability": 0.884521484375}, {"start": 2846.83, "end": 2847.17, "word": " x", "probability": 0.9130859375}, {"start": 2847.17, "end": 2847.87, "word": " تقول", "probability": 0.963134765625}, {"start": 2847.87, "end": 2848.43, "word": " إلى", "probability": 0.437255859375}, {"start": 2848.43, "end": 2849.19, "word": " سفر", "probability": 0.8865559895833334}, {"start": 2849.19, "end": 2850.63, "word": " does", "probability": 0.7373046875}, {"start": 2850.63, "end": 2850.93, "word": " not", "probability": 0.9716796875}, {"start": 2850.93, "end": 2851.49, "word": " exist", "probability": 0.97119140625}], "temperature": 1.0}, {"id": 103, "seek": 288758, "start": 2860.56, "end": 2887.58, "text": "اللي هي ان ال limit لل signal function عند السفر does not exist شوفنا ان ال limit من اليمين واحد عند السفر و ال limit و ال limit عند السفر مليار ساعة سالف واحد وبالتالي مش متساوي اتين اذا ال limit عند السفر does not exist okay تمام اذا ال ال function هذه ماهياش متصلة عند السفر لعدم نظرا لعدم وجود ال limit عند السفر", "tokens": [6027, 20292, 39896, 16472, 2423, 4948, 24976, 6358, 2445, 43242, 21136, 5172, 2288, 775, 406, 2514, 13412, 38688, 8315, 16472, 2423, 4948, 9154, 45595, 2304, 9957, 36764, 24401, 43242, 21136, 5172, 2288, 4032, 2423, 4948, 4032, 2423, 4948, 43242, 21136, 5172, 2288, 3714, 20292, 9640, 8608, 995, 27884, 8608, 6027, 5172, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 37893, 44650, 3794, 995, 45865, 1975, 2655, 9957, 1975, 15730, 2423, 4948, 43242, 21136, 5172, 2288, 775, 406, 2514, 1392, 46811, 10943, 1975, 15730, 2423, 2423, 2445, 29538, 19446, 3224, 1829, 33599, 44650, 36520, 3660, 43242, 21136, 5172, 2288, 5296, 22488, 2304, 8717, 19913, 23557, 5296, 22488, 2304, 49610, 23328, 2423, 4948, 43242, 21136, 5172, 2288], "avg_logprob": -0.2604619674060656, "compression_ratio": 2.238532110091743, "no_speech_prob": 0.0, "words": [{"start": 2860.56, "end": 2860.84, "word": "اللي", "probability": 0.3155517578125}, {"start": 2860.84, "end": 2860.94, "word": " هي", "probability": 0.469482421875}, {"start": 2860.94, "end": 2861.08, "word": " ان", "probability": 0.394287109375}, {"start": 2861.08, "end": 2861.2, "word": " ال", "probability": 0.59326171875}, {"start": 2861.2, "end": 2861.44, "word": " limit", "probability": 0.82763671875}, {"start": 2861.44, "end": 2861.66, "word": " 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" من", "probability": 0.9736328125}, {"start": 2865.64, "end": 2866.06, "word": " اليمين", "probability": 0.8009440104166666}, {"start": 2866.06, "end": 2866.58, "word": " واحد", "probability": 0.8359375}, {"start": 2866.58, "end": 2866.8, "word": " عند", "probability": 0.98486328125}, {"start": 2866.8, "end": 2867.34, "word": " السفر", "probability": 0.9860026041666666}, {"start": 2867.34, "end": 2867.98, "word": " و", "probability": 0.51611328125}, {"start": 2867.98, "end": 2868.06, "word": " ال", "probability": 0.482666015625}, {"start": 2868.06, "end": 2868.4, "word": " limit", "probability": 0.96044921875}, {"start": 2868.4, "end": 2868.62, "word": " و", "probability": 0.35595703125}, {"start": 2868.62, "end": 2868.74, "word": " ال", "probability": 0.80224609375}, {"start": 2868.74, "end": 2868.96, "word": " limit", "probability": 0.9765625}, {"start": 2868.96, "end": 2869.24, "word": " عند", "probability": 0.9853515625}, {"start": 2869.24, "end": 2869.68, "word": " السفر", 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عند", "probability": 0.951171875}, {"start": 2873.74, "end": 2874.04, "word": " السفر", "probability": 0.9900716145833334}, {"start": 2874.04, "end": 2874.26, "word": " does", "probability": 0.8193359375}, {"start": 2874.26, "end": 2874.52, "word": " not", "probability": 0.96435546875}, {"start": 2874.52, "end": 2874.98, "word": " exist", "probability": 0.955078125}, {"start": 2874.98, "end": 2876.82, "word": " okay", "probability": 0.392822265625}, {"start": 2876.82, "end": 2878.64, "word": " تمام", "probability": 0.948486328125}, {"start": 2878.64, "end": 2879.06, "word": " اذا", "probability": 0.6634521484375}, {"start": 2879.06, "end": 2879.46, "word": " ال", "probability": 0.95751953125}, {"start": 2879.46, "end": 2880.0, "word": " ال", "probability": 0.560546875}, {"start": 2880.0, "end": 2880.38, "word": " function", "probability": 0.85546875}, {"start": 2880.38, "end": 2880.68, "word": " هذه", "probability": 0.7138671875}, {"start": 2880.68, "end": 2881.2, "word": " ماهياش", "probability": 0.9508056640625}, {"start": 2881.2, "end": 2881.84, "word": " متصلة", "probability": 0.9246419270833334}, {"start": 2881.84, "end": 2882.04, "word": " عند", "probability": 0.98876953125}, {"start": 2882.04, "end": 2882.58, "word": " السفر", "probability": 0.9933268229166666}, {"start": 2882.58, "end": 2883.86, "word": " لعدم", "probability": 0.9368489583333334}, {"start": 2883.86, "end": 2884.24, "word": " نظرا", "probability": 0.70751953125}, {"start": 2884.24, "end": 2884.7, "word": " لعدم", "probability": 0.94921875}, {"start": 2884.7, "end": 2885.12, "word": " وجود", "probability": 0.955078125}, {"start": 2885.12, "end": 2885.28, "word": " ال", "probability": 0.59716796875}, {"start": 2885.28, "end": 2885.64, "word": " limit", "probability": 0.98681640625}, {"start": 2885.64, "end": 2886.92, "word": " عند", "probability": 0.96630859375}, {"start": 2886.92, "end": 2887.58, "word": " السفر", "probability": 0.9923502604166666}], "temperature": 1.0}, {"id": 104, "seek": 291285, "start": 2888.83, "end": 2912.85, "text": "رغم أن الدالة هذه معرفة عند السفر، الـSignum للسفر هي معرفة عند السفر بساوي سفر تمام؟ طيب، لكن ممكن اثبات أن الـSignum function متصلة عند كل X لا يساوي سفر", "tokens": [2288, 17082, 2304, 14739, 32748, 6027, 3660, 29538, 20449, 28480, 3660, 43242, 21136, 5172, 2288, 12399, 2423, 39184, 50, 788, 449, 24976, 3794, 5172, 2288, 39896, 20449, 28480, 3660, 43242, 21136, 5172, 2288, 4724, 3794, 995, 45865, 8608, 5172, 2288, 46811, 10943, 22807, 23032, 1829, 3555, 12399, 44381, 3714, 43020, 1975, 12984, 3555, 9307, 14739, 2423, 39184, 50, 788, 449, 2445, 44650, 36520, 3660, 43242, 28242, 1783, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.24259868813188454, "compression_ratio": 1.6560509554140128, "no_speech_prob": 0.0, "words": [{"start": 2888.83, "end": 2889.29, "word": "رغم", "probability": 0.83251953125}, {"start": 2889.29, "end": 2889.53, "word": " أن", "probability": 0.60546875}, {"start": 2889.53, "end": 2890.03, 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2897.31, "end": 2897.73, "word": " سفر", "probability": 0.85546875}, {"start": 2897.73, "end": 2904.93, "word": " تمام؟", "probability": 0.6940104166666666}, {"start": 2904.93, "end": 2906.37, "word": " طيب،", "probability": 0.851806640625}, {"start": 2906.37, "end": 2907.35, "word": " لكن", "probability": 0.76123046875}, {"start": 2907.35, "end": 2907.77, "word": " ممكن", "probability": 0.96240234375}, {"start": 2907.77, "end": 2908.41, "word": " اثبات", "probability": 0.83892822265625}, {"start": 2908.41, "end": 2908.67, "word": " أن", "probability": 0.6484375}, {"start": 2908.67, "end": 2909.19, "word": " الـSignum", "probability": 0.8892578125}, {"start": 2909.19, "end": 2909.57, "word": " function", "probability": 0.74755859375}, {"start": 2909.57, "end": 2910.71, "word": " متصلة", "probability": 0.892578125}, {"start": 2910.71, "end": 2911.11, "word": " عند", "probability": 0.8701171875}, {"start": 2911.11, "end": 2911.43, "word": " كل", "probability": 0.99072265625}, {"start": 2911.43, "end": 2911.81, "word": " X", "probability": 0.75048828125}, {"start": 2911.81, "end": 2912.05, "word": " لا", "probability": 0.8154296875}, {"start": 2912.05, "end": 2912.49, "word": " يساوي", "probability": 0.83575439453125}, {"start": 2912.49, "end": 2912.85, "word": " سفر", "probability": 0.9840494791666666}], "temperature": 1.0}, {"id": 105, "seek": 294946, "start": 2925.1, "end": 2949.46, "text": "However، الـ signum الـ signum function is continuous at every x لا يساوي سفر لأنه", "tokens": [6462, 1054, 12399, 2423, 39184, 1465, 449, 2423, 39184, 1465, 449, 2445, 307, 10957, 412, 633, 2031, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 5296, 33456, 3224], "avg_logprob": -0.34455817732317695, "compression_ratio": 1.0842105263157895, "no_speech_prob": 0.0, "words": [{"start": 2925.1, "end": 2926.42, "word": "However،", "probability": 0.4700113932291667}, {"start": 2926.42, "end": 2926.5, "word": " الـ", "probability": 0.798095703125}, {"start": 2926.5, "end": 2927.02, "word": " signum", "probability": 0.59619140625}, {"start": 2927.02, "end": 2929.38, "word": " الـ", "probability": 0.448974609375}, {"start": 2929.38, "end": 2929.72, "word": " signum", "probability": 0.793212890625}, {"start": 2929.72, "end": 2930.28, "word": " function", "probability": 0.9375}, {"start": 2930.28, "end": 2932.44, "word": " is", "probability": 0.71923828125}, {"start": 2932.44, "end": 2933.46, "word": " continuous", "probability": 0.92138671875}, {"start": 2933.46, "end": 2939.28, "word": " at", "probability": 0.869140625}, {"start": 2939.28, "end": 2939.94, "word": " every", "probability": 0.8046875}, {"start": 2939.94, "end": 2941.7, "word": " x", "probability": 0.703125}, {"start": 2941.7, "end": 2942.04, "word": " لا", "probability": 0.84326171875}, {"start": 2942.04, "end": 2942.66, "word": " يساوي", "probability": 0.898193359375}, {"start": 2942.66, "end": 2943.1, "word": " سفر", "probability": 0.8219401041666666}, {"start": 2943.1, "end": 2949.46, "word": " لأنه", "probability": 0.88037109375}], "temperature": 1.0}, {"id": 106, "seek": 298929, "start": 2962.23, "end": 2989.29, "text": "proof fix c لا تنتمي لار وc لا يساوي ستة تمام then absolute signum x minus signum", "tokens": [15690, 3191, 269, 20193, 6055, 29399, 2304, 1829, 5296, 9640, 4032, 66, 20193, 7251, 3794, 995, 45865, 8608, 2655, 3660, 46811, 10943, 550, 8236, 1465, 449, 2031, 3175, 1465, 449], "avg_logprob": -0.37676410328957344, "compression_ratio": 1.127659574468085, "no_speech_prob": 0.0, "words": [{"start": 2962.23, "end": 2963.63, "word": "proof", "probability": 0.26513671875}, {"start": 2963.63, "end": 2965.03, "word": " fix", "probability": 0.40771484375}, {"start": 2965.03, "end": 2967.33, "word": " c", "probability": 0.367431640625}, {"start": 2967.33, "end": 2967.97, "word": " لا", "probability": 0.78955078125}, {"start": 2967.97, "end": 2970.17, "word": " تنتمي", "probability": 0.8448486328125}, {"start": 2970.17, "end": 2970.85, "word": " لار", "probability": 0.456298828125}, {"start": 2970.85, "end": 2971.65, "word": " وc", "probability": 0.568359375}, {"start": 2971.65, "end": 2971.91, "word": " لا", "probability": 0.8974609375}, {"start": 2971.91, "end": 2972.57, "word": " يساوي", "probability": 0.9361572265625}, {"start": 2972.57, "end": 2973.51, "word": " ستة", "probability": 0.7147623697916666}, {"start": 2973.51, "end": 2975.31, "word": " تمام", "probability": 0.6937255859375}, {"start": 2975.31, "end": 2982.61, "word": " then", "probability": 0.408203125}, {"start": 2982.61, "end": 2984.67, "word": " absolute", "probability": 0.8701171875}, {"start": 2984.67, "end": 2985.93, "word": " signum", "probability": 0.778076171875}, {"start": 2985.93, "end": 2986.93, "word": " x", "probability": 0.734375}, {"start": 2986.93, "end": 2988.33, "word": " minus", "probability": 0.97314453125}, {"start": 2988.33, "end": 2989.29, "word": " signum", "probability": 0.9765625}], "temperature": 1.0}, {"id": 107, "seek": 301516, "start": 2990.32, "end": 3015.16, "text": "الـ C بساوي absolute X على absolute X أو بلاش", "tokens": [6027, 39184, 383, 4724, 3794, 995, 45865, 8236, 1783, 15844, 8236, 1783, 34051, 4724, 1211, 33599], "avg_logprob": -0.18772977941176472, "compression_ratio": 1.0508474576271187, "no_speech_prob": 0.0, "words": [{"start": 2990.32, "end": 2990.8, "word": "الـ", "probability": 0.898681640625}, {"start": 2990.8, "end": 2991.24, "word": " C", "probability": 0.58251953125}, {"start": 2991.24, "end": 2992.28, "word": " بساوي", "probability": 0.74005126953125}, {"start": 2992.28, "end": 2993.46, "word": " absolute", "probability": 0.6279296875}, {"start": 2993.46, "end": 2997.42, "word": " X", "probability": 0.84619140625}, {"start": 2997.42, "end": 2997.92, "word": " على", "probability": 0.7666015625}, {"start": 2997.92, "end": 2998.74, "word": " absolute", "probability": 0.94580078125}, {"start": 2998.74, "end": 2999.96, "word": " X", "probability": 0.6279296875}, {"start": 2999.96, "end": 3014.64, "word": " أو", "probability": 0.8681640625}, {"start": 3014.64, "end": 3015.16, "word": " بلاش", "probability": 0.9685872395833334}], "temperature": 1.0}, {"id": 108, "seek": 304563, "start": 3019.85, "end": 3045.63, "text": "then ال limit ل sigma x لما x تقول إلى c بساوي لما x تقول إلى c فهذا عبارة عن limit x على absolute x لما x تقول إلى c", "tokens": [19096, 2423, 4948, 5296, 12771, 2031, 5296, 15042, 2031, 6055, 39648, 30731, 269, 4724, 3794, 995, 45865, 5296, 15042, 2031, 6055, 39648, 30731, 269, 6156, 3224, 15730, 6225, 3555, 9640, 3660, 18871, 4948, 2031, 15844, 8236, 2031, 5296, 15042, 2031, 6055, 39648, 30731, 269], "avg_logprob": -0.24739583598242865, "compression_ratio": 1.6095238095238096, "no_speech_prob": 0.0, "words": [{"start": 3019.85, "end": 3020.33, "word": "then", "probability": 0.1683349609375}, {"start": 3020.33, "end": 3020.65, "word": " ال", "probability": 0.354248046875}, {"start": 3020.65, "end": 3021.03, "word": " limit", "probability": 0.90380859375}, {"start": 3021.03, "end": 3022.67, "word": " ل", "probability": 0.904296875}, {"start": 3022.67, "end": 3023.51, "word": " sigma", "probability": 0.3125}, {"start": 3023.51, "end": 3026.73, "word": " x", "probability": 0.7685546875}, {"start": 3026.73, "end": 3027.09, "word": " لما", "probability": 0.87060546875}, {"start": 3027.09, "end": 3027.43, "word": " x", "probability": 0.96630859375}, {"start": 3027.43, "end": 3027.99, "word": " تقول", "probability": 0.885009765625}, {"start": 3027.99, "end": 3028.49, "word": " إلى", "probability": 0.59423828125}, {"start": 3028.49, "end": 3029.21, "word": " c", "probability": 0.7431640625}, {"start": 3029.21, "end": 3034.39, "word": " بساوي", "probability": 0.80908203125}, {"start": 3034.39, "end": 3037.99, "word": " لما", "probability": 0.6973876953125}, {"start": 3037.99, "end": 3038.39, "word": " x", "probability": 0.2783203125}, {"start": 3038.39, "end": 3038.77, "word": " تقول", "probability": 0.97998046875}, {"start": 3038.77, "end": 3039.01, "word": " إلى", "probability": 0.720703125}, {"start": 3039.01, "end": 3039.29, "word": " c", "probability": 0.9287109375}, {"start": 3039.29, "end": 3039.83, "word": " فهذا", "probability": 0.8347981770833334}, {"start": 3039.83, "end": 3040.15, "word": " عبارة", "probability": 0.991455078125}, {"start": 3040.15, "end": 3040.37, "word": " عن", "probability": 0.99560546875}, {"start": 3040.37, "end": 3040.81, "word": " limit", "probability": 0.98046875}, {"start": 3040.81, "end": 3042.21, "word": " x", "probability": 0.9365234375}, {"start": 3042.21, "end": 3042.47, "word": " على", "probability": 0.6787109375}, {"start": 3042.47, "end": 3043.05, "word": " absolute", "probability": 0.62890625}, {"start": 3043.05, "end": 3043.67, "word": " x", "probability": 0.97998046875}, {"start": 3043.67, "end": 3044.03, "word": " لما", "probability": 0.97265625}, {"start": 3044.03, "end": 3044.39, "word": " x", "probability": 0.990234375}, {"start": 3044.39, "end": 3044.83, "word": " تقول", "probability": 0.9638671875}, {"start": 3044.83, "end": 3045.13, "word": " إلى", "probability": 0.96044921875}, {"start": 3045.13, "end": 3045.63, "word": " c", "probability": 0.97119140625}], "temperature": 1.0}, {"id": 109, "seek": 307801, "start": 3063.05, "end": 3078.01, "text": "فده كانت ال X لا تساوي سفر فاما ال X موجة بقى أو سالي بقى then C أكبر من السفر or C أصغر من سفر", "tokens": [5172, 3215, 3224, 25961, 2655, 2423, 1783, 20193, 6055, 3794, 995, 45865, 8608, 5172, 2288, 6156, 10943, 995, 2423, 1783, 3714, 29245, 3660, 4724, 4587, 7578, 34051, 8608, 6027, 1829, 4724, 4587, 7578, 550, 383, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 420, 383, 5551, 9381, 17082, 2288, 9154, 8608, 5172, 2288], "avg_logprob": -0.20931603211276936, "compression_ratio": 1.3220338983050848, "no_speech_prob": 0.0, "words": [{"start": 3063.0499999999997, "end": 3063.97, "word": "فده", "probability": 0.5834147135416666}, {"start": 3063.97, "end": 3064.33, "word": " كانت", "probability": 0.93017578125}, {"start": 3064.33, "end": 3064.49, "word": " ال", "probability": 0.8671875}, {"start": 3064.49, "end": 3064.67, "word": " X", "probability": 0.70263671875}, {"start": 3064.67, "end": 3064.93, "word": " لا", "probability": 0.79052734375}, {"start": 3064.93, "end": 3065.47, "word": " تساوي", "probability": 0.9249267578125}, {"start": 3065.47, "end": 3066.41, "word": " سفر", "probability": 0.755859375}, {"start": 3066.41, "end": 3067.67, "word": " فاما", "probability": 0.7452799479166666}, {"start": 3067.67, "end": 3067.77, "word": " ال", "probability": 0.95751953125}, {"start": 3067.77, "end": 3067.99, "word": " X", "probability": 0.97705078125}, {"start": 3067.99, "end": 3068.47, "word": " موجة", "probability": 0.9334309895833334}, {"start": 3068.47, "end": 3068.59, "word": " بقى", "probability": 0.8416341145833334}, {"start": 3068.59, "end": 3068.75, "word": " أو", "probability": 0.6474609375}, {"start": 3068.75, "end": 3069.23, "word": " سالي", "probability": 0.66845703125}, {"start": 3069.23, "end": 3072.89, "word": " بقى", "probability": 0.9837239583333334}, {"start": 3072.89, "end": 3073.75, "word": " then", "probability": 0.677734375}, {"start": 3073.75, "end": 3074.55, "word": " C", "probability": 0.80126953125}, {"start": 3074.55, "end": 3075.13, "word": " أكبر", "probability": 0.9597981770833334}, {"start": 3075.13, "end": 3075.33, "word": " من", "probability": 0.99462890625}, {"start": 3075.33, "end": 3075.97, "word": " السفر", "probability": 0.90234375}, {"start": 3075.97, "end": 3076.35, "word": " or", "probability": 0.361083984375}, {"start": 3076.35, "end": 3076.79, "word": " C", "probability": 0.91357421875}, {"start": 3076.79, "end": 3077.35, "word": " أصغر", "probability": 0.9912109375}, {"start": 3077.35, "end": 3077.55, "word": " من", "probability": 0.998046875}, {"start": 3077.55, "end": 3078.01, "word": " سفر", "probability": 0.9514973958333334}], "temperature": 1.0}, {"id": 110, "seek": 311098, "start": 3083.04, "end": 3110.98, "text": "الـ C هتكون أكبر من السفر الـ C هنا لأ تساوي سفر إذا أما C أكبر من السفر أو أصغر من السفر case one لو كانت C أكبر من سفر فهذا بقد أنه limit signum X as X tends to C بساوي limit X على absolute X", "tokens": [6027, 39184, 383, 8032, 2655, 30544, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 2423, 39184, 383, 34105, 5296, 10721, 6055, 3794, 995, 45865, 8608, 5172, 2288, 11933, 15730, 5551, 15042, 383, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 34051, 5551, 9381, 17082, 2288, 9154, 21136, 5172, 2288, 1389, 472, 45164, 25961, 2655, 383, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 6156, 3224, 15730, 4724, 4587, 3215, 14739, 3224, 4948, 1465, 449, 1783, 382, 1783, 12258, 281, 383, 4724, 3794, 995, 45865, 4948, 1783, 15844, 8236, 1783], "avg_logprob": -0.2194683849811554, "compression_ratio": 1.6628571428571428, "no_speech_prob": 0.0, "words": [{"start": 3083.04, "end": 3083.64, "word": "الـ", "probability": 0.690185546875}, {"start": 3083.64, "end": 3083.82, "word": " C", "probability": 0.40869140625}, {"start": 3083.82, "end": 3084.32, "word": " هتكون", "probability": 0.7647298177083334}, {"start": 3084.32, "end": 3084.7, "word": " أكبر", "probability": 0.8645833333333334}, {"start": 3084.7, "end": 3084.9, "word": " من", "probability": 0.99072265625}, {"start": 3084.9, "end": 3085.38, "word": " السفر", "probability": 0.7901204427083334}, {"start": 3085.38, "end": 3085.82, "word": " الـ", "probability": 0.6046142578125}, {"start": 3085.82, "end": 3086.0, "word": " C", "probability": 0.7392578125}, {"start": 3086.0, "end": 3086.2, "word": " هنا", "probability": 0.958984375}, {"start": 3086.2, "end": 3086.4, "word": " لأ", "probability": 0.687255859375}, {"start": 3086.4, "end": 3086.8, "word": " تساوي", "probability": 0.875732421875}, {"start": 3086.8, "end": 3087.12, "word": " سفر", "probability": 0.95068359375}, {"start": 3087.12, "end": 3087.34, "word": " إذا", "probability": 0.64111328125}, {"start": 3087.34, "end": 3087.62, "word": " أما", "probability": 0.677734375}, {"start": 3087.62, "end": 3087.92, "word": " C", "probability": 0.8173828125}, {"start": 3087.92, "end": 3088.3, "word": " أكبر", "probability": 0.9763997395833334}, {"start": 3088.3, "end": 3088.48, "word": " من", "probability": 0.98876953125}, {"start": 3088.48, "end": 3088.96, "word": " السفر", "probability": 0.9484049479166666}, {"start": 3088.96, "end": 3089.58, "word": " أو", "probability": 0.9306640625}, {"start": 3089.58, "end": 3089.96, "word": " أصغر", "probability": 0.98291015625}, {"start": 3089.96, "end": 3090.14, "word": " من", "probability": 0.98828125}, {"start": 3090.14, "end": 3090.54, "word": " السفر", "probability": 0.9581705729166666}, {"start": 3090.54, "end": 3092.1, "word": " case", "probability": 0.5703125}, {"start": 3092.1, "end": 3093.24, "word": " one", "probability": 0.55126953125}, {"start": 3093.24, "end": 3094.8, "word": " لو", "probability": 0.9052734375}, {"start": 3094.8, "end": 3095.34, "word": " كانت", "probability": 0.96435546875}, {"start": 3095.34, "end": 3095.54, "word": " C", "probability": 0.90625}, {"start": 3095.54, "end": 3096.0, "word": " أكبر", "probability": 0.9778645833333334}, {"start": 3096.0, "end": 3096.22, "word": " من", "probability": 0.998046875}, {"start": 3096.22, "end": 3096.64, "word": " سفر", "probability": 0.8639322916666666}, {"start": 3096.64, "end": 3097.02, "word": " فهذا", "probability": 0.9314778645833334}, {"start": 3097.02, "end": 3097.4, "word": " بقد", "probability": 0.6541341145833334}, {"start": 3097.4, "end": 3097.68, "word": " أنه", "probability": 0.686279296875}, {"start": 3097.68, "end": 3098.0, "word": " limit", "probability": 0.953125}, {"start": 3098.0, "end": 3100.54, "word": " signum", "probability": 0.528533935546875}, {"start": 3100.54, "end": 3101.0, "word": " X", "probability": 0.685546875}, {"start": 3101.0, "end": 3101.74, "word": " as", "probability": 0.77783203125}, {"start": 3101.74, "end": 3102.14, "word": " X", "probability": 0.9873046875}, {"start": 3102.14, "end": 3102.5, "word": " tends", "probability": 0.8173828125}, {"start": 3102.5, "end": 3102.68, "word": " to", "probability": 0.97412109375}, {"start": 3102.68, "end": 3103.94, "word": " C", "probability": 0.87255859375}, {"start": 3103.94, "end": 3106.24, "word": " بساوي", "probability": 0.8160400390625}, {"start": 3106.24, "end": 3107.12, "word": " limit", "probability": 0.97265625}, {"start": 3107.12, "end": 3109.38, "word": " X", "probability": 0.9599609375}, {"start": 3109.38, "end": 3109.7, "word": " على", "probability": 0.88525390625}, {"start": 3109.7, "end": 3110.28, "word": " absolute", "probability": 0.87939453125}, {"start": 3110.28, "end": 3110.98, "word": " X", "probability": 0.9931640625}], "temperature": 1.0}, {"id": 111, "seek": 314470, "start": 3119.94, "end": 3144.7, "text": "و طبعا ال X أكبر من ال C أكبر من السفر ف absolute .. فهذا بيساوي واحد بيساوي limit واحد as X tends to C بيساوي واحد بيساوي F and C", "tokens": [2407, 23032, 3555, 3615, 995, 2423, 1783, 5551, 4117, 26890, 9154, 2423, 383, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 6156, 8236, 4386, 6156, 3224, 15730, 4724, 1829, 3794, 995, 45865, 36764, 24401, 4724, 1829, 3794, 995, 45865, 4948, 36764, 24401, 382, 1783, 12258, 281, 383, 4724, 1829, 3794, 995, 45865, 36764, 24401, 4724, 1829, 3794, 995, 45865, 479, 293, 383], "avg_logprob": -0.16078629657145468, "compression_ratio": 1.5634920634920635, "no_speech_prob": 0.0, "words": [{"start": 3119.9399999999996, "end": 3120.74, "word": "و", "probability": 0.62841796875}, {"start": 3120.74, "end": 3121.08, "word": " طبعا", "probability": 0.8746337890625}, {"start": 3121.08, "end": 3121.22, "word": " ال", "probability": 0.904296875}, {"start": 3121.22, "end": 3121.5, "word": " X", "probability": 0.60888671875}, {"start": 3121.5, "end": 3122.12, "word": " أكبر", "probability": 0.9422200520833334}, {"start": 3122.12, "end": 3122.44, "word": " من", "probability": 0.98095703125}, {"start": 3122.44, "end": 3125.66, "word": " ال", "probability": 0.85400390625}, {"start": 3125.66, "end": 3125.9, "word": " C", "probability": 0.6611328125}, {"start": 3125.9, "end": 3126.36, "word": " أكبر", "probability": 0.9703776041666666}, {"start": 3126.36, "end": 3126.56, "word": " من", "probability": 0.99658203125}, {"start": 3126.56, "end": 3127.12, "word": " السفر", "probability": 0.8352864583333334}, {"start": 3127.12, "end": 3129.76, "word": " ف", "probability": 0.935546875}, {"start": 3129.76, "end": 3130.38, "word": " absolute", "probability": 0.56591796875}, {"start": 3130.38, "end": 3130.64, "word": " ..", "probability": 0.254150390625}, {"start": 3130.64, "end": 3130.96, "word": " فهذا", "probability": 0.8434244791666666}, {"start": 3130.96, "end": 3131.46, "word": " بيساوي", "probability": 0.85263671875}, {"start": 3131.46, "end": 3131.86, "word": " واحد", "probability": 0.9775390625}, {"start": 3131.86, "end": 3132.44, "word": " بيساوي", "probability": 0.90224609375}, {"start": 3132.44, "end": 3132.86, "word": " limit", "probability": 0.95166015625}, {"start": 3132.86, "end": 3134.54, "word": " واحد", "probability": 0.916748046875}, {"start": 3134.54, "end": 3134.92, "word": " as", "probability": 0.9189453125}, {"start": 3134.92, "end": 3135.26, "word": " X", "probability": 0.755859375}, {"start": 3135.26, "end": 3135.52, "word": " tends", "probability": 0.245361328125}, {"start": 3135.52, "end": 3135.64, "word": " to", "probability": 0.9599609375}, {"start": 3135.64, "end": 3136.18, "word": " C", "probability": 0.876953125}, {"start": 3136.18, "end": 3138.02, "word": " بيساوي", "probability": 0.97119140625}, {"start": 3138.02, "end": 3141.44, "word": " واحد", "probability": 0.9443359375}, {"start": 3141.44, "end": 3143.64, "word": " بيساوي", "probability": 0.964453125}, {"start": 3143.64, "end": 3144.04, "word": " F", "probability": 0.9345703125}, {"start": 3144.04, "end": 3144.38, "word": " and", "probability": 0.9384765625}, {"start": 3144.38, "end": 3144.7, "word": " C", "probability": 0.9853515625}], "temperature": 1.0}, {"id": 112, "seek": 316391, "start": 3147.29, "end": 3163.91, "text": "أو signum C لأن ال C موجبة فلما ال C تكون موجبة ف absolute ال C بساوي ال C بطلع المخضر هذا بطلع واحد", "tokens": [10721, 2407, 1465, 449, 383, 5296, 33456, 2423, 383, 3714, 29245, 49401, 6156, 1211, 15042, 2423, 383, 6055, 30544, 3714, 29245, 49401, 6156, 8236, 2423, 383, 4724, 3794, 995, 45865, 2423, 383, 4724, 9566, 1211, 3615, 9673, 9778, 11242, 2288, 23758, 4724, 9566, 1211, 3615, 36764, 24401], "avg_logprob": -0.2887369903425376, "compression_ratio": 1.3166666666666667, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 3147.29, "end": 3147.63, "word": "أو", "probability": 0.5682373046875}, {"start": 3147.63, "end": 3148.19, "word": " signum", "probability": 0.6112060546875}, {"start": 3148.19, "end": 3148.51, "word": " C", "probability": 0.5791015625}, {"start": 3148.51, "end": 3152.49, "word": " لأن", "probability": 0.76513671875}, {"start": 3152.49, "end": 3152.71, "word": " ال", "probability": 0.544921875}, {"start": 3152.71, "end": 3152.87, "word": " C", "probability": 0.37255859375}, {"start": 3152.87, "end": 3153.59, "word": " موجبة", "probability": 0.9586588541666666}, {"start": 3153.59, "end": 3154.77, "word": " فلما", "probability": 0.7957356770833334}, {"start": 3154.77, "end": 3155.01, "word": " ال", "probability": 0.56640625}, {"start": 3155.01, "end": 3156.65, "word": " C", "probability": 0.935546875}, {"start": 3156.65, "end": 3158.09, "word": " تكون", "probability": 0.9326171875}, {"start": 3158.09, "end": 3158.73, "word": " موجبة", "probability": 0.98974609375}, {"start": 3158.73, "end": 3158.99, "word": " ف", "probability": 0.96728515625}, {"start": 3158.99, "end": 3159.45, "word": " absolute", "probability": 0.64501953125}, {"start": 3159.45, "end": 3159.81, "word": " ال", "probability": 0.4697265625}, {"start": 3159.81, "end": 3160.05, "word": " C", "probability": 0.95703125}, {"start": 3160.05, "end": 3160.69, "word": " بساوي", "probability": 0.8609619140625}, {"start": 3160.69, "end": 3161.17, "word": " ال", "probability": 0.7568359375}, {"start": 3161.17, "end": 3161.49, "word": " C", "probability": 0.9384765625}, {"start": 3161.49, "end": 3162.11, "word": " بطلع", "probability": 0.74859619140625}, {"start": 3162.11, "end": 3163.09, "word": " المخضر", "probability": 0.7437744140625}, {"start": 3163.09, "end": 3163.27, "word": " هذا", "probability": 0.8837890625}, {"start": 3163.27, "end": 3163.63, "word": " بطلع", "probability": 0.9365234375}, {"start": 3163.63, "end": 3163.91, "word": " واحد", "probability": 0.97802734375}], "temperature": 1.0}, {"id": 113, "seek": 319195, "start": 3165.97, "end": 3191.95, "text": "و بالتالي إذا ال signal x is continuous at c case 2 إذا كانت ال c أصغر من سفر ف similar to case 1 في الحالة هذه", "tokens": [2407, 20666, 2655, 6027, 1829, 11933, 15730, 2423, 6358, 2031, 307, 10957, 412, 269, 1389, 568, 11933, 15730, 25961, 2655, 2423, 269, 5551, 9381, 17082, 2288, 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{"start": 3203.06, "end": 3203.24, "word": " c", "probability": 0.58154296875}, {"start": 3203.24, "end": 3204.1, "word": " إذا", "probability": 0.697509765625}, {"start": 3204.1, "end": 3204.32, "word": " ال", "probability": 0.6796875}, {"start": 3204.32, "end": 3204.46, "word": " sign", "probability": 0.13916015625}, {"start": 3204.46, "end": 3204.62, "word": " and", "probability": 0.65283203125}, {"start": 3204.62, "end": 3205.0, "word": " function", "probability": 0.9736328125}, {"start": 3205.0, "end": 3205.26, "word": " مش", "probability": 0.9794921875}, {"start": 3205.26, "end": 3205.78, "word": " متصلة", "probability": 0.9500325520833334}, {"start": 3205.78, "end": 3205.92, "word": " عند", "probability": 0.982421875}, {"start": 3205.92, "end": 3206.32, "word": " الصفر", "probability": 0.759521484375}, {"start": 3206.32, "end": 3206.78, "word": " لكنها", "probability": 0.731201171875}, {"start": 3206.78, "end": 3207.46, "word": " متصلة", "probability": 0.9703776041666666}, {"start": 3207.46, "end": 3208.3, "word": " عن", "probability": 0.89453125}, {"start": 3208.3, "end": 3208.66, "word": " كل", "probability": 0.9765625}, {"start": 3208.66, "end": 3209.22, "word": " الأعداد", "probability": 0.9278971354166666}, {"start": 3209.22, "end": 3209.88, "word": " الحقيقية", "probability": 0.978271484375}, {"start": 3209.88, "end": 3210.6, "word": " المختلفة", "probability": 0.9930419921875}, {"start": 3210.6, "end": 3210.8, "word": " عن", "probability": 0.958984375}, {"start": 3210.8, "end": 3211.22, "word": " الصفر", "probability": 0.9796549479166666}], "temperature": 1.0}, {"id": 115, "seek": 323835, "start": 3212.85, "end": 3238.35, "text": "Okay بنكتفي بهذا القدر و بنكمل طبعا إن شاء الله في المحاضرة القادمة هنعطيكم إن شاء الله break خمس دقائق و بعدين نواصل المحاضرة التانية اللي هناخد فيها discussion أو مناقشة لل chapter أربعة section أربعة واحد و أربعة اتنين", "tokens": [8297, 44945, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288, 4032, 44945, 24793, 1211, 23032, 3555, 3615, 995, 36145, 13412, 16606, 21984, 8978, 9673, 5016, 46958, 25720, 25062, 18513, 46007, 8032, 1863, 3615, 9566, 1829, 24793, 36145, 13412, 16606, 21984, 1821, 16490, 2304, 3794, 11778, 4587, 16373, 4587, 4032, 39182, 9957, 8717, 2407, 33546, 1211, 9673, 5016, 46958, 25720, 16712, 7649, 10632, 13672, 1829, 8032, 1863, 47283, 3215, 8978, 11296, 5017, 34051, 3714, 8315, 4587, 8592, 3660, 24976, 7187, 5551, 25513, 27884, 3541, 5551, 25513, 27884, 36764, 24401, 4032, 5551, 25513, 27884, 1975, 2655, 1863, 9957], "avg_logprob": -0.11009986867609713, "compression_ratio": 1.6818181818181819, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 3212.85, "end": 3213.27, "word": "Okay", "probability": 0.900390625}, {"start": 3213.27, "end": 3213.81, "word": " بنكتفي", "probability": 0.840576171875}, {"start": 3213.81, "end": 3214.17, "word": " بهذا", "probability": 0.960205078125}, {"start": 3214.17, "end": 3214.77, "word": " القدر", "probability": 0.99609375}, {"start": 3214.77, "end": 3216.21, "word": " و", "probability": 0.80615234375}, {"start": 3216.21, "end": 3217.11, "word": " بنكمل", "probability": 0.9396158854166666}, {"start": 3217.11, "end": 3217.53, "word": " طبعا", "probability": 0.9783935546875}, {"start": 3217.53, "end": 3217.71, "word": " إن", "probability": 0.2607421875}, {"start": 3217.71, "end": 3217.91, "word": " شاء", "probability": 0.979248046875}, {"start": 3217.91, "end": 3218.05, "word": " الله", "probability": 0.9541015625}, {"start": 3218.05, "end": 3218.49, "word": " في", "probability": 0.9638671875}, {"start": 3218.49, "end": 3219.39, "word": " المحاضرة", "probability": 0.974609375}, {"start": 3219.39, "end": 3221.01, "word": " القادمة", "probability": 0.99462890625}, {"start": 3221.01, "end": 3223.41, "word": " هنعطيكم", "probability": 0.8848470052083334}, {"start": 3223.41, "end": 3223.61, "word": " إن", "probability": 0.8759765625}, {"start": 3223.61, "end": 3223.75, "word": " شاء", "probability": 0.990478515625}, {"start": 3223.75, "end": 3223.91, "word": " الله", "probability": 0.966796875}, {"start": 3223.91, "end": 3224.39, "word": " break", "probability": 0.990234375}, {"start": 3224.39, "end": 3224.73, "word": " خمس", "probability": 0.8828125}, {"start": 3224.73, "end": 3225.33, "word": " دقائق", "probability": 0.85888671875}, {"start": 3225.33, "end": 3226.39, "word": " و", "probability": 0.87451171875}, {"start": 3226.39, "end": 3226.87, "word": " بعدين", "probability": 0.92724609375}, {"start": 3226.87, "end": 3227.61, "word": " نواصل", "probability": 0.90966796875}, {"start": 3227.61, "end": 3228.23, "word": " المحاضرة", "probability": 0.9945068359375}, {"start": 3228.23, "end": 3228.83, "word": " التانية", "probability": 0.9806315104166666}, {"start": 3228.83, "end": 3229.35, "word": " اللي", "probability": 0.850341796875}, {"start": 3229.35, "end": 3229.81, "word": " هناخد", "probability": 0.97119140625}, {"start": 3229.81, "end": 3230.17, "word": " فيها", "probability": 0.993408203125}, {"start": 3230.17, "end": 3231.27, "word": " discussion", "probability": 0.9853515625}, {"start": 3231.27, "end": 3231.67, "word": " أو", "probability": 0.64892578125}, {"start": 3231.67, "end": 3232.71, "word": " مناقشة", "probability": 0.960546875}, {"start": 3232.71, "end": 3234.83, "word": " لل", "probability": 0.3310546875}, {"start": 3234.83, "end": 3235.41, "word": " chapter", "probability": 0.94873046875}, {"start": 3235.41, "end": 3236.09, "word": " أربعة", "probability": 0.7698567708333334}, {"start": 3236.09, "end": 3236.95, "word": " section", "probability": 0.8017578125}, {"start": 3236.95, "end": 3237.31, "word": " أربعة", "probability": 0.8863932291666666}, {"start": 3237.31, "end": 3237.63, "word": " واحد", "probability": 0.95849609375}, {"start": 3237.63, "end": 3237.77, "word": " و", "probability": 0.87060546875}, {"start": 3237.77, "end": 3237.99, "word": " أربعة", "probability": 0.9130859375}, {"start": 3237.99, "end": 3238.35, "word": " اتنين", "probability": 0.9254150390625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3239.741, "duration_after_vad": 2800.675624999989}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e9277334a6bb81561e479de8762ab5c5e1894991
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Gx7j9GpXuiI_raw.srt
@@ -0,0 +1,1812 @@
+1
+00:00:21,580 --> 00:00:26,880
+بسم الله الرحمن الرحيم اليوم ان شاء الله هنبدأ
+
+2
+00:00:26,880 --> 00:00:34,000
+chapter خمسة و هذا اخر chapter هناخده في ال course
+
+3
+00:00:34,000 --> 00:00:50,080
+فانواع ال chapter هذا continuous
+
+4
+00:00:53,880 --> 00:01:01,820
+functions الدوالة المتصلة و
+
+5
+00:01:01,820 --> 00:01:08,460
+أول section برضه section خمسة واحد في هذا ال
+
+6
+00:01:08,460 --> 00:01:16,320
+chapter برضه عنوانه continuous functions
+
+7
+00:01:24,100 --> 00:01:29,280
+الدولة المتصلة فنعرف شو معنى الدولة تكون متصلة عن
+
+8
+00:01:29,280 --> 00:01:35,160
+نقطة definition let
+
+9
+00:01:35,160 --> 00:01:49,280
+f be function from a to r and c be element of a we
+
+10
+00:01:49,280 --> 00:02:00,630
+sayإنه ال function if is continuous if
+
+11
+00:02:00,630 --> 00:02:05,770
+is continuous at
+
+12
+00:02:05,770 --> 00:02:18,950
+x بساوي c if إذا تحقق الشرط التالي for
+
+13
+00:02:18,950 --> 00:02:20,470
+every
+
+14
+00:02:22,680 --> 00:02:29,400
+إبسلون أكبر من السفر نقدر نرد عليها delta تعتمد
+
+15
+00:02:29,400 --> 00:02:37,840
+على إبسلون positive number بحيث أنه لكل X لكل
+
+16
+00:02:37,840 --> 00:02:44,090
+X في Aو ال absolute value ل x minus c أصغر من
+
+17
+00:02:44,090 --> 00:02:52,170
+delta فهذا بتضمن ان absolute f of x minus f of c
+
+18
+00:02:52,170 --> 00:03:01,630
+أصغر من ال epsilon فهذا
+
+19
+00:03:01,630 --> 00:03:13,010
+بنسميه this is calledthis is called epsilon delta
+
+20
+00:03:13,010 --> 00:03:18,770
+definition of
+
+21
+00:03:18,770 --> 00:03:31,170
+continuity لأن
+
+22
+00:03:31,170 --> 00:03:36,790
+هذا تعريف epsilon delta للاتصال لحظو هذا التعريف
+
+23
+00:03:36,790 --> 00:03:44,530
+تقريبا هوهو تعريف انه limit ال function f of x لما
+
+24
+00:03:44,530 --> 00:03:52,310
+x تقوله c بساوي f of c هدد
+
+25
+00:03:52,310 --> 00:04:07,210
+كانت c is a cluster point طب
+
+26
+00:04:07,210 --> 00:04:13,930
+لحظة انتوالما عرفنا احنا ما معناه انه ال limit ل
+
+27
+00:04:13,930 --> 00:04:18,710
+function and x بيساوي C و C cluster point للمجموع
+
+28
+00:04:18,710 --> 00:04:24,570
+A بيساوي عدد L بدلنا L هنا ب F و C صح؟ معناه كان
+
+29
+00:04:24,570 --> 00:04:30,290
+لكل إبسلون فيه Delta بحيث لكل X في A و ال X هذه
+
+30
+00:04:30,290 --> 00:04:37,540
+كانت مختلفة لا تساوي C فكنا نحط هنا أكبر من 0فإذا
+
+31
+00:04:37,540 --> 00:04:41,480
+كانت المسافة هذه أصغر من دلتا تطلع المسافة من f of
+
+32
+00:04:41,480 --> 00:04:46,040
+x وال L اللي هي ال limit هنا طبعا احنا بدلنا ال L
+
+33
+00:04:46,040 --> 00:04:50,940
+ب F of C فبين هذا يطلع أصغر من X هنا تقريبا نفس
+
+34
+00:04:50,940 --> 00:04:56,480
+التعريف if
+
+35
+00:04:56,480 --> 00:05:00,460
+if
+
+36
+00:05:00,460 --> 00:05:09,090
+is not continuousلو كانت ال F ليست متصلة عند
+
+37
+00:05:09,090 --> 00:05:14,910
+النقطة C فبنقول if
+
+38
+00:05:14,910 --> 00:05:31,810
+F fails to be continuous at C we say ان F is
+
+39
+00:05:31,810 --> 00:05:32,990
+discontinuous
+
+40
+00:05:38,310 --> 00:05:46,350
+discontinuous at c إذا لو كانت الدالة مش متصلة عن
+
+41
+00:05:46,350 --> 00:05:52,710
+c يعني شرط الاتصال هذا مش متحقق فبنقول أن الدالة
+
+42
+00:05:52,710 --> 00:05:57,610
+discontinuous منفصلة عند النقطة c okay تمام
+
+43
+00:06:09,660 --> 00:06:17,360
+بنلاحظ انه ال .. زي ما شوفنا في section 4-1 تعريف
+
+44
+00:06:17,360 --> 00:06:21,840
+epsilon delta لل limits of functions في بكافة
+
+45
+00:06:21,840 --> 00:06:26,600
+neighborhood definition وهنا برضه تعريف ال epsilon
+
+46
+00:06:26,600 --> 00:06:31,760
+delta definition للاتصال عن النقطة في بكافة
+
+47
+00:06:31,760 --> 00:06:36,400
+neighborhood definition فنكتب ال neighborhood
+
+48
+00:06:36,400 --> 00:06:37,340
+definition هذا
+
+49
+00:06:46,200 --> 00:06:53,400
+لت if دي function from a to r و c belong to a then
+
+50
+00:06:53,400 --> 00:07:02,480
+the following statements are equivalent واحد
+
+51
+00:07:02,480 --> 00:07:11,180
+ال function if is continuous is continuous at x
+
+52
+00:07:11,180 --> 00:07:12,540
+بساوي z
+
+53
+00:07:20,900 --> 00:07:26,360
+إتنين هذا طبعا إتنين نسميه in labor hood
+
+54
+00:07:26,360 --> 00:07:31,940
+definition of continuity
+
+55
+00:07:45,120 --> 00:07:48,580
+الـ neighborhood definition للـ continuity ايش
+
+56
+00:07:48,580 --> 00:07:57,920
+بيقول لكل for every epsilon neighborhood v epsilon
+
+57
+00:07:57,920 --> 00:08:05,700
+لنقطة f of c there
+
+58
+00:08:05,700 --> 00:08:18,440
+exist delta neighborhood v deltaof C لنقطة C طبعا
+
+59
+00:08:18,440 --> 00:08:26,200
+هذا epsilon neighborhood ل F of C يوجد delta
+
+60
+00:08:26,200 --> 00:08:38,660
+neighborhood V Delta of C بحيث انه لكل X تنتمي إلى
+
+61
+00:08:38,660 --> 00:08:47,830
+A تقاطع الـ Deltaneighborhood ل C لازم هذا يضمن ان
+
+62
+00:08:47,830 --> 00:08:53,050
+صورة ال X تنتمي
+
+63
+00:08:53,050 --> 00:09:04,590
+الى D epsilon ل F of C that
+
+64
+00:09:04,590 --> 00:09:08,630
+is that
+
+65
+00:09:08,630 --> 00:09:11,910
+is هذا يعني ان ال
+
+66
+00:09:14,980 --> 00:09:23,060
+الـ image للست A تقاطع V Delta of C is contained
+
+67
+00:09:23,060 --> 00:09:34,140
+in الـ Epsilon neighbourhood لـ F of C
+
+68
+00:09:34,140 --> 00:09:40,100
+هاي
+
+69
+00:09:40,100 --> 00:09:47,330
+كان في عنديزي هيك مثلا يكون في اندي فانكشن زي هذه
+
+70
+00:09:47,330 --> 00:09:57,210
+y بساوي f of x وقلنا
+
+71
+00:09:57,210 --> 00:10:03,810
+انه لو كانت x او c c
+
+72
+00:10:03,810 --> 00:10:07,670
+نقطة ال dial عندها متصلة هي f of c
+
+73
+00:10:11,410 --> 00:10:17,830
+ما معناه ان الدالة متصلة عند X بساوي C معناه لو
+
+74
+00:10:17,830 --> 00:10:23,770
+أخدت لأي
+
+75
+00:10:23,770 --> 00:10:30,850
+إبسلون أكبر من سفر فيه Delta أو لو أخدت أي إبسلون
+
+76
+00:10:30,850 --> 00:10:31,290
+neighborhood
+
+77
+00:10:34,530 --> 00:10:38,270
+يعني النقطة هذه F of C زاد Epsilon النقطة هذه
+
+78
+00:10:38,270 --> 00:10:48,430
+المسافة هذه Epsilon فهذه F of C سالب Epsilon فهذه
+
+79
+00:10:48,430 --> 00:10:53,610
+الفترة المفتوحة عبارة عن Epsilon neighborhood ل F
+
+80
+00:10:53,610 --> 00:10:54,150
+of C
+
+81
+00:10:57,200 --> 00:11:01,620
+فلأي إبسلون أكبر من السفر ممكن أكوّن إبسلون
+
+82
+00:11:01,620 --> 00:11:06,420
+neighborhood ل F of C وبالتالي بقدر أرد على الـ
+
+83
+00:11:06,420 --> 00:11:14,580
+Epsilon neighborhood هذا بـ Delta يعني
+
+84
+00:11:14,580 --> 00:11:20,980
+أكوّن Delta neighborhood هنا C minus Delta C موجة
+
+85
+00:11:20,980 --> 00:11:21,460
+بDelta
+
+86
+00:11:26,200 --> 00:11:37,060
+إذاً هذا عبارة عن V Delta V Delta ل C إذاً
+
+87
+00:11:37,060 --> 00:11:43,200
+لأي إبسلون لأي إبسلون neighborhood ل F of C بقدر
+
+88
+00:11:43,200 --> 00:11:52,720
+ألاقي Delta neighborhood للمقطة C بحيث أنه لكل Xلو
+
+89
+00:11:52,720 --> 00:12:01,620
+أخدت x نقطة في الـ delta neighborhood فصورتها f of
+
+90
+00:12:01,620 --> 00:12:09,060
+x هتطلع تنتمي لل epsilon neighborhood لل F of C
+
+91
+00:12:09,060 --> 00:12:17,140
+okay تمام فهذا هو نفسه هذا بكافي التعريف هذا بكافي
+
+92
+00:12:17,140 --> 00:12:20,660
+التعريف ال epsilon delta definition لل continuity
+
+93
+00:12:24,390 --> 00:12:29,850
+هي لكل إبسلون لكل إبسلون أكبر من الصفر يعني كأني
+
+94
+00:12:29,850 --> 00:12:36,450
+بقول لكل إبسلون نبرهود ل F و C يوجد Delta عدد موجب
+
+95
+00:12:36,450 --> 00:12:44,290
+فهذا معناه يوجد Delta نبرهود لل C بحيث أنه لكل X
+
+96
+00:12:44,290 --> 00:12:50,560
+المسافر لكل X تنتمي لكل X في Aو X بالتحقق
+
+97
+00:12:50,560 --> 00:12:55,980
+المتباينة دي معناته X سنتمي المسافة بين X و C أصغر
+
+98
+00:12:55,980 --> 00:12:56,380
+من Delta
+
+99
+00:13:02,120 --> 00:13:07,000
+فهذا بيقدي ان المسافة بين F of X و F of C هي F of
+
+100
+00:13:07,000 --> 00:13:12,160
+X و F of C أصغر من Epsilon يعني ال F of X هذه
+
+101
+00:13:12,160 --> 00:13:17,900
+تنتمي لل Epsilon برهود ل F of C إذن التعريفين هذول
+
+102
+00:13:17,900 --> 00:13:24,800
+متكافئين وهذا واضح من الرسم وبالتالي البرهان جاهز
+
+103
+00:13:24,800 --> 00:13:32,000
+من .. بس ترجمتهالحاجات هذه الى لغة ال neighborhood
+
+104
+00:13:32,000 --> 00:13:39,600
+اذا في لان تعريفين للاتصال على النقطة واحد epsilon
+
+105
+00:13:39,600 --> 00:13:45,400
+delta definition والتاني اللي بكافه neighborhood
+
+106
+00:13:45,400 --> 00:13:50,360
+definition طيب
+
+107
+00:13:50,360 --> 00:13:55,260
+ناخد بعض الملاحظات على تعريف الاتصال
+
+108
+00:14:16,000 --> 00:14:22,640
+إذا C هو مقاومة مقاومة
+
+109
+00:14:22,640 --> 00:14:30,180
+A ثم
+
+110
+00:14:30,180 --> 00:14:38,200
+F مستمر في X بساوي
+
+111
+00:14:42,830 --> 00:14:47,530
+لو كانت الـ C هذه cluster point فالاتصال ان C
+
+112
+00:14:47,530 --> 00:14:55,730
+بكافئ بكافئ ان ال limit ل F of X من تعريف ال
+
+113
+00:14:55,730 --> 00:15:03,570
+limits ان C بساوي F of C وهذا
+
+114
+00:15:03,570 --> 00:15:06,790
+طبعاً
+
+115
+00:15:06,790 --> 00:15:09,090
+this condition
+
+116
+00:15:12,780 --> 00:15:19,680
+is three in
+
+117
+00:15:19,680 --> 00:15:24,800
+one ال
+
+118
+00:15:24,800 --> 00:15:30,480
+definition هذا بكافئ تلت او الشرط هذا بكافئ تلت
+
+119
+00:15:30,480 --> 00:15:37,600
+شروط او هو تلت شروط في واحد اول شرط ان ال function
+
+120
+00:15:37,600 --> 00:15:39,540
+f and c is defined
+
+121
+00:15:43,900 --> 00:15:49,540
+يعني هذا عبارة عن عدد حقيقي name ال limit ل f of x
+
+122
+00:15:49,540 --> 00:15:56,180
+لما x تقول إلى c exist يعني عدد حقيقي والشرط
+
+123
+00:15:56,180 --> 00:16:04,880
+التالت أنه لازم ال limit لل function f and c بساوة
+
+124
+00:16:04,880 --> 00:16:09,980
+قيمة الدالة and c يعني عشان الدالة تكون متصلة عند
+
+125
+00:16:09,980 --> 00:16:16,020
+النقطة c في مجالهاو لو كانت الـ C هي cluster point
+
+126
+00:16:16,020 --> 00:16:21,790
+طبعاأو حتى لو ماكنتش cluster point فلازم التلاتة
+
+127
+00:16:21,790 --> 00:16:25,250
+صوروطها تتحقق الدالة معرفة عن C طبعا هذا لأن C
+
+128
+00:16:25,250 --> 00:16:30,450
+نقطة في مجال الدالة فلازم تكون معرفة عن ده لازم ال
+
+129
+00:16:30,450 --> 00:16:34,830
+limit ل F عن C تكون موجودة وقيمة ال limit بساوي
+
+130
+00:16:34,830 --> 00:16:39,290
+قيمة الدالة عند النقطة C لو أي واحد ماليش صوروط
+
+131
+00:16:39,290 --> 00:16:43,830
+التلاتة هدول اختل فبنقول ان ال function مش متصلة
+
+132
+00:16:43,830 --> 00:16:49,410
+عند النقطة COkay تمام واضح اذا لو كانت ال C هي دي
+
+133
+00:16:49,410 --> 00:16:53,510
+cluster point فتعريف الاتصال النقطة هو بالظبط
+
+134
+00:16:53,510 --> 00:16:58,470
+تعريف ان limited دالة ان C تكون موجودة و بتساوي
+
+135
+00:16:58,470 --> 00:17:02,570
+قيمتها ان C وهذا الشرط هو تلات شروط و ال C في ال A
+
+136
+00:17:02,570 --> 00:17:09,510
+نعم ال C تنتمي ل A اه طبعا ال C تنتمي ل A ال C
+
+137
+00:17:09,510 --> 00:17:11,130
+دائما تنتمي ل A
+
+138
+00:17:17,100 --> 00:17:22,120
+طب لو ماكناش ال c cluster point الملاحظة التانية
+
+139
+00:17:22,120 --> 00:17:29,440
+if c is not يعني لو كان ال c تنتمي طبعا دايما ال c
+
+140
+00:17:29,440 --> 00:17:40,980
+تنتمي ل a is not a cluster point is
+
+141
+00:17:40,980 --> 00:17:44,100
+not cluster point of a
+
+142
+00:17:48,950 --> 00:17:54,070
+then من تعريف ال cluster point لازم نلاقي delta
+
+143
+00:17:54,070 --> 00:18:05,430
+أكبر من السفر such that a تقاطع v delta of c بساوي
+
+144
+00:18:05,430 --> 00:18:06,850
+singleton c
+
+145
+00:18:11,300 --> 00:18:14,580
+ما معناه ان النقطة C الموجودة في A مايعنياش
+
+146
+00:18:14,580 --> 00:18:18,460
+cluster point او ما معناه ان C تنتمي ل A cluster
+
+147
+00:18:18,460 --> 00:18:24,380
+point معناها ان كل delta neighborhood لل C بيتقاطع
+
+148
+00:18:24,380 --> 00:18:30,400
+مع A في نقطة مختلفة عن ال C على الأقلمعناه ان الـ
+
+149
+00:18:30,400 --> 00:18:34,040
+C ما تكونش cluster point معناه ان يوجد delta
+
+150
+00:18:34,040 --> 00:18:37,040
+neighborhood واحد يعني يوجد delta عدد موجب
+
+151
+00:18:37,040 --> 00:18:40,780
+وبالتالي يوجد على الأقل delta neighborhood للـ C
+
+152
+00:18:40,780 --> 00:18:46,620
+وهذا ال delta neighborhoodمابتقاطعش مع a في أي
+
+153
+00:18:46,620 --> 00:18:50,660
+نقطة مختلفة عن ال c يعني التقاطع هذا بس في نقطة
+
+154
+00:18:50,660 --> 00:18:55,300
+واحدة c لأن ال c هي مركز ال neighborhood و c تنتمي
+
+155
+00:18:55,300 --> 00:19:03,320
+ل a فالتقاطع هذا مافيش فيه أي x مختلفة عن ال c في
+
+156
+00:19:03,320 --> 00:19:09,740
+الحالة هذه in this case in
+
+157
+00:19:09,740 --> 00:19:10,580
+this case
+
+158
+00:19:14,230 --> 00:19:23,970
+if is automatically continuous
+
+159
+00:19:23,970 --> 00:19:34,940
+at cالدولة في الحالة هذه بتكون متصلة تلقائيًا عند
+
+160
+00:19:34,940 --> 00:19:39,360
+النقطة C أو التعريف متحقق تلقائيًا ليه؟ لأنه
+
+161
+00:19:39,360 --> 00:19:44,060
+تعالوا نرجع للتعريف ما معناه أن F تكون متصلة عند
+
+162
+00:19:44,060 --> 00:19:49,660
+النقطة C معناه لأي epsilon neighborhood ل F و C
+
+163
+00:19:49,660 --> 00:19:53,680
+نقدر
+
+164
+00:19:53,680 --> 00:19:57,020
+نلاقي يوجد delta neighborhood ل C فخد ال delta
+
+165
+00:19:57,020 --> 00:20:00,040
+neighborhoodفي التعريف هذا خد الـ delta
+
+166
+00:20:00,040 --> 00:20:07,900
+neighborhood هو هذا ففي الحالة هذه لكل x تنتمي إلى
+
+167
+00:20:07,900 --> 00:20:12,340
+a تقاطع v delta و c ما التقاطع هذا مافيش فيه إلا
+
+168
+00:20:12,340 --> 00:20:17,100
+نقطة واحدة اللي هي c صح فلكل x موجود في التقاطع
+
+169
+00:20:17,100 --> 00:20:21,950
+هذا مافيش إلا x بالساوية cفصورة ال X هذه هي صورة
+
+170
+00:20:21,950 --> 00:20:28,210
+ال C وبالتالي صورة ال X هذه هي صورة ال C فهذه أكيد
+
+171
+00:20:28,210 --> 00:20:33,310
+تنتمي لإمسلون برهود ل F of C لأن ال F of C هي
+
+172
+00:20:33,310 --> 00:20:38,850
+المركز تبع الفترة هذه صح فهذا شرط متحقق trivially
+
+173
+00:20:38,850 --> 00:20:44,870
+تلقايا وبالتالي إذا السواء
+
+174
+00:20:46,570 --> 00:20:49,730
+سواء الـ C هنا كانت cluster point أو ماكنتش
+
+175
+00:20:49,730 --> 00:20:55,630
+cluster point فممكن نعتبر أن التعريف لاتصال النقطة
+
+176
+00:20:55,630 --> 00:21:00,450
+هو التعريف هذا لأن لو كانت ال C cluster point
+
+177
+00:21:00,450 --> 00:21:04,190
+فتعريف لاتصال النقطة هو هذا التعريف لو كانت ال C
+
+178
+00:21:04,190 --> 00:21:07,750
+ماهياش cluster point فهذا التعريف متحقق ال trivia
+
+179
+00:21:07,750 --> 00:21:12,380
+اللي بدهيوبالتالي مافيش داعي ان احنا نقول .. لما
+
+180
+00:21:12,380 --> 00:21:14,840
+نيجي نفحص الاتصال على النقطة C نقول هل ال C
+
+181
+00:21:14,840 --> 00:21:18,840
+cluster point او مش cluster point سواء كانت
+
+182
+00:21:18,840 --> 00:21:24,380
+cluster point او ماكانتش cluster point فالاتصال
+
+183
+00:21:24,380 --> 00:21:33,020
+and ال C بيصير هو .. يعني هل هذا شرط بتحقق او لا
+
+184
+00:21:41,130 --> 00:21:44,890
+طبعا زي ما اخدنا احنا ايام ما خدنا دراسنا ال
+
+185
+00:21:44,890 --> 00:21:54,950
+limits لل functions فكان
+
+186
+00:21:54,950 --> 00:21:57,590
+في عندي sequential criterion for limits
+
+187
+00:22:02,270 --> 00:22:06,810
+بنفس الطريقة في هنا sequential criterion for
+
+188
+00:22:06,810 --> 00:22:15,990
+continuity للاتصال إذا في عندي هنا sequential
+
+189
+00:22:15,990 --> 00:22:21,130
+criterion
+
+190
+00:22:21,130 --> 00:22:24,150
+for
+
+191
+00:22:24,150 --> 00:22:25,110
+continuity
+
+192
+00:22:35,670 --> 00:22:44,430
+let f be function from a to r و c نقطة في a then
+
+193
+00:22:44,430 --> 00:22:56,170
+the following statements are equivalent واحد
+
+194
+00:22:56,170 --> 00:23:08,010
+if is continuousif is continuous at c for
+
+195
+00:23:08,010 --> 00:23:11,910
+every for
+
+196
+00:23:11,910 --> 00:23:22,050
+every sequence x in contained in a with
+
+197
+00:23:22,050 --> 00:23:25,370
+limit
+
+198
+00:23:25,370 --> 00:23:41,270
+x in بساوي cنحن لدينا ان ال limit ل f of x n as n
+
+199
+00:23:41,270 --> 00:23:45,790
+tensor infinity بسوي f of c
+
+200
+00:23:51,740 --> 00:23:54,940
+الان الـ sequential criterion for continuity بتقول
+
+201
+00:23:54,940 --> 00:24:00,380
+عشان اثبت ان الدالة F continuous عند نقطة يكفي ان
+
+202
+00:24:00,380 --> 00:24:04,900
+انا اثبت ان لو اخدت اي sequence نهايتها اي
+
+203
+00:24:04,900 --> 00:24:07,660
+sequence في مجال الدالة طبعا كنا في ال limits
+
+204
+00:24:07,660 --> 00:24:13,020
+نشترط ان X in كل انصر في ال sequence مختلف عن ال C
+
+205
+00:24:13,020 --> 00:24:17,200
+هنا لأ ممكن يساوي ال C مش مشكلة هاي الاختلاف بس
+
+206
+00:24:17,200 --> 00:24:21,430
+بين ال sequential criterion for limitsو Sequential
+
+207
+00:24:21,430 --> 00:24:26,030
+criterion for continuity إنه لكل sequence x in في
+
+208
+00:24:26,030 --> 00:24:32,550
+مجال الدالة و نهايتها بتساوي c لازم اطلع عندي
+
+209
+00:24:32,550 --> 00:24:37,990
+نهاية ال image تبعت ال sequence x in بتساوي العدد
+
+210
+00:24:37,990 --> 00:24:42,860
+f و cوبرهان النظرية هذه زي برهان sequential
+
+211
+00:24:42,860 --> 00:24:49,120
+criterion for limits مع تعديلات طفيفة مع التعديلات
+
+212
+00:24:49,120 --> 00:24:58,580
+الطفيفة في التعريفين او في التعريف تبع الاتصال اذا
+
+213
+00:24:58,580 --> 00:25:11,090
+ال proof similar to proof ofsequential criterion
+
+214
+00:25:11,090 --> 00:25:19,570
+for limits for limits sequential criterion for
+
+215
+00:25:19,570 --> 00:25:34,190
+limits of functions in section أربعة واحد with
+
+216
+00:25:34,190 --> 00:25:38,030
+slight modification
+
+217
+00:25:45,120 --> 00:25:51,780
+مع تعديل بسيط مع تعديل بسيط التعديل هنا انه ال هنا
+
+218
+00:25:51,780 --> 00:25:58,180
+كنا نطلب ال X لا تساوي C وكمان كنا هناك نطلب انه C
+
+219
+00:25:58,180 --> 00:26:02,740
+تكون cluster point لكن شوفنا حتى لو C ماكنتش
+
+220
+00:26:02,740 --> 00:26:10,940
+cluster point فهذا برضه متحقق تلقائيا برضه
+
+221
+00:26:10,940 --> 00:26:11,700
+أخدنا
+
+222
+00:26:14,550 --> 00:26:18,230
+بعد ما أخدنا الـ sequential criterion for limits
+
+223
+00:26:18,230 --> 00:26:22,410
+of functions في section 4-1 أخدنا بعدها على طول
+
+224
+00:26:22,410 --> 00:26:29,850
+مباشرة divergence criterion for limits فهنا بقابل
+
+225
+00:26:29,850 --> 00:26:38,190
+ال divergence criterion اللي هو
+
+226
+00:26:38,190 --> 00:26:39,910
+discontinuity criterion
+
+227
+00:26:46,180 --> 00:26:48,980
+discontinuity criterion
+
+228
+00:27:00,500 --> 00:27:10,940
+لت if بي function from a to r و c نقطة في a و d
+
+229
+00:27:10,940 --> 00:27:15,440
+then the
+
+230
+00:27:15,440 --> 00:27:23,000
+following statements are equivalent واحد if is
+
+231
+00:27:23,000 --> 00:27:24,160
+discontinuous
+
+232
+00:27:26,370 --> 00:27:36,730
+إذا كان الـ discontinuous at x بساوي c ثم يوجد
+
+233
+00:27:36,730 --> 00:27:49,070
+سيكوينس x in contained in a with limit x in بساوي
+
+234
+00:27:49,070 --> 00:27:49,610
+c
+
+235
+00:27:53,460 --> 00:28:01,180
+but limit الـ image للـ sequence x in لا يساوي f
+
+236
+00:28:01,180 --> 00:28:08,260
+of z وبرهان
+
+237
+00:28:08,260 --> 00:28:13,100
+النظرية هذه بيجي من النظرية الـ sequential
+
+238
+00:28:13,100 --> 00:28:16,980
+criterion أنا
+
+239
+00:28:16,980 --> 00:28:21,400
+عندي واحد one بكافي اتنين one if and only if two
+
+240
+00:28:24,600 --> 00:28:29,660
+وبالتالي not one نفي one بكافئ نفي two طيب تعالى
+
+241
+00:28:29,660 --> 00:28:35,400
+نشوف نفي one if is discontinuous at c نفي two for
+
+242
+00:28:35,400 --> 00:28:40,420
+every sequence بتحقق الشرط هذا نهايت صورتها بساوي
+
+243
+00:28:40,420 --> 00:28:45,160
+f of c ان في الشرط العبارة هذه فبصير there exist a
+
+244
+00:28:45,160 --> 00:28:50,380
+sequence x in contained in a ونهايتها c لكن نهايت
+
+245
+00:28:50,380 --> 00:28:56,550
+صورتها لا تساوي f of cOkay تمام إذا البرهان نظرية
+
+246
+00:28:56,550 --> 00:29:05,170
+هذه جاي من نفي أو ينتج من النظرية السابقة طب
+
+247
+00:29:05,170 --> 00:29:15,350
+نرجع ناخد قبل ما ناخد أمثلة بدنا ناخد بس تعريف
+
+248
+00:29:15,350 --> 00:29:20,170
+الاتصال على مجموعة definition
+
+249
+00:29:24,990 --> 00:29:32,690
+استخدم الفرصة let f be a function from a to r and
+
+250
+00:29:32,690 --> 00:29:38,050
+let
+
+251
+00:29:38,050 --> 00:29:47,090
+b be a subset of a نقول
+
+252
+00:29:47,090 --> 00:29:50,890
+ان الفرصة is continuous
+
+253
+00:29:54,760 --> 00:30:05,060
+if is continuous on الـ set B on the
+
+254
+00:30:05,060 --> 00:30:16,640
+set B if is continuous on the set B if if is
+
+255
+00:30:16,640 --> 00:30:32,720
+continuous at every at everyما ينتمي إلى دي إذا
+
+256
+00:30:32,720 --> 00:30:38,880
+الإتصال على مجموعة معناه إن الدالة تكون متصلة عند
+
+257
+00:30:38,880 --> 00:30:47,520
+كل نقطة في المجموعة، عند كل نقطة في المجموعة طيب
+
+258
+00:30:47,520 --> 00:30:49,080
+ناخد بعض الأمثلة
+
+259
+00:31:06,780 --> 00:31:17,520
+الـ function f of x بتساوي k و
+
+260
+00:31:17,520 --> 00:31:30,460
+x belong to R is continuous on R الدالة
+
+261
+00:31:30,460 --> 00:31:43,300
+ثابت k continuous على كل الـ Rاحنا شفنا proof fix
+
+262
+00:31:43,300 --> 00:31:46,240
+c أنتمي الار
+
+263
+00:31:51,650 --> 00:32:02,150
+Since limit ل F of X as X Sin C بساوي K احنا
+
+264
+00:32:02,150 --> 00:32:07,850
+اثمتنا قبلين ان limit اي ده لثابته بساوي ثابت K
+
+265
+00:32:07,850 --> 00:32:15,690
+وهذا بساوي F and C فال
+
+266
+00:32:15,690 --> 00:32:29,850
+F is continuousat every c ينتمي إلى r فاكرين
+
+267
+00:32:29,850 --> 00:32:34,430
+احنا هدف بقناه باستخدام تعريف epsilon delta قولنا
+
+268
+00:32:34,430 --> 00:32:39,930
+لأي epsilon أكبر من السفر choose أي delta أكبر من
+
+269
+00:32:39,930 --> 00:32:43,690
+السفر فتعريف
+
+270
+00:32:43,690 --> 00:32:47,670
+ال limit بتحقق
+
+271
+00:32:47,670 --> 00:32:48,790
+وهنا نفس الحاجة
+
+272
+00:33:16,050 --> 00:33:25,330
+طيب المثال تاني لو أخدت f of x بساوي x لكل x ينتمي
+
+273
+00:33:25,330 --> 00:33:31,570
+إلى R ال identity function فبرضه
+
+274
+00:33:31,570 --> 00:33:39,350
+أثبتنا احنا ان ال function هذه is continuous if is
+
+275
+00:33:39,350 --> 00:33:44,290
+continuousعلى مجموعة الأعداد الحقيقية
+
+276
+00:34:07,950 --> 00:34:17,850
+فممكن أن نثبت C ينتمي إلى R و أثبتنا احنا في
+
+277
+00:34:17,850 --> 00:34:24,390
+section أربعة واحد ان limit F of X لما X تقول C
+
+278
+00:34:24,390 --> 00:34:32,530
+طلعت بساوي C صح؟ وهذا عبارة عن F of C فالـ F is
+
+279
+00:34:32,530 --> 00:34:35,610
+continuous at C
+
+280
+00:34:39,860 --> 00:34:48,180
+و بما انه c arbitrary element اذا
+
+281
+00:34:48,180 --> 00:34:55,720
+ال F يكون continuous at every c ينتمي ال R
+
+282
+00:34:55,720 --> 00:35:03,220
+وبالتالي continuous على كل ال R ممكن
+
+283
+00:35:03,220 --> 00:35:08,760
+برضهنستخدم تعريف epsilon دلتا مباشرة بلاش نقول ان
+
+284
+00:35:08,760 --> 00:35:13,440
+احنا اثبتنا ان ال limit ل ال function f and c
+
+285
+00:35:13,440 --> 00:35:17,020
+بالساوية c في section اربعة واحدة انا ممكن اثبت
+
+286
+00:35:17,020 --> 00:35:22,520
+يعني استخدم تعريف epsilon دلتا مباشرة و اقول let
+
+287
+00:35:22,520 --> 00:35:32,180
+if fix اول حاجة fix c تنتمي ل R to showif is
+
+288
+00:35:32,180 --> 00:35:39,820
+continuous at c let epsilon أكبر من السفر be given
+
+289
+00:35:39,820 --> 00:35:44,720
+it
+
+290
+00:35:44,720 --> 00:35:49,540
+shows .. زي ما عملنا في ال limits it shows delta
+
+291
+00:35:49,540 --> 00:35:54,640
+بساوي epsilon لذن
+
+292
+00:35:54,640 --> 00:36:00,160
+هي يوجد delta تعتمد على epsilonThen لهذه الـ Delta
+
+293
+00:36:00,160 --> 00:36:06,600
+لو كان X ينتمي إلى A A هنا اللي هي R و Absolute X
+
+294
+00:36:06,600 --> 00:36:12,360
+minus C أصغر من Delta فهذا بتضمن أنه Absolute F of
+
+295
+00:36:12,360 --> 00:36:20,080
+X Absolute F of X minus F of C هذا بيطلع بساوي
+
+296
+00:36:20,080 --> 00:36:28,590
+Absolute X minus F of X بساويX و F of C بساوي C
+
+297
+00:36:28,590 --> 00:36:33,010
+وهذا أصغر من Delta ماخدين المسافة هذه أصغر من
+
+298
+00:36:33,010 --> 00:36:38,250
+Delta وأنا اختارت Delta بساوي Epsilon إذن هذه
+
+299
+00:36:38,250 --> 00:36:42,110
+أثبتت لكل Epsilon يوجد Delta تعتمد على Epsilon
+
+300
+00:36:42,110 --> 00:36:46,150
+بحيث لكل X في مجال الدالة المسافة بينها و بين C
+
+301
+00:36:46,150 --> 00:36:50,650
+أصغر من Delta طلع المسافة بين F of X و F of C أصغر
+
+302
+00:36:50,650 --> 00:36:58,390
+من Epsilon إذن هذا معناه أن F is continuousat C
+
+303
+00:36:58,390 --> 00:37:06,010
+since C تنتمي ل R was arbitrary اذا F is
+
+304
+00:37:06,010 --> 00:37:12,750
+continuous على كل الأعداد الحقيقية تمام؟ اذا هذا
+
+305
+00:37:12,750 --> 00:37:15,890
+ممكن استخدم تعريف Epsilon Delta مباشرة
+
+306
+00:37:19,990 --> 00:37:23,390
+دون الاعتماد على النتائج اللي عملناها تابعة
+
+307
+00:37:23,390 --> 00:37:28,390
+النهاية في section أربعة واحد بالمثل ممكن مثال زي
+
+308
+00:37:28,390 --> 00:37:35,290
+هذا برضه ال function f
+
+309
+00:37:35,290 --> 00:37:43,790
+of x بساوي x سربية is continuous على كل الأعداد
+
+310
+00:37:43,790 --> 00:37:44,570
+الحقيقية
+
+311
+00:38:05,350 --> 00:38:08,350
+الدالة متصلة عند النقطة C
+
+312
+00:38:13,110 --> 00:38:18,330
+نفس تعريف epsilon دلتا زي ما عملنا في اثبات ان ال
+
+313
+00:38:18,330 --> 00:38:24,490
+limit لل function f of x and x بساوي c بساوي c
+
+314
+00:38:24,490 --> 00:38:30,110
+تربيه اللي هو f of c وذلك
+
+315
+00:38:30,110 --> 00:38:35,710
+بياخد اي epsilon اكبر من صفر و بنجيب دلتا زي ما
+
+316
+00:38:35,710 --> 00:38:38,510
+عملنا في section اربعة واحد دلتا بساوي ال minimum
+
+317
+00:38:38,510 --> 00:38:45,380
+لقمتينو نثبت أنه لكل x المسافة بينها و بين الـC
+
+318
+00:38:45,380 --> 00:38:47,960
+أصغر من الـDelta بيطلع المسافة هذه أصغر من الـC
+
+319
+00:38:47,960 --> 00:38:53,120
+نعيد يعني إيش نفس البرمجة، إذن هذا لو طلب منكم
+
+320
+00:38:53,120 --> 00:38:56,460
+استخدام تعريف epsilon delta لإثبات أن الدالة هذه
+
+321
+00:38:56,460 --> 00:39:00,420
+مقتصرة على R فبتقول لأي epsilon أكبر من السفر
+
+322
+00:39:00,420 --> 00:39:05,060
+choose delta زي ما عملنا في section 4-1 في إثبات
+
+323
+00:39:05,060 --> 00:39:08,900
+أن limit للدالة هذه عن C بساوي C تربية
+
+324
+00:39:12,370 --> 00:39:18,470
+أو ممكن تقولي we should اذا ما طلبش منك استخدم
+
+325
+00:39:18,470 --> 00:39:23,590
+التعريف epsilon دلتا فبتقولي we should أثبتنا in
+
+326
+00:39:23,590 --> 00:39:33,970
+section أربع واحد that limit ل F of X لما X تقول
+
+327
+00:39:33,970 --> 00:39:42,230
+إلى C بساوي C تربية اللي هي F of Cحسب تعريف
+
+328
+00:39:42,230 --> 00:39:45,470
+الاتصال على النقطة بيطلع أي شرط تلاتة في واحد
+
+329
+00:39:45,470 --> 00:39:54,190
+متحقق وبالتاني if is continuous at c okay تمام
+
+330
+00:39:57,190 --> 00:40:00,230
+وطبعاً بما أن الـ C تنتمي الـ R was arbitrary إذن
+
+331
+00:40:00,230 --> 00:40:03,970
+الدالة F continuous على كل الـ R okay إذا دامت
+
+332
+00:40:03,970 --> 00:40:11,050
+ياندي إما نستخدم نتائج section 4-1 أو نعيد البرهان
+
+333
+00:40:11,050 --> 00:40:15,250
+باستخدام تعريف epsilon Delta زي ما عملنا في المثال
+
+334
+00:40:15,250 --> 00:40:22,770
+الأخير أو زي ما عملنا في section 4-1 الدالة كمان
+
+335
+00:40:22,770 --> 00:40:23,930
+عندي الدالة
+
+336
+00:40:32,140 --> 00:40:41,000
+لو أخدت five X بيساوي واحد على X فهذه الدالة is
+
+337
+00:40:41,000 --> 00:40:46,280
+continuous on ال set A
+
+338
+00:40:58,940 --> 00:41:04,860
+اللي هي كل ال X ينتمي إلى R حقيته X أكبر من السفر
+
+339
+00:41:04,860 --> 00:41:11,380
+فاحنا
+
+340
+00:41:11,380 --> 00:41:20,880
+أثبتنا في X C تنتمي إلى A هذا بقدر انه C أكبر من
+
+341
+00:41:20,880 --> 00:41:23,280
+سفر و أثبتنا
+
+342
+00:41:28,820 --> 00:41:35,560
+In section أربع
+
+343
+00:41:35,560 --> 00:41:44,320
+واحد ذات limit لـ function phi of x لما x تقول إلى
+
+344
+00:41:44,320 --> 00:41:52,240
+c بسوى واحد على c بسوى phi of c باستخدام تعريف
+
+345
+00:41:52,240 --> 00:41:58,070
+epsilon دلتا اما نعيدالبرهان هداك لأي epsilon في
+
+346
+00:41:58,070 --> 00:42:03,450
+ديلتا بساوي minimum لقمتين او نقول انه احنا اثبتنا
+
+347
+00:42:03,450 --> 00:42:06,890
+ان limit الدالة هدا عند اي عدد c موجد بساوي واحد
+
+348
+00:42:06,890 --> 00:42:12,590
+على c اللي هو قيمة الدالة عن c وبالتالي اذا الدالة
+
+349
+00:42:12,590 --> 00:42:19,830
+في is continuous at c بما ان ال c تنتمي ل a was
+
+350
+00:42:19,830 --> 00:42:26,450
+arbitrary اذا ال في continuousعلى المجموعة A
+
+351
+00:42:26,450 --> 00:42:30,370
+بالمثل
+
+352
+00:42:30,370 --> 00:42:35,050
+ممكن نثبت ان الدالة دي continuous كمان على
+
+353
+00:42:35,050 --> 00:42:44,530
+المجموعة B اللي هي كل ال X ينتمي ل R حيث X أصغر من
+
+354
+00:42:44,530 --> 00:42:48,990
+0 الدالة
+
+355
+00:42:48,990 --> 00:42:54,190
+دي متصلة عند كل الأعداد الحقيقية مع عدد 0فهي متصلة
+
+356
+00:42:54,190 --> 00:42:57,610
+عند الأعداد الحقيقية الموجبة وعند الأعداد الحقيقية
+
+357
+00:42:57,610 --> 00:43:07,950
+السالبة طيب
+
+358
+00:43:07,950 --> 00:43:13,370
+الدالة five
+
+359
+00:43:13,370 --> 00:43:19,950
+x نفسها برضه بساوي واحد على x is not is
+
+360
+00:43:19,950 --> 00:43:33,190
+discontinuousis discontinuous at c بساوي سفر proof
+
+361
+00:43:33,190 --> 00:43:39,090
+one الدالة
+
+362
+00:43:39,090 --> 00:43:44,530
+هذه ليست متصلة عند السفر فالبرهان ذلك ممكن نقول
+
+363
+00:43:44,530 --> 00:43:49,610
+أنه في في
+
+364
+00:43:52,850 --> 00:43:59,250
+is undefined is undefined is undefined is
+
+365
+00:43:59,250 --> 00:44:05,090
+undefined is undefined is undefined is undefined
+
+366
+00:44:05,090 --> 00:44:05,090
+is undefined is undefined is undefined is
+
+367
+00:44:05,090 --> 00:44:05,970
+undefined is undefined is undefined is undefined
+
+368
+00:44:05,970 --> 00:44:07,390
+is undefined is undefined is undefined is
+
+369
+00:44:07,390 --> 00:44:07,390
+undefined is undefined is undefined is undefined
+
+370
+00:44:07,390 --> 00:44:07,390
+is undefined is undefined is undefined is
+
+371
+00:44:07,390 --> 00:44:07,470
+undefined is undefined is undefined is undefined
+
+372
+00:44:07,470 --> 00:44:07,830
+is undefined is undefined is undefined is
+
+373
+00:44:07,830 --> 00:44:07,830
+undefined is undefined is undefined is undefined
+
+374
+00:44:07,830 --> 00:44:07,830
+is undefined is undefined is undefined is
+
+375
+00:44:07,830 --> 00:44:07,830
+undefined is undefined is undefined is undefined
+
+376
+00:44:07,830 --> 00:44:07,830
+is undefined is undefined is undefined is
+
+377
+00:44:07,830 --> 00:44:07,830
+undefined is undefined is undefined is undefined
+
+378
+00:44:07,830 --> 00:44:09,950
+is undefined is undefined is undefined is
+
+379
+00:44:09,950 --> 00:44:16,950
+undefined is undefined is undefined is undefined
+
+380
+00:44:18,990 --> 00:44:25,170
+can't be continuous at x بساوي سفر لأن عشان هي
+
+381
+00:44:25,170 --> 00:44:28,550
+تكون متصلة عند سفر لازم تلات شروط يتحققوا أنها
+
+382
+00:44:28,550 --> 00:44:32,790
+تكون أول chart معرفة عند السفر فده هي مش معرفة عند
+
+383
+00:44:32,790 --> 00:44:38,390
+السفر فكيف تلات شروط هيتحققوا هذا برهان تاني برهان
+
+384
+00:44:38,390 --> 00:44:45,850
+آخر ان ما احنا شوفنا we should
+
+385
+00:44:48,290 --> 00:44:52,870
+in section أربع
+
+386
+00:44:52,870 --> 00:44:57,990
+واحد أو أربع اتنين that
+
+387
+00:44:57,990 --> 00:45:08,290
+limit لفاي of x as x tends to zero does not exist
+
+388
+00:45:08,290 --> 00:45:12,850
+أثبتنا إن الـ function هذه ما لهاش limit عند السفر
+
+389
+00:45:15,830 --> 00:45:21,510
+فا استخدمنا ال divergence criterion ا شفنا ان هناك
+
+390
+00:45:21,510 --> 00:45:27,450
+sequence اللى هى واحد عال ان converge للسفر but
+
+391
+00:45:27,450 --> 00:45:34,690
+limit ال image لل sequence واحد على ان as n tends
+
+392
+00:45:34,690 --> 00:45:40,170
+to infinity بساوي limit in بساوي infinity does not
+
+393
+00:45:40,170 --> 00:45:47,950
+exist in Rوبالتالي by divergence criterion ال
+
+394
+00:45:47,950 --> 00:45:51,270
+function هذه مالهاش limit وبالتالي مش ممكن تكون
+
+395
+00:45:51,270 --> 00:46:02,990
+continuous so if I can't be continuous at x بساوي
+
+396
+00:46:02,990 --> 00:46:09,510
+سفر تمام؟ لأن واحد من الشروط التلاتة تبعت الاتصال
+
+397
+00:46:09,510 --> 00:46:12,650
+عن نقطة غير متحققة تمام؟
+
+398
+00:46:22,580 --> 00:46:28,520
+في كمان مثال أخدناه في section
+
+399
+00:46:28,520 --> 00:46:36,020
+4-1 الـ
+
+400
+00:46:36,020 --> 00:46:42,220
+signum function اللي
+
+401
+00:46:42,220 --> 00:46:52,050
+كان تعريفهابتساوي سفر if x بساوي سفر و x على
+
+402
+00:46:52,050 --> 00:47:00,370
+absolute x إذا كان x لا يساوي سفر is discontinuous
+
+403
+00:47:00,370 --> 00:47:09,170
+is discontinuous at x بساوي سفر why
+
+404
+00:47:18,170 --> 00:47:23,550
+لأنه اثبتنا احنا في section أربعة واحد انه limit ل
+
+405
+00:47:23,550 --> 00:47:31,490
+signum x لما x تقول إلى سفر does not exist
+
+406
+00:47:40,560 --> 00:47:43,240
+اللي هي ان ال limit لل signal function عند السفر
+
+407
+00:47:43,240 --> 00:47:46,580
+does not exist شوفنا ان ال limit من اليمين واحد
+
+408
+00:47:46,580 --> 00:47:50,020
+عند السفر و ال limit و ال limit عند السفر مليار
+
+409
+00:47:50,020 --> 00:47:53,340
+ساعة سالف واحد وبالتالي مش متساوي اتين اذا ال
+
+410
+00:47:53,340 --> 00:48:00,000
+limit عند السفر does not exist okay تمام اذا ال ال
+
+411
+00:48:00,000 --> 00:48:04,700
+function هذه ماهياش متصلة عند السفر لعدم نظرا لعدم
+
+412
+00:48:04,700 --> 00:48:10,970
+وجود ال limit عند السفررغم أن الدالة هذه معرفة عند
+
+413
+00:48:10,970 --> 00:48:17,310
+السفر، الـSignum للسفر هي معرفة عند السفر بساوي
+
+414
+00:48:17,310 --> 00:48:24,930
+سفر تمام؟
+
+415
+00:48:24,930 --> 00:48:30,710
+طيب، لكن ممكن اثبات أن الـSignum function متصلة
+
+416
+00:48:30,710 --> 00:48:32,850
+عند كل X لا يساوي سفر
+
+417
+00:48:45,100 --> 00:48:52,440
+However، الـ signum الـ signum function is
+
+418
+00:48:52,440 --> 00:48:59,280
+continuous at
+
+419
+00:48:59,280 --> 00:49:09,460
+every x لا يساوي سفر لأنه
+
+420
+00:49:22,230 --> 00:49:42,610
+proof fix c لا تنتمي لار وc لا يساوي ستة تمام then
+
+421
+00:49:42,610 --> 00:49:53,460
+absolute signum x minus signumالـ C بساوي absolute
+
+422
+00:49:53,460 --> 00:49:57,420
+X
+
+423
+00:49:57,420 --> 00:50:14,640
+على absolute X أو
+
+424
+00:50:14,640 --> 00:50:15,160
+بلاش
+
+425
+00:50:19,850 --> 00:50:26,730
+then ال limit ل sigma x
+
+426
+00:50:26,730 --> 00:50:34,390
+لما x تقول إلى c بساوي
+
+427
+00:50:34,390 --> 00:50:37,990
+لما
+
+428
+00:50:37,990 --> 00:50:43,670
+x تقول إلى c فهذا عبارة عن limit x على absolute x
+
+429
+00:50:43,670 --> 00:50:45,630
+لما x تقول إلى c
+
+430
+00:51:03,050 --> 00:51:08,750
+فده كانت ال X لا تساوي سفر فاما ال X موجة بقى أو
+
+431
+00:51:08,750 --> 00:51:12,890
+سالي بقى
+
+432
+00:51:12,890 --> 00:51:18,010
+then C أكبر من السفر or C أصغر من سفر
+
+433
+00:51:23,040 --> 00:51:27,120
+الـ C هتكون أكبر من السفر الـ C هنا لأ تساوي سفر
+
+434
+00:51:27,120 --> 00:51:33,240
+إذا أما C أكبر من السفر أو أصغر من السفر case one
+
+435
+00:51:33,240 --> 00:51:41,000
+لو كانت C أكبر من سفر فهذا بقد أنه limit signum X
+
+436
+00:51:41,000 --> 00:51:50,980
+as X tends to C بساوي limit X على absolute X
+
+437
+00:51:59,940 --> 00:52:05,660
+و طبعا ال X أكبر من ال
+
+438
+00:52:05,660 --> 00:52:11,860
+C أكبر من السفر ف absolute .. فهذا بيساوي واحد
+
+439
+00:52:11,860 --> 00:52:21,440
+بيساوي limit واحد as X tends to C بيساوي واحد
+
+440
+00:52:21,440 --> 00:52:32,490
+بيساوي F and Cأو signum C لأن
+
+441
+00:52:32,490 --> 00:52:40,050
+ال C موجبة فلما ال C تكون موجبة ف absolute ال C
+
+442
+00:52:40,050 --> 00:52:47,250
+بساوي ال C بطلع المخضر هذا بطلع واحدو بالتالي إذا
+
+443
+00:52:47,250 --> 00:52:57,970
+ال signal x is continuous at c case 2 إذا كانت ال
+
+444
+00:52:57,970 --> 00:53:11,210
+c أصغر من سفر ف similar to case 1 في
+
+445
+00:53:11,210 --> 00:53:17,600
+الحالة هذهقيمة ال function هتطلع سالب واحد عند c و
+
+446
+00:53:17,600 --> 00:53:22,820
+limit عند c هتطلع سالب واحد وبالتالي في اتصال عند
+
+447
+00:53:22,820 --> 00:53:26,320
+ال c إذا ال sign and function مش متصلة عند الصفر
+
+448
+00:53:26,320 --> 00:53:30,800
+لكنها متصلة عن كل الأعداد الحقيقية المختلفة عن
+
+449
+00:53:30,800 --> 00:53:37,910
+الصفرOkay بنكتفي بهذا القدر و بنكمل طبعا إن شاء
+
+450
+00:53:37,910 --> 00:53:44,390
+الله في المحاضرة القادمة هنعطيكم إن شاء الله break
+
+451
+00:53:44,390 --> 00:53:49,350
+خمس دقائق و بعدين نواصل المحاضرة التانية اللي
+
+452
+00:53:49,350 --> 00:53:56,090
+هناخد فيها discussion أو مناقشة لل chapter أربعة
+
+453
+00:53:56,090 --> 00:53:58,350
+section أربعة واحد و أربعة اتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4.srt
new file mode 100644
index 0000000000000000000000000000000000000000..64bff5e0a4180213f05de03472324f6fea34f5c2
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4.srt
@@ -0,0 +1,1471 @@
+1
+00:00:21,380 --> 00:00:26,860
+بسم الله الرحمن الرحيم اليوم هنكمل ... في الجزء
+
+2
+00:00:26,860 --> 00:00:31,720
+الأول من المحاضرة هنكمل section خمسة واحد و في 
+
+3
+00:00:31,720 --> 00:00:37,160
+الوقت المتبقي من المحاضرة هنعطي مجال لكم ل ...
+
+4
+00:00:37,160 --> 00:00:43,850
+لأسئلة على ال homework أو أسئلة في امتحانات سابقة،
+
+5
+00:00:43,850 --> 00:00:49,390
+ماشي الحال، فناخد ال ... في ... أخدنا المرة اللي
+
+6
+00:00:49,390 --> 00:00:56,930
+فاتت أمثلة على الاتصال ووقفنا عند المثال التالي
+
+7
+00:01:13,460 --> 00:01:31,840
+Let A be a subset of R and define f a function from 
+
+8
+00:01:31,840 --> 00:01:38,320
+R to R by f of x
+
+9
+00:01:41,550 --> 00:01:49,110
+بتساوي واحد إذا كان x is rational وبتساوي صفر إذا
+
+10
+00:01:49,110 --> 00:02:00,410
+كان x is irrational show
+
+11
+00:02:00,410 --> 00:02:09,570
+that show if f is discontinuous is discontinuous at
+
+12
+00:02:13,630 --> 00:02:21,890
+every x تنتمي إلى r إذا ال function هذه اللي
+
+13
+00:02:21,890 --> 00:02:27,310
+معرفها بالطريقة كما هو موضح على هذه ال function
+
+14
+00:02:27,310 --> 00:02:33,450
+بتكون discontinuous ليست متصلة عند أي x في R ليه
+
+15
+00:02:33,450 --> 00:02:34,930
+برهان ذلك proof
+
+16
+00:02:42,420 --> 00:02:50,400
+fix c تنتمي إلى R then
+
+17
+00:02:50,400 --> 00:02:57,220
+فما احنا عارفين ان ال real
+
+18
+00:02:57,220 --> 00:03:00,980
+numbers عبارة عن ال union disjoint union لل
+
+19
+00:03:00,980 --> 00:03:03,320
+rational numbers وال irrational numbers
+
+20
+00:03:09,680 --> 00:03:22,500
+then X أو C تنتمي إلى Q أو C تنتمي إلى R minus Q
+
+21
+00:03:22,500 --> 00:03:25,680
+إذا الـ C هذا إما هتكون rational number أو
+
+22
+00:03:25,680 --> 00:03:32,160
+irrational number الحالة الأولى case one لو كانت C
+
+23
+00:03:32,160 --> 00:03:33,180
+rational number
+
+24
+00:03:37,870 --> 00:03:49,730
+استخدم الـ Corollary to Density
+
+25
+00:03:49,730 --> 00:03:58,310
+Theorem لتختار
+
+26
+00:03:58,310 --> 00:04:01,830
+سيكوينس
+
+27
+00:04:01,830 --> 00:04:06,630
+XN من أعداد غير عقلية
+
+28
+00:04:19,740 --> 00:04:23,560
+أعتقد أننا اثبتنا حاجة زي هذه سابقا
+
+29
+00:04:28,770 --> 00:04:33,750
+حتى لو كان rational فby ال corollary النتيجة تبع
+
+30
+00:04:33,750 --> 00:04:38,630
+ال density theorem ممكن اثبات ان يوجد sequence of
+
+31
+00:04:38,630 --> 00:04:46,150
+irrationals ونهيتها العدد cلأ هذا حسب النتيجة تبع
+
+32
+00:04:46,150 --> 00:04:50,250
+ال density term لأن النتيجة هذه بتقول between any
+
+33
+00:04:50,250 --> 00:04:54,030
+two real numbers there is irrational او اي open
+
+34
+00:04:54,030 --> 00:04:58,990
+interval contains an irrational number فشوفنا احنا
+
+35
+00:04:58,990 --> 00:05:06,210
+البرهان بالتفصيل وبالتالي so نلاحظ
+
+36
+00:05:06,210 --> 00:05:14,530
+أنه ال limitل ... ال ... أو ال ... ال function قيمة
+
+37
+00:05:14,530 --> 00:05:21,230
+ال function f عند xn بيساوي ال xn is irrational و
+
+38
+00:05:21,230 --> 00:05:24,370
+من ال definition تبع ال function f عند اي
+
+39
+00:05:24,370 --> 00:05:30,310
+irrational بيساوي صفر هذا صحيح لكل n هذا بيقدي ان
+
+40
+00:05:30,310 --> 00:05:41,200
+ال limit ل f of xn as n tends to infinityبساوي
+
+41
+00:05:41,200 --> 00:05:44,140
+limit ثابت صفر بساوي صفر
+
+42
+00:05:49,060 --> 00:05:55,480
+وهذا لا يساوي واحد اللي هو f of c ال c rational
+
+43
+00:05:55,480 --> 00:06:00,360
+number ف f and c بيساوي واحد اذا هاي اندي انا
+
+44
+00:06:00,360 --> 00:06:05,940
+أثبتت ان يوجد sequence xn في المجال تبع الدولة
+
+45
+00:06:05,940 --> 00:06:11,340
+اللي هو R و ال sequence هذه converge ل c لكن ال
+
+46
+00:06:11,340 --> 00:06:16,660
+limit لل image لل sequence لا تساوي f of c اذا by
+
+47
+00:06:16,660 --> 00:06:22,490
+divergence criterionأو by discontinuity criterion
+
+48
+00:06:22,490 --> 00:06:28,450
+إذا by discontinuity
+
+49
+00:06:28,450 --> 00:06:32,690
+criterion if
+
+50
+00:06:32,690 --> 00:06:43,570
+f is discontinuous is discontinuous at C في
+
+51
+00:06:43,570 --> 00:06:47,750
+الحالة اللي هي C rational number في الحالة التانية
+
+52
+00:06:52,270 --> 00:07:01,110
+لو كان الـ c irrational number فممكن
+
+53
+00:07:01,110 --> 00:07:05,330
+نستخدم ال density theorem نفسها use density
+
+54
+00:07:05,330 --> 00:07:09,130
+theorem to
+
+55
+00:07:09,130 --> 00:07:13,710
+choose a
+
+56
+00:07:13,710 --> 00:07:21,280
+sequence x in contained of rational numbers
+
+57
+00:07:21,280 --> 00:07:23,760
+contained in Q يعني ال sequence هذه عناصرها
+
+58
+00:07:23,760 --> 00:07:30,440
+rational numbers بحيث انه ال limit لل sequence x
+
+59
+00:07:30,440 --> 00:07:36,260
+in as n tends to infinity بساوي c هذا أثبتناه في
+
+60
+00:07:36,260 --> 00:07:42,480
+المحاضرة السابقة وبالتالي
+
+61
+00:07:42,480 --> 00:07:43,920
+so
+
+62
+00:07:46,980 --> 00:07:52,620
+أنا عندي بما أنه ال Xn rationals ف F of Xn بتساوي
+
+63
+00:07:52,620 --> 00:08:00,280
+واحد لكل N هذا بيقدي انه limit ل F of Xn as N
+
+64
+00:08:00,280 --> 00:08:04,950
+tends to infinityبساوي limit ثابت واحد بطلع واحد
+
+65
+00:08:04,950 --> 00:08:12,710
+واحد لا يساوي صفر اللي هو ال image ل ال C ال C هنا
+
+66
+00:08:12,710 --> 00:08:17,170
+irrational و من ال definition of the function if
+
+67
+00:08:17,170 --> 00:08:21,730
+and irrational بساوي صفر اذا انا هنا اثبتت ان يوجد
+
+68
+00:08:21,730 --> 00:08:23,610
+sequence
+
+69
+00:08:25,710 --> 00:08:32,350
+في مجال الدالة نهايتها c لكن نهاية صورتها لا تساوي
+
+70
+00:08:32,350 --> 00:08:41,750
+صورة ال c كمان مرة hence by
+
+71
+00:08:41,750 --> 00:08:47,790
+discontinuity by
+
+72
+00:08:47,790 --> 00:08:50,150
+discontinuity criterion
+
+73
+00:08:52,890 --> 00:09:03,530
+الـ function f is discontinuous at c لأن بما أن c
+
+74
+00:09:03,530 --> 00:09:09,410
+was arbitrary فالـ function is discontinuous عن كل
+
+75
+00:09:09,410 --> 00:09:15,050
+real number c طبعا في الحالتين لأن هذا بكمل
+
+76
+00:09:15,050 --> 00:09:22,090
+البرهانة لأن هذه ال function هي function من R لRو
+
+77
+00:09:22,090 --> 00:09:26,950
+discontinuous ليست متصلة عند أي x فارق هذه ال
+
+78
+00:09:26,950 --> 00:09:31,030
+function لها اسم ومشهورة ومعروفة اسمها Dirichlet
+
+79
+00:09:31,030 --> 00:09:35,210
+function اذا
+
+80
+00:09:35,210 --> 00:09:41,110
+ال function هذه remark
+
+81
+00:09:41,110 --> 00:09:46,870
+the
+
+82
+00:09:46,870 --> 00:09:49,610
+above function
+
+83
+00:09:55,060 --> 00:10:03,000
+This function is well-known as
+
+84
+00:10:03,000 --> 00:10:10,040
+Dirichlet Dirichlet
+
+85
+00:10:10,040 --> 00:10:17,540
+دا عالم رياضيات Dirichlet's function أو Dirichlet
+
+86
+00:10:17,540 --> 00:10:20,000
+is a discontinuous function
+
+87
+00:10:25,450 --> 00:10:34,010
+This discontinuous function ده
+
+88
+00:10:34,010 --> 00:10:41,070
+اللي مهمة ويلها meta في ال real analysis okay تمام
+
+89
+00:10:41,070 --> 00:10:47,630
+طيب ناخد بعض الملاحظات واضح في أي سؤال واضح هنا
+
+90
+00:10:47,630 --> 00:10:49,750
+البرهن في أي سلسلة
+
+91
+00:11:06,590 --> 00:11:15,470
+طيب ناخد شوية remarks ال
+
+92
+00:11:15,470 --> 00:11:19,410
+remark
+
+93
+00:11:19,410 --> 00:11:27,290
+الأولى sometimes a
+
+94
+00:11:27,290 --> 00:11:34,590
+function خلينا احنا نرجع لل definition تبع الاتصال
+
+95
+00:11:34,590 --> 00:11:35,770
+عن النقطة definition
+
+96
+00:11:56,710 --> 00:12:04,230
+الشرط الثاني أن ال limitلأ F of X لما X تقول إلى C
+
+97
+00:12:04,230 --> 00:12:11,490
+exists و بساوي F of C اللي هو الشرط هذا هذا الشرط
+
+98
+00:12:11,490 --> 00:12:16,730
+سمنها تلاتة في واحد هذا يتضمن تلات شروط ال limit
+
+99
+00:12:16,730 --> 00:12:22,690
+and C exist F is defined at C و اتنين متساوين تلات
+
+100
+00:12:22,690 --> 00:12:28,970
+شروط الان لو اي واحد من التلات الشروط هدول اختل فال
+
+101
+00:12:28,970 --> 00:12:32,210
+function بتكونش continuous and النقطة c ف
+
+102
+00:12:32,210 --> 00:12:42,490
+sometimes the function can be discontinuous
+
+103
+00:12:42,490 --> 00:12:46,970
+at x بساوي c because
+
+104
+00:12:49,450 --> 00:13:06,610
+f is undefined is undefined at C لكن
+
+105
+00:13:06,610 --> 00:13:10,750
+أحيانا أخرى وفي الحالة اللي بقدرش أعمل حاجة بقدرش
+
+106
+00:13:10,750 --> 00:13:17,030
+أعمل حاجة however
+
+107
+00:13:20,670 --> 00:13:28,290
+لو كانت if ال limit لل function f of x as x tends
+
+108
+00:13:28,290 --> 00:13:39,010
+to c موجودة exists exists
+
+109
+00:13:39,010 --> 00:13:46,490
+and equals L ينتمي ل R ففي
+
+110
+00:13:46,490 --> 00:13:49,510
+الحالة هذه we can
+
+111
+00:13:53,080 --> 00:13:58,780
+we can define we
+
+112
+00:13:58,780 --> 00:14:09,440
+can define a function capital F من a اتحاد ال
+
+113
+00:14:09,440 --> 00:14:20,800
+singleton set C إلى R by capital F of X بساوي
+
+114
+00:14:20,800 --> 00:14:21,980
+العدد L
+
+115
+00:14:24,670 --> 00:14:32,970
+fx بتساوي c و بتساوي ال function f of x إذا كان ال
+
+116
+00:14:32,970 --> 00:14:44,530
+x ينتمي إلى a ولا يساوي c in this case in this
+
+117
+00:14:44,530 --> 00:14:54,830
+case the function capital F is continuous at x
+
+118
+00:14:54,830 --> 00:15:00,530
+بساوي c indeed
+
+119
+00:15:00,530 --> 00:15:07,490
+في حقيقة الأمر indeed في حقيقة الأمر تعالى نشوف
+
+120
+00:15:07,490 --> 00:15:14,630
+high limit capital f of x as x tends to c بساوي x
+
+121
+00:15:14,630 --> 00:15:19,510
+تقول ل c إذا ال x بالتأكيد بتسويش c لما x ما
+
+122
+00:15:19,510 --> 00:15:26,390
+بتسويش c يعني x and time ل aفcapital F هي نفسها
+
+123
+00:15:26,390 --> 00:15:35,430
+حسب تعريفها هي small f طب احنا فرضين ان ال limit ل
+
+124
+00:15:35,430 --> 00:15:41,150
+F of X exist و بساوي and C و بساوي L، إذن هذه تطلع
+
+125
+00:15:41,150 --> 00:15:43,070
+موجودة و بساوي L
+
+126
+00:15:46,940 --> 00:15:50,740
+و احنا من ال definition تبع ال function ال ال
+
+127
+00:15:50,740 --> 00:15:53,800
+ماخدينها هي عبارة عن قيمة ال function capital F
+
+128
+00:15:53,800 --> 00:16:00,180
+and C اذا هاي شرط الاتصال للدالة capital F and C
+
+129
+00:16:00,180 --> 00:16:10,320
+متحقق وبالتالي اذا capital F is continuous at C ال
+
+130
+00:16:10,320 --> 00:16:20,300
+function capital F the function capital F is called
+
+131
+00:16:20,300 --> 00:16:30,700
+a continuous extension
+
+132
+00:16:30,700 --> 00:16:36,980
+of
+
+133
+00:16:36,980 --> 00:16:43,900
+small f عبارة عن continuous extension يعني توسعة
+
+134
+00:16:43,900 --> 00:16:49,530
+متصلة توسعة متصلة لدالة small f لحظة انتوا هنا ان
+
+135
+00:16:49,530 --> 00:16:59,350
+الدالة capital F الدالة
+
+136
+00:16:59,350 --> 00:17:05,550
+capital F function من A union single to C إلى R
+
+137
+00:17:05,550 --> 00:17:11,810
+طبعا هنا C لا تنتمي إلى A وعندي small f هي function
+
+138
+00:17:11,810 --> 00:17:19,850
+من A إلى Rفلاحظوا ان ال function capital F لما
+
+139
+00:17:19,850 --> 00:17:28,250
+أعمل restriction لل domain تبعها على a فقط فهي نفس
+
+140
+00:17:28,250 --> 00:17:35,490
+f يعني لما اقيد او احصر الدولة capital F على 
+
+141
+00:17:35,490 --> 00:17:41,610
+المجموعة a فقط فهي نفس الـ F هي لكل x  capital F
+
+142
+00:17:41,610 --> 00:17:47,020
+هي small fالزيادة أن capital F معرفة عن C و F مش 
+
+143
+00:17:47,020 --> 00:17:51,720
+معرفة عن C فهذه الدالة capital F بنسميها توسعة على
+
+144
+00:17:51,720 --> 00:17:56,120
+small f التوسعة هذه متصلة عند النقطة C
+
+145
+00:17:59,220 --> 00:18:03,740
+تمام؟ إذا هذه أول ملاحظة إذا لو كانت الدالة مش 
+
+146
+00:18:03,740 --> 00:18:10,580
+معرفة لو كانت small f مش معرفة عن c لكن نهايتها عن
+
+147
+00:18:10,580 --> 00:18:15,580
+c موجودة وبالساوي عدد L فبقدر أعرف دالة جديدة
+
+148
+00:18:15,580 --> 00:18:20,080
+capital F اللي هي توسعة extension ل small f و ال
+
+149
+00:18:20,080 --> 00:18:24,640
+extension الدالة هذه اللي هي توسعة بتكون متصلة عند
+
+150
+00:18:24,640 --> 00:18:28,920
+الـ C اننا ناخد قيمتها عند الـ C بساوي قيمة ال
+
+151
+00:18:28,920 --> 00:18:36,540
+limit و عند النقاط ال A هي نفس small f لكن
+
+152
+00:18:36,540 --> 00:18:39,620
+الملاحظة الثانية
+
+153
+00:18:51,350 --> 00:18:55,490
+لو كانت الدالة .. ممكن تكون الدالة معرفة عن C
+
+154
+00:18:55,490 --> 00:19:03,670
+يعني هنا sometimes a
+
+155
+00:19:03,670 --> 00:19:09,670
+function g is discontinuous
+
+156
+00:19:09,670 --> 00:19:16,190
+a function g from A to R is discontinuous at C
+
+157
+00:19:19,590 --> 00:19:25,970
+بسبب و ال c ممكن تكون موجودة في a أو حتى لو مش 
+
+158
+00:19:25,970 --> 00:19:35,630
+موجودة في a is discontinuous at c because ال limit
+
+159
+00:19:35,630 --> 00:19:43,210
+ل g of x as x tends to c does not exist يعني ممكن 
+
+160
+00:19:43,210 --> 00:19:48,330
+ال g تكون معرفة and ال c لكن النهايتها عند الـ c مش 
+
+161
+00:19:48,330 --> 00:19:51,690
+موجودة فطبعا في الحالة دي ال function بتكون 
+
+162
+00:19:51,690 --> 00:19:57,830
+discontinuous at c وفي الحالة دي ماقدرش أعرف in
+
+163
+00:19:57,830 --> 00:20:08,410
+this case in this case we can't لا نستطيع we can't
+
+164
+00:20:08,410 --> 00:20:14,990
+define a continuous extension
+
+165
+00:20:23,400 --> 00:20:33,900
+ممكن مانقدرش نعرف continuous extension لال .. 
+
+166
+00:20:33,900 --> 00:20:42,000
+اللي هو capital G from A union singleton C إلى R
+
+167
+00:20:42,000 --> 00:20:45,820
+أو
+
+168
+00:20:45,820 --> 00:20:52,800
+we cannot define an extension and extension g من a
+
+169
+00:20:52,800 --> 00:21:02,520
+union c لr by g of x بساوي عدد
+
+170
+00:21:02,520 --> 00:21:11,400
+capital C if x بساوي c و بساوي g of x إذا كان x
+
+171
+00:21:11,400 --> 00:21:18,500
+ينتمي and
+
+172
+00:21:20,360 --> 00:21:25,060
+جي بي continuous at c
+
+173
+00:21:39,530 --> 00:21:44,470
+طبعا هنا الـ G باخدها عند C بيساوي capital C هاد
+
+174
+00:21:44,470 --> 00:21:50,550
+عيرنا ال number ع شوية و عند X بيساوي A لازم تساوي
+
+175
+00:21:50,550 --> 00:21:55,970
+G of X عشان تكون توسعة أو extension ل small g ف to
+
+176
+00:21:55,970 --> 00:22:05,990
+see this لبرهان ذلك to see this to see that such
+
+177
+00:22:05,990 --> 00:22:18,380
+G can't be continuous at c assume خلّيني
+
+178
+00:22:18,380 --> 00:22:25,200
+أعمل برهان بالتناقض assume on contrary assume on
+
+179
+00:22:25,200 --> 00:22:33,900
+contrary أن counter G is continuous at c then هذا
+
+180
+00:22:33,900 --> 00:22:40,660
+معناه أن ال limit ل capital G of X as X tends to C
+
+181
+00:22:40,660 --> 00:22:48,420
+بساوي capital G اللي
+
+182
+00:22:48,420 --> 00:22:56,380
+هي بساوي limit بتطلع
+
+183
+00:22:56,380 --> 00:23:06,900
+بساوي capital G of C بساوي capital C و هذه بتساوي
+
+184
+00:23:06,900 --> 00:23:11,700
+limit g of x لما x تقول ل c هي عبارة عن limit من
+
+185
+00:23:11,700 --> 00:23:18,320
+تعريف ال g limit g of x لما x تقول ل c لإن لما x
+
+186
+00:23:18,320 --> 00:23:22,960
+تقول ل c x بستويش ال c x بستويش ال c معناته x 
+
+187
+00:23:22,960 --> 00:23:29,600
+تنتمي ل a لإن g of x هي عبارة عن small g of x لإن
+
+188
+00:23:29,600 --> 00:23:32,260
+في الحالة هذه limit
+
+189
+00:23:34,430 --> 00:23:40,110
+إذا limit small g of x لما x تقولها c exist اه إذا
+
+190
+00:23:40,110 --> 00:23:46,110
+هذا بتطلع exist and equals c هي بالساوية c
+
+191
+00:23:46,110 --> 00:23:51,370
+contradiction هذا تناقض لإن احنا فرضين إن ال limit
+
+192
+00:23:51,370 --> 00:23:53,970
+ل g of x and c مش موجودة
+
+193
+00:23:58,090 --> 00:24:02,390
+لما تكون الدالة مش متصلة على النقطة لعدم وجود
+
+194
+00:24:02,390 --> 00:24:06,830
+نهايتها عند النقطة فماقدرش أعرف continuous
+
+195
+00:24:06,830 --> 00:24:15,610
+extension لدالة و النقطة C لأن هنا فرضنا أن هناك 
+
+196
+00:24:15,610 --> 00:24:19,030
+extension هذا ال extension مش ممكن يكون continuous
+
+197
+00:24:19,030 --> 00:24:23,050
+عند النقطة C لأنه لو كان continuous عند النقطة C
+
+198
+00:24:23,050 --> 00:24:27,750
+سيقدر ان الملمت الدالة small g and c exist وهذا
+
+199
+00:24:27,750 --> 00:24:34,830
+يتناقض مع الفرض تمام، هذا النوع من ال
+
+200
+00:24:34,830 --> 00:24:40,570
+discontinuity لما تكون الدالة discontinuous لعدم
+
+201
+00:24:42,160 --> 00:24:43,920
+لما تكون الدالة discontinuous
+
+202
+00:24:46,760 --> 00:24:51,500
+السبب في ذلك أنها مش معرفة عند النقطة C لكن نهيتها
+
+203
+00:24:51,500 --> 00:24:55,560
+موجودة عند الـ C فهذا النوع من ال discontinuity من
+
+204
+00:24:55,560 --> 00:25:00,360
+عدم الاتصال بنسميه removable يعني ممكن إزالته أو 
+
+205
+00:25:00,360 --> 00:25:05,300
+التخلص منه وهي فعلا اتخلصنا من عدم الاتصال للدالة 
+
+206
+00:25:05,300 --> 00:25:09,480
+small f and c بتعريف دالة capital F بالطريقة هذه
+
+207
+00:25:09,480 --> 00:25:12,670
+وشوفنا أن الدالة الجديدة continuous عند ال C إذا
+
+208
+00:25:12,670 --> 00:25:16,190
+هذا النوع من عدم الاتصال أو ال discontinuity is
+
+209
+00:25:16,190 --> 00:25:19,990
+called removable يمكن إزالته يمكن التخلص منه أما إذا
+
+210
+00:25:19,990 --> 00:25:25,610
+كانت الدالة discontinuous عند النقطة C لأن نهايتها
+
+211
+00:25:25,610 --> 00:25:30,050
+in C مش موجودة فهذا النوع من عدم الاتصال بنسميه
+
+212
+00:25:30,050 --> 00:25:34,950
+essential يعني أساسي لا يمكن التخلص منه زي ما شفنا 
+
+213
+00:25:34,950 --> 00:25:38,890
+في التحليل تحت okay تمام إن هذه أنواع ال
+
+214
+00:25:38,890 --> 00:25:42,570
+discontinuity ال discontinuity أو عدم الاتصال
+
+215
+00:25:42,570 --> 00:25:47,030
+نوعين نوع removable ممكن إزالته ممكن التخلص منه و
+
+216
+00:25:47,030 --> 00:25:55,210
+نوع ثاني essential أساسي لايمكن التخلص منه ممكن
+
+217
+00:25:55,210 --> 00:25:58,090
+ناخد بعض الأمثلة على ذلك
+
+218
+00:26:14,950 --> 00:26:21,050
+نأخد الـ function consider الـ
+
+219
+00:26:21,050 --> 00:26:28,750
+function g of x بتساوي sin واحد على x حيث x لا
+
+220
+00:26:28,750 --> 00:26:31,970
+يساوي صفر طبعا احنا شفنا 
+
+221
+00:26:35,380 --> 00:26:41,220
+في مثال صادق أن limit g of x as x tends to zero
+
+222
+00:26:41,220 --> 00:26:46,880
+does not exist limit
+
+223
+00:26:46,880 --> 00:26:57,520
+الدالة هذه غير موجودة وبالتالي
+
+224
+00:26:57,520 --> 00:27:04,540
+كمان برضه كمان الدالة برضه جي عند صفر مش معرفة إذا
+
+225
+00:27:04,540 --> 00:27:17,220
+.. اذا in this case we can't .. we can't
+
+226
+00:27:17,220 --> 00:27:28,580
+.. we can't define a continuous extension
+
+227
+00:27:30,680 --> 00:27:41,180
+of small g at الصفر هذا النوع الثاني من ال
+
+228
+00:27:41,180 --> 00:27:47,520
+discontinuity لكن لو أخدت دالة زي هذه 
+
+229
+00:27:59,300 --> 00:28:11,080
+لو أخدت f of x بساوي x في ال sign واحد على x فطبعا 
+
+230
+00:28:11,080 --> 00:28:16,220
+هنا و x لا يساوي صفر فطبعا واضح أن f عند الصفر is
+
+231
+00:28:16,220 --> 00:28:17,240
+undefined
+
+232
+00:28:20,560 --> 00:28:26,040
+الـ limit شوفنا أنه limit ل f of x لما x تقول ل 0
+
+233
+00:28:26,040 --> 00:28:30,520
+by squeeze theorem أثبتنا باستخدام squeeze theorem
+
+234
+00:28:30,520 --> 00:28:38,100
+أن limit هذه بيساوي 0 exist بساوي 0 طبعا إذا هنا
+
+235
+00:28:38,100 --> 00:28:47,020
+f is discontinuous discontinuous at x بساوي 0 لإن
+
+236
+00:28:47,020 --> 00:28:51,920
+أنا مش معرف عند الصفر لكن بما ان ال limit تبقى عند
+
+237
+00:28:51,920 --> 00:28:58,320
+الصفر موجودة we can define
+
+238
+00:28:58,320 --> 00:29:07,320
+a continuous extension of
+
+239
+00:29:07,320 --> 00:29:11,220
+f at صفر as follows
+
+240
+00:29:14,800 --> 00:29:21,980
+فعندي capital F of X بنعرفها على أنها بالساوى
+
+241
+00:29:21,980 --> 00:29:26,040
+الصفر اللي هو limit لل function عند الصفر إذا كان
+
+242
+00:29:26,040 --> 00:29:34,960
+ال X بساوي الصفر و بالساوى small f of X إذا كان X
+
+243
+00:29:34,960 --> 00:29:42,260
+تنتمي ل domain الدالة اللي هو اللي هو ر مع ده صفر ر
+
+244
+00:29:42,260 --> 00:29:48,760
+مع ده صفر مش هذا هو ال domain تبع الدالة الانف
+
+245
+00:29:48,760 --> 00:29:58,000
+دالة if now you can verify أن
+
+246
+00:29:58,000 --> 00:30:05,040
+capital F is continuous is continuous at x بساوي
+
+247
+00:30:05,040 --> 00:30:12,580
+صفر بينما small f is not continuous از صفر حيا
+
+248
+00:30:12,580 --> 00:30:17,220
+عندي limit capital 
+
+249
+00:30:17,220 --> 00:30:23,560
+F of X as X tends to zero بساوي لما X ساوي للصفر X 
+
+250
+00:30:23,560 --> 00:30:29,800
+بساويش صفر لما X ماتساويش صفر ف capital F هي small f
+
+251
+00:30:32,400 --> 00:30:37,700
+و limit small f بتساوي صفر اللي هي capital F معرفة 
+
+252
+00:30:37,700 --> 00:30:43,620
+عند الصفر okay إذا هي شرط الاتصال عند الصفر متحقق 
+
+253
+00:30:43,620 --> 00:30:48,740
+وبالتالي capital F متصلة عند الصفر okay تمام هفهم 
+
+254
+00:30:48,740 --> 00:30:54,740
+okay بنوقف هنا و نتيح الآن المجال لكم إذا كان في 
+
+255
+00:30:54,740 --> 00:30:58,920
+عندكم أي أسئلة بخصوص ال homework أو الامتحان 
+
+256
+00:30:58,920 --> 00:31:03,560
+النصفي اللي هناخده بكرا إن شاء الله في عندكم أي
+
+257
+00:31:03,560 --> 00:31:11,680
+سؤال أو استفسار؟ في حد عنده أي سؤال؟
+
+258
+00:31:11,680 --> 00:31:24,080
+في
+
+259
+00:31:24,080 --> 00:31:29,970
+أي سؤال؟ أي سؤال؟ دكتور أنا ممكن استخدم limit ال K
+
+260
+00:31:29,970 --> 00:31:33,830
+لما ال X تقول ال C و سوى K و limit ال X لما ال X 
+
+261
+00:31:33,830 --> 00:31:38,490
+تقول ال C و سوى C ممكن استخدمهم أنا بحثت صح صح لأن 
+
+262
+00:31:38,490 --> 00:31:42,490
+هذه صارت حاجات ال trivial و يعني اثبتناها إلا ده
+
+263
+00:31:42,490 --> 00:31:47,150
+طول منك اثبتها باستخدام تعريف epsilon دلت اكيد أي
+
+264
+00:31:47,150 --> 00:31:53,810
+سؤال بدك ممكن الثاني الثاني اللي مش محلول اه اللي
+
+265
+00:31:53,810 --> 00:31:54,830
+مش محلول نعم
+
+266
+00:32:04,200 --> 00:32:13,180
+طيب suppose question suppose
+
+267
+00:32:13,180 --> 00:32:25,460
+أن xn أكبر من أو يساوي صفر لكل n في n and .. and ال
+
+268
+00:32:25,460 --> 00:32:25,820
+limit
+
+269
+00:32:28,550 --> 00:32:34,910
+لسالب واحد أس n في xn ال sequence هذه لما n تقول
+
+270
+00:32:34,910 --> 00:32:46,590
+ل infinity بساوي x ينتمي إلى r show أن ال limit ل
+
+271
+00:32:46,590 --> 00:32:56,630
+xn لما n تقول ل infinity بساوي صفر فضلي
+
+272
+00:33:21,960 --> 00:33:28,720
+فنشوف الآن أنا عندي ال limit لل sequence للحد 
+
+273
+00:33:28,720 --> 00:33:33,900
+العام تبعها سالب واحد قوّة ان في x in لما ان تقول
+
+274
+00:33:33,900 --> 00:33:44,020
+infinity بساوي x اذا ان and السالب واحد قوّة اتنين N
+
+275
+00:33:44,020 --> 00:33:52,460
+في X اتنين N هذه عبارة عن subsequence subsequence
+
+276
+00:33:52,460 --> 00:33:58,540
+of السيكوانس الحد العام تبعها السالب واحد قوّة N X
+
+277
+00:33:58,540 --> 00:34:07,730
+X N هذه subsequence خط الحدود الزوجية صح؟ طيب ال
+
+278
+00:34:07,730 --> 00:34:12,950
+subsequence هذه هي الحد العام تبعها x اتنين n
+
+279
+00:34:12,950 --> 00:34:16,890
+مفروض converge خلّينا احنا في نظرية ان كانت ال
+
+280
+00:34:16,890 --> 00:34:21,350
+sequence convergent ل x فأي subsequence منها بتكون 
+
+281
+00:34:21,350 --> 00:34:27,810
+convergent لنفس الـ x تمام؟ في نفس الوجود
+
+282
+00:34:38,530 --> 00:34:44,370
+في نفس الوقت الـ sequence سالب واحد أس اتنين in
+
+283
+00:34:44,370 --> 00:34:48,750
+سالب واحد في
+
+284
+00:34:48,750 --> 00:34:56,810
+x أس اتنين in سالب واحد هذا عبارة عن sub sequence من
+
+285
+00:34:56,810 --> 00:35:02,070
+الـ sequence الأصلي المعطاة اللي هي الحد العام
+
+286
+00:35:02,070 --> 00:35:07,020
+تبعها سالب واحد to the int x in يعني أنا أخذت هنا
+
+287
+00:35:07,020 --> 00:35:10,680
+الحدود الفردية من الـ sequence هذه طبعا هذه أكيد
+
+288
+00:35:10,680 --> 00:35:17,600
+subsequence فالحد العام هذا عبارة عن سالب X أس اتنين
+
+289
+00:35:17,600 --> 00:35:23,140
+in سالب واحد الآن الـ subsequence هذه المفروض أنها
+
+290
+00:35:23,140 --> 00:35:28,180
+converge إلى X لأن الـ sequence الأصلية convergent
+
+291
+00:35:28,180 --> 00:35:34,560
+لـ X تمام؟ إذا أنا في عندي هنا limit
+
+292
+00:35:38,370 --> 00:35:46,990
+x أس اتنين n سالب واحد بساوي سالب x لأن limit سالب الـ
+
+293
+00:35:46,990 --> 00:35:53,070
+sequence هذه بساوي x إذا أضربها في سالب واحد طبعا
+
+294
+00:35:53,070 --> 00:36:02,030
+ونأخذ من هنا من هناك limit x أس اتنين n لما n تؤول
+
+295
+00:36:02,030 --> 00:36:05,810
+to infinity بساوي
+
+296
+00:36:05,810 --> 00:36:06,190
+x
+
+297
+00:36:17,310 --> 00:36:28,710
+نِجمعهم؟ لا لا ما نجمعهمش الناس ما
+
+298
+00:36:28,710 --> 00:36:29,450
+بنجمعهم
+
+299
+00:36:37,410 --> 00:36:42,950
+ما بنجمع مش عاملين نجمع الآن في عندي أنا لحظة أنت 
+
+300
+00:36:42,950 --> 00:36:50,530
+عندك من الفرض xn أكبر من أو يساوي صفر لكل n في n
+
+301
+00:36:55,480 --> 00:37:03,280
+فبالتالي إذا x2n سالب واحد أكبر من أو يساوي صفر لكل n
+
+302
+00:37:03,280 --> 00:37:15,820
+وكذلك x2n برضه أكبر من أو يساوي صفر لكل n صح؟ أصحبت؟
+
+303
+00:37:15,820 --> 00:37:22,240
+وبالتالي إذا الـ limit لـ x2n سالب واحد تطلع أكبر من
+
+304
+00:37:22,240 --> 00:37:30,430
+أو يساوي صفر والـ limit لـ x2n تطلع أكبر من أو يساوي
+
+305
+00:37:30,430 --> 00:37:34,890
+صفر هذه نظرية أخذناها صح؟ أخذناها نظرية بتقول لو
+
+306
+00:37:34,890 --> 00:37:38,550
+كانت الـ sequence كل حدودها غير سالبة والـ limit
+
+307
+00:37:38,550 --> 00:37:44,330
+تبعها exist فالـ limit تبعها تطلع غير سالبة صح؟
+
+308
+00:37:44,330 --> 00:37:50,750
+فالـ limit هنا هي حسبناها سالب x والـ limit هنا
+
+309
+00:37:50,750 --> 00:37:59,060
+طلعت x إذا أنا في عندي سالب X أكبر من أو يساوي صفر
+
+310
+00:37:59,060 --> 00:38:06,420
+and X أكبر من أو يساوي صفر هذا بيؤدي إلى أن X أصغر من
+
+311
+00:38:06,420 --> 00:38:13,680
+أو يساوي صفر and X أكبر من أو يساوي صفر هذا بيؤدي إلى أن
+
+312
+00:38:13,680 --> 00:38:15,040
+X بساوي صفر
+
+313
+00:38:19,470 --> 00:38:23,930
+ما يعني x أصغر من 0 وأكبر من 0 يعني x أصغر من 0
+
+314
+00:38:23,930 --> 00:38:27,470
+العدد الوحيد اللي بيمتلك الخاصيتين هدول في نفس
+
+315
+00:38:27,470 --> 00:38:34,090
+الوقت هو 0 إذا أنا أثبتت أن الـ limit لـ الـ
+
+316
+00:38:34,090 --> 00:38:37,550
+sequence
+
+317
+00:38:37,550 --> 00:38:42,870
+أثبتت
+
+318
+00:38:42,870 --> 00:38:44,170
+أن x أصغر من 0
+
+319
+00:38:47,480 --> 00:38:52,500
+طبعا احنا ما أثبتناش لحد الآن أن الـ limit للـ
+
+320
+00:38:52,500 --> 00:38:59,260
+sequence إن أنا عندي .. إذا أنا أصبح عندي لأن الـ
+
+321
+00:38:59,260 --> 00:39:05,920
+limit للـ sequence سالب واحد أس n في xn لما n تؤول
+
+322
+00:39:05,920 --> 00:39:13,090
+to infinity طلعت بساوي صفر وبالتالي هذا بيؤدي في
+
+323
+00:39:13,090 --> 00:39:25,710
+exercise أخذناه بيقول إذا كان if
+
+324
+00:39:25,710 --> 00:39:34,750
+limit xn as n tends to infinity بيساوي 7 then
+
+325
+00:39:34,750 --> 00:39:42,200
+limit absolute xn as n tends to infinity بيساوي صفر
+
+326
+00:39:42,200 --> 00:39:48,200
+والعكس كمان فباستخدام الـ exercise هذا هذا بيؤدي
+
+327
+00:39:48,200 --> 00:39:54,600
+إلى أن limit absolute سالب واحد أس n في xn as n
+
+328
+00:39:54,600 --> 00:39:58,660
+tends to infinity بيساوي
+
+329
+00:39:58,660 --> 00:40:01,260
+صفر اللي هو بيساوي limit
+
+330
+00:40:11,310 --> 00:40:15,930
+أنا مش عارف أنا ليش رفضت إن نجمع مش هو عبارة الـ
+
+331
+00:40:15,930 --> 00:40:21,010
+XN هي عبارة عن حدود زوجية وفردية لو جمعناهم بدون
+
+332
+00:40:21,010 --> 00:40:22,370
+الـ N مش sequence
+
+333
+00:40:30,350 --> 00:40:37,410
+هذا تفكير ضحل مع احترام طبعا لسؤالك هاي عندك أنت 
+
+334
+00:40:37,410 --> 00:40:46,250
+هاي
+
+335
+00:40:46,250 --> 00:40:54,950
+عندك sequence مثلا هاي الـ sequence سالب واحد أس
+
+336
+00:40:54,950 --> 00:40:55,710
+اتنين in
+
+337
+00:40:59,480 --> 00:41:07,080
+حدودها واحد واحد إلى آخرها صح؟ هتلاحظ؟ وسالب واحد
+
+338
+00:41:07,080 --> 00:41:15,380
+أس اتنين in سالب واحد حدودها سالب واحد تمام؟
+
+339
+00:41:15,380 --> 00:41:23,480
+لما أجمعهم add أجمعهم فهي الـ sequence الأول إن هي
+
+340
+00:41:23,480 --> 00:41:30,690
+زائد الـ sequence التانية لما بجمعهم بجمع الحد الأول
+
+341
+00:41:30,690 --> 00:41:34,610
+على الأول التاني على التاني وهكذا صح؟ إيش هيطلع
+
+342
+00:41:34,610 --> 00:41:41,090
+عندك؟ صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+343
+00:41:41,090 --> 00:41:43,030
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+344
+00:41:43,030 --> 00:41:46,190
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+345
+00:41:46,190 --> 00:41:51,350
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+346
+00:41:51,350 --> 00:41:53,750
+صفر ص
+
+347
+00:42:00,230 --> 00:42:07,010
+المشكلة عندك إنك بتقول X sequence XN ممكن تعتبرها
+
+348
+00:42:07,010 --> 00:42:15,250
+عبارة عن مجموعة sub sequence X2N زائد sub sequence
+
+349
+00:42:15,250 --> 00:42:24,310
+X2N-1 هذا غلط هذا غلط ليس صحيح وهي مثال واضح خطأ
+
+350
+00:42:24,310 --> 00:42:30,270
+okay تمام؟ في أي أسئلة تانية لو سمحتوا؟ مين عندها
+
+351
+00:42:30,270 --> 00:42:44,050
+سؤال تاني؟ ما لديها سؤال؟ في عندكم أسئلة؟ تمام؟
+
+352
+00:42:44,050 --> 00:42:50,950
+مش مقتنعة أه؟ هات مثال هي قدامك أمامك افحص المثال
+
+353
+00:42:50,950 --> 00:42:56,510
+كويس هتقتنعي لكلامك صح لأن هذا المجموع بساوي
+
+354
+00:42:56,510 --> 00:43:03,010
+سالب واحد أصلا صح ليش
+
+355
+00:43:03,010 --> 00:43:10,730
+أسئلة تانية عندكم سؤال تسعة
+
+356
+00:43:10,730 --> 00:43:14,890
+أربعة
+
+357
+00:43:14,890 --> 00:43:20,710
+واحد سؤال
+
+358
+00:43:20,710 --> 00:43:26,430
+تسعة الفرع ديهذا شبيه بالأفراد التانية، بيديك يعني
+
+359
+00:43:26,430 --> 00:43:30,570
+تحاول .. أنا وصلت إن الـ absolute لـ 2x نفس الواحد
+
+360
+00:43:30,570 --> 00:43:33,950
+على 2 absolute الـ x زائد الواحد، لأن كل الـ answer
+
+361
+00:43:33,950 --> 00:43:37,190
+اللي فاتت كان ما يكونش .. ما يكونش إيش في الـ bust
+
+362
+00:43:37,190 --> 00:43:42,950
+يعني، أنا أجيب علاقة الـ x زائد الواحد يعني delta
+
+363
+00:43:42,950 --> 00:43:45,650
+في النهاية هتكون هي الـ minimum لـ .. لـ .. لقيم
+
+364
+00:43:45,650 --> 00:43:47,070
+تانية لقيم تانية، لأن مثلا أنا مش عارفة ..
+
+365
+00:43:47,070 --> 00:43:52,290
+ما طلعتيش؟ أنا مش عارفة القيمة التانية، يعني عرفت
+
+366
+00:43:52,290 --> 00:43:53,010
+القيمة الأولى
+
+367
+00:44:00,180 --> 00:44:05,940
+نعم طيب عشان بس الـ .. الـ .. الوقت يعني انتهى خلينا
+
+368
+00:44:05,940 --> 00:44:13,320
+نقول إن يعني نوقف هنا وهيك المحاضرة بتكون انتهت 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6ff3df5702a305c2c959dcdc98fcd99b0ce3fc8a
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4_postprocess.srt
@@ -0,0 +1,1472 @@
+1
+00:00:21,380 --> 00:00:26,860
+بسم الله الرحمن الرحيم اليوم هنكمل .. في الجزء
+
+2
+00:00:26,860 --> 00:00:31,720
+الأول من المحاضرة هنكمل section خمسة واحد و في
+
+3
+00:00:31,720 --> 00:00:37,160
+الوقت المتبقي من المحاضرة هنعطي مجال لكم ل ..
+
+4
+00:00:37,160 --> 00:00:43,850
+لأسئلة على ال homework أوأسلة في امتحانات سابقة،
+
+5
+00:00:43,850 --> 00:00:49,390
+ماشي الحال، فناخد ال .. في .. أخدنا المرة اللي
+
+6
+00:00:49,390 --> 00:00:56,930
+فاتت أمثلة على الاتصال ووقفنا عند المثال التالي
+
+7
+00:01:13,460 --> 00:01:31,840
+لتألي subset of R and define f function from
+
+8
+00:01:31,840 --> 00:01:38,320
+R to R by f of x
+
+9
+00:01:41,550 --> 00:01:49,110
+بتساوي واحد إذا كان x is rational وبتساوي سفر إذا
+
+10
+00:01:49,110 --> 00:02:00,410
+كان x is irrational show
+
+11
+00:02:00,410 --> 00:02:09,570
+that show if is discontinuous is discontinuous at
+
+12
+00:02:13,630 --> 00:02:21,890
+every x تنتمي إلى r إذا ال function هذه اللي
+
+13
+00:02:21,890 --> 00:02:27,310
+معرفها بالطريقة كما هو موضح على هذه ال function
+
+14
+00:02:27,310 --> 00:02:33,450
+بتكون discontinuous ليست متصلة عند أي x في R ليه
+
+15
+00:02:33,450 --> 00:02:34,930
+برهان ذلك proof
+
+16
+00:02:42,420 --> 00:02:50,400
+fix C تنتمي إلى R then
+
+17
+00:02:50,400 --> 00:02:57,220
+فما احنا عارفين ان ال real
+
+18
+00:02:57,220 --> 00:03:00,980
+numbers عبارة عن ال union disjoint union لل
+
+19
+00:03:00,980 --> 00:03:03,320
+rational numbers وال irrational numbers
+
+20
+00:03:09,680 --> 00:03:22,500
+then X أو C تنتمي إلى Q أو C تنتمي إلى R minus Q
+
+21
+00:03:22,500 --> 00:03:25,680
+إذا الـ C هذا إما هتكون rational number أو
+
+22
+00:03:25,680 --> 00:03:32,160
+irrational number الحالة الأولى case one لو كانت C
+
+23
+00:03:32,160 --> 00:03:33,180
+rational number
+
+24
+00:03:37,870 --> 00:03:49,730
+استخدم الـ Corollary to Density
+
+25
+00:03:49,730 --> 00:03:58,310
+Theorem لتختار
+
+26
+00:03:58,310 --> 00:04:01,830
+سيكوينس
+
+27
+00:04:01,830 --> 00:04:06,630
+XN من أعداد غير عقلية
+
+28
+00:04:19,740 --> 00:04:23,560
+اعتقد ان احنا اثبتنا حاجة زي هذه سابقا
+
+29
+00:04:28,770 --> 00:04:33,750
+حتى لو كان rational فby ال corollary النتيجة تبع
+
+30
+00:04:33,750 --> 00:04:38,630
+ال density theorem ممكن اثبات ان يوجد sequence of
+
+31
+00:04:38,630 --> 00:04:46,150
+irrationals ونهيتها العدد cلأ هذا حسب النتيجة تبع
+
+32
+00:04:46,150 --> 00:04:50,250
+ال density term لأن النتيجة هذه بتقول between any
+
+33
+00:04:50,250 --> 00:04:54,030
+two real numbers there is irrational او اي open
+
+34
+00:04:54,030 --> 00:04:58,990
+interval contains an irrational number فشوفنا احنا
+
+35
+00:04:58,990 --> 00:05:06,210
+البرهان بالتفصيل وبالتالي so نلاحظ
+
+36
+00:05:06,210 --> 00:05:14,530
+انه ال limitل .. ال .. او ال .. ال function قيمة
+
+37
+00:05:14,530 --> 00:05:21,230
+ال function f عند xn بيساوي ال xn is irrational و
+
+38
+00:05:21,230 --> 00:05:24,370
+من ال definition تبع ال function f عند اي
+
+39
+00:05:24,370 --> 00:05:30,310
+irrational بيساوي سفر هذا صحيح لكل n هذا بيقدي ان
+
+40
+00:05:30,310 --> 00:05:41,200
+ال limit ل f of xn as n tends to infinityبساوي
+
+41
+00:05:41,200 --> 00:05:44,140
+limit ثابت سفر بساوي سفر
+
+42
+00:05:49,060 --> 00:05:55,480
+وهذا لا يساوي واحد اللي هو f of c ال c rational
+
+43
+00:05:55,480 --> 00:06:00,360
+number ف f and c بيساوي واحد اذا هاي اندي انا
+
+44
+00:06:00,360 --> 00:06:05,940
+اثبتت ان يوجد sequence xn في المجال تبع الدولة
+
+45
+00:06:05,940 --> 00:06:11,340
+اللي هو R و ال sequence هذه converge ل c لكن ال
+
+46
+00:06:11,340 --> 00:06:16,660
+limit لل image لل sequence لا تساوي f of c اذا by
+
+47
+00:06:16,660 --> 00:06:22,490
+divergence criterionأو by discontinuity criterion
+
+48
+00:06:22,490 --> 00:06:28,450
+اذا by discontinuity
+
+49
+00:06:28,450 --> 00:06:32,690
+criterion if
+
+50
+00:06:32,690 --> 00:06:43,570
+is discontinuous is discontinuous at C في
+
+51
+00:06:43,570 --> 00:06:47,750
+الحالة اللي هي C rational number في الحالة التانية
+
+52
+00:06:52,270 --> 00:07:01,110
+لو كان الـ c irrational number فممكن
+
+53
+00:07:01,110 --> 00:07:05,330
+نستخدم ال density theorem نفسها use density
+
+54
+00:07:05,330 --> 00:07:09,130
+theorem to
+
+55
+00:07:09,130 --> 00:07:13,710
+choose a
+
+56
+00:07:13,710 --> 00:07:21,280
+sequence x in containedof rational numbers
+
+57
+00:07:21,280 --> 00:07:23,760
+contained in Q يعني ال sequence هذه عناصرها
+
+58
+00:07:23,760 --> 00:07:30,440
+rational numbers بحيث انه ال limit لل sequence x
+
+59
+00:07:30,440 --> 00:07:36,260
+in as n tends to infinity بساوي c هذا أثبتناه في
+
+60
+00:07:36,260 --> 00:07:42,480
+المحاضرة السابقة وبالتالي
+
+61
+00:07:42,480 --> 00:07:43,920
+so
+
+62
+00:07:46,980 --> 00:07:52,620
+أنا عندي بما أنه ال Xn rationals ف F of Xn بتساوي
+
+63
+00:07:52,620 --> 00:08:00,280
+واحد لكل N هذا بيقدي انه limit ل F of Xn as N
+
+64
+00:08:00,280 --> 00:08:04,950
+tends to infinityبساوي limit ثابت واحد بطلع واحد
+
+65
+00:08:04,950 --> 00:08:12,710
+واحد لا يساوي سفر اللي هو ال image ل ال C ال C هنا
+
+66
+00:08:12,710 --> 00:08:17,170
+irrational و من ال definition of the function if
+
+67
+00:08:17,170 --> 00:08:21,730
+and irrational بساوي سفر اذا انا هنا اثبتت ان يوجد
+
+68
+00:08:21,730 --> 00:08:23,610
+sequence
+
+69
+00:08:25,710 --> 00:08:32,350
+في مجال الدالة نهايتها c لكن نهاية صورتها لا تساوي
+
+70
+00:08:32,350 --> 00:08:41,750
+صورة ال c كمان مرة hence by
+
+71
+00:08:41,750 --> 00:08:47,790
+discontinuity by
+
+72
+00:08:47,790 --> 00:08:50,150
+discontinuity criterion
+
+73
+00:08:52,890 --> 00:09:03,530
+الـ function f is discontinuous at c لأن بما أن c
+
+74
+00:09:03,530 --> 00:09:09,410
+was arbitrary فالـ function is discontinuous عن كل
+
+75
+00:09:09,410 --> 00:09:15,050
+real number c طبعا في الحالتين لأن هذا بكمل
+
+76
+00:09:15,050 --> 00:09:22,090
+البرهانة لأن هذه ال function هي function من R لRو
+
+77
+00:09:22,090 --> 00:09:26,950
+discontinuous ليست متصلة عند أي x فارق هذه ال
+
+78
+00:09:26,950 --> 00:09:31,030
+function لها اسم ومشهورة ومعروفة اسمها Dirichlet
+
+79
+00:09:31,030 --> 00:09:35,210
+function اذا
+
+80
+00:09:35,210 --> 00:09:41,110
+ال function هذه remark
+
+81
+00:09:41,110 --> 00:09:46,870
+the
+
+82
+00:09:46,870 --> 00:09:49,610
+above function
+
+83
+00:09:55,060 --> 00:10:03,000
+function is well-known as
+
+84
+00:10:03,000 --> 00:10:10,040
+Dirichlet Dirichlet
+
+85
+00:10:10,040 --> 00:10:17,540
+دا عالم رياضيات Dirichlet is function او Dirichlet
+
+86
+00:10:17,540 --> 00:10:20,000
+is discontinuous function
+
+87
+00:10:25,450 --> 00:10:34,010
+this continuous function ده
+
+88
+00:10:34,010 --> 00:10:41,070
+اللي مهمة ويلها meta في ال real analysis okay تمام
+
+89
+00:10:41,070 --> 00:10:47,630
+طيب ناخد بعض الملاحظات واضح في أي سؤال واضح هنا
+
+90
+00:10:47,630 --> 00:10:49,750
+البرهن في أي سلسلة
+
+91
+00:11:06,590 --> 00:11:15,470
+طيب ناخد شوية remarks ال
+
+92
+00:11:15,470 --> 00:11:19,410
+remark
+
+93
+00:11:19,410 --> 00:11:27,290
+الأولى sometimes a
+
+94
+00:11:27,290 --> 00:11:34,590
+function خلينا احنا نرجع لل definition تبع الاتصال
+
+95
+00:11:34,590 --> 00:11:35,770
+عن النقطة definition
+
+96
+00:11:56,710 --> 00:12:04,230
+الشرط التاني ان ال limitلأ F of X لما X تقول إلى C
+
+97
+00:12:04,230 --> 00:12:11,490
+exist و بساوي F of C اللي هو الشرط هذا هذا الشرط
+
+98
+00:12:11,490 --> 00:12:16,730
+سمنها تلاتة في واحد هذا يتضمن تلات شروط ال limit
+
+99
+00:12:16,730 --> 00:12:22,690
+and C exist F is defined at C و اتنين متساوين تلات
+
+100
+00:12:22,690 --> 00:12:28,970
+شروطالان لو اي واحد من التلات الشروط هدول اختل فال
+
+101
+00:12:28,970 --> 00:12:32,210
+function بتكونش continuous and النقطة c ف
+
+102
+00:12:32,210 --> 00:12:42,490
+sometimes the function can be discontinuous
+
+103
+00:12:42,490 --> 00:12:46,970
+at x بساوي c because
+
+104
+00:12:49,450 --> 00:13:06,610
+if is undefined is undefined at C لكن
+
+105
+00:13:06,610 --> 00:13:10,750
+أحيانا أخرى وفي الحالة اللي بقدرش أعمل حاجة بقدرش
+
+106
+00:13:10,750 --> 00:13:17,030
+أعمل حاجة however
+
+107
+00:13:20,670 --> 00:13:28,290
+لو كانت if ال limit لل function f of x as x tends
+
+108
+00:13:28,290 --> 00:13:39,010
+to c موجودة exists exists
+
+109
+00:13:39,010 --> 00:13:46,490
+and equals L ينتمي ل R ففي
+
+110
+00:13:46,490 --> 00:13:49,510
+الحالة هذه we can
+
+111
+00:13:53,080 --> 00:13:58,780
+we can define we
+
+112
+00:13:58,780 --> 00:14:09,440
+can define a function capital F من a اتحاد ال
+
+113
+00:14:09,440 --> 00:14:20,800
+singleton set C إلى R by capital F of X بساوي
+
+114
+00:14:20,800 --> 00:14:21,980
+العدد L
+
+115
+00:14:24,670 --> 00:14:32,970
+fx بتساوي c و بتساوي ال function f of x إذا كان ال
+
+116
+00:14:32,970 --> 00:14:44,530
+x ينتمي إلى a ولا يساوي c in this case in this
+
+117
+00:14:44,530 --> 00:14:54,830
+case the function capital F is continuousat x
+
+118
+00:14:54,830 --> 00:15:00,530
+بساوي c indeed
+
+119
+00:15:00,530 --> 00:15:07,490
+في حقيقة القمر indeed في حقيقة القمر تعالى نشوف
+
+120
+00:15:07,490 --> 00:15:14,630
+high limit capital f of x as x tends to c بساوي x
+
+121
+00:15:14,630 --> 00:15:19,510
+تقول ل c إذا ال x بالتأكيد بتسويش c لما x ما
+
+122
+00:15:19,510 --> 00:15:26,390
+بتسويش c يعني x and time ل aفcapital F هي نفسها
+
+123
+00:15:26,390 --> 00:15:35,430
+حسب تعريفها هي small f طب احنا فرضين ان ال limit ل
+
+124
+00:15:35,430 --> 00:15:41,150
+F of X exist و بساوي and C و بساوي L، إذن هذه تطلع
+
+125
+00:15:41,150 --> 00:15:43,070
+موجودة و بساوي L
+
+126
+00:15:46,940 --> 00:15:50,740
+و احنا من ال definition تبع ال function ال ال
+
+127
+00:15:50,740 --> 00:15:53,800
+ماخدينها هي عبارة عن قيمة ال function capital F
+
+128
+00:15:53,800 --> 00:16:00,180
+and C اذا هاي شرط الاتصال للدالة capital F and C
+
+129
+00:16:00,180 --> 00:16:10,320
+متحقق وبالتالي اذا capital F is continuous at C ال
+
+130
+00:16:10,320 --> 00:16:20,300
+function capital F the functioncapital F is called
+
+131
+00:16:20,300 --> 00:16:30,700
+a continuous extension
+
+132
+00:16:30,700 --> 00:16:36,980
+of
+
+133
+00:16:36,980 --> 00:16:43,900
+small f عبارة عن continuous extension يعني توسعة
+
+134
+00:16:43,900 --> 00:16:49,530
+متصلةتوسعة متصلة لدالة small f لحظة انتوا هنا ان
+
+135
+00:16:49,530 --> 00:16:59,350
+الدالة capital F الدالة
+
+136
+00:16:59,350 --> 00:17:05,550
+capital F function من A union single to C إلى R
+
+137
+00:17:05,550 --> 00:17:11,810
+طبعا هنا C لا تمتم إلى A وعندي small f هي function
+
+138
+00:17:11,810 --> 00:17:19,850
+من A إلى Rفلاحظوا ان ال function capital F لما
+
+139
+00:17:19,850 --> 00:17:28,250
+اعمل restriction لل domain تبعها على a فقط فهي نفس
+
+140
+00:17:28,250 --> 00:17:35,490
+f يعني لما اقيد او احصر الدولة capital F على
+
+141
+00:17:35,490 --> 00:17:41,610
+المجموعة a فقط فهي نفس ال F هي لكل x a capital F
+
+142
+00:17:41,610 --> 00:17:47,020
+هي small fالزيادة ان capital F معرفة عن C و F مش
+
+143
+00:17:47,020 --> 00:17:51,720
+معرفة عن C فهذه الدالة capital F بنسميها توسعة على
+
+144
+00:17:51,720 --> 00:17:56,120
+small f التوسعة هذه متصلة عند النقطة C
+
+145
+00:17:59,220 --> 00:18:03,740
+تمام؟ إذا هذه أول ملاحظة إذا لو كانت الدالة مش
+
+146
+00:18:03,740 --> 00:18:10,580
+معرفة لو كانت small f مش معرفة عن c لكن نهايتها عن
+
+147
+00:18:10,580 --> 00:18:15,580
+c موجودة وبالساوي عدد L فبقدر أعرف دالة جديدة
+
+148
+00:18:15,580 --> 00:18:20,080
+capital F اللي هي توسعة extension ل small f و ال
+
+149
+00:18:20,080 --> 00:18:24,640
+extension الدالة هذه اللي هي توسعةبتكون متصلة عند
+
+150
+00:18:24,640 --> 00:18:28,920
+الـ C اننا ناخد قيمتها عند الـ C بساوي قيمة ال
+
+151
+00:18:28,920 --> 00:18:36,540
+limit و عند النقاط ال A هي نفس small if لكن
+
+152
+00:18:36,540 --> 00:18:39,620
+الملاحظة التانية
+
+153
+00:18:51,350 --> 00:18:55,490
+لو كانت الادالة .. ممكن تكون الادالة معرفة عن C
+
+154
+00:18:55,490 --> 00:19:03,670
+يعني هنا sometimes a
+
+155
+00:19:03,670 --> 00:19:09,670
+function g is discontinuous
+
+156
+00:19:09,670 --> 00:19:16,190
+a function g from A to R is discontinuous at C
+
+157
+00:19:19,590 --> 00:19:25,970
+بسبب و ال c ممكن تكون موجودة في a او حتى لو مش
+
+158
+00:19:25,970 --> 00:19:35,630
+موجودة في a is discontinuous at c because ال limit
+
+159
+00:19:35,630 --> 00:19:43,210
+ل g of x as x tends to c does not exist يعني ممكن
+
+160
+00:19:43,210 --> 00:19:48,330
+ال g تكون معرفة and ال c لكنالنهايتها عند الـ c مش
+
+161
+00:19:48,330 --> 00:19:51,690
+موجودة فطبعا في الحالة دي ال function بتكون
+
+162
+00:19:51,690 --> 00:19:57,830
+discontinuous at c وفي الحالة دي ماقدرش اعرف in
+
+163
+00:19:57,830 --> 00:20:08,410
+this case in this case we can't لا نستطيع we can't
+
+164
+00:20:08,410 --> 00:20:14,990
+define a continuous extension
+
+165
+00:20:23,400 --> 00:20:33,900
+ممكن مانقدرش نعرف continuous extension لال ..
+
+166
+00:20:33,900 --> 00:20:42,000
+اللي هو capital G from A union singleton C إلى R
+
+167
+00:20:42,000 --> 00:20:45,820
+أو
+
+168
+00:20:45,820 --> 00:20:52,800
+we cannot define an extensionand extension g من a
+
+169
+00:20:52,800 --> 00:21:02,520
+union c لr by g of x بساوي عدد
+
+170
+00:21:02,520 --> 00:21:11,400
+capital c if x بساوي c و بساوي g of x إذا كان x
+
+171
+00:21:11,400 --> 00:21:18,500
+ينتمي and
+
+172
+00:21:20,360 --> 00:21:25,060
+جي بي continuous at c
+
+173
+00:21:39,530 --> 00:21:44,470
+طبعا هنا الـ G باخدها عند C بيساوي capital C هاد
+
+174
+00:21:44,470 --> 00:21:50,550
+عيرنا ال number ع شوية و عند X بيساوي A لازم تساوي
+
+175
+00:21:50,550 --> 00:21:55,970
+G of X عشان تكون توسعة أو extension ل small g ف to
+
+176
+00:21:55,970 --> 00:22:05,990
+see this لبرهان ذلك to see this to see that such
+
+177
+00:22:05,990 --> 00:22:18,380
+Gcan't be continuous at c assume خلّيني
+
+178
+00:22:18,380 --> 00:22:25,200
+أعمل برهان بالتناقض assume on contrary assume on
+
+179
+00:22:25,200 --> 00:22:33,900
+contrary أن counter G is continuous at c then هذا
+
+180
+00:22:33,900 --> 00:22:40,660
+معناه أن ال limitلـ capital G of X as X tends to C
+
+181
+00:22:40,660 --> 00:22:48,420
+بساوي capital G اللي
+
+182
+00:22:48,420 --> 00:22:56,380
+هي بساوي limit بتطلع
+
+183
+00:22:56,380 --> 00:23:06,900
+بساوي capital G of C بساوي capital Cو هذه بتساوي
+
+184
+00:23:06,900 --> 00:23:11,700
+limit g of x لما x تقول ل c هي عبارة عن limit من
+
+185
+00:23:11,700 --> 00:23:18,320
+تعريف ال g limit g of x لما x تقول ل c لإن لما x
+
+186
+00:23:18,320 --> 00:23:22,960
+تقول ل c x بستويش ال c x بستويش ال c معناته x
+
+187
+00:23:22,960 --> 00:23:29,600
+تنتمي ل a لإن g of x هي عبارة عن small g of x لإن
+
+188
+00:23:29,600 --> 00:23:32,260
+في الحالة هذه limit
+
+189
+00:23:34,430 --> 00:23:40,110
+إذا limit small g of x لما x تقولها c exist اه إذا
+
+190
+00:23:40,110 --> 00:23:46,110
+هذا بتطلع exist and equals c هي بالساوية c
+
+191
+00:23:46,110 --> 00:23:51,370
+contradiction هذا تناقض لإن احنا فرضين إن ال limit
+
+192
+00:23:51,370 --> 00:23:53,970
+ل g of x and c مش موجودة
+
+193
+00:23:58,090 --> 00:24:02,390
+لما تكون الدالة مش متصلة على النقطة لعدم وجود
+
+194
+00:24:02,390 --> 00:24:06,830
+نهايتها عند النقطة فماقدرش أعرف continuous
+
+195
+00:24:06,830 --> 00:24:15,610
+extension لدالة and النقطة Cلأن هنا فرضنا أن هناك
+
+196
+00:24:15,610 --> 00:24:19,030
+extension هذا ال extension مش ممكن يكون continuous
+
+197
+00:24:19,030 --> 00:24:23,050
+عند النقطة C لأنه لو كان continuous عند النقطة C
+
+198
+00:24:23,050 --> 00:24:27,750
+سيقدر ان الملمت الدالة small g and c exist وهذا
+
+199
+00:24:27,750 --> 00:24:34,830
+يتناقض مع الفرض تمام، هذا النوع من ال
+
+200
+00:24:34,830 --> 00:24:40,570
+discontinuity لما تكون الدالة discontinuous لعدم
+
+201
+00:24:42,160 --> 00:24:43,920
+لما تكون الدالة discontinuous
+
+202
+00:24:46,760 --> 00:24:51,500
+السبب في ذلك أنها مش معرفة عند النقطة C لكن نهيتها
+
+203
+00:24:51,500 --> 00:24:55,560
+موجودة عند الـ C فهذا النوع من ال discontinuity من
+
+204
+00:24:55,560 --> 00:25:00,360
+عدم الاتصال بنسميه removable يعني ممكن إزالته أو
+
+205
+00:25:00,360 --> 00:25:05,300
+التخلص منه وهي فعلا اتخلصنا من عدم الاتصال للدالة
+
+206
+00:25:05,300 --> 00:25:09,480
+small f and c بتعريف دالة capital F بالطريقة هذه
+
+207
+00:25:09,480 --> 00:25:12,670
+وشوفنا أن الدالة الجديدة continuous عند ال Cإذا
+
+208
+00:25:12,670 --> 00:25:16,190
+هذا النوع من عدم الاتصال أو ال discontinuity is
+
+209
+00:25:16,190 --> 00:25:19,990
+called removable يمكن إزالته يمكن تخلص منه أما إذا
+
+210
+00:25:19,990 --> 00:25:25,610
+كانت الدالة discontinuous عند النقطة C لأن نهايتها
+
+211
+00:25:25,610 --> 00:25:30,050
+in C مش موجودة فهذا النوع من عدم الاتصال بنسميه
+
+212
+00:25:30,050 --> 00:25:34,950
+essential يعني أساسي لا يمكن تخلص منه زي ما شفنا
+
+213
+00:25:34,950 --> 00:25:38,890
+في التحليل التحت okay تمامإن هذه أنواع ال
+
+214
+00:25:38,890 --> 00:25:42,570
+discontinuity ال discontinuity أو عدم الاتصال
+
+215
+00:25:42,570 --> 00:25:47,030
+نوعين نوع removable ممكن يزالته ممكن تخلص منه و
+
+216
+00:25:47,030 --> 00:25:55,210
+نوع تاني essential أساسي لايمكن تخلص منه ممكن
+
+217
+00:25:55,210 --> 00:25:58,090
+ناخد بعض الأمثلة على ذلك
+
+218
+00:26:14,950 --> 00:26:21,050
+نأخد الـ function consider الـ
+
+219
+00:26:21,050 --> 00:26:28,750
+function g of x بتساوي sin واحد على x حيث x لا
+
+220
+00:26:28,750 --> 00:26:31,970
+يساوي سفر طبعا احنا شفنا
+
+221
+00:26:35,380 --> 00:26:41,220
+في مثال صادق انه limit g of x as x tends to zero
+
+222
+00:26:41,220 --> 00:26:46,880
+does not exist limit
+
+223
+00:26:46,880 --> 00:26:57,520
+الدالة هذه غير موجودة وبالتالي
+
+224
+00:26:57,520 --> 00:27:04,540
+كمان برضهوكمان الدالة برضه جي عند سفر مش معرفة اذا
+
+225
+00:27:04,540 --> 00:27:17,220
+.. اذا in this case we can't .. we can't
+
+226
+00:27:17,220 --> 00:27:28,580
+.. we can't define a continuous extension
+
+227
+00:27:30,680 --> 00:27:41,180
+of small g at السفر هذا النوع التاني من ال
+
+228
+00:27:41,180 --> 00:27:47,520
+discontinuity لكن لو أخدت دالة زي هذه
+
+229
+00:27:59,300 --> 00:28:11,080
+لو أخدت f of x بساوي x في ال sign واحد على x فطبعا
+
+230
+00:28:11,080 --> 00:28:16,220
+هنا و x لا يساوي سفر فطبعا واضح أن f عند السفر is
+
+231
+00:28:16,220 --> 00:28:17,240
+undefined
+
+232
+00:28:20,560 --> 00:28:26,040
+الـ limit شوفنا أنه limit ل f of x لما x تقول ل 0
+
+233
+00:28:26,040 --> 00:28:30,520
+by squeeze theorem أثبتنا باستخدام squeeze theorem
+
+234
+00:28:30,520 --> 00:28:38,100
+أنه limit هذه بيساوي 0 exist بساوي 0 طبعا إذا هنا
+
+235
+00:28:38,100 --> 00:28:47,020
+f is discontinuous discontinuous at x بساوي 0 لإن
+
+236
+00:28:47,020 --> 00:28:51,920
+أنا مش معرف عند السفرلكن بما ان ال limit تبقى عند
+
+237
+00:28:51,920 --> 00:28:58,320
+السفر موجودة we can define
+
+238
+00:28:58,320 --> 00:29:07,320
+a continuous extension of
+
+239
+00:29:07,320 --> 00:29:11,220
+f at سفر as follows
+
+240
+00:29:14,800 --> 00:29:21,980
+فعندى capital F of X بنعرفها على أنها بالساوى
+
+241
+00:29:21,980 --> 00:29:26,040
+السفر اللى هو limit لل function عند السفر إذا كان
+
+242
+00:29:26,040 --> 00:29:34,960
+ال X بساوى السفر و بالساوى small f of X إذا كان X
+
+243
+00:29:34,960 --> 00:29:42,260
+تنتمي ل domain الدالة اللى هواللي هو ر مع ده سفر ر
+
+244
+00:29:42,260 --> 00:29:48,760
+مع ده سفر مش هذا هو ال domain تبع الدالة الانف
+
+245
+00:29:48,760 --> 00:29:58,000
+دالة if now you can verify انه
+
+246
+00:29:58,000 --> 00:30:05,040
+capital F is continuous is continuous at x بساوي
+
+247
+00:30:05,040 --> 00:30:12,580
+سفربينما small f is not continuous از سفر حيا
+
+248
+00:30:12,580 --> 00:30:17,220
+عندي limit capital
+
+249
+00:30:17,220 --> 00:30:23,560
+F of X as X tends to zero بساوي لما X ساوي للسفر X
+
+250
+00:30:23,560 --> 00:30:29,800
+بسويش سفر لما X ماتساويش سفر فcapital F هي small f
+
+251
+00:30:32,400 --> 00:30:37,700
+و limit small f بتساوي سفر اللي هي capital F معرفة
+
+252
+00:30:37,700 --> 00:30:43,620
+عند السفر okay إذا هي شرط الاتصال عند السفر متحقق
+
+253
+00:30:43,620 --> 00:30:48,740
+وبالتالي capital F متصلة عند السفر okay تمام هفهم
+
+254
+00:30:48,740 --> 00:30:54,740
+okay بنوقف هنا و بنتيح الآن المجال إلكم إذا كان في
+
+255
+00:30:54,740 --> 00:30:58,920
+عندكم أي أسئلة بخصوص ال homework أو الامتحان
+
+256
+00:30:58,920 --> 00:31:03,560
+النصفي اللي هناخده بكرا إن شاء اللهفي عندكم أي
+
+257
+00:31:03,560 --> 00:31:11,680
+سؤال أو استفسار؟ في حد عنده أي سؤال؟
+
+258
+00:31:11,680 --> 00:31:24,080
+في
+
+259
+00:31:24,080 --> 00:31:29,970
+أي سؤال؟ أي سؤال؟دكتور انا ممكن استخدم limit ال K
+
+260
+00:31:29,970 --> 00:31:33,830
+لما ال X تقول ال C و سوى K و limit ال X لما ال X
+
+261
+00:31:33,830 --> 00:31:38,490
+تقول ال C و سوى C ممكن استخدمهم انا بحثت صح صح لان
+
+262
+00:31:38,490 --> 00:31:42,490
+هذه صارت حاجات ال trivial و يعني اثبتناها الا ده
+
+263
+00:31:42,490 --> 00:31:47,150
+طول منك اثبتها باستخدام تعريف epsilon دلت اكيد اي
+
+264
+00:31:47,150 --> 00:31:53,810
+سؤال بدك ممكن التاني التاني اللي مش محلول اه اللي
+
+265
+00:31:53,810 --> 00:31:54,830
+مش محلول نعم
+
+266
+00:32:04,200 --> 00:32:13,180
+طيب suppose question suppose
+
+267
+00:32:13,180 --> 00:32:25,460
+أن xn أكبر من أو ساوى سفر لكل n في n and .. and ال
+
+268
+00:32:25,460 --> 00:32:25,820
+limit
+
+269
+00:32:28,550 --> 00:32:34,910
+لسالب واحد أُس n في xn ال sequence هذه لما n تقول
+
+270
+00:32:34,910 --> 00:32:46,590
+ل infinity بساوي x ينتمي إلى r show أن ال limit ل
+
+271
+00:32:46,590 --> 00:32:56,630
+xn لما n تقول ل infinity بساوي سفر فضلي
+
+272
+00:33:21,960 --> 00:33:28,720
+فنشوف الان انا عندي ال limit لل sequence للحد
+
+273
+00:33:28,720 --> 00:33:33,900
+العام تبعها سالب واحد قص ان في x in لما ان تقول
+
+274
+00:33:33,900 --> 00:33:44,020
+infinity بساوي x اذا ان andالسالب واحد قصة اتنين N
+
+275
+00:33:44,020 --> 00:33:52,460
+في X اتنين N هذه عبارة عن subsequence subsequence
+
+276
+00:33:52,460 --> 00:33:58,540
+of السيكوانس الحد العام تبعها السالب واحد قصة N X
+
+277
+00:33:58,540 --> 00:34:07,730
+X N هذه subsequence خط الحدود الزوجية صح؟طيب ال
+
+278
+00:34:07,730 --> 00:34:12,950
+subsequence هذه هي الحد العام تبعها x اتنين n
+
+279
+00:34:12,950 --> 00:34:16,890
+مفروض converge اخلنا احنا في نظرية ان كانت ال
+
+280
+00:34:16,890 --> 00:34:21,350
+sequence convergent ل x فأي subsequence منها بتكون
+
+281
+00:34:21,350 --> 00:34:27,810
+convergent لنفس ال x تمام؟ في نفس الوجد
+
+282
+00:34:38,530 --> 00:34:44,370
+في نفس الوقت ال sequence سالب واحد قصة اتنين in
+
+283
+00:34:44,370 --> 00:34:48,750
+سالب واحد في
+
+284
+00:34:48,750 --> 00:34:56,810
+x اتنين in سالب واحد هذا عبارة عن sub sequence من
+
+285
+00:34:56,810 --> 00:35:02,070
+ال sequence الأصلي المعطاة اللي هي الحد العام
+
+286
+00:35:02,070 --> 00:35:07,020
+تبعها سالب واحد to int x inيعني انا اخدت هنا
+
+287
+00:35:07,020 --> 00:35:10,680
+الحدود الفردية من الـ sequence هذه طبعا هذه اكيد
+
+288
+00:35:10,680 --> 00:35:17,600
+subsequence فالحد العام هذا عبارة عن سالب X اتنين
+
+289
+00:35:17,600 --> 00:35:23,140
+M سالب واحد الآن ال subsequence هذه المفروض انها
+
+290
+00:35:23,140 --> 00:35:28,180
+converge الى X لان ال sequence الاصلية convergent
+
+291
+00:35:28,180 --> 00:35:34,560
+ل X تمام؟ اذا انا في عندي هنا limit
+
+292
+00:35:38,370 --> 00:35:46,990
+x اتنين n سالب واحد بساوي سالب x لأن limit سالب ال
+
+293
+00:35:46,990 --> 00:35:53,070
+sequence هذه بساوي x اذا اضربيها في سالب واحد طبعا
+
+294
+00:35:53,070 --> 00:36:02,030
+واندي من هنا من هناك limit x اتنين n لما n تقول
+
+295
+00:36:02,030 --> 00:36:05,810
+infinity بساوي
+
+296
+00:36:05,810 --> 00:36:06,190
+x
+
+297
+00:36:17,310 --> 00:36:28,710
+نجمعهم؟ لأ لأ منجمعوش الناس ما
+
+298
+00:36:28,710 --> 00:36:29,450
+بنجمعهم
+
+299
+00:36:37,410 --> 00:36:42,950
+ما بنجمع مش عاملة اتجمع الان في اندي انا لحظة انت
+
+300
+00:36:42,950 --> 00:36:50,530
+عندك من الفرض xn أكبر من أو سوى سفر لكل n في n
+
+301
+00:36:55,480 --> 00:37:03,280
+فبالتالي إذا x2n سالب واحد أكبر من أوسعه سفر لكل n
+
+302
+00:37:03,280 --> 00:37:15,820
+وكذلك x2n برضه أكبر من أوسعه سفر لكل n صح؟ أصبت؟
+
+303
+00:37:15,820 --> 00:37:22,240
+وبالتالي إذا ال limit ل x2n سالب واحد تطلع أكبر من
+
+304
+00:37:22,240 --> 00:37:30,430
+أوسعه سفرو ال limit ل x2n تطلع أكبر من أوي ساوي
+
+305
+00:37:30,430 --> 00:37:34,890
+سفر هذه نظرية أخدناها صح؟ أخدناها نظرية بتقول لو
+
+306
+00:37:34,890 --> 00:37:38,550
+كانت ال sequence كل حدوة ده غير سالبة و ال limit
+
+307
+00:37:38,550 --> 00:37:44,330
+تبعتها exist فال limit تبعتها تطلع غير سالبة صح؟
+
+308
+00:37:44,330 --> 00:37:50,750
+فال limit هنا هي حسبناها سالم x و ال limit هنا
+
+309
+00:37:50,750 --> 00:37:59,060
+طلعت xإذا أنا في عندي سالب X أكبر من أو ساوي سفر
+
+310
+00:37:59,060 --> 00:38:06,420
+and X أكبر من أو ساوي سفر هذا بيقدر ان X أصغر من
+
+311
+00:38:06,420 --> 00:38:13,680
+أو ساوي سفر and X أكبر من أو ساوي سفر هذا بيقدر ان
+
+312
+00:38:13,680 --> 00:38:15,040
+X بساوي سفر
+
+313
+00:38:19,470 --> 00:38:23,930
+مايعني x أصغر من 0 و أكبر من 0 يعني x أصغر من 0
+
+314
+00:38:23,930 --> 00:38:27,470
+العدد الوحيد اللي بتمتع بالخاصيتين هدول في نفس
+
+315
+00:38:27,470 --> 00:38:34,090
+الواجت هو 0 إذا هاني أثبتت أن ال limit ل ال
+
+316
+00:38:34,090 --> 00:38:37,550
+sequence
+
+317
+00:38:37,550 --> 00:38:42,870
+أثبتت
+
+318
+00:38:42,870 --> 00:38:44,170
+أن x أصغر من 0
+
+319
+00:38:47,480 --> 00:38:52,500
+طبعا احنا ما اثبتناش لحد تلان ان ال limit لل
+
+320
+00:38:52,500 --> 00:38:59,260
+sequence ان انا عندي .. اذا انا اصبح عندي لان ال
+
+321
+00:38:59,260 --> 00:39:05,920
+limit لل sequence سارب واحد أس ن في xn لما n تقول
+
+322
+00:39:05,920 --> 00:39:13,090
+ال infinity طلعت بالساوية سفر وبالتالي هذا بيقديفي
+
+323
+00:39:13,090 --> 00:39:25,710
+exercise أخدناه بيقول إذا كان if
+
+324
+00:39:25,710 --> 00:39:34,750
+limit xn as n tends to infinity بيساوي 7 then
+
+325
+00:39:34,750 --> 00:39:42,200
+limitabsolute xn as n tends to infinity بيساوي سفر
+
+326
+00:39:42,200 --> 00:39:48,200
+والعكس كمان فباستخدام ال exercise هذا هذا بيقدي
+
+327
+00:39:48,200 --> 00:39:54,600
+انه limit absolute سالب واحد اص ان في xn as n
+
+328
+00:39:54,600 --> 00:39:58,660
+tends to infinity بيساوي
+
+329
+00:39:58,660 --> 00:40:01,260
+سفر اللي هو بيساوي limit
+
+330
+00:40:11,310 --> 00:40:15,930
+أنا مش عارف أنا ليش رفضت اللي نجمع مش هو عبارة ال
+
+331
+00:40:15,930 --> 00:40:21,010
+XN هي عبارة عن حدود زوجية و فردية لو جمعناهم بدون
+
+332
+00:40:21,010 --> 00:40:22,370
+ال N مش sequence
+
+333
+00:40:30,350 --> 00:40:37,410
+هذا تفكير ضحل مع احترام طبعا لسؤالك هاي عندك انت
+
+334
+00:40:37,410 --> 00:40:46,250
+هاي
+
+335
+00:40:46,250 --> 00:40:54,950
+عندك sequence مثلا هاي ال sequence سالب واحد قصة
+
+336
+00:40:54,950 --> 00:40:55,710
+اتنين in
+
+337
+00:40:59,480 --> 00:41:07,080
+حدودها واحد واحد إلى آخرها صح؟ هزبوت؟ و سالب واحد
+
+338
+00:41:07,080 --> 00:41:15,380
+قصة اتنين in سالب واحد حدودها سالب واحد تمام؟
+
+339
+00:41:15,380 --> 00:41:23,480
+لما اجمعهم add اجمعيهم فهي ال sequence الأول ان هي
+
+340
+00:41:23,480 --> 00:41:30,690
+زائد ال sequence التانيةلما بجمعهم بجمع الحد الأول
+
+341
+00:41:30,690 --> 00:41:34,610
+على الأول التاني على التاني و هكذا صح؟ إيش هيطلع
+
+342
+00:41:34,610 --> 00:41:41,090
+عندك؟ صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+343
+00:41:41,090 --> 00:41:43,030
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+344
+00:41:43,030 --> 00:41:46,190
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+345
+00:41:46,190 --> 00:41:51,350
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+346
+00:41:51,350 --> 00:41:53,750
+صفر صف
+
+347
+00:42:00,230 --> 00:42:07,010
+المشكلة عندك انك بتقول X sequence XN ممكن تعتبرها
+
+348
+00:42:07,010 --> 00:42:15,250
+ابارعا مجموعة sub sequence X2N زاد sub sequence
+
+349
+00:42:15,250 --> 00:42:24,310
+X2N-1 هذا غلط هذا غلط ليس صحيحوهي مثالي واضح خطأ
+
+350
+00:42:24,310 --> 00:42:30,270
+okay تمام؟ في أي أسئلة تانية لو سمحتوا؟ مين عندها
+
+351
+00:42:30,270 --> 00:42:44,050
+سؤال تاني؟ مال لديها سؤال؟ في عندكم أسئلة؟ تمام؟
+
+352
+00:42:44,050 --> 00:42:50,950
+مش مقتنعة أه؟ هد مثال هي قدامك أمامك افحص المثال
+
+353
+00:42:50,950 --> 00:42:56,510
+كويس هتقتنعيلو كلامك صح لكان هذا المجموع بساوي
+
+354
+00:42:56,510 --> 00:43:03,010
+سالب واحد اصلا صح ليش
+
+355
+00:43:03,010 --> 00:43:10,730
+اسئلة تانية عندكم سؤال تسعة
+
+356
+00:43:10,730 --> 00:43:14,890
+اربعة
+
+357
+00:43:14,890 --> 00:43:20,710
+واحد سؤال
+
+358
+00:43:20,710 --> 00:43:26,430
+تسعة الفرع ديهذا شبيه بالأفراد التانية، بديك يعني
+
+359
+00:43:26,430 --> 00:43:30,570
+تحاول .. انا وصلت ان ال absolute ل 2x نفس الواحد
+
+360
+00:43:30,570 --> 00:43:33,950
+على 2 absolute ال x زائد الواحد، لإن كل ال answer
+
+361
+00:43:33,950 --> 00:43:37,190
+اللي فاتت كان مايكونش .. مايكونش إيش في ال bust
+
+362
+00:43:37,190 --> 00:43:42,950
+يعني، انا أجيب علاقة ال x زائد الواحد يعني delta
+
+363
+00:43:42,950 --> 00:43:45,650
+في النهاية هتكون هي ال minimum ل .. ل .. لقيم
+
+364
+00:43:45,650 --> 00:43:47,070
+تانية لقيم تانية، لإن مثلا انا مش عارفة ..
+
+365
+00:43:47,070 --> 00:43:52,290
+ماطلعتيش؟ انا مش عارفة القيمة التانية، يعني عرفت
+
+366
+00:43:52,290 --> 00:43:53,010
+القيمة الأولى
+
+367
+00:44:00,180 --> 00:44:05,940
+نعم طيب عشان بس ال .. ال .. الوقت يعني انتهى خلينا
+
+368
+00:44:05,940 --> 00:44:13,320
+نقول انه يعني نواجف هنا و هيك المحاضرة بتكون انتهت
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8e2c9dc11251d2a8c4ac51c6b80fb096b52dcfba
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Ij2H9eVnog4_raw.srt
@@ -0,0 +1,1476 @@
+1
+00:00:21,380 --> 00:00:26,860
+بسم الله الرحمن الرحيم اليوم هنكمل .. في الجزء
+
+2
+00:00:26,860 --> 00:00:31,720
+الأول من المحاضرة هنكمل section خمسة واحد و في
+
+3
+00:00:31,720 --> 00:00:37,160
+الوقت المتبقي من المحاضرة هنعطي مجال لكم ل ..
+
+4
+00:00:37,160 --> 00:00:43,850
+لأسئلة على ال homework أوأسلة في امتحانات سابقة،
+
+5
+00:00:43,850 --> 00:00:49,390
+ماشي الحال، فناخد ال .. في .. أخدنا المرة اللي
+
+6
+00:00:49,390 --> 00:00:56,930
+فاتت أمثلة على الاتصال ووقفنا عند المثال التالي
+
+7
+00:01:13,460 --> 00:01:31,840
+لتألي subset of R and define f function from
+
+8
+00:01:31,840 --> 00:01:38,320
+R to R by f of x
+
+9
+00:01:41,550 --> 00:01:49,110
+بتساوي واحد إذا كان x is rational وبتساوي سفر إذا
+
+10
+00:01:49,110 --> 00:02:00,410
+كان x is irrational show
+
+11
+00:02:00,410 --> 00:02:09,570
+that show if is discontinuous is discontinuous at
+
+12
+00:02:13,630 --> 00:02:21,890
+every x تنتمي إلى r إذا ال function هذه اللي
+
+13
+00:02:21,890 --> 00:02:27,310
+معرفها بالطريقة كما هو موضح على هذه ال function
+
+14
+00:02:27,310 --> 00:02:33,450
+بتكون discontinuous ليست متصلة عند أي x في R ليه
+
+15
+00:02:33,450 --> 00:02:34,930
+برهان ذلك proof
+
+16
+00:02:42,420 --> 00:02:50,400
+fix C تنتمي إلى R then
+
+17
+00:02:50,400 --> 00:02:57,220
+فما احنا عارفين ان ال real
+
+18
+00:02:57,220 --> 00:03:00,980
+numbers عبارة عن ال union disjoint union لل
+
+19
+00:03:00,980 --> 00:03:03,320
+rational numbers وال irrational numbers
+
+20
+00:03:09,680 --> 00:03:22,500
+then X أو C تنتمي إلى Q أو C تنتمي إلى R minus Q
+
+21
+00:03:22,500 --> 00:03:25,680
+إذا الـ C هذا إما هتكون rational number أو
+
+22
+00:03:25,680 --> 00:03:32,160
+irrational number الحالة الأولى case one لو كانت C
+
+23
+00:03:32,160 --> 00:03:33,180
+rational number
+
+24
+00:03:37,870 --> 00:03:49,730
+استخدم الـ Corollary to Density
+
+25
+00:03:49,730 --> 00:03:58,310
+Theorem لتختار
+
+26
+00:03:58,310 --> 00:04:01,830
+سيكوينس
+
+27
+00:04:01,830 --> 00:04:06,630
+XN من أعداد غير عقلية
+
+28
+00:04:19,740 --> 00:04:23,560
+اعتقد ان احنا اثبتنا حاجة زي هذه سابقا
+
+29
+00:04:28,770 --> 00:04:33,750
+حتى لو كان rational فby ال corollary النتيجة تبع
+
+30
+00:04:33,750 --> 00:04:38,630
+ال density theorem ممكن اثبات ان يوجد sequence of
+
+31
+00:04:38,630 --> 00:04:46,150
+irrationals ونهيتها العدد cلأ هذا حسب النتيجة تبع
+
+32
+00:04:46,150 --> 00:04:50,250
+ال density term لأن النتيجة هذه بتقول between any
+
+33
+00:04:50,250 --> 00:04:54,030
+two real numbers there is irrational او اي open
+
+34
+00:04:54,030 --> 00:04:58,990
+interval contains an irrational number فشوفنا احنا
+
+35
+00:04:58,990 --> 00:05:06,210
+البرهان بالتفصيل وبالتالي so نلاحظ
+
+36
+00:05:06,210 --> 00:05:14,530
+انه ال limitل .. ال .. او ال .. ال function قيمة
+
+37
+00:05:14,530 --> 00:05:21,230
+ال function f عند xn بيساوي ال xn is irrational و
+
+38
+00:05:21,230 --> 00:05:24,370
+من ال definition تبع ال function f عند اي
+
+39
+00:05:24,370 --> 00:05:30,310
+irrational بيساوي سفر هذا صحيح لكل n هذا بيقدي ان
+
+40
+00:05:30,310 --> 00:05:41,200
+ال limit ل f of xn as n tends to infinityبساوي
+
+41
+00:05:41,200 --> 00:05:44,140
+limit ثابت سفر بساوي سفر
+
+42
+00:05:49,060 --> 00:05:55,480
+وهذا لا يساوي واحد اللي هو f of c ال c rational
+
+43
+00:05:55,480 --> 00:06:00,360
+number ف f and c بيساوي واحد اذا هاي اندي انا
+
+44
+00:06:00,360 --> 00:06:05,940
+اثبتت ان يوجد sequence xn في المجال تبع الدولة
+
+45
+00:06:05,940 --> 00:06:11,340
+اللي هو R و ال sequence هذه converge ل c لكن ال
+
+46
+00:06:11,340 --> 00:06:16,660
+limit لل image لل sequence لا تساوي f of c اذا by
+
+47
+00:06:16,660 --> 00:06:22,490
+divergence criterionأو by discontinuity criterion
+
+48
+00:06:22,490 --> 00:06:28,450
+اذا by discontinuity
+
+49
+00:06:28,450 --> 00:06:32,690
+criterion if
+
+50
+00:06:32,690 --> 00:06:43,570
+is discontinuous is discontinuous at C في
+
+51
+00:06:43,570 --> 00:06:47,750
+الحالة اللي هي C rational number في الحالة التانية
+
+52
+00:06:52,270 --> 00:07:01,110
+لو كان الـ c irrational number فممكن
+
+53
+00:07:01,110 --> 00:07:05,330
+نستخدم ال density theorem نفسها use density
+
+54
+00:07:05,330 --> 00:07:09,130
+theorem to
+
+55
+00:07:09,130 --> 00:07:13,710
+choose a
+
+56
+00:07:13,710 --> 00:07:21,280
+sequence x in containedof rational numbers
+
+57
+00:07:21,280 --> 00:07:23,760
+contained in Q يعني ال sequence هذه عناصرها
+
+58
+00:07:23,760 --> 00:07:30,440
+rational numbers بحيث انه ال limit لل sequence x
+
+59
+00:07:30,440 --> 00:07:36,260
+in as n tends to infinity بساوي c هذا أثبتناه في
+
+60
+00:07:36,260 --> 00:07:42,480
+المحاضرة السابقة وبالتالي
+
+61
+00:07:42,480 --> 00:07:43,920
+so
+
+62
+00:07:46,980 --> 00:07:52,620
+أنا عندي بما أنه ال Xn rationals ف F of Xn بتساوي
+
+63
+00:07:52,620 --> 00:08:00,280
+واحد لكل N هذا بيقدي انه limit ل F of Xn as N
+
+64
+00:08:00,280 --> 00:08:04,950
+tends to infinityبساوي limit ثابت واحد بطلع واحد
+
+65
+00:08:04,950 --> 00:08:12,710
+واحد لا يساوي سفر اللي هو ال image ل ال C ال C هنا
+
+66
+00:08:12,710 --> 00:08:17,170
+irrational و من ال definition of the function if
+
+67
+00:08:17,170 --> 00:08:21,730
+and irrational بساوي سفر اذا انا هنا اثبتت ان يوجد
+
+68
+00:08:21,730 --> 00:08:23,610
+sequence
+
+69
+00:08:25,710 --> 00:08:32,350
+في مجال الدالة نهايتها c لكن نهاية صورتها لا تساوي
+
+70
+00:08:32,350 --> 00:08:41,750
+صورة ال c كمان مرة hence by
+
+71
+00:08:41,750 --> 00:08:47,790
+discontinuity by
+
+72
+00:08:47,790 --> 00:08:50,150
+discontinuity criterion
+
+73
+00:08:52,890 --> 00:09:03,530
+الـ function f is discontinuous at c لأن بما أن c
+
+74
+00:09:03,530 --> 00:09:09,410
+was arbitrary فالـ function is discontinuous عن كل
+
+75
+00:09:09,410 --> 00:09:15,050
+real number c طبعا في الحالتين لأن هذا بكمل
+
+76
+00:09:15,050 --> 00:09:22,090
+البرهانة لأن هذه ال function هي function من R لRو
+
+77
+00:09:22,090 --> 00:09:26,950
+discontinuous ليست متصلة عند أي x فارق هذه ال
+
+78
+00:09:26,950 --> 00:09:31,030
+function لها اسم ومشهورة ومعروفة اسمها Dirichlet
+
+79
+00:09:31,030 --> 00:09:35,210
+function اذا
+
+80
+00:09:35,210 --> 00:09:41,110
+ال function هذه remark
+
+81
+00:09:41,110 --> 00:09:46,870
+the
+
+82
+00:09:46,870 --> 00:09:49,610
+above function
+
+83
+00:09:55,060 --> 00:10:03,000
+function is well-known as
+
+84
+00:10:03,000 --> 00:10:10,040
+Dirichlet Dirichlet
+
+85
+00:10:10,040 --> 00:10:17,540
+دا عالم رياضيات Dirichlet is function او Dirichlet
+
+86
+00:10:17,540 --> 00:10:20,000
+is discontinuous function
+
+87
+00:10:25,450 --> 00:10:34,010
+this continuous function ده
+
+88
+00:10:34,010 --> 00:10:41,070
+اللي مهمة ويلها meta في ال real analysis okay تمام
+
+89
+00:10:41,070 --> 00:10:47,630
+طيب ناخد بعض الملاحظات واضح في أي سؤال واضح هنا
+
+90
+00:10:47,630 --> 00:10:49,750
+البرهن في أي سلسلة
+
+91
+00:11:06,590 --> 00:11:15,470
+طيب ناخد شوية remarks ال
+
+92
+00:11:15,470 --> 00:11:19,410
+remark
+
+93
+00:11:19,410 --> 00:11:27,290
+الأولى sometimes a
+
+94
+00:11:27,290 --> 00:11:34,590
+function خلينا احنا نرجع لل definition تبع الاتصال
+
+95
+00:11:34,590 --> 00:11:35,770
+عن النقطة definition
+
+96
+00:11:56,710 --> 00:12:04,230
+الشرط التاني ان ال limitلأ F of X لما X تقول إلى C
+
+97
+00:12:04,230 --> 00:12:11,490
+exist و بساوي F of C اللي هو الشرط هذا هذا الشرط
+
+98
+00:12:11,490 --> 00:12:16,730
+سمنها تلاتة في واحد هذا يتضمن تلات شروط ال limit
+
+99
+00:12:16,730 --> 00:12:22,690
+and C exist F is defined at C و اتنين متساوين تلات
+
+100
+00:12:22,690 --> 00:12:28,970
+شروطالان لو اي واحد من التلات الشروط هدول اختل فال
+
+101
+00:12:28,970 --> 00:12:32,210
+function بتكونش continuous and النقطة c ف
+
+102
+00:12:32,210 --> 00:12:42,490
+sometimes the function can be discontinuous
+
+103
+00:12:42,490 --> 00:12:46,970
+at x بساوي c because
+
+104
+00:12:49,450 --> 00:13:06,610
+if is undefined is undefined at C لكن
+
+105
+00:13:06,610 --> 00:13:10,750
+أحيانا أخرى وفي الحالة اللي بقدرش أعمل حاجة بقدرش
+
+106
+00:13:10,750 --> 00:13:17,030
+أعمل حاجة however
+
+107
+00:13:20,670 --> 00:13:28,290
+لو كانت if ال limit لل function f of x as x tends
+
+108
+00:13:28,290 --> 00:13:39,010
+to c موجودة exists exists
+
+109
+00:13:39,010 --> 00:13:46,490
+and equals L ينتمي ل R ففي
+
+110
+00:13:46,490 --> 00:13:49,510
+الحالة هذه we can
+
+111
+00:13:53,080 --> 00:13:58,780
+we can define we
+
+112
+00:13:58,780 --> 00:14:09,440
+can define a function capital F من a اتحاد ال
+
+113
+00:14:09,440 --> 00:14:20,800
+singleton set C إلى R by capital F of X بساوي
+
+114
+00:14:20,800 --> 00:14:21,980
+العدد L
+
+115
+00:14:24,670 --> 00:14:32,970
+fx بتساوي c و بتساوي ال function f of x إذا كان ال
+
+116
+00:14:32,970 --> 00:14:44,530
+x ينتمي إلى a ولا يساوي c in this case in this
+
+117
+00:14:44,530 --> 00:14:54,830
+case the function capital F is continuousat x
+
+118
+00:14:54,830 --> 00:15:00,530
+بساوي c indeed
+
+119
+00:15:00,530 --> 00:15:07,490
+في حقيقة القمر indeed في حقيقة القمر تعالى نشوف
+
+120
+00:15:07,490 --> 00:15:14,630
+high limit capital f of x as x tends to c بساوي x
+
+121
+00:15:14,630 --> 00:15:19,510
+تقول ل c إذا ال x بالتأكيد بتسويش c لما x ما
+
+122
+00:15:19,510 --> 00:15:26,390
+بتسويش c يعني x and time ل aفcapital F هي نفسها
+
+123
+00:15:26,390 --> 00:15:35,430
+حسب تعريفها هي small f طب احنا فرضين ان ال limit ل
+
+124
+00:15:35,430 --> 00:15:41,150
+F of X exist و بساوي and C و بساوي L، إذن هذه تطلع
+
+125
+00:15:41,150 --> 00:15:43,070
+موجودة و بساوي L
+
+126
+00:15:46,940 --> 00:15:50,740
+و احنا من ال definition تبع ال function ال ال
+
+127
+00:15:50,740 --> 00:15:53,800
+ماخدينها هي عبارة عن قيمة ال function capital F
+
+128
+00:15:53,800 --> 00:16:00,180
+and C اذا هاي شرط الاتصال للدالة capital F and C
+
+129
+00:16:00,180 --> 00:16:10,320
+متحقق وبالتالي اذا capital F is continuous at C ال
+
+130
+00:16:10,320 --> 00:16:20,300
+function capital F the functioncapital F is called
+
+131
+00:16:20,300 --> 00:16:30,700
+a continuous extension
+
+132
+00:16:30,700 --> 00:16:36,980
+of
+
+133
+00:16:36,980 --> 00:16:43,900
+small f عبارة عن continuous extension يعني توسعة
+
+134
+00:16:43,900 --> 00:16:49,530
+متصلةتوسعة متصلة لدالة small f لحظة انتوا هنا ان
+
+135
+00:16:49,530 --> 00:16:59,350
+الدالة capital F الدالة
+
+136
+00:16:59,350 --> 00:17:05,550
+capital F function من A union single to C إلى R
+
+137
+00:17:05,550 --> 00:17:11,810
+طبعا هنا C لا تمتم إلى A وعندي small f هي function
+
+138
+00:17:11,810 --> 00:17:19,850
+من A إلى Rفلاحظوا ان ال function capital F لما
+
+139
+00:17:19,850 --> 00:17:28,250
+اعمل restriction لل domain تبعها على a فقط فهي نفس
+
+140
+00:17:28,250 --> 00:17:35,490
+f يعني لما اقيد او احصر الدولة capital F على
+
+141
+00:17:35,490 --> 00:17:41,610
+المجموعة a فقط فهي نفس ال F هي لكل x a capital F
+
+142
+00:17:41,610 --> 00:17:47,020
+هي small fالزيادة ان capital F معرفة عن C و F مش
+
+143
+00:17:47,020 --> 00:17:51,720
+معرفة عن C فهذه الدالة capital F بنسميها توسعة على
+
+144
+00:17:51,720 --> 00:17:56,120
+small f التوسعة هذه متصلة عند النقطة C
+
+145
+00:17:59,220 --> 00:18:03,740
+تمام؟ إذا هذه أول ملاحظة إذا لو كانت الدالة مش
+
+146
+00:18:03,740 --> 00:18:10,580
+معرفة لو كانت small f مش معرفة عن c لكن نهايتها عن
+
+147
+00:18:10,580 --> 00:18:15,580
+c موجودة وبالساوي عدد L فبقدر أعرف دالة جديدة
+
+148
+00:18:15,580 --> 00:18:20,080
+capital F اللي هي توسعة extension ل small f و ال
+
+149
+00:18:20,080 --> 00:18:24,640
+extension الدالة هذه اللي هي توسعةبتكون متصلة عند
+
+150
+00:18:24,640 --> 00:18:28,920
+الـ C اننا ناخد قيمتها عند الـ C بساوي قيمة ال
+
+151
+00:18:28,920 --> 00:18:36,540
+limit و عند النقاط ال A هي نفس small if لكن
+
+152
+00:18:36,540 --> 00:18:39,620
+الملاحظة التانية
+
+153
+00:18:51,350 --> 00:18:55,490
+لو كانت الادالة .. ممكن تكون الادالة معرفة عن C
+
+154
+00:18:55,490 --> 00:19:03,670
+يعني هنا sometimes a
+
+155
+00:19:03,670 --> 00:19:09,670
+function g is discontinuous
+
+156
+00:19:09,670 --> 00:19:16,190
+a function g from A to R is discontinuous at C
+
+157
+00:19:19,590 --> 00:19:25,970
+بسبب و ال c ممكن تكون موجودة في a او حتى لو مش
+
+158
+00:19:25,970 --> 00:19:35,630
+موجودة في a is discontinuous at c because ال limit
+
+159
+00:19:35,630 --> 00:19:43,210
+ل g of x as x tends to c does not exist يعني ممكن
+
+160
+00:19:43,210 --> 00:19:48,330
+ال g تكون معرفة and ال c لكنالنهايتها عند الـ c مش
+
+161
+00:19:48,330 --> 00:19:51,690
+موجودة فطبعا في الحالة دي ال function بتكون
+
+162
+00:19:51,690 --> 00:19:57,830
+discontinuous at c وفي الحالة دي ماقدرش اعرف in
+
+163
+00:19:57,830 --> 00:20:08,410
+this case in this case we can't لا نستطيع we can't
+
+164
+00:20:08,410 --> 00:20:14,990
+define a continuous extension
+
+165
+00:20:23,400 --> 00:20:33,900
+ممكن مانقدرش نعرف continuous extension لال ..
+
+166
+00:20:33,900 --> 00:20:42,000
+اللي هو capital G from A union singleton C إلى R
+
+167
+00:20:42,000 --> 00:20:45,820
+أو
+
+168
+00:20:45,820 --> 00:20:52,800
+we cannot define an extensionand extension g من a
+
+169
+00:20:52,800 --> 00:21:02,520
+union c لr by g of x بساوي عدد
+
+170
+00:21:02,520 --> 00:21:11,400
+capital c if x بساوي c و بساوي g of x إذا كان x
+
+171
+00:21:11,400 --> 00:21:18,500
+ينتمي and
+
+172
+00:21:20,360 --> 00:21:25,060
+جي بي continuous at c
+
+173
+00:21:39,530 --> 00:21:44,470
+طبعا هنا الـ G باخدها عند C بيساوي capital C هاد
+
+174
+00:21:44,470 --> 00:21:50,550
+عيرنا ال number ع شوية و عند X بيساوي A لازم تساوي
+
+175
+00:21:50,550 --> 00:21:55,970
+G of X عشان تكون توسعة أو extension ل small g ف to
+
+176
+00:21:55,970 --> 00:22:05,990
+see this لبرهان ذلك to see this to see that such
+
+177
+00:22:05,990 --> 00:22:18,380
+Gcan't be continuous at c assume خلّيني
+
+178
+00:22:18,380 --> 00:22:25,200
+أعمل برهان بالتناقض assume on contrary assume on
+
+179
+00:22:25,200 --> 00:22:33,900
+contrary أن counter G is continuous at c then هذا
+
+180
+00:22:33,900 --> 00:22:40,660
+معناه أن ال limitلـ capital G of X as X tends to C
+
+181
+00:22:40,660 --> 00:22:48,420
+بساوي capital G اللي
+
+182
+00:22:48,420 --> 00:22:56,380
+هي بساوي limit بتطلع
+
+183
+00:22:56,380 --> 00:23:06,900
+بساوي capital G of C بساوي capital Cو هذه بتساوي
+
+184
+00:23:06,900 --> 00:23:11,700
+limit g of x لما x تقول ل c هي عبارة عن limit من
+
+185
+00:23:11,700 --> 00:23:18,320
+تعريف ال g limit g of x لما x تقول ل c لإن لما x
+
+186
+00:23:18,320 --> 00:23:22,960
+تقول ل c x بستويش ال c x بستويش ال c معناته x
+
+187
+00:23:22,960 --> 00:23:29,600
+تنتمي ل a لإن g of x هي عبارة عن small g of x لإن
+
+188
+00:23:29,600 --> 00:23:32,260
+في الحالة هذه limit
+
+189
+00:23:34,430 --> 00:23:40,110
+إذا limit small g of x لما x تقولها c exist اه إذا
+
+190
+00:23:40,110 --> 00:23:46,110
+هذا بتطلع exist and equals c هي بالساوية c
+
+191
+00:23:46,110 --> 00:23:51,370
+contradiction هذا تناقض لإن احنا فرضين إن ال limit
+
+192
+00:23:51,370 --> 00:23:53,970
+ل g of x and c مش موجودة
+
+193
+00:23:58,090 --> 00:24:02,390
+لما تكون الدالة مش متصلة على النقطة لعدم وجود
+
+194
+00:24:02,390 --> 00:24:06,830
+نهايتها عند النقطة فماقدرش أعرف continuous
+
+195
+00:24:06,830 --> 00:24:15,610
+extension لدالة and النقطة Cلأن هنا فرضنا أن هناك
+
+196
+00:24:15,610 --> 00:24:19,030
+extension هذا ال extension مش ممكن يكون continuous
+
+197
+00:24:19,030 --> 00:24:23,050
+عند النقطة C لأنه لو كان continuous عند النقطة C
+
+198
+00:24:23,050 --> 00:24:27,750
+سيقدر ان الملمت الدالة small g and c exist وهذا
+
+199
+00:24:27,750 --> 00:24:34,830
+يتناقض مع الفرض تمام، هذا النوع من ال
+
+200
+00:24:34,830 --> 00:24:40,570
+discontinuity لما تكون الدالة discontinuous لعدم
+
+201
+00:24:42,160 --> 00:24:43,920
+لما تكون الدالة discontinuous
+
+202
+00:24:46,760 --> 00:24:51,500
+السبب في ذلك أنها مش معرفة عند النقطة C لكن نهيتها
+
+203
+00:24:51,500 --> 00:24:55,560
+موجودة عند الـ C فهذا النوع من ال discontinuity من
+
+204
+00:24:55,560 --> 00:25:00,360
+عدم الاتصال بنسميه removable يعني ممكن إزالته أو
+
+205
+00:25:00,360 --> 00:25:05,300
+التخلص منه وهي فعلا اتخلصنا من عدم الاتصال للدالة
+
+206
+00:25:05,300 --> 00:25:09,480
+small f and c بتعريف دالة capital F بالطريقة هذه
+
+207
+00:25:09,480 --> 00:25:12,670
+وشوفنا أن الدالة الجديدة continuous عند ال Cإذا
+
+208
+00:25:12,670 --> 00:25:16,190
+هذا النوع من عدم الاتصال أو ال discontinuity is
+
+209
+00:25:16,190 --> 00:25:19,990
+called removable يمكن إزالته يمكن تخلص منه أما إذا
+
+210
+00:25:19,990 --> 00:25:25,610
+كانت الدالة discontinuous عند النقطة C لأن نهايتها
+
+211
+00:25:25,610 --> 00:25:30,050
+in C مش موجودة فهذا النوع من عدم الاتصال بنسميه
+
+212
+00:25:30,050 --> 00:25:34,950
+essential يعني أساسي لا يمكن تخلص منه زي ما شفنا
+
+213
+00:25:34,950 --> 00:25:38,890
+في التحليل التحت okay تمامإن هذه أنواع ال
+
+214
+00:25:38,890 --> 00:25:42,570
+discontinuity ال discontinuity أو عدم الاتصال
+
+215
+00:25:42,570 --> 00:25:47,030
+نوعين نوع removable ممكن يزالته ممكن تخلص منه و
+
+216
+00:25:47,030 --> 00:25:55,210
+نوع تاني essential أساسي لايمكن تخلص منه ممكن
+
+217
+00:25:55,210 --> 00:25:58,090
+ناخد بعض الأمثلة على ذلك
+
+218
+00:26:14,950 --> 00:26:21,050
+نأخد الـ function consider الـ
+
+219
+00:26:21,050 --> 00:26:28,750
+function g of x بتساوي sin واحد على x حيث x لا
+
+220
+00:26:28,750 --> 00:26:31,970
+يساوي سفر طبعا احنا شفنا
+
+221
+00:26:35,380 --> 00:26:41,220
+في مثال صادق انه limit g of x as x tends to zero
+
+222
+00:26:41,220 --> 00:26:46,880
+does not exist limit
+
+223
+00:26:46,880 --> 00:26:57,520
+الدالة هذه غير موجودة وبالتالي
+
+224
+00:26:57,520 --> 00:27:04,540
+كمان برضهوكمان الدالة برضه جي عند سفر مش معرفة اذا
+
+225
+00:27:04,540 --> 00:27:17,220
+.. اذا in this case we can't .. we can't
+
+226
+00:27:17,220 --> 00:27:28,580
+.. we can't define a continuous extension
+
+227
+00:27:30,680 --> 00:27:41,180
+of small g at السفر هذا النوع التاني من ال
+
+228
+00:27:41,180 --> 00:27:47,520
+discontinuity لكن لو أخدت دالة زي هذه
+
+229
+00:27:59,300 --> 00:28:11,080
+لو أخدت f of x بساوي x في ال sign واحد على x فطبعا
+
+230
+00:28:11,080 --> 00:28:16,220
+هنا و x لا يساوي سفر فطبعا واضح أن f عند السفر is
+
+231
+00:28:16,220 --> 00:28:17,240
+undefined
+
+232
+00:28:20,560 --> 00:28:26,040
+الـ limit شوفنا أنه limit ل f of x لما x تقول ل 0
+
+233
+00:28:26,040 --> 00:28:30,520
+by squeeze theorem أثبتنا باستخدام squeeze theorem
+
+234
+00:28:30,520 --> 00:28:38,100
+أنه limit هذه بيساوي 0 exist بساوي 0 طبعا إذا هنا
+
+235
+00:28:38,100 --> 00:28:47,020
+f is discontinuous discontinuous at x بساوي 0 لإن
+
+236
+00:28:47,020 --> 00:28:51,920
+أنا مش معرف عند السفرلكن بما ان ال limit تبقى عند
+
+237
+00:28:51,920 --> 00:28:58,320
+السفر موجودة we can define
+
+238
+00:28:58,320 --> 00:29:07,320
+a continuous extension of
+
+239
+00:29:07,320 --> 00:29:11,220
+f at سفر as follows
+
+240
+00:29:14,800 --> 00:29:21,980
+فعندى capital F of X بنعرفها على أنها بالساوى
+
+241
+00:29:21,980 --> 00:29:26,040
+السفر اللى هو limit لل function عند السفر إذا كان
+
+242
+00:29:26,040 --> 00:29:34,960
+ال X بساوى السفر و بالساوى small f of X إذا كان X
+
+243
+00:29:34,960 --> 00:29:42,260
+تنتمي ل domain الدالة اللى هواللي هو ر مع ده سفر ر
+
+244
+00:29:42,260 --> 00:29:48,760
+مع ده سفر مش هذا هو ال domain تبع الدالة الانف
+
+245
+00:29:48,760 --> 00:29:58,000
+دالة if now you can verify انه
+
+246
+00:29:58,000 --> 00:30:05,040
+capital F is continuous is continuous at x بساوي
+
+247
+00:30:05,040 --> 00:30:12,580
+سفربينما small f is not continuous از سفر حيا
+
+248
+00:30:12,580 --> 00:30:17,220
+عندي limit capital
+
+249
+00:30:17,220 --> 00:30:23,560
+F of X as X tends to zero بساوي لما X ساوي للسفر X
+
+250
+00:30:23,560 --> 00:30:29,800
+بسويش سفر لما X ماتساويش سفر فcapital F هي small f
+
+251
+00:30:32,400 --> 00:30:37,700
+و limit small f بتساوي سفر اللي هي capital F معرفة
+
+252
+00:30:37,700 --> 00:30:43,620
+عند السفر okay إذا هي شرط الاتصال عند السفر متحقق
+
+253
+00:30:43,620 --> 00:30:48,740
+وبالتالي capital F متصلة عند السفر okay تمام هفهم
+
+254
+00:30:48,740 --> 00:30:54,740
+okay بنوقف هنا و بنتيح الآن المجال إلكم إذا كان في
+
+255
+00:30:54,740 --> 00:30:58,920
+عندكم أي أسئلة بخصوص ال homework أو الامتحان
+
+256
+00:30:58,920 --> 00:31:03,560
+النصفي اللي هناخده بكرا إن شاء اللهفي عندكم أي
+
+257
+00:31:03,560 --> 00:31:11,680
+سؤال أو استفسار؟ في حد عنده أي سؤال؟
+
+258
+00:31:11,680 --> 00:31:24,080
+في
+
+259
+00:31:24,080 --> 00:31:29,970
+أي سؤال؟ أي سؤال؟دكتور انا ممكن استخدم limit ال K
+
+260
+00:31:29,970 --> 00:31:33,830
+لما ال X تقول ال C و سوى K و limit ال X لما ال X
+
+261
+00:31:33,830 --> 00:31:38,490
+تقول ال C و سوى C ممكن استخدمهم انا بحثت صح صح لان
+
+262
+00:31:38,490 --> 00:31:42,490
+هذه صارت حاجات ال trivial و يعني اثبتناها الا ده
+
+263
+00:31:42,490 --> 00:31:47,150
+طول منك اثبتها باستخدام تعريف epsilon دلت اكيد اي
+
+264
+00:31:47,150 --> 00:31:53,810
+سؤال بدك ممكن التاني التاني اللي مش محلول اه اللي
+
+265
+00:31:53,810 --> 00:31:54,830
+مش محلول نعم
+
+266
+00:32:04,200 --> 00:32:13,180
+طيب suppose question suppose
+
+267
+00:32:13,180 --> 00:32:25,460
+أن xn أكبر من أو ساوى سفر لكل n في n and .. and ال
+
+268
+00:32:25,460 --> 00:32:25,820
+limit
+
+269
+00:32:28,550 --> 00:32:34,910
+لسالب واحد أُس n في xn ال sequence هذه لما n تقول
+
+270
+00:32:34,910 --> 00:32:46,590
+ل infinity بساوي x ينتمي إلى r show أن ال limit ل
+
+271
+00:32:46,590 --> 00:32:56,630
+xn لما n تقول ل infinity بساوي سفر فضلي
+
+272
+00:33:21,960 --> 00:33:28,720
+فنشوف الان انا عندي ال limit لل sequence للحد
+
+273
+00:33:28,720 --> 00:33:33,900
+العام تبعها سالب واحد قص ان في x in لما ان تقول
+
+274
+00:33:33,900 --> 00:33:44,020
+infinity بساوي x اذا ان andالسالب واحد قصة اتنين N
+
+275
+00:33:44,020 --> 00:33:52,460
+في X اتنين N هذه عبارة عن subsequence subsequence
+
+276
+00:33:52,460 --> 00:33:58,540
+of السيكوانس الحد العام تبعها السالب واحد قصة N X
+
+277
+00:33:58,540 --> 00:34:07,730
+X N هذه subsequence خط الحدود الزوجية صح؟طيب ال
+
+278
+00:34:07,730 --> 00:34:12,950
+subsequence هذه هي الحد العام تبعها x اتنين n
+
+279
+00:34:12,950 --> 00:34:16,890
+مفروض converge اخلنا احنا في نظرية ان كانت ال
+
+280
+00:34:16,890 --> 00:34:21,350
+sequence convergent ل x فأي subsequence منها بتكون
+
+281
+00:34:21,350 --> 00:34:27,810
+convergent لنفس ال x تمام؟ في نفس الوجد
+
+282
+00:34:38,530 --> 00:34:44,370
+في نفس الوقت ال sequence سالب واحد قصة اتنين in
+
+283
+00:34:44,370 --> 00:34:48,750
+سالب واحد في
+
+284
+00:34:48,750 --> 00:34:56,810
+x اتنين in سالب واحد هذا عبارة عن sub sequence من
+
+285
+00:34:56,810 --> 00:35:02,070
+ال sequence الأصلي المعطاة اللي هي الحد العام
+
+286
+00:35:02,070 --> 00:35:07,020
+تبعها سالب واحد to int x inيعني انا اخدت هنا
+
+287
+00:35:07,020 --> 00:35:10,680
+الحدود الفردية من الـ sequence هذه طبعا هذه اكيد
+
+288
+00:35:10,680 --> 00:35:17,600
+subsequence فالحد العام هذا عبارة عن سالب X اتنين
+
+289
+00:35:17,600 --> 00:35:23,140
+M سالب واحد الآن ال subsequence هذه المفروض انها
+
+290
+00:35:23,140 --> 00:35:28,180
+converge الى X لان ال sequence الاصلية convergent
+
+291
+00:35:28,180 --> 00:35:34,560
+ل X تمام؟ اذا انا في عندي هنا limit
+
+292
+00:35:38,370 --> 00:35:46,990
+x اتنين n سالب واحد بساوي سالب x لأن limit سالب ال
+
+293
+00:35:46,990 --> 00:35:53,070
+sequence هذه بساوي x اذا اضربيها في سالب واحد طبعا
+
+294
+00:35:53,070 --> 00:36:02,030
+واندي من هنا من هناك limit x اتنين n لما n تقول
+
+295
+00:36:02,030 --> 00:36:05,810
+infinity بساوي
+
+296
+00:36:05,810 --> 00:36:06,190
+x
+
+297
+00:36:17,310 --> 00:36:28,710
+نجمعهم؟ لأ لأ منجمعوش الناس ما
+
+298
+00:36:28,710 --> 00:36:29,450
+بنجمعهم
+
+299
+00:36:37,410 --> 00:36:42,950
+ما بنجمع مش عاملة اتجمع الان في اندي انا لحظة انت
+
+300
+00:36:42,950 --> 00:36:50,530
+عندك من الفرض xn أكبر من أو سوى سفر لكل n في n
+
+301
+00:36:55,480 --> 00:37:03,280
+فبالتالي إذا x2n سالب واحد أكبر من أوسعه سفر لكل n
+
+302
+00:37:03,280 --> 00:37:15,820
+وكذلك x2n برضه أكبر من أوسعه سفر لكل n صح؟ أصبت؟
+
+303
+00:37:15,820 --> 00:37:22,240
+وبالتالي إذا ال limit ل x2n سالب واحد تطلع أكبر من
+
+304
+00:37:22,240 --> 00:37:30,430
+أوسعه سفرو ال limit ل x2n تطلع أكبر من أوي ساوي
+
+305
+00:37:30,430 --> 00:37:34,890
+سفر هذه نظرية أخدناها صح؟ أخدناها نظرية بتقول لو
+
+306
+00:37:34,890 --> 00:37:38,550
+كانت ال sequence كل حدوة ده غير سالبة و ال limit
+
+307
+00:37:38,550 --> 00:37:44,330
+تبعتها exist فال limit تبعتها تطلع غير سالبة صح؟
+
+308
+00:37:44,330 --> 00:37:50,750
+فال limit هنا هي حسبناها سالم x و ال limit هنا
+
+309
+00:37:50,750 --> 00:37:59,060
+طلعت xإذا أنا في عندي سالب X أكبر من أو ساوي سفر
+
+310
+00:37:59,060 --> 00:38:06,420
+and X أكبر من أو ساوي سفر هذا بيقدر ان X أصغر من
+
+311
+00:38:06,420 --> 00:38:13,680
+أو ساوي سفر and X أكبر من أو ساوي سفر هذا بيقدر ان
+
+312
+00:38:13,680 --> 00:38:15,040
+X بساوي سفر
+
+313
+00:38:19,470 --> 00:38:23,930
+مايعني x أصغر من 0 و أكبر من 0 يعني x أصغر من 0
+
+314
+00:38:23,930 --> 00:38:27,470
+العدد الوحيد اللي بتمتع بالخاصيتين هدول في نفس
+
+315
+00:38:27,470 --> 00:38:34,090
+الواجت هو 0 إذا هاني أثبتت أن ال limit ل ال
+
+316
+00:38:34,090 --> 00:38:37,550
+sequence
+
+317
+00:38:37,550 --> 00:38:42,870
+أثبتت
+
+318
+00:38:42,870 --> 00:38:44,170
+أن x أصغر من 0
+
+319
+00:38:47,480 --> 00:38:52,500
+طبعا احنا ما اثبتناش لحد تلان ان ال limit لل
+
+320
+00:38:52,500 --> 00:38:59,260
+sequence ان انا عندي .. اذا انا اصبح عندي لان ال
+
+321
+00:38:59,260 --> 00:39:05,920
+limit لل sequence سارب واحد أس ن في xn لما n تقول
+
+322
+00:39:05,920 --> 00:39:13,090
+ال infinity طلعت بالساوية سفر وبالتالي هذا بيقديفي
+
+323
+00:39:13,090 --> 00:39:25,710
+exercise أخدناه بيقول إذا كان if
+
+324
+00:39:25,710 --> 00:39:34,750
+limit xn as n tends to infinity بيساوي 7 then
+
+325
+00:39:34,750 --> 00:39:42,200
+limitabsolute xn as n tends to infinity بيساوي سفر
+
+326
+00:39:42,200 --> 00:39:48,200
+والعكس كمان فباستخدام ال exercise هذا هذا بيقدي
+
+327
+00:39:48,200 --> 00:39:54,600
+انه limit absolute سالب واحد اص ان في xn as n
+
+328
+00:39:54,600 --> 00:39:58,660
+tends to infinity بيساوي
+
+329
+00:39:58,660 --> 00:40:01,260
+سفر اللي هو بيساوي limit
+
+330
+00:40:11,310 --> 00:40:15,930
+أنا مش عارف أنا ليش رفضت اللي نجمع مش هو عبارة ال
+
+331
+00:40:15,930 --> 00:40:21,010
+XN هي عبارة عن حدود زوجية و فردية لو جمعناهم بدون
+
+332
+00:40:21,010 --> 00:40:22,370
+ال N مش sequence
+
+333
+00:40:30,350 --> 00:40:37,410
+هذا تفكير ضحل مع احترام طبعا لسؤالك هاي عندك انت
+
+334
+00:40:37,410 --> 00:40:46,250
+هاي
+
+335
+00:40:46,250 --> 00:40:54,950
+عندك sequence مثلا هاي ال sequence سالب واحد قصة
+
+336
+00:40:54,950 --> 00:40:55,710
+اتنين in
+
+337
+00:40:59,480 --> 00:41:07,080
+حدودها واحد واحد إلى آخرها صح؟ هزبوت؟ و سالب واحد
+
+338
+00:41:07,080 --> 00:41:15,380
+قصة اتنين in سالب واحد حدودها سالب واحد تمام؟
+
+339
+00:41:15,380 --> 00:41:23,480
+لما اجمعهم add اجمعيهم فهي ال sequence الأول ان هي
+
+340
+00:41:23,480 --> 00:41:30,690
+زائد ال sequence التانيةلما بجمعهم بجمع الحد الأول
+
+341
+00:41:30,690 --> 00:41:34,610
+على الأول التاني على التاني و هكذا صح؟ إيش هيطلع
+
+342
+00:41:34,610 --> 00:41:41,090
+عندك؟ صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+343
+00:41:41,090 --> 00:41:43,030
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+344
+00:41:43,030 --> 00:41:43,030
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+345
+00:41:43,030 --> 00:41:46,190
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+346
+00:41:46,190 --> 00:41:51,350
+صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر صفر
+
+347
+00:41:51,350 --> 00:41:53,750
+صفر صف
+
+348
+00:42:00,230 --> 00:42:07,010
+المشكلة عندك انك بتقول X sequence XN ممكن تعتبرها
+
+349
+00:42:07,010 --> 00:42:15,250
+ابارعا مجموعة sub sequence X2N زاد sub sequence
+
+350
+00:42:15,250 --> 00:42:24,310
+X2N-1 هذا غلط هذا غلط ليس صحيحوهي مثالي واضح خطأ
+
+351
+00:42:24,310 --> 00:42:30,270
+okay تمام؟ في أي أسئلة تانية لو سمحتوا؟ مين عندها
+
+352
+00:42:30,270 --> 00:42:44,050
+سؤال تاني؟ مال لديها سؤال؟ في عندكم أسئلة؟ تمام؟
+
+353
+00:42:44,050 --> 00:42:50,950
+مش مقتنعة أه؟ هد مثال هي قدامك أمامك افحص المثال
+
+354
+00:42:50,950 --> 00:42:56,510
+كويس هتقتنعيلو كلامك صح لكان هذا المجموع بساوي
+
+355
+00:42:56,510 --> 00:43:03,010
+سالب واحد اصلا صح ليش
+
+356
+00:43:03,010 --> 00:43:10,730
+اسئلة تانية عندكم سؤال تسعة
+
+357
+00:43:10,730 --> 00:43:14,890
+اربعة
+
+358
+00:43:14,890 --> 00:43:20,710
+واحد سؤال
+
+359
+00:43:20,710 --> 00:43:26,430
+تسعة الفرع ديهذا شبيه بالأفراد التانية، بديك يعني
+
+360
+00:43:26,430 --> 00:43:30,570
+تحاول .. انا وصلت ان ال absolute ل 2x نفس الواحد
+
+361
+00:43:30,570 --> 00:43:33,950
+على 2 absolute ال x زائد الواحد، لإن كل ال answer
+
+362
+00:43:33,950 --> 00:43:37,190
+اللي فاتت كان مايكونش .. مايكونش إيش في ال bust
+
+363
+00:43:37,190 --> 00:43:42,950
+يعني، انا أجيب علاقة ال x زائد الواحد يعني delta
+
+364
+00:43:42,950 --> 00:43:45,650
+في النهاية هتكون هي ال minimum ل .. ل .. لقيم
+
+365
+00:43:45,650 --> 00:43:47,070
+تانية لقيم تانية، لإن مثلا انا مش عارفة ..
+
+366
+00:43:47,070 --> 00:43:52,290
+ماطلعتيش؟ انا مش عارفة القيمة التانية، يعني عرفت
+
+367
+00:43:52,290 --> 00:43:53,010
+القيمة الأولى
+
+368
+00:44:00,180 --> 00:44:05,940
+نعم طيب عشان بس ال .. ال .. الوقت يعني انتهى خلينا
+
+369
+00:44:05,940 --> 00:44:13,320
+نقول انه يعني نواجف هنا و هيك المحاضرة بتكون انتهت
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/JXFFuyzuuqA_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/JXFFuyzuuqA_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..24224e19f3e49a0a0f7c1485abf48966b370298c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/JXFFuyzuuqA_postprocess.srt
@@ -0,0 +1,1860 @@
+1
+00:00:20,220 --> 00:00:25,360
+بسم الله الرحمن الرحيم هندرس اليوم ان شاء الله مع
+
+2
+00:00:25,360 --> 00:00:32,000
+بعض ال section خمسة أربعةاللي بيتحدث عن موضوع ال
+
+3
+00:00:32,000 --> 00:00:36,720
+uniform continuity أو الاتصال المنظم للدوال
+
+4
+00:00:36,720 --> 00:00:40,600
+هحنحاول ناخد أكبر جزء ممكن من ال section الجزء
+
+5
+00:00:40,600 --> 00:00:44,860
+المتبقي ممكن نكمله في المحاضرة الجاية يوم الأتنين
+
+6
+00:00:44,860 --> 00:00:49,820
+فال
+
+7
+00:00:49,820 --> 00:00:54,540
+.. خلّينا الأول نراجع .. نراجع تعريف الاتصال
+
+8
+00:00:54,540 --> 00:00:59,270
+العاديال continuity على مجموعة فلو كان في handy
+
+9
+00:00:59,270 --> 00:01:04,170
+function f من a ل r فالعبارات التالية بتكون
+
+10
+00:01:04,170 --> 00:01:13,410
+متكافئة if is continuous at at
+
+11
+00:01:13,410 --> 00:01:20,810
+every at every u ينتمي إلى a اللي هو مجال الدالة
+
+12
+00:01:20,810 --> 00:01:24,370
+العبارة التانية given
+
+13
+00:01:27,500 --> 00:01:36,300
+epsilon أكبر من السفر and given u ينتمي إلى a يوجد
+
+14
+00:01:36,300 --> 00:01:41,160
+.. بيقدر نلاقي delta و ال delta هذه تعتمد على ال
+
+15
+00:01:41,160 --> 00:01:51,590
+epsilon و على ال u عدد موجببحيث أنه لكل x ينتمي
+
+16
+00:01:51,590 --> 00:01:59,250
+إلى a و absolute x minus u أصغر من delta فهذا
+
+17
+00:01:59,250 --> 00:02:07,830
+بتضمن إلى absolute f of x minus f of u أصغر من
+
+18
+00:02:07,830 --> 00:02:08,310
+epsilon
+
+19
+00:02:19,690 --> 00:02:30,650
+خلّينا بس ناخد المثال التالي consider
+
+20
+00:02:30,650 --> 00:02:41,910
+ال function f of xبتساوي واحد على X و X ينتبه لايه
+
+21
+00:02:41,910 --> 00:02:45,890
+اللي هي الفترة
+
+22
+00:02:45,890 --> 00:02:56,270
+كل ال X في R حيث X أكبر من الصفر إذا ال function F
+
+23
+00:02:56,270 --> 00:03:02,770
+معرفة على كل الأعداد الموجبة احنا
+
+24
+00:03:02,770 --> 00:03:05,770
+أثبتنا قبل هيك و proved
+
+25
+00:03:10,640 --> 00:03:14,920
+earlier فيما سبق في دراساتنا السابقة في section
+
+26
+00:03:14,920 --> 00:03:21,540
+اربعة خمسة ثلاثة او خمسة اتنين اثبتنا ان ال
+
+27
+00:03:21,540 --> 00:03:30,700
+function f is continuous على المجموعة a وخلنا
+
+28
+00:03:30,700 --> 00:03:36,580
+نراجع مع بعض ان مع بعض نراجع البرهان fix
+
+29
+00:03:39,080 --> 00:03:46,920
+fix u ينتمي إلى a given إبصر
+
+30
+00:03:46,920 --> 00:03:49,760
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+31
+00:03:49,760 --> 00:03:50,560
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+32
+00:03:50,560 --> 00:03:53,060
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+33
+00:03:53,060 --> 00:03:56,600
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+34
+00:03:56,600 --> 00:03:57,260
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+35
+00:03:57,260 --> 00:03:57,360
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+36
+00:03:57,360 --> 00:04:00,020
+أكبر من صفر أكبر من صفر أكبر من صفر أكبر من صفر
+
+37
+00:04:00,020 --> 00:04:06,790
+أكبر من صفر أكبر من صفربتطبيق تعريف epsilon delta
+
+38
+00:04:06,790 --> 00:04:12,110
+للاتصال ان نقطة given epsilon اذا بيطلع ارجعه we
+
+39
+00:04:12,110 --> 00:04:19,350
+found delta و ال delta هذه كانت ال minimum لقنتين
+
+40
+00:04:19,350 --> 00:04:24,470
+u ع اتنين او كانت هناك c ع اتنين بدل u كانت النقطة
+
+41
+00:04:24,470 --> 00:04:33,350
+بيسميها c فعندي u ع اتنين و u تربيه على اتنين في
+
+42
+00:04:33,350 --> 00:04:40,450
+epsilonطبعا هذا عدد موجب واضح ان ال delta هذه عدد
+
+43
+00:04:40,450 --> 00:04:44,530
+موجب لان هذا عدد موجب و هذا عدد موجب و بعدين ال
+
+44
+00:04:44,530 --> 00:04:50,530
+delta لاحظوا انها بتعتمد على ال epsilon و على ال U
+
+45
+00:04:52,480 --> 00:04:55,840
+الـ Delta بتعتمد على الـ Epsilon وعلى الـ U مش بس
+
+46
+00:04:55,840 --> 00:04:58,280
+على الـ Epsilon وعلى النقطة U اللى احنا بدنا نفحص
+
+47
+00:04:58,280 --> 00:05:05,020
+عندها الاتصال فشوفنا بعد هيك انه .. اذا for this
+
+48
+00:05:05,020 --> 00:05:11,880
+Delta اذا لو أخدنا X ينتمي إلى A و Absolute X
+
+49
+00:05:11,880 --> 00:05:19,560
+minus U أصغر من Delta فطبعا هذا قدهذا أدى أن الـ
+
+50
+00:05:19,560 --> 00:05:26,240
+delta هنا أصغر من أو يساوي U ع 2 وبالتالي هذا
+
+51
+00:05:26,240 --> 00:05:35,600
+بيقدر أن X أصغر من 3U ع 2 أكبر من U ع 2 لما نحل
+
+52
+00:05:35,600 --> 00:05:42,720
+المعادلة المتبينة هذه في U وهذا
+
+53
+00:05:42,720 --> 00:05:44,520
+بيقدر بدوره
+
+54
+00:05:46,640 --> 00:05:59,580
+أبسلوت f of x minus f of u طالع بيساوي أبسلوت واحد
+
+55
+00:05:59,580 --> 00:06:06,580
+على x minus واحد على u هذا بيساوي أبسلوت u minus x
+
+56
+00:06:06,580 --> 00:06:13,390
+على x في u المفروض أحط هنا أبسلوتأكس في U لكن ال X
+
+57
+00:06:13,390 --> 00:06:17,290
+و ال U عناصر في A و A عناصرها كل أعداد موجبة فلا
+
+58
+00:06:17,290 --> 00:06:21,950
+داعي ال absolute value الأن absolute أنا عندي هنا
+
+59
+00:06:21,950 --> 00:06:31,390
+من المتباينة هذه بيطلع عندي المفروض أنه أنا عندي
+
+60
+00:06:31,390 --> 00:06:43,100
+بيطلع U على 2 أصغر من X صح فهذا بيقدي أنه X فيأضرب
+
+61
+00:06:43,100 --> 00:06:47,420
+في U، U عدد موجب فبطلع U تربيع اتنين اصغر من X
+
+62
+00:06:47,420 --> 00:06:55,520
+وبالتالي واحد مقلوب XU بطلع اصغر من اتنين على U
+
+63
+00:06:55,520 --> 00:07:02,200
+تربيع اذا مقلوب XU اصغر من اتنين على U تربيع في
+
+64
+00:07:02,200 --> 00:07:08,790
+absolute U minus Xو هذي أصغر من دلتا إذاً هذي أصغر
+
+65
+00:07:08,790 --> 00:07:13,830
+من اتنين على U تربية في دلتا طيب الدلتا أنا
+
+66
+00:07:13,830 --> 00:07:18,390
+اختارها ال minimum للقيمة هذه وهذه فبالتالي الدلتا
+
+67
+00:07:18,390 --> 00:07:22,890
+هذه تطلع أصغر من أو ساوي القيمة التانيةإذن اتنين
+
+68
+00:07:22,890 --> 00:07:28,850
+على U تربية ضرب U تربية على اتنين في Epsilon و
+
+69
+00:07:28,850 --> 00:07:33,490
+طبعا هذولا بيروحوا مع بعض و بيظل Epsilon وبالتالي
+
+70
+00:07:33,490 --> 00:07:38,290
+بما أن Epsilon was arbitrary إذا ال F is
+
+71
+00:07:38,290 --> 00:07:48,110
+continuous at U ولمّا كانت U arbitrary since U
+
+72
+00:07:48,110 --> 00:07:49,770
+belonged to A was
+
+73
+00:07:52,720 --> 00:08:00,980
+arbitrary if is continuous على كل المجموعة ايه هذا
+
+74
+00:08:00,980 --> 00:08:05,740
+كان برهانة خلناها قبل هيك طيب ما الغرض مش ايش
+
+75
+00:08:05,740 --> 00:08:10,200
+النقطة ان احنا نعيد البرهان النقطة هي عايزين نفكز
+
+76
+00:08:10,200 --> 00:08:16,160
+او نأكد انه في اثبات الاتصال عند النقطة U لاحظنا
+
+77
+00:08:16,160 --> 00:08:20,330
+ان ال delta بتعتمد على ال epsilon و على ال Uهذا
+
+78
+00:08:20,330 --> 00:08:24,510
+معناه ان الـ delta بتتغير قيمتها مع تغير ال U
+
+79
+00:08:24,510 --> 00:08:28,070
+فمثلا
+
+80
+00:08:28,070 --> 00:08:40,890
+لو جينا نعمل هاي الدالة دي لو جينا رسمناها هاي
+
+81
+00:08:40,890 --> 00:08:47,730
+الدالة واحد على X لو جيت اخدت انا X لو كان هذا
+
+82
+00:08:47,730 --> 00:08:59,250
+واحد هذا اتنينفو هذا نص لو كانت ال U تبعتي لو كانت
+
+83
+00:08:59,250 --> 00:09:07,750
+ال U بساوي نص ف
+
+84
+00:09:07,750 --> 00:09:17,810
+F لنص بساوي هيطلع اتنين هذا بساوي F لنص طب لو جيت
+
+85
+00:09:17,810 --> 00:09:25,470
+أخدتأبسلون نيبرهود لاتنين اذا هذا عبارة عن بي
+
+86
+00:09:25,470 --> 00:09:32,310
+ابسلون لاتنين اللي هو صورة النص فهذا الابسلون
+
+87
+00:09:32,310 --> 00:09:38,130
+نيبرهود هيقابله delta
+
+88
+00:09:38,130 --> 00:09:43,350
+neighborhood هيقابله
+
+89
+00:09:43,350 --> 00:09:44,150
+delta
+
+90
+00:09:50,400 --> 00:09:59,440
+هذا عبارة عن delta neighborhood للنص باللاحظ هنا
+
+91
+00:09:59,440 --> 00:10:02,680
+ان ال delta هي قيمتها
+
+92
+00:10:20,550 --> 00:10:25,830
+هذه اتنين لو اخدت U بساوة اتنين لو اخدت U بساوة
+
+93
+00:10:25,830 --> 00:10:30,230
+اتنين احنا اثبتنا ان الدالة متصلة على الاتنين وهذه
+
+94
+00:10:30,230 --> 00:10:37,730
+ال function شكلها هيكون زي هيك يعني
+
+95
+00:10:37,730 --> 00:10:41,770
+هون ف F لتنين
+
+96
+00:10:44,810 --> 00:10:49,990
+بساوي نص او صورة اتنين بطلع نص اللي هي صورة
+
+97
+00:10:49,990 --> 00:10:54,470
+الاتنين الان لو انا اخدت كوانة epsilon
+
+98
+00:10:54,470 --> 00:11:01,750
+neighborhood لنقطة نص هذه ال epsilon هنا نفس قيمة
+
+99
+00:11:01,750 --> 00:11:06,890
+ال epsilon اللي هنا نفس القيمة وبالتالي الان اذا
+
+100
+00:11:06,890 --> 00:11:13,680
+في عندي انا دي epsilon لن نصفطبعاً لكل epsilon
+
+101
+00:11:13,680 --> 00:11:16,480
+neighborhood للنص بما أن الدلة متصلة عند اتنين
+
+102
+00:11:16,480 --> 00:11:22,480
+هيوجد V Delta يوجد
+
+103
+00:11:22,480 --> 00:11:28,800
+V Delta okay
+
+104
+00:11:28,800 --> 00:11:32,960
+هذا هيكون V Delta
+
+105
+00:11:39,350 --> 00:11:43,010
+هذا عبارة عن V Delta او Delta neighborhood للإفنين
+
+106
+00:11:43,010 --> 00:11:48,190
+فبلاحظ انه رغم ان ال epsilon هنا نفس قيمة ال
+
+107
+00:11:48,190 --> 00:11:52,890
+epsilon هنا الا ان ال delta هنا شوف جدش صغيرة
+
+108
+00:11:52,890 --> 00:12:00,400
+بينما ال delta هنا شايفين ما اكبرها؟تغيرت مين اللي
+
+109
+00:12:00,400 --> 00:12:05,220
+غير ال delta ال U لما ال U كانت نص ال delta كانت
+
+110
+00:12:05,220 --> 00:12:11,340
+صغيرة لما ال U كانت اتنين ال U كبرت اذا ال delta
+
+111
+00:12:11,340 --> 00:12:15,600
+هنا او ال delta نبرهود بيعتمد على ال epsilon او ال
+
+112
+00:12:15,600 --> 00:12:19,200
+delta بتعتمد على ال مش بس على ال epsilon و على ال
+
+113
+00:12:19,200 --> 00:12:23,840
+U و على النقطة نفسها okay واضح اذا هنا ال delta
+
+114
+00:12:23,840 --> 00:12:31,210
+تغيرت مع تغير ال UOkay تمام وبالتالي ال delta لأي
+
+115
+00:12:31,210 --> 00:12:34,470
+epsilon ال delta ده بتعتمد على ال u على ال epsilon
+
+116
+00:12:34,470 --> 00:12:39,410
+أو على النقطة وعلى ال epsilon تمام واضحة النقطة
+
+117
+00:12:39,410 --> 00:12:45,370
+هذه طيب احنا خلينا نقبل ناشية ده المثال خلينا ناخد
+
+118
+00:12:45,370 --> 00:12:54,770
+مثال تاني example
+
+119
+00:12:54,770 --> 00:12:56,210
+2
+
+120
+00:12:59,420 --> 00:13:09,840
+خلّينا ناخد الـ function f of x بساوي 2x و x ينتمي
+
+121
+00:13:09,840 --> 00:13:13,780
+إلى R Note
+
+122
+00:13:13,780 --> 00:13:20,620
+that .. خلّينا نلاحظ أول أن absolute f of x minus
+
+123
+00:13:20,620 --> 00:13:29,440
+f of uبساوي absolute اتنين X minus اتنين U بساوي
+
+124
+00:13:29,440 --> 00:13:38,420
+اتنين في absolute X minus U لكل X و U ينتمي ال R
+
+125
+00:13:38,420 --> 00:13:44,880
+مظبوط هيك؟ طيب
+
+126
+00:13:44,880 --> 00:13:51,760
+الدالة هذه معروفة انها متصلة على R المجال تبعها
+
+127
+00:13:51,760 --> 00:13:52,200
+صح؟
+
+128
+00:14:03,920 --> 00:14:13,000
+على الـ set R فكيف بنعمل فكس بنثبت U في R بنثبت أن
+
+129
+00:14:13,000 --> 00:14:22,180
+F متصل عند الـ U صح؟ and let أكبر من السفر be
+
+130
+00:14:22,180 --> 00:14:22,780
+given
+
+131
+00:14:28,810 --> 00:14:36,250
+تختار دلتا نختار دلتا بساوي أبسلون ع اتنين أكبر من
+
+132
+00:14:36,250 --> 00:14:45,010
+السفر فلهذه الدلتا then لو كان x ينتمي إلى ال a
+
+133
+00:14:45,010 --> 00:14:51,490
+اللي هي r و absolute x minus u أصغر من الدلتا فهذا
+
+134
+00:14:51,490 --> 00:14:58,840
+هيديني absolute f of x minus f of uبتقول إن هذا
+
+135
+00:14:58,840 --> 00:15:03,440
+بيطلع بساوية أصغر من أو ساوية اتنين في absolute x
+
+136
+00:15:03,440 --> 00:15:09,940
+minus u أو بساوية بالأعلى، صح؟ طيب مانا ال X هذه
+
+137
+00:15:09,940 --> 00:15:14,660
+ماخدها بحيث أن absolute x minus u أصغر من ال
+
+138
+00:15:14,660 --> 00:15:20,160
+delta، صح؟عشان ذلك انا اخترت delta بساوي epsilon ع
+
+139
+00:15:20,160 --> 00:15:24,500
+اتنين اه شوفت ايش خدنا delta بساوي epsilon ع اتنين
+
+140
+00:15:24,500 --> 00:15:30,740
+طيب و هذا بساوي epsilon حسب اختيارنا لل delta
+
+141
+00:15:30,740 --> 00:15:37,460
+وبالتالي هيك اذا ال function بما انه epsilon was
+
+142
+00:15:37,460 --> 00:15:44,800
+arbitrarily اذا f is continuousat الـ U وبما أن U
+
+143
+00:15:44,800 --> 00:15:48,060
+belong to R وزر فبتره إذا if continuous على كل ال
+
+144
+00:15:48,060 --> 00:15:55,240
+R كمان مرة النقطة هنا اللي عايزين أن أكد عليها هو
+
+145
+00:15:55,240 --> 00:16:01,520
+إن ال Delta لأي إبسلون و لأي U و لأي إبسلون ال
+
+146
+00:16:01,520 --> 00:16:06,160
+Delta هنا تعتمد على إبسلون فقط مالهاش دعوة في ال U
+
+147
+00:16:06,160 --> 00:16:11,790
+بمعنى آخر لو أنا ال U هذه غيرتهاأخذت U تانية لو
+
+148
+00:16:11,790 --> 00:16:14,670
+كانت تانية مثلا U بالساعة و سفر او واحد او اتنين
+
+149
+00:16:14,670 --> 00:16:19,310
+او تلاتة او اي عدد حقيقي فكل مرة ال delta نفس ال
+
+150
+00:16:19,310 --> 00:16:25,800
+delta F2 will work لل U لكل Uلأي إبسن خدي نفس ال
+
+151
+00:16:25,800 --> 00:16:28,340
+delta إبسن على اتنين هتعطيهم إيه ال implication
+
+152
+00:16:28,340 --> 00:16:33,640
+هذه بغض النظر عن ال U okay؟ وبالتالي هنا في ال ..
+
+153
+00:16:33,640 --> 00:16:37,320
+في ال .. في الاتصال هذا ال delta هنا تعتمد على
+
+154
+00:16:37,320 --> 00:16:40,540
+إبسن فقط و لا تعتمد على U بينما في المثال السابق
+
+155
+00:16:40,540 --> 00:16:45,240
+شوفنا ال delta بتعتمد على Uهذا النوع من الاتصال
+
+156
+00:16:45,240 --> 00:16:48,860
+بنسميه اتصال منتظم اللي فيه ال delta تعتمد على
+
+157
+00:16:48,860 --> 00:16:52,760
+epsilon فقط اتصال منتظم او uniform continuity
+
+158
+00:16:52,760 --> 00:16:55,540
+اتصال اللي جابله اللي ال delta تعتمد على ال
+
+159
+00:16:55,540 --> 00:17:00,510
+epsilon و على النقطة Uهذا نسميه continuity عادية
+
+160
+00:17:00,510 --> 00:17:04,230
+او نقول continuity اتصال اما هذا uniform
+
+161
+00:17:04,230 --> 00:17:08,770
+continuity هنشوف ال gate من التعريف ان ال uniform
+
+162
+00:17:08,770 --> 00:17:13,990
+continuity اقوى و اشمل من ال continuity العادية
+
+163
+00:17:13,990 --> 00:17:22,670
+okay تمام اذا خليني اضع تعريف ال uniform
+
+164
+00:17:22,670 --> 00:17:25,590
+continuity definition
+
+165
+00:17:28,670 --> 00:17:40,970
+فنشطة f من a الى r هي عامة عامة
+
+166
+00:17:40,970 --> 00:17:49,130
+مستمرة عامة مستمرة عامة عامة عامة عامة عامة عامة
+
+167
+00:17:49,130 --> 00:17:49,170
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+168
+00:17:49,170 --> 00:17:54,610
+عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+169
+00:17:54,610 --> 00:17:55,930
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+170
+00:17:55,930 --> 00:17:55,950
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+171
+00:17:55,950 --> 00:17:56,070
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+172
+00:18:00,400 --> 00:18:06,760
+إبسلون أكبر من السفر يوجد Delta تعتمد على إبسلون
+
+173
+00:18:06,760 --> 00:18:13,920
+فقط، عدد موجب بحيث أنه لكل X و U تنتمي إلى A
+
+174
+00:18:13,920 --> 00:18:20,620
+وأبسليوت X minus U أصغر من Delta هذا بتضمن أن
+
+175
+00:18:20,620 --> 00:18:29,420
+أبسليوت F of X minus F of U أصغر من الإبسلون
+
+176
+00:18:31,760 --> 00:18:35,660
+إذا هنا لأي أبسلون أكبر من السفر في دلتة واحدة
+
+177
+00:18:35,660 --> 00:18:40,100
+تعتمد على أبسلون فقط والدلتة هذه تلف على كل ال X
+
+178
+00:18:40,100 --> 00:18:44,620
+وكل ال U أو لكل ال U مرة واحدة فهنا لكل X و لأي U
+
+179
+00:18:44,620 --> 00:18:48,300
+إذا المسافة بينهم أصغر من الدلتة فالمسافة بين
+
+180
+00:18:48,300 --> 00:18:54,140
+أصغرهم أصغر من X تمام؟ الآن واضح من التعريفات
+
+181
+00:18:58,880 --> 00:19:05,820
+remarks المراحبة الأولى uniform
+
+182
+00:19:05,820 --> 00:19:13,760
+continuity
+
+183
+00:19:13,760 --> 00:19:23,440
+uniform continuity implies continuity
+
+184
+00:19:27,240 --> 00:19:35,720
+الاتصال المنتظر بيؤدي للاتصال العادى و البرهان
+
+185
+00:19:35,720 --> 00:19:39,960
+واضح يعني بمعنى اخر لو في عندي function f from a
+
+186
+00:19:39,960 --> 00:19:46,460
+to r و ال function كانت uniformly continuous فهذا
+
+187
+00:19:46,460 --> 00:19:54,350
+بيؤدي ان f continuous فالبرهان ذلكأفرضي أن F
+
+188
+00:19:54,350 --> 00:20:00,770
+uniformly continuous إذا اشترتها تتحقق تبع الـ
+
+189
+00:20:00,770 --> 00:20:05,750
+Uniform Continuous الآن لإثبات أن F continuous على
+
+190
+00:20:05,750 --> 00:20:11,210
+A بتثبت أن F continuous at every U ينتمي لـ A يعني
+
+191
+00:20:11,210 --> 00:20:17,090
+بتثبت أنه لأي epsilon و لأي U فـ let epsilon be
+
+192
+00:20:17,090 --> 00:20:20,030
+given و let U be fixed element في A
+
+193
+00:20:22,930 --> 00:20:26,390
+من هنا لهذه الـ Epsilon من هنا بما أن هذا الشرط
+
+194
+00:20:26,390 --> 00:20:32,670
+متحقق لأن خد ال Delta لأي ال Epsilon هادي given خد
+
+195
+00:20:32,670 --> 00:20:34,950
+ال Delta اللي هي هذه موجودة في ال uniform
+
+196
+00:20:34,950 --> 00:20:38,650
+continuous اللي بتعتمد على Epsilon فقط خديها هي ال
+
+197
+00:20:38,650 --> 00:20:43,730
+Delta هذه فطبعا هذه ال Delta بتخلي ال implication
+
+198
+00:20:43,730 --> 00:20:51,850
+هذه تتحقق نظرت؟ لو هذه متحققة فهذه متحققةإذن هيك
+
+199
+00:20:51,850 --> 00:20:55,270
+واضح إن ال uniform continuous إذا الواحدة بيقدر أن
+
+200
+00:20:55,270 --> 00:20:58,930
+f continuous and u لما أن u واظهر بترة إذا f
+
+201
+00:20:58,930 --> 00:21:05,310
+continuous and كل على كل المجموعية لكن
+
+202
+00:21:05,310 --> 00:21:11,690
+العكس مش صحيح إذا العكس المواحدة التانية العكس مش
+
+203
+00:21:11,690 --> 00:21:17,570
+صحيح but not conversely
+
+204
+00:21:22,160 --> 00:21:26,380
+العكس مش صحيح، يعني ال continuity لا تؤدي إلى ال
+
+205
+00:21:26,380 --> 00:21:37,360
+uniform continuity و على سبيل المثال for
+
+206
+00:21:37,360 --> 00:21:39,220
+example على سبيل المثال
+
+207
+00:21:46,610 --> 00:21:51,610
+أحنا شوفنا قبل شوية في بداية المحاضرة الـ function
+
+208
+00:21:51,610 --> 00:21:56,870
+f of x بساوي واحد على x و x ينتمي إلى a اللي هي
+
+209
+00:21:56,870 --> 00:22:03,870
+الفترة مفتوحة من سفر لماء لنهاية is continuous on
+
+210
+00:22:03,870 --> 00:22:11,550
+a أثبتت أنها continuous على المجموعة a but
+
+211
+00:22:15,440 --> 00:22:28,480
+but if is not uniformly continuous on a as we
+
+212
+00:22:28,480 --> 00:22:34,140
+shall see in
+
+213
+00:22:34,140 --> 00:22:39,100
+a few minutes
+
+214
+00:22:39,100 --> 00:22:46,160
+كما سنرى بعد لحظات الدالة هذه ليست متصلةإتصالا
+
+215
+00:22:46,160 --> 00:22:52,940
+منتظم هنأخر المرحلة ده شوية و هنبرهنه فلكن في
+
+216
+00:22:52,940 --> 00:22:59,560
+الأول خلينا من التعريف تبع ال uniform continuity
+
+217
+00:22:59,560 --> 00:23:09,720
+نستنتج non uniform continuity criterion من
+
+218
+00:23:09,720 --> 00:23:13,120
+هنا non uniform
+
+219
+00:23:15,380 --> 00:23:22,560
+non uniform continuity criteria
+
+220
+00:23:22,560 --> 00:23:33,940
+let
+
+221
+00:23:33,940 --> 00:23:41,240
+f from a to r be a function then
+
+222
+00:23:44,150 --> 00:23:53,730
+the following statements are equivalent واحد if is
+
+223
+00:23:53,730 --> 00:23:58,810
+not uniformly
+
+224
+00:23:58,810 --> 00:24:09,510
+continuous على المجال تبعها نين there exists
+
+225
+00:24:09,510 --> 00:24:17,380
+epsilon zero أكبر من السفرsuch that for every
+
+226
+00:24:17,380 --> 00:24:26,620
+delta أكبر من السفر يوجد x delta و u delta أناصر
+
+227
+00:24:26,620 --> 00:24:36,220
+في a such that absolute x delta minus u delta أصغر
+
+228
+00:24:36,220 --> 00:24:45,160
+من delta and absolute f of x delta-f of u دلتا
+
+229
+00:24:45,160 --> 00:24:53,160
+أكبر من أو يساوي epsilon zero الأبارع
+
+230
+00:24:53,160 --> 00:25:00,020
+التالتة there exist epsilon zero أكبر من الصفر and
+
+231
+00:25:00,020 --> 00:25:06,200
+two sequences متتاليتين xn
+
+232
+00:25:07,630 --> 00:25:14,930
+و un موجودين في مجال الدالة a such that بحيث ان
+
+233
+00:25:14,930 --> 00:25:23,910
+limit xn minus un بساوي سفر as n tends to infinity
+
+234
+00:25:23,910 --> 00:25:25,690
+and
+
+235
+00:25:27,050 --> 00:25:35,910
+absolute f of xn minus f of un أكبر من أو ساوي
+
+236
+00:25:35,910 --> 00:25:42,350
+epsilon zero هذا صحيح لكل n ينتبه للأعداد الطبيعية
+
+237
+00:25:42,350 --> 00:25:51,070
+okay تمام طيب نشوف البرهان تبع النظرية هذه البرهان
+
+238
+00:25:51,070 --> 00:25:55,440
+تبع النظرية هذه ينتج مباشرة منتعريف ال uniform
+
+239
+00:25:55,440 --> 00:26:01,980
+continuity تعالى نشوف واحد بكافئ اتنين طيب ما
+
+240
+00:26:01,980 --> 00:26:07,300
+معناه if uniform continuous على المجموعة ايه؟
+
+241
+00:26:07,300 --> 00:26:12,920
+معناه الشرط هذا بتحقق طيب ما معناه ان if not
+
+242
+00:26:12,920 --> 00:26:16,540
+uniform continuous على ايه؟ معناه ال negation تبع
+
+243
+00:26:16,540 --> 00:26:19,720
+العبارة دي بتحقق تعالى ننفذ العبارة انفذ العبارة
+
+244
+00:26:20,730 --> 00:26:25,250
+بدل لكل epsilon يوجد epsilon zero بدل يوجد delta
+
+245
+00:26:25,250 --> 00:26:31,550
+لكل delta موجبة بدل لكل x و u يوجد x و u يعتمد كل
+
+246
+00:26:31,550 --> 00:26:36,730
+واحد منهم يعتمد على ال delta بحيث لو كان هذا أصغر
+
+247
+00:26:36,730 --> 00:26:41,950
+من delta فلازم هذا يقدر انه الأصغر هذا أكبر من أو
+
+248
+00:26:41,950 --> 00:26:45,910
+ساوى ال epsilon zeroلأن واضح أن العبارة الأولى
+
+249
+00:26:45,910 --> 00:26:50,330
+بتكافئ التانية لأنه نفي التعريف بكافئ العبارة
+
+250
+00:26:50,330 --> 00:26:55,570
+التانية طيب التانية بتكافئ التالتة وهذا برضه صح
+
+251
+00:26:55,570 --> 00:27:01,650
+افرض أن التانية صحيحة تعني نثبت أن التالتة صحيحة
+
+252
+00:27:01,650 --> 00:27:05,170
+طيب هي التانية يوجد epsilon zero يوجد و هكذا
+
+253
+00:27:12,050 --> 00:27:16,770
+بحيث لكل delta خدي delta بالساوية واحد على n يعني
+
+254
+00:27:16,770 --> 00:27:21,370
+معنى أخر لكل n يوجد delta بالساوية واحد على n عدد
+
+255
+00:27:21,370 --> 00:27:26,730
+موجةوبالتالي يوجد X يعتمد على الـ Delta اللي هي
+
+256
+00:27:26,730 --> 00:27:31,670
+واحد على N اللي بتعتمد على N إذا لكل N لكل N يوجد
+
+257
+00:27:31,670 --> 00:27:37,310
+XN و UN صح؟ وبالتالي يوجد two sequences و ال two
+
+258
+00:27:37,310 --> 00:27:41,470
+sequences هدول بيحققوا أن absolute X واحدة XN
+
+259
+00:27:41,470 --> 00:27:46,310
+minus UN أصغر من واحد على N اللي هي ال Delta و هذا
+
+260
+00:27:46,310 --> 00:27:52,660
+صحيح لكل N إذا ال limitإذا كان هذا أصغر من واحد
+
+261
+00:27:52,660 --> 00:27:55,900
+على xn minus un على absolute أصغر من واحد على n
+
+262
+00:27:55,900 --> 00:28:00,020
+حصّم نظرية اتنين أربعة هذا معناه limit xn minus un
+
+263
+00:28:00,020 --> 00:28:06,420
+بساوة سفر وهذا هي absolute f of xn minus f of un
+
+264
+00:28:06,420 --> 00:28:12,180
+أكبر من أوسع okay فهو واضح وطبعا العكس نفس الحاجة
+
+265
+00:28:12,180 --> 00:28:16,020
+إذن البرهانة النظرية هذه ينتج مباشرة من ال
+
+266
+00:28:16,020 --> 00:28:20,340
+definition تبع ال uniform continuity
+
+267
+00:28:22,600 --> 00:28:27,400
+الان دعونا نرجع للمثال
+
+268
+00:28:27,400 --> 00:28:38,560
+هذا اذا هنا example to
+
+269
+00:28:38,560 --> 00:28:46,710
+show ان ال functionf of x بالساوي واحد على x is
+
+270
+00:28:46,710 --> 00:28:51,190
+not uniformly
+
+271
+00:28:51,190 --> 00:28:58,750
+continuous على المجموعة a اللي هي الفافرة مفتوحة
+
+272
+00:28:58,750 --> 00:29:07,010
+من صفر لما لا نهاية we use non
+
+273
+00:29:07,010 --> 00:29:09,270
+uniform
+
+274
+00:29:11,050 --> 00:29:16,390
+Non-uniform continuity
+
+275
+00:29:16,390 --> 00:29:21,890
+criteria
+
+276
+00:29:37,150 --> 00:29:47,310
+يوجد أبسلون زيرو يوجد
+
+277
+00:29:47,310 --> 00:29:49,870
+عدد أبسلون زيرو موجد
+
+278
+00:30:07,550 --> 00:30:16,570
+تختار خيار Xm بساوي واحد على ان اكيد هذه ال
+
+279
+00:30:16,570 --> 00:30:19,630
+sequence contain في الفترة المفتوحة من سفر للملا
+
+280
+00:30:19,630 --> 00:30:28,210
+نهاية صح؟ and كمان تختار خيار ثاني UN بساوي واحد
+
+281
+00:30:28,210 --> 00:30:33,370
+على ان زايد واحدبرضه هذه ال sequence حدودها كلها
+
+282
+00:30:33,370 --> 00:30:37,730
+موزبة وبالتالي مجموعة جزئية من الفترة المفتوحة من
+
+283
+00:30:37,730 --> 00:30:41,830
+سفر لملنغا Clearly
+
+284
+00:30:41,830 --> 00:30:45,290
+واضح
+
+285
+00:30:45,290 --> 00:30:54,330
+ان ال limit ل xn minus un as n times infinity
+
+286
+00:30:54,330 --> 00:31:04,720
+بساوي limit1 على n minus 1 على n زاد 1 as n equals
+
+287
+00:31:04,720 --> 00:31:11,660
+infinity ف limit الأولى ساوي سفر limit ال sequence
+
+288
+00:31:11,660 --> 00:31:18,200
+التانية سفر وبالتالي بيطلع سفر لأن هنا حققت كل
+
+289
+00:31:18,200 --> 00:31:24,020
+شروط ضايل بس المتبينة هادية also
+
+290
+00:31:28,610 --> 00:31:38,510
+أنا عندي absolute f of x in minus f of u in هذا
+
+291
+00:31:38,510 --> 00:31:46,990
+المفروض بيطلع بيساوي absolute in minus in زد واحد،
+
+292
+00:31:46,990 --> 00:31:53,430
+أزبوتك؟ وهذا بيساوي واحد، واحد أصغر من أوي، بيساوي
+
+293
+00:31:53,430 --> 00:32:00,000
+واحد اللي هو epsilon zeroو هذا صحيح لكل n في n
+
+294
+00:32:00,000 --> 00:32:08,940
+أصبوت هنا هاني انا ايش عملت ال criterion رقم تلاتة
+
+295
+00:32:08,940 --> 00:32:15,660
+اتحققتها اتحققت انها متحققة ها يوجد epsilon zero
+
+296
+00:32:15,660 --> 00:32:21,600
+واحد لاحظوا الواحد علشان انا اختارت واحد ممكن اخد
+
+297
+00:32:21,600 --> 00:32:25,040
+برضه epsilon zero بساوة اتنين لان الواحد اصغر من
+
+298
+00:32:25,040 --> 00:32:29,380
+الاتنينمافي مشكلة بس مش أقل من واحد يعني نص من
+
+299
+00:32:29,380 --> 00:32:32,840
+فعشان لأن هي أثبتت يوجد epsilon zero عدد موجب
+
+300
+00:32:32,840 --> 00:32:36,280
+ويوجد two sequences انا اختارتهم انا اوجدتهم بنفسي
+
+301
+00:32:36,280 --> 00:32:39,780
+واحد على n واحد على n زيادة واحد كلهم موجودين في
+
+302
+00:32:39,780 --> 00:32:45,380
+مجال الدالة ايه و limit الفرق بينهم سفر لكن
+
+303
+00:32:45,380 --> 00:32:52,960
+absolute الفرق بين صورهممش أقوى هذا هيكون بساوي
+
+304
+00:32:52,960 --> 00:32:59,860
+واحد أكبر من أو ساوي .. مش أصغر من أو ساوي بدي
+
+305
+00:32:59,860 --> 00:33:06,140
+أكبر من أو ساوي واحد اللي هو epsilon خليني
+
+306
+00:33:06,140 --> 00:33:09,760
+أنا أسحب الكلام اللي حكيته سابقا و أقول هنا ممكن
+
+307
+00:33:09,760 --> 00:33:13,140
+أخد ال epsilon zero بساوي واحد أو أي حاجة أصغر
+
+308
+00:33:13,140 --> 00:33:19,750
+يعني نص بنفعيعني أبسلون زيرو بساوي نص منفع لكن أي
+
+309
+00:33:19,750 --> 00:33:23,630
+شيء أكبر من واحد منفعش لأن أنا بدي واحد يكون أكبر
+
+310
+00:33:23,630 --> 00:33:28,270
+من أو ساوي أبسلون زيرو تمام؟ إذا هذه أثبتنا
+
+311
+00:33:28,270 --> 00:33:31,750
+وبالتالي حسب ال non-uniform continuity criterion
+
+312
+00:33:31,750 --> 00:33:35,710
+ال .. ال function هذه is not uniform ل continuous
+
+313
+00:33:35,710 --> 00:33:42,250
+تمام؟ لكن أثبتنا سابق جابليك أنها is continuous
+
+314
+00:33:42,250 --> 00:33:48,150
+على المجال تبعهاإذا لو قلنا لكم prove or disprove
+
+315
+00:33:48,150 --> 00:33:51,330
+continuity
+
+316
+00:33:51,330 --> 00:33:55,010
+implies continuity .. ال uniform .. continuity
+
+317
+00:33:55,010 --> 00:33:58,970
+implies uniform continuity هتقولي هذا ال statement
+
+318
+00:33:58,970 --> 00:34:04,150
+false و ال counter example هو هذا هذا مثال على
+
+319
+00:34:04,150 --> 00:34:07,570
+function continuous لكن ليست uniformly continuous
+
+320
+00:34:07,570 --> 00:34:17,820
+تمام؟ طيب، كويسخلّينا الآن نثبت بعض النظريات
+
+321
+00:34:17,820 --> 00:34:24,300
+المهمة اللي بتخص uniform continuity ومن أهم
+
+322
+00:34:24,300 --> 00:34:32,680
+النظريات التي هي النظرية التالية theorem اسمها
+
+323
+00:34:32,680 --> 00:34:36,660
+uniform continuity
+
+324
+00:34:36,660 --> 00:34:40,160
+continuity theorem
+
+325
+00:34:49,430 --> 00:34:56,770
+let I بساوي be
+
+326
+00:34:56,770 --> 00:35:05,570
+a closed and bounded interval
+
+327
+00:35:05,570 --> 00:35:09,350
+اذا
+
+328
+00:35:09,350 --> 00:35:17,360
+I عبارة عن closed and bounded interval لو كانلو
+
+329
+00:35:17,360 --> 00:35:22,980
+كانت الـ function f continuous، if f from I to R
+
+330
+00:35:22,980 --> 00:35:34,040
+is continuous on I، then f is uniformly ..
+
+331
+00:35:34,040 --> 00:35:43,060
+uniformly continuous on
+
+332
+00:35:43,060 --> 00:35:43,620
+I
+
+333
+00:35:46,190 --> 00:35:51,870
+والبرهان السهل prove by contradiction اذا ان بكل
+
+334
+00:35:51,870 --> 00:35:57,070
+بساطة نظرية هذه رغم بساطة بساطة ال statement تبعها
+
+335
+00:35:57,070 --> 00:36:01,710
+اللي انا من اهم النظريات بكل بساطة النظرية اللي
+
+336
+00:36:01,710 --> 00:36:04,850
+بيقول لو كان في عندك function متصل على المجال
+
+337
+00:36:04,850 --> 00:36:08,550
+تبعها والمجال تبعها closed bounded interval اذا
+
+338
+00:36:08,550 --> 00:36:13,970
+الاتصال العادي يصبح اتصال منتظمإن ان هذه الحالة
+
+339
+00:36:13,970 --> 00:36:18,030
+الوحيدة اللي او يعني احد الحالات اللي فيها بيكون
+
+340
+00:36:18,030 --> 00:36:22,650
+الاتصال العادى بقدر الاتصال المنظم ان احنا اضافنا
+
+341
+00:36:22,650 --> 00:36:26,630
+شرط ان مجال تبع الدالة مايكونش اي set لازم يكون
+
+342
+00:36:26,630 --> 00:36:31,090
+closed bounded interval لبرهان ذلك بال
+
+343
+00:36:31,090 --> 00:36:39,670
+contradiction assume on contrary that
+
+344
+00:36:41,290 --> 00:36:55,010
+if is not uniformly continuous on I then by non
+
+345
+00:36:55,010 --> 00:37:03,550
+uniform continuity criteria النظرية
+
+346
+00:37:03,550 --> 00:37:10,620
+اللي فوقيوجد إبسلون زيرو أكبر من السفر و two
+
+347
+00:37:10,620 --> 00:37:15,620
+sequences and
+
+348
+00:37:15,620 --> 00:37:25,040
+two sequences واحدة نسميها x in والتانية un
+
+349
+00:37:25,040 --> 00:37:37,510
+contained in I بحيث أنهabsolute xn minus un أصغر
+
+350
+00:37:37,510 --> 00:37:46,390
+من واحد على n لكل n and absolute f of xn minus f
+
+351
+00:37:46,390 --> 00:37:56,420
+of unأكبر من أو ساوي epsilon zero لكل n في n كل
+
+352
+00:37:56,420 --> 00:38:01,300
+هذا ناخده من ال non uniform continuity criterion
+
+353
+00:38:01,300 --> 00:38:11,500
+الآن بدنا نصل لتناقض طيب
+
+354
+00:38:11,500 --> 00:38:15,980
+عشان نصل لتناقض since
+
+355
+00:38:18,370 --> 00:38:25,750
+I is bounded الفترة دي احنا فرضين انها bounded و
+
+356
+00:38:25,750 --> 00:38:32,550
+ال sequence x in contained in I then ال sequence x
+
+357
+00:38:32,550 --> 00:38:35,450
+in is bounded
+
+358
+00:38:41,210 --> 00:38:57,810
+هنا باستخدام حسب bolzano
+
+359
+00:38:57,810 --> 00:39:01,890
+weierstrass
+
+360
+00:39:01,890 --> 00:39:02,350
+firm
+
+361
+00:39:11,180 --> 00:39:23,360
+السيكوينس هناك سبسيكوينس سميها xnk of xn such that
+
+362
+00:39:23,360 --> 00:39:28,740
+السيكوينس had a convergence limit xnk as k tends
+
+363
+00:39:28,740 --> 00:39:33,840
+to infinity as
+
+364
+00:39:33,840 --> 00:39:40,030
+k tends to infinity بساوي z ينتمي إلى rبالنسبة لـ
+
+365
+00:39:40,030 --> 00:39:45,090
+some z and some r بلزانو فيروس عسكرية كل sequence
+
+366
+00:39:45,090 --> 00:39:48,570
+لها convergence subsequence سم السبسيكوينس هكذا
+
+367
+00:39:48,570 --> 00:39:50,330
+وسم ال limit تبعتها هكذا
+
+368
+00:39:54,530 --> 00:40:00,450
+الـ sub-sequence X in K contained in I التي هي
+
+369
+00:40:00,450 --> 00:40:05,450
+الفترة المغلقة من A إلى B فأحنا أخدنا نظرية تقول
+
+370
+00:40:05,450 --> 00:40:08,290
+أن لو كان هناك sequence حدودها محصورة بين A وB
+
+371
+00:40:08,290 --> 00:40:13,230
+ومتقاربة فنهايتها أيضًا محصورة بين A وB فهذا سيؤدي
+
+372
+00:40:13,230 --> 00:40:18,230
+إلى أن Z تنتمي إلى الفترة المغلقة من A إلى B التي
+
+373
+00:40:18,230 --> 00:40:18,890
+هي I
+
+374
+00:40:24,140 --> 00:40:28,340
+الذي يدفع الاتصال
+
+375
+00:40:28,340 --> 00:40:31,620
+الاتصال
+
+376
+00:40:31,620 --> 00:40:34,420
+الاتصال الاتصال الاتصال الاتصال الاتصال الاتصال
+
+377
+00:40:34,420 --> 00:40:44,240
+الاتصال الاتصال
+
+378
+00:40:51,890 --> 00:41:02,550
+موجودين في I موجودين في I موجودين
+
+379
+00:41:02,550 --> 00:41:12,990
+في I موجودين في Iالـ subsequence UN برضه لها
+
+380
+00:41:12,990 --> 00:41:16,970
+subsequence مشابهة وconvergent لنفس الـ Z هذا مش
+
+381
+00:41:16,970 --> 00:41:25,390
+واضح لثبته لثباته to see this to see this note
+
+382
+00:41:25,390 --> 00:41:28,250
+that
+
+383
+00:41:31,680 --> 00:41:37,040
+بنقدر اخل الفرق بين
+
+384
+00:41:37,040 --> 00:41:47,580
+unk و z أصغر من أي epsilon فهذا
+
+385
+00:41:47,580 --> 00:41:58,020
+أصغر من أو ساوي unk minus xnk زاد absolute xnk
+
+386
+00:41:58,020 --> 00:42:03,380
+minus zهو في الأصل أن أنا المفروض أكتب انا اشعر
+
+387
+00:42:03,380 --> 00:42:07,840
+بالإضطراحة x in k و رجعتها و استخدمت ال triangle
+
+388
+00:42:07,840 --> 00:42:18,420
+inequality طيب هذا صحيح لكل k ينتمي إلى n طيب أنا
+
+389
+00:42:18,420 --> 00:42:22,220
+عندي limit
+
+390
+00:42:22,220 --> 00:42:30,290
+x in minus u in بالساوية سفرلأن هذا صحيح لكل n ف
+
+391
+00:42:30,290 --> 00:42:36,130
+limit u in k minus x in k برضه بيساوي سفر فهذا
+
+392
+00:42:36,130 --> 00:42:43,750
+بيروح لسفر as k tends to infinity وعندي أنا برضه u
+
+393
+00:42:43,750 --> 00:42:50,290
+ال x in k جلنا تقول إلى z فبالتالي ال absolute
+
+394
+00:42:50,290 --> 00:42:56,630
+value هذه بتروح لسفر as k tends to infinityوهذا
+
+395
+00:42:56,630 --> 00:43:08,170
+أكبر من سفر، إذن by squeeze theorem ال sequence
+
+396
+00:43:08,170 --> 00:43:13,030
+هذه محصورة بين ال sequence هذه بالسفر ومجموعة two
+
+397
+00:43:13,030 --> 00:43:18,270
+sequences بيقولوا للسفرإذا من ال limit ل absolute
+
+398
+00:43:18,270 --> 00:43:25,570
+u in k minus z as k tends to infinity بساوي سفر و
+
+399
+00:43:25,570 --> 00:43:31,270
+منها بطلع ال limit u in k as k tends to infinity
+
+400
+00:43:31,270 --> 00:43:38,230
+بساوي z وبالتالي هذا بثبت ال claim تمام؟ إذا هنا
+
+401
+00:43:38,230 --> 00:43:43,830
+أثبتنا ال claim الآن بعد ما أثبتنا ال claim
+
+402
+00:43:57,160 --> 00:44:04,320
+طيب طيب now انا
+
+403
+00:44:04,320 --> 00:44:12,300
+اندي قولنا اثبتنا انه النقطة z تنتمي .. z تنتمي ل
+
+404
+00:44:12,300 --> 00:44:16,880
+I ال limit تبعت ال subsequence تنتمي ل I وال F
+
+405
+00:44:16,880 --> 00:44:17,460
+continuous
+
+406
+00:44:21,950 --> 00:44:25,850
+إن الهدف بقدم if is continuous لأن if continuous
+
+407
+00:44:25,850 --> 00:44:32,210
+على I وبالتالي continuous عند أي نقطة في ال I ولا
+
+408
+00:44:32,210 --> 00:44:36,990
+تكن ال Z hence
+
+409
+00:44:36,990 --> 00:44:40,730
+by
+
+410
+00:44:40,730 --> 00:44:46,510
+sequential criterion by sequential criterion for
+
+411
+00:44:46,510 --> 00:44:50,500
+continuous functionالـ function continuous عند
+
+412
+00:44:50,500 --> 00:44:54,640
+النقطة z وفي عندي sequence x in k converged ل z
+
+413
+00:44:54,640 --> 00:45:01,260
+اذا ال limit لصورة ال sequence او ال subsequence
+
+414
+00:45:01,260 --> 00:45:10,180
+لما كتره ل infinity بساوي f of z و كذلك ايضاAnd
+
+415
+00:45:10,180 --> 00:45:13,760
+برضه ال limit أنا عندي برضه ال sequence هذي
+
+416
+00:45:13,760 --> 00:45:20,220
+converge ل z فنهاية صورة ال subsequence u in k as
+
+417
+00:45:20,220 --> 00:45:27,260
+k tends to infinity برضه بيساوي f of z تمام
+
+418
+00:45:27,260 --> 00:45:31,520
+طيب
+
+419
+00:45:31,520 --> 00:45:35,000
+لكن
+
+420
+00:45:35,000 --> 00:45:43,090
+أنا عنديأنا عندي المتباينة الـ but أنا عندي
+
+421
+00:45:43,090 --> 00:45:47,170
+absolute f of x in
+
+422
+00:45:55,850 --> 00:46:01,570
+من الفرض هيها من الفرض ان ال function not
+
+423
+00:46:01,570 --> 00:46:07,070
+uniformly continuous انا عندي هذا اكبر من او يساوي
+
+424
+00:46:07,070 --> 00:46:10,050
+epsilon zero لكل n لكل حدود ال sequences
+
+425
+00:46:14,300 --> 00:46:20,060
+فهذا بيقدّي .. هذا بدوره بيقدّي انه epsilon zero
+
+426
+00:46:20,060 --> 00:46:27,180
+هي epsilon zero أصغر من أو ساوي absolute f of x in
+
+427
+00:46:27,180 --> 00:46:36,220
+k minus f of u in k تمام؟
+
+428
+00:46:37,840 --> 00:46:41,720
+هذا صحيح لـ sequence x in و لـ sequence u in إذا
+
+429
+00:46:41,720 --> 00:46:46,200
+صحيح للـ subsequence للـ subsequences إذا هذه جاية
+
+430
+00:46:46,200 --> 00:46:50,620
+من هنا طيب و by triangle inequality by triangle
+
+431
+00:46:50,620 --> 00:46:56,760
+inequality ممكن أخلي هذا أصغر لو ساوي f of x nk
+
+432
+00:46:56,760 --> 00:47:10,090
+minus f of z زاد absolute f of z-F of U in K انا
+
+433
+00:47:10,090 --> 00:47:14,610
+شو انا عاملة اتراحت من هنا F of Z و رجعتها اه و
+
+434
+00:47:14,610 --> 00:47:17,810
+استخدمت ال triangle equality فصار اندي اصلا مجموعة
+
+435
+00:47:17,810 --> 00:47:24,070
+two absolute values طيب
+
+436
+00:47:24,070 --> 00:47:30,660
+ما انا ممكن اخليأنا عندي limit ال sequence هذه
+
+437
+00:47:30,660 --> 00:47:36,800
+بساوي f of z فلأي given epsilon أكبر من الصفر ممكن
+
+438
+00:47:36,800 --> 00:47:42,300
+أخلي absolute الفرخ هذا أصغر من epsilon على 2 ونفس
+
+439
+00:47:42,300 --> 00:47:47,260
+الحاجة أنا عندي ال sequence f of u and k converged
+
+440
+00:47:47,260 --> 00:47:51,840
+ل f of z إذا ممكن أخلي ال absolute value للفرخ هذه
+
+441
+00:47:51,840 --> 00:47:59,540
+أصغر من epsilon على 2وبالتالي بيطلع المجموعة
+
+442
+00:47:59,540 --> 00:48:06,420
+epsilon هذا صحيح لكل K أكبر من أو ساوي كابتل K أو
+
+443
+00:48:06,420 --> 00:48:12,360
+كابتل N واضح تمام؟ يعني لا أي epsilon أكبر من صفر
+
+444
+00:48:12,360 --> 00:48:15,820
+أو لا عفو ان ال epsilon نفس ال epsilon zero هذه ال
+
+445
+00:48:15,820 --> 00:48:19,380
+epsilon هي نفس ال epsilon zeroهي عندي الـ Epsilon
+
+446
+00:48:19,380 --> 00:48:23,780
+Zero given لما ان ال sequence هي ال converge إذا
+
+447
+00:48:23,780 --> 00:48:28,400
+يوجد capital N واحد يعتمد على Epsilon Zero بحيث أن
+
+448
+00:48:28,400 --> 00:48:33,580
+أبسلوت الفرق هذا أصغر من أو ساوي Epsilon على اتنين
+
+449
+00:48:33,580 --> 00:48:37,140
+لكل K أكبر من أو ساوي capital K واحد او capital N
+
+450
+00:48:37,140 --> 00:48:42,560
+واحد ونفس الحاجة لنفس ال Epsilon Zero يوجد N اتنين
+
+451
+00:48:43,830 --> 00:48:47,730
+بحيث انه بما انه هذه ال sequence converge اذا
+
+452
+00:48:47,730 --> 00:48:52,310
+الفرخة ده بقدر اخليه لكل n اكبر من او لكل k اكبر
+
+453
+00:48:52,310 --> 00:48:56,630
+من او ساوي n اتنين اصغر من يبسلون اتنين الان خدي n
+
+454
+00:48:56,630 --> 00:49:05,410
+بساوي ال maximum ل n واحد و n اتنين فبقدر
+
+455
+00:49:05,410 --> 00:49:11,930
+اخلي هذا اصغر من يبسلون زيرو لكل k اكبر من او ساوي
+
+456
+00:49:11,930 --> 00:49:16,590
+nففي النهاية بيطلع عندى epsilon zero أقل من
+
+457
+00:49:16,590 --> 00:49:19,650
+epsilon أصغر من epsilon zero هذا مديني
+
+458
+00:49:19,650 --> 00:49:23,590
+contradiction لأن هذا التناقض بيقول لي أن ال
+
+459
+00:49:23,590 --> 00:49:28,150
+assumption تبعنا أن ال function not uniformly
+
+460
+00:49:28,150 --> 00:49:32,050
+continuous كان assumption خطأ لأن الصح أن ال F
+
+461
+00:49:32,050 --> 00:49:37,810
+تكون uniformly continuous okay تمام واضح؟Okay إذا
+
+462
+00:49:37,810 --> 00:49:44,230
+بنوقف ان شاء الله هنا عند نهاية البرهان هذا و
+
+463
+00:49:44,230 --> 00:49:51,690
+بيكون هيك احنا يعني خلصنا جزء مش بسيط في section
+
+464
+00:49:51,690 --> 00:49:56,730
+خمسة أربعة و نكتفي بهذا القدر و يعطيكم ألف عافية و
+
+465
+00:49:56,730 --> 00:49:58,590
+شكرا لحصن أصغائكم
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM.srt
new file mode 100644
index 0000000000000000000000000000000000000000..20585784739089606c37ee19351ef7b73edd6087
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM.srt
@@ -0,0 +1,1731 @@
+1
+00:00:21,210 --> 00:00:28,030
+لنراجع مع بعض the order properties of R أو خواص 
+
+2
+00:00:28,030 --> 00:00:33,790
+الترتيب للأعداد الحقيقية، احنا من بداية الـ chapter 
+
+3
+00:00:33,790 --> 00:00:37,630
+قلنا إنّ الـ real number system نظام الأعداد 
+
+4
+00:00:37,630 --> 00:00:43,250
+الحقيقية، نظام 
+
+5
+00:00:43,250 --> 00:00:52,230
+الأعداد الحقيقية يتكون من مجموعة R، boldface R مع 
+
+6
+00:00:52,230 --> 00:00:57,370
+عمليتين ثنائيتين، واحدة عملية الجمع، واحدة عملية 
+
+7
+00:00:57,370 --> 00:01:04,390
+الضرب، و افترضنا أنّ العمليات هذه بتحقق خمس خواص اللي 
+
+8
+00:01:04,390 --> 00:01:08,910
+هي خواص الـ field اللي هو الـ commutative law,
+
+9
+00:01:09,050 --> 00:01:17,690
+associative law, distributive laws, existence of 
+
+10
+00:01:17,690 --> 00:01:23,340
+identities, existence of inverses. بعدين ضفنا على 
+
+11
+00:01:23,340 --> 00:01:28,300
+ذلك أنّ افترضنا أنّ الـ real number system R بتحقق 
+
+12
+00:01:28,300 --> 00:01:33,020
+برضه خاصية الترتيب أو خواص الترتيب اللي هي الخاصية 
+
+13
+00:01:33,020 --> 00:01:38,440
+السادسة، هذه الخاصية السادسة هذه تجزأت، يعني تنص على 
+
+14
+00:01:38,440 --> 00:01:43,560
+ما يلي: نفترض أنّ يوجد مجموعة جزئية من R غير خالية 
+
+15
+00:01:43,560 --> 00:01:49,360
+و المجموعة الجزئية هذه بنسميها P، اللي هو أول حرف 
+
+16
+00:01:49,360 --> 00:01:56,080
+في positive عشان نسميها بعد هيك the set of positive
+
+17
+00:01:56,080 --> 00:02:02,020
+real numbers. فالـ set P هذه closed under addition 
+
+18
+00:02:02,020 --> 00:02:08,540
+and under multiplication. كمان نفترض أنّ الـ set P 
+
+19
+00:02:08,540 --> 00:02:13,180
+هذه بتحقق الخاصية الثلاثية، الـ trichotomy property
+
+20
+00:02:14,240 --> 00:02:18,920
+which means that for any real number a, exactly one 
+
+21
+00:02:18,920 --> 00:02:27,060
+of the three possibilities holds: either a belongs 
+
+22
+00:02:27,060 --> 00:02:34,420
+to P, or a equals zero, or negative a belongs to P. 
+
+23
+00:02:34,420 --> 00:02:42,230
+بناءً على هذه الخاصية، شفنا أنّ الأعداد الحقيقية gets 
+
+24
+00:02:42,230 --> 00:02:47,350
+partitioned to three mutually disjoint sets، يعني 
+
+25
+00:02:47,350 --> 00:02:54,450
+الخاصية هذه بتجزّئ، بتخليني أجزّئ الأعداد الحقيقية إلى 
+
+26
+00:02:54,450 --> 00:03:00,110
+ثلاث مجموعات منفصلة مثنى مثنى، pair-wise disjoint
+
+27
+00:03:00,110 --> 00:03:05,650
+يعني إنّ لو أخدت أي مجموعتين عشوائيتين من الثلاث 
+
+28
+00:03:05,650 --> 00:03:09,310
+تقطعهم بيساوي فاي، ما فيش بينهم عناصر، ومش ثلاث 
+
+29
+00:03:10,410 --> 00:03:15,170
+واتحادهم بيساوي الـ R، لأنّ هذا بشكل تجزئة على الـ R، 
+
+30
+00:03:15,170 --> 00:03:19,630
+تجزئة على الـ R، أو بنسميها في الرياضيات partition of 
+
+31
+00:03:19,630 --> 00:03:25,210
+R. الـ set P هذه سميناها set of positive real numbers
+
+32
+00:03:25,210 --> 00:03:35,210
+وعرفنا negative P على أنّها negative عناصر الـ set P
+
+33
+00:03:44,210 --> 00:03:49,430
+Okay. فهي معرفة negative P هي كلّ الـ elements 
+
+34
+00:03:49,430 --> 00:03:56,850
+negative A such that A element in P. بعدين 
+
+35
+00:03:56,850 --> 00:04:02,430
+عرفنا علاقة الترتيب، الآن بنعرف اللي هو order 
+
+36
+00:04:02,430 --> 00:04:08,340
+relation على R. ما معنى أنّ a لو في عندي two real 
+
+37
+00:04:08,340 --> 00:04:12,720
+numbers، ما معنى a أصغر من b أو b أكبر من a؟ قولنا 
+
+38
+00:04:12,720 --> 00:04:19,820
+معناها أنّ الفرق بين b و a is a positive real number 
+
+39
+00:04:19,820 --> 00:04:24,480
+أو ينتمي لمجموعة الأعداد الموجبة. طب ما معناه a 
+
+40
+00:04:24,480 --> 00:04:28,760
+أصغر من أو يساوي b أو b أكبر من أو يساوي a؟ معناته 
+
+41
+00:04:28,760 --> 00:04:33,240
+الفرق بين b و a ينتمي للأعداد الموجبة، يعني الفرق 
+
+42
+00:04:33,240 --> 00:04:40,120
+موجب أو يساوي صفر، أو يساوي صفر، إذن معناه تاني طيب
+
+43
+00:04:40,120 --> 00:04:50,940
+و أعتقد أنّ احنا بعد هيك أثبتنا آه
+
+44
+00:04:50,940 --> 00:04:52,780
+وقفنا عند النظرية هذه
+
+45
+00:04:57,810 --> 00:05:02,310
+النظرية 1-5، قلنا إنّه لأي لو أخدت أي ثلاث أعداد 
+
+46
+00:05:02,310 --> 00:05:08,730
+حقيقية، فعند الخواص التالية تتحقق، مجموعة الخواص هذه 
+
+47
+00:05:08,730 --> 00:05:17,210
+تتحقق، فالخواص 
+
+48
+00:05:17,210 --> 00:05:21,630
+هذه هذا 
+
+49
+00:05:21,630 --> 00:05:27,330
+هي أمامكم، transitivity، خاصية التعدي. إيه يعني 
+
+50
+00:05:27,330 --> 00:05:35,310
+التعدي؟ يعني إذا أنا في عندي ثلاث أعداد حقيقية في A و 
+
+51
+00:05:35,310 --> 00:05:38,770
+B و C
+
+52
+00:05:43,970 --> 00:05:52,530
+وكان B هنا أكبر .. B أكبر من A and C أكبر من B 
+
+53
+00:05:52,530 --> 00:05:55,790
+فهذا 
+
+54
+00:05:55,790 --> 00:06:09,770
+بيؤدي أنّ C أكبر من A. خليني 
+
+55
+00:06:09,770 --> 00:06:15,290
+أنا أكتبهم عشان .. كلهم زي .. مهمة موجودة هناك هي a 
+
+56
+00:06:15,290 --> 00:06:27,090
+أكبر من b، هي a أكبر من b، و b أكبر من c، فبيطلع 
+
+57
+00:06:27,090 --> 00:06:38,470
+c، أو a، بيطلع أكبر من c. هذه a أكبر من b، و b أكبر من c.
+
+58
+00:06:38,470 --> 00:06:44,750
+إذا نقدر نتعدى و نقول a أكبر من c. فهذه بيسموها في 
+
+59
+00:06:44,750 --> 00:06:50,730
+الرياضيات transitivity، أو خاصية التعدي، الخاصية 
+
+60
+00:06:50,730 --> 00:06:53,990
+التانية 
+
+61
+00:06:53,990 --> 00:06:59,930
+بنسميها trichotomy، برضه خاصية ثلاثية جاية من 
+
+62
+00:06:59,930 --> 00:07:05,770
+الخاصية الثلاثية اللي شفناها قبل شوية، فبتقول 
+
+63
+00:07:05,770 --> 00:07:08,650
+exactly one of the following holds، واحد من ثلاث 
+
+64
+00:07:08,650 --> 00:07:19,190
+احتمالات بتحصل، إما a أكبر من b، أو a بتساوي b، أو a 
+
+65
+00:07:19,190 --> 00:07:28,260
+أصغر من b، لأي عددين حقيقيين A و B، واحد فقط من 
+
+66
+00:07:28,260 --> 00:07:32,800
+الاحتمالات الثلاثة بيكون صحيح، وهو إما A أكبر من B 
+
+67
+00:07:32,800 --> 00:07:38,860
+أو A بيساوي B، أو A أصغر من B. الـ Antisymmetry 
+
+68
+00:07:38,860 --> 00:07:43,640
+property، علاقة أكبر من أو يساويها دي بنسميها 
+
+69
+00:07:43,640 --> 00:07:48,400
+Antisymmetric، يعني إيه؟ بتحقق خاصية تضاد التماثل 
+
+70
+00:07:49,760 --> 00:07:54,960
+إيه يعني؟ مع أنّ لو كانت A على علاقة مع B، و B على 
+
+71
+00:07:54,960 --> 00:08:00,940
+علاقة مع A، فلازم يطلع A بيساوي B، A أكبر من أو يساوي 
+
+72
+00:08:00,940 --> 00:08:05,300
+B، و B أكبر من أو يساوي A، فلازم A يساوي B، هذه 
+
+73
+00:08:05,300 --> 00:08:11,640
+بنسميها Anti-symmetry property. هنا الخاصية هذه لو 
+
+74
+00:08:11,640 --> 00:08:18,140
+كان a أكبر من b، ووضفت للطرفين أي عدد c، فالمتباينة 
+
+75
+00:08:18,140 --> 00:08:22,040
+تبقى زي ما هي، شريطة زي ما هي. طيب لو في عندي 
+
+76
+00:08:22,040 --> 00:08:26,100
+متباينة a أكبر من b، لأنّ نتحدث عن متباينات 
+
+77
+00:08:26,100 --> 00:08:31,960
+inequalities. لو كان a أكبر من b، و c عدد موجب، وضربت 
+
+78
+00:08:31,960 --> 00:08:35,960
+الطرفين في عدد الموجب c، فإشارة المتباينة تبقى كما 
+
+79
+00:08:35,960 --> 00:08:40,380
+هي. لكن لو ضربت المتباينة في عدد سالب، إشارة 
+
+80
+00:08:40,380 --> 00:08:46,900
+المتباينة بتتغير. الخاصية f بتقول أنّ لأي عدد حقيقي 
+
+81
+00:08:46,900 --> 00:08:51,300
+لا يساوي صفر، مربع أي عدد حقيقي لا يساوي صفر دائمًا 
+
+82
+00:08:51,300 --> 00:08:55,400
+بيكون عدد موجب. الواحد 
+
+83
+00:08:55,900 --> 00:08:59,780
+الـ Distinguished elements في R أو في الـ real 
+
+84
+00:08:59,780 --> 00:09:02,940
+number system، اللي هم الصفر والواحد، اللي هو الـ 
+
+85
+00:09:02,940 --> 00:09:07,280
+identity elements، سميناهم، بيحققوا أنّ واحد دائمًا 
+
+86
+00:09:07,280 --> 00:09:13,400
+أكبر من الصفر، وسالب واحد أصغر من الصفر. كمان لأي 
+
+87
+00:09:13,400 --> 00:09:16,840
+عدد طبيعي، هذه the set of natural numbers، أي عدد 
+
+88
+00:09:16,840 --> 00:09:23,210
+طبيعي بيكون دائمًا موجب، أي عدد طبيعي بيطلع موجب. لو 
+
+89
+00:09:23,210 --> 00:09:27,450
+كان a عدد حقيقي موجب، فمقلوبه موجب، لو كان a عدد 
+
+90
+00:09:27,450 --> 00:09:38,990
+حقيقي سالب، مقلوبه بيطلع سالب. الخاصية 
+
+91
+00:09:38,990 --> 00:09:45,610
+الأخيرة، لو كان a أصغر من b، واثنين موجبين 
+
+92
+00:09:45,610 --> 00:09:53,160
+فمقلوب الصغير أكبر من مقلوب الكبير، أو مقلوب الكبير 
+
+93
+00:09:53,160 --> 00:09:56,680
+أصغر من مقلوب الصغير. بصراحة اثنين اللي هم نفس 
+
+94
+00:09:56,680 --> 00:10:00,980
+الإشارة. لكن لو كان واحد موجب بواحد سالب، فالكلام 
+
+95
+00:10:00,980 --> 00:10:08,060
+هذا مش صحيح، خدوا بالكم. طيب نشوف، نمر بسرعة على 
+
+96
+00:10:08,060 --> 00:10:15,500
+البرهان، قرأتم البرهان أنتم؟ طيب
+
+97
+00:10:30,910 --> 00:10:38,550
+خاصية التعدي، خاصية التعدي. أنا كان عندي a أكبر من b 
+
+98
+00:10:38,550 --> 00:10:44,710
+and b أكبر من c، بدنا نثبت أنّ هذا بيعدي أنّ a أكبر من 
+
+99
+00:10:44,710 --> 00:10:52,250
+c. فالبرهان لذلك يكفي نثبت أنّ الفرق بين c و a موجب 
+
+100
+00:10:52,990 --> 00:10:56,990
+يعني ينتمي للـ set P of positive real numbers
+
+101
+00:10:56,990 --> 00:11:02,450
+فتعالوا نثبت الكلام هذا، أنا عندي من المعطيات أو من 
+
+102
+00:11:02,450 --> 00:11:08,370
+الفرض، الفرق هذا موجب، والفرق هذا موجب من المعطيات 
+
+103
+00:11:09,240 --> 00:11:13,680
+طيب، set P closed under addition، مغلقة تحت عملية 
+
+104
+00:11:13,680 --> 00:11:18,820
+الجمع، إذا مجموعة عنصرين في P بيطلع عنصر ثالث في P
+
+105
+00:11:18,820 --> 00:11:22,660
+هذا العنصر الثالث اللي بيقول المجموعة، طلع A سالب C 
+
+106
+00:11:22,660 --> 00:11:28,560
+هذا معناه، مادام الفرق هذا ينتمي لـ P، معناته الفرق هذا 
+
+107
+00:11:28,560 --> 00:11:33,900
+موجب، أو A أكبر من C، as required، كما هو مطلوب، 
+
+108
+00:11:33,900 --> 00:11:38,040
+مظبوط؟ واضح؟ طيب
+
+109
+00:11:42,860 --> 00:11:49,940
+أي عدد حقيقي له واحد من ثلاث احتمالات، إما موجب أو 
+
+110
+00:11:49,940 --> 00:11:56,720
+صفر أو سالب. الآن بناءً على هذه الخاصية، ممكن نثبت 
+
+111
+00:11:56,720 --> 00:12:01,940
+الخاصية الثلاثية، الخاصية 
+
+112
+00:12:01,940 --> 00:12:05,600
+الثانية. 
+
+113
+00:12:09,230 --> 00:12:15,030
+قلنا إنّ لو كان لأي عددين حقيقيين، لأي عددين حقيقيين 
+
+114
+00:12:15,030 --> 00:12:19,750
+a و b، a أكبر من b أو a بيساوي b أو a أصغر من b.
+
+115
+00:12:19,750 --> 00:12:22,910
+فالبرهان 
+
+116
+00:12:22,910 --> 00:12:27,350
+ذلك بيعتمد على الـ trichotomy property اللي شفناها 
+
+117
+00:12:27,350 --> 00:12:33,470
+قبل شوية، فأنا 
+
+118
+00:12:33,470 --> 00:12:33,870
+عندي 
+
+119
+00:12:37,890 --> 00:12:41,310
+حسب الـ trichotomy property، لو أخدت الفرق هذا، 
+
+120
+00:12:41,310 --> 00:12:46,850
+هذا real number، فأي real number إمّا positive أو 
+
+121
+00:12:46,850 --> 00:12:54,390
+بيساوى صفر أو negative، صح؟ وهذا بكافئ، الكلام هذا 
+
+122
+00:12:54,390 --> 00:13:01,210
+بكافئ A سالب B ينتمي لـ P، بكافئ أنّ الـ A أكبر من B. 
+
+123
+00:13:02,260 --> 00:13:07,220
+طب وهذا ينتمي لـ 0، بكافئ أنّ a بيساوي b، أو الفرق 
+
+124
+00:13:07,220 --> 00:13:11,660
+بيساوى 0، وبالتالي a بيساوي b، والفرق هذا ينتمي لـ 
+
+125
+00:13:11,660 --> 00:13:16,180
+negative P، معناته الفرق هذا سالب، يعني معناه أنّ a 
+
+126
+00:13:16,180 --> 00:13:21,760
+أصغر من b، وهذا اللي بدنا إياه، هذا اللي بدنا إياه
+
+127
+00:13:21,760 --> 00:13:25,620
+طيب 
+
+128
+00:13:25,620 --> 00:13:32,390
+الجزء C، قلنا اللي هو الـ antisymmetry property، الـ 
+
+129
+00:13:32,390 --> 00:13:37,990
+Anti-symmetry property، نفكركم فيها، بتقول لو كان a 
+
+130
+00:13:37,990 --> 00:13:46,590
+أكبر من أو يساوي b and b أكبر من أو يساوي a، فهذا 
+
+131
+00:13:46,590 --> 00:13:52,210
+بيؤدي أنّ a بيساوي b، بظبط؟ طيب
+
+132
+00:13:57,150 --> 00:14:02,570
+أنا بدي أثبت أنّ A بيساوي B، هذه النتيجة، فبدأ أعمل 
+
+133
+00:14:02,570 --> 00:14:07,750
+برهان بالتناقض، فبرهان بالتناقض دائمًا نفرض ما فيه 
+
+134
+00:14:07,750 --> 00:14:12,670
+النتيجة هو الصح، وبنوصلها إلى التناقض، فـ assume أنّ 
+
+135
+00:14:12,670 --> 00:14:21,500
+A لا تساوي B. إذا حسب الخاصية الثلاثية، هذا بيؤدي أنّ 
+
+136
+00:14:21,500 --> 00:14:30,800
+إمّا a أصغر من b or b أصغر من a، مظبوط؟ طيب إذا هنا 
+
+137
+00:14:30,800 --> 00:14:39,720
+.. الآن لو أخدت .. لو أخدت الـ a أكبر من b اللي هو 
+
+138
+00:14:39,720 --> 00:14:46,880
+الاحتمال هذا. لو أخدت .. لو قلت أنّ a أكبر من b، فهذا 
+
+139
+00:14:46,880 --> 00:14:54,140
+بتناقض مع الفرض .. بتناقض مع الفرض أنّ a أصغر من .. 
+
+140
+00:14:54,140 --> 00:15:01,160
+a أصغر من أو يساوي b، هدول اثنين بيعطون التناقض طيب 
+
+141
+00:15:01,160 --> 00:15:06,400
+لو افترضت الاحتمال الثاني أن a أصغر من b فهذا
+
+142
+00:15:06,400 --> 00:15:15,210
+بتناقض مع الفرض أن a أكبر من أو يساوي الـ B إذا في
+
+143
+00:15:15,210 --> 00:15:20,790
+الحالتين لو فرضت هذا صح بتناقض مع هذا الجزء لو
+
+144
+00:15:20,790 --> 00:15:25,410
+فرضت هذا صح بتناقض مع هذا الجزء اللي هو جزء من
+
+145
+00:15:25,410 --> 00:15:29,970
+الفرض وبالتالي في كلتا الحالتين الفرض أن A لا 
+
+146
+00:15:29,970 --> 00:15:35,050
+يساوي B أدى إلى تناقض إذا الصح أن A لازم تساوي B
+
+147
+00:15:35,050 --> 00:15:40,600
+كما هو مطلوب okay هذا برهان بالتناقض واضح تمام
+
+148
+00:15:40,600 --> 00:15:47,740
+مفهوم فاهمين ولا هيك يعني أمور
+
+149
+00:15:47,740 --> 00:15:53,040
+سهلة وبسيطة وكلها يعني مبادئ رياضيات احنا هنا يعني
+
+150
+00:15:53,040 --> 00:15:58,840
+مراجعة لمبادئ رياضيات أو طرق البرهان في مبادئ
+
+151
+00:15:58,840 --> 00:16:03,680
+رياضيات طيب
+
+152
+00:16:03,680 --> 00:16:07,220
+الآن بنثبت القضية F
+
+153
+00:16:13,430 --> 00:16:18,050
+لأي عدد حقيقي لا يساوي صفر دائماً مربعه بيطلع موجب
+
+154
+00:16:18,050 --> 00:16:22,330
+فعشان أثبت مربع ال a موجب لازم أثبت أن مربع ال a
+
+155
+00:16:22,330 --> 00:16:30,890
+ينتمي لفئة أو مجموعة الأعداد الموجبة طيب احنا فرضنا
+
+156
+00:16:30,890 --> 00:16:34,950
+a لا يساوي صفر إذا by trichotomy property بالخاصية
+
+157
+00:16:34,950 --> 00:16:40,470
+الثلاثية أما a موجب أو سالب يعني معناه هذا أو هذا
+
+158
+00:16:40,470 --> 00:16:51,220
+الآن لو كانت ال A موجبة فمربعها و P مغلقة تحت
+
+159
+00:16:51,220 --> 00:16:56,440
+عملية الضرب فحاصل ضرب A في A اللي هو A تربيع بيطلع
+
+160
+00:16:56,440 --> 00:17:03,580
+ينتمي يعني هذا بيساوي A تربيع ال
+
+161
+00:17:03,580 --> 00:17:07,660
+A ينتمي ل P إذا حاصل الضرب ينتمي ل P وبالتالي A 
+
+162
+00:17:07,660 --> 00:17:12,080
+تربيع موجبة okay وهذا اللي احنا عايزينه الحالة
+
+163
+00:17:12,080 --> 00:17:17,600
+الثانية طب افرض أنه negative A تنتمي ل P أو A
+
+164
+00:17:17,600 --> 00:17:23,200
+تنتمي ل negative P يعني A سالب ففي الحالة هذه لو 
+
+165
+00:17:23,200 --> 00:17:29,220
+ضربت هذا العنصر في نفسه بيطلع ينتمي إلى ال P بيطلع
+
+166
+00:17:29,220 --> 00:17:33,480
+ينتمي إلى ال P وهذا بيطلع بيساوي من الخواص اللي
+
+167
+00:17:33,480 --> 00:17:37,510
+أخذناها قبل هيك يعني هذا عبارة عن هذا سالب أيه
+
+168
+00:17:37,510 --> 00:17:40,970
+بكتبه سالب واحد في أيه و سالب أيه الثاني نفس
+
+169
+00:17:40,970 --> 00:17:45,650
+الحاجة سالب واحد في أيه فبيطلع سالب واحد في سالب
+
+170
+00:17:45,650 --> 00:17:50,170
+واحد في أيه تربيع و هذا واحد فبيطلع أيه تربيع تنتمي
+
+171
+00:17:50,170 --> 00:17:54,830
+لدي وبالتالي أيه تربيع موجبة إذا هنا أثبتنا أن أي
+
+172
+00:17:54,830 --> 00:17:59,690
+عدد حقيقي مختلف عن الصفر دائماً مربعه موجب
+
+173
+00:18:13,230 --> 00:18:23,990
+خاصية G الخاصية
+
+174
+00:18:23,990 --> 00:18:24,510
+G
+
+175
+00:18:30,830 --> 00:18:36,810
+احنا بنثبت أن الواحد أكبر من الصفر فبكل بساطة واحد
+
+176
+00:18:36,810 --> 00:18:42,710
+بيساوي واحد ضرب نفسه وهذا بيطلع واحد تربيع وقبل
+
+177
+00:18:42,710 --> 00:18:46,710
+شوية شوفنا والواحد مختلف عن الصفر إذا المربع
+
+178
+00:18:46,710 --> 00:18:54,870
+بيطلع موجب حسب الخاصية السابقة، أثبتنا؟ هذا معناه إذا
+
+179
+00:18:54,870 --> 00:18:59,770
+هيُثبتنا واحد أكبر من الصفر وبالتالي واحد ينتمي لل
+
+180
+00:18:59,770 --> 00:19:04,670
+positive real numbers إذا سالب واحد ينتمي ل
+
+181
+00:19:04,670 --> 00:19:07,610
+negative two يعني سالب واحد أصغر من الصفر
+
+182
+00:19:07,610 --> 00:19:20,050
+عملية بسيطة طيب احنا الآن بدنا نثبت أن كل
+
+183
+00:19:22,970 --> 00:19:31,370
+عدد حقيقي موجب مقلوبه موجب اه فبنعمل برهان
+
+184
+00:19:31,370 --> 00:19:36,190
+بالتناقض إذا هنا هندي اللي عايز نثبته هنا بس هنذكر
+
+185
+00:19:36,190 --> 00:19:41,430
+ال statement اللي بدنا نثبته يعني ال statement
+
+186
+00:19:41,430 --> 00:19:46,710
+اللي عايز نثبته لو كان a موجب ف reciprocal تبعه
+
+187
+00:19:46,710 --> 00:19:51,920
+بيطلع موجب أو مقلوبه بيطلع موجبلبرهان ذلك نعمل برهان
+
+188
+00:19:51,920 --> 00:20:01,560
+بالتناقض نفرض أن واحد على a أقل من الصفر وطبعاً
+
+189
+00:20:01,560 --> 00:20:06,380
+عندي أنا من الفرض هذا الفرض لازال قائم a أكبر من 
+
+190
+00:20:06,380 --> 00:20:13,520
+الصفر عندي الفرضين هدول فعندي a أكبر من الصفر و
+
+191
+00:20:13,520 --> 00:20:17,640
+واحد على a أصغر من الصفر فهذا بيؤدي
+
+192
+00:20:20,870 --> 00:20:28,190
+لو ضربت المتباينة هذه في a اللي هو عدد موجب فهيصبح
+
+193
+00:20:28,190 --> 00:20:32,450
+أن واحد على a في a أصغر من صفر في a اللي هو
+
+194
+00:20:32,450 --> 00:20:37,290
+بيساوي صفر طب هذا بيساوي واحد ان هك بيطلع واحد
+
+195
+00:20:37,290 --> 00:20:42,530
+أصغر من صفر وبالتالي هذا يعطيني تناقض لأن الواحد
+
+196
+00:20:42,530 --> 00:20:47,530
+أكبر من صفر لسه مثبتينه قبل شوية أن هذا بيؤدي إلى
+
+197
+00:20:47,530 --> 00:20:54,640
+تناقض وبالتالي مقلوب الـ A لازم يكون موجب بالمثل لو 
+
+198
+00:20:54,640 --> 00:21:00,980
+كان مقلوب الـ A سالب فممكن نثبت أنه مقلوبه أيضاً
+
+199
+00:21:00,980 --> 00:21:05,900
+بيطلع سالب فالبرهان مشابه هأسيبكم أنتم تكتبوا
+
+200
+00:21:05,900 --> 00:21:07,720
+تمام؟
+
+201
+00:21:21,570 --> 00:21:23,670
+أنا مش عارف لسه أنا هيك بعمل
+
+202
+00:21:50,000 --> 00:21:59,840
+طيب ال .. الجزء هذا الأخير إيش كان هذا؟ إيش كنا
+
+203
+00:21:59,840 --> 00:22:14,980
+بدنا نثبت هناك؟
+
+204
+00:22:17,920 --> 00:22:26,800
+اه إذا كان a عدد موجب و أصغر من b فهذا بيؤدي أن
+
+205
+00:22:26,800 --> 00:22:34,100
+مقلوب الكبير أصغر من مقلوب الصغير بظبط
+
+206
+00:22:34,100 --> 00:22:39,080
+وطبعاً هذا موجب فلإثبات
+
+207
+00:22:39,080 --> 00:22:43,360
+أن واحد على b أصغر من واحد على a بتثبت أن الفرق
+
+208
+00:22:43,360 --> 00:22:52,860
+بين واحد على a و1 على b ينتمي إلى P
+
+209
+00:22:52,860 --> 00:22:59,900
+أو موجب طيب الآن هاي ناخد 1 على A ناقص 1 على B
+
+210
+00:22:59,900 --> 00:23:05,140
+فأخذنا خاصية ناخذ مقام مشترك A B وبعدين بيصير
+
+211
+00:23:05,140 --> 00:23:11,160
+عندي هذا بتحول لحاصل ضرب الآن هذا positive number
+
+212
+00:23:11,160 --> 00:23:15,420
+لأن احنا فرضنا أن ال B أكبر من A فالفرق هذا
+
+213
+00:23:15,420 --> 00:23:23,780
+positive و A B فبيطلع
+
+214
+00:23:23,780 --> 00:23:29,200
+هذا مقلوب ال positive بيطلع positive فهذا 
+
+215
+00:23:29,200 --> 00:23:33,120
+positive وهذا positive و P دي closed under
+
+216
+00:23:33,120 --> 00:23:36,400
+multiplication إذن حاصل الضرب ده بيطلع positive
+
+217
+00:23:36,400 --> 00:23:45,000
+لكون حاصل الضرب هنا العناصر فيه موجبة وبالتالي
+
+218
+00:23:46,170 --> 00:23:53,450
+إذا .. إذا هذا بيطلع أكبر من الصفر هذا بيطلع الفرق
+
+219
+00:23:53,450 --> 00:23:58,030
+أكبر من أو موجب وبالتالي واحد على A أكبر من واحد
+
+220
+00:23:58,030 --> 00:24:03,770
+على B okay الأجزاء المتبقية D و E و H ممكن برهانها
+
+221
+00:24:03,770 --> 00:24:10,030
+بالمثل فاحنا دائماً بنسيب للطالب شوية حاجات يثبتها
+
+222
+00:24:11,300 --> 00:24:15,740
+يعني عشان أن الطالب يشارك شوية وإلا بيصير عملية
+
+223
+00:24:15,740 --> 00:24:20,020
+التدريس مملة لو احنا بدنا نشرح لكم كل حاجة وما نخليش
+
+224
+00:24:20,020 --> 00:24:25,880
+ولا إيش للطالب فبيصير عملية مملة وبعدين الفهم بيكون
+
+225
+00:24:25,880 --> 00:24:31,860
+ماخص كل ما أنت شاركت أكثر كل ما شعرتِ أو حسيتِ
+
+226
+00:24:31,860 --> 00:24:36,900
+بالمعلومة أكثر وكل ما فهمتيها أكثر فالحاجات هذه
+
+227
+00:24:36,900 --> 00:24:40,500
+بالإضافة للتمارين اللي في نهاية كل section في
+
+228
+00:24:40,500 --> 00:24:47,560
+الكتاب حالها كتير بساعد في فهم المادة بدون ذلك يظل
+
+229
+00:24:47,560 --> 00:24:57,420
+فهمكم ناقص ننتقل إلى نظرية أخرى نظرية 1.6
+
+230
+00:24:57,420 --> 00:25:02,680
+نظرية هذه نظرية يعني بسيطة ومهمة
+
+231
+00:25:04,730 --> 00:25:09,830
+رغم بساطتها لكن مهمة إيش بتقول النظرية هذه بتقول
+
+232
+00:25:09,830 --> 00:25:16,250
+لو أخدت أي عددين حقيقين و a أكبر من b فلازم يكون a
+
+233
+00:25:16,250 --> 00:25:26,310
+أكبر من متوسط a و b وأكبر من b البرهان بسيط هي
+
+234
+00:25:26,310 --> 00:25:32,330
+عندي الفرض أنا فارض أن a أكبر من b بتثبت أن a أكبر
+
+235
+00:25:32,330 --> 00:25:39,330
+من نصف مجموع a و b ونصف مجموع a و b أكبر من b طيب
+
+236
+00:25:39,330 --> 00:25:44,490
+نثبت المتباينة الأولى هذه نثبت المتباينة الأولى
+
+237
+00:25:44,490 --> 00:25:49,230
+الأولى بعدين نثبت الثانية
+
+238
+00:25:52,810 --> 00:25:59,530
+فالإثبات الجزء الأول فهي عندي a أكبر من b إذا لو
+
+239
+00:25:59,530 --> 00:26:07,110
+جمعت a على نفسها ده اثنين a لو جمعت على الطرفين a
+
+240
+00:26:07,110 --> 00:26:11,630
+فبيطلع عندي a زائد a أكبر من b زائد a هذه خاصية
+
+241
+00:26:11,630 --> 00:26:15,810
+أخذناها قبل a إذا اثنين a بيطلع أكبر من a زائد b
+
+242
+00:26:16,680 --> 00:26:22,900
+كذلك لو جمعت على الطرفين هنا B فبيطلع A زائد B أكبر
+
+243
+00:26:22,900 --> 00:26:26,320
+من B زائد B A زائد B أكبر من B زائد B اللي هو 
+
+244
+00:26:26,320 --> 00:26:32,320
+اثنين B إذا أنا في عندي الآن متباينتين اثنين A
+
+245
+00:26:32,320 --> 00:26:39,960
+أكبر من A زائد B هي اثنين A أكبر من A زائد B وA 
+
+246
+00:26:39,960 --> 00:26:46,700
+زائد B أكبر من اثنين B إذا by transitivity خاصية
+
+247
+00:26:46,700 --> 00:26:52,740
+التعدي ممكن استنتج أن اثنين a أكبر من a زائد b
+
+248
+00:26:52,740 --> 00:26:59,440
+أكبر من اثنين b الآن العدد اثنين عدد طبيعي وشوفنا
+
+249
+00:26:59,440 --> 00:27:03,360
+في الخاصية بتقول أي عدد طبيعي هو عدد موجب في
+
+250
+00:27:03,360 --> 00:27:09,520
+النظرية اللي فاتت كذلك أي عدد موجب مقلوبه موجب إذا 
+
+251
+00:27:09,520 --> 00:27:14,520
+النصف عدد موجب الآن لو ضربت المتباينة هذه في النصف
+
+252
+00:27:14,520 --> 00:27:19,340
+اللي هو عدد موجب إشاراتها تبقى زي ما هي هذه خاصية
+
+253
+00:27:19,340 --> 00:27:27,740
+أخذناها في النظرية هذه تمام؟ إذا أنا بضرب في النصف هي 
+
+254
+00:27:27,740 --> 00:27:33,960
+ضربت طبعاً هذا بيساوي a وهذا بيساوي b وبالتالي نحصل
+
+255
+00:27:33,960 --> 00:27:40,520
+على المطلوب إذا يعني براهين سهلة وبسيطة النظرية
+
+256
+00:27:40,520 --> 00:27:46,960
+هذه مهمة لأن نتيجة اللي بعدها أو أهميتها تظهر في
+
+257
+00:27:46,960 --> 00:27:53,580
+النتيجة اللي بعدها اللي هي corollary 1.7 corollary
+
+258
+00:27:53,580 --> 00:28:04,620
+1.7 بيقول أن
+
+259
+00:28:04,620 --> 00:28:12,040
+أي عدد موجب بيكون أكبر من نصفه اللي هو موجب أي عدد
+
+260
+00:28:12,040 --> 00:28:18,520
+حقيقي موجب دائماً أكبر من نصفه وبالتالي هذا معناه في
+
+261
+00:28:18,520 --> 00:28:23,200
+رياضيات أن الأعداد الحقيقية الموجبة مالهاش
+
+262
+00:28:23,200 --> 00:28:27,960
+smallest element مافيش .. لو أخدت الأعداد الحقيقية
+
+263
+00:28:27,960 --> 00:28:35,200
+الموجبة اللي هي set P فهذا ال set ما أقدرش أحط أصبعي
+
+264
+00:28:35,200 --> 00:28:42,360
+على أصغر عنصر فيها مالهاش أصغر عنصر has no smallest
+
+265
+00:28:42,360 --> 00:28:48,580
+element لأن لو أخدت أي عنصر موجب وسميته a فبقدر
+
+266
+00:28:48,580 --> 00:28:54,200
+ألاقي عدد موجب آخر أصغر منه اللي هو نصفه وبالتالي
+
+267
+00:28:54,200 --> 00:28:59,800
+ال set of positive numbers has no strictly
+
+268
+00:28:59,800 --> 00:29:04,500
+positive element تمام؟ البرهان تبع الكورلاري هذا
+
+269
+00:29:04,500 --> 00:29:09,360
+ينتج من النظرية يعني خد b بيساوي صفر في النظرية اللي
+
+270
+00:29:09,360 --> 00:29:26,280
+فاتت نظرية 1.6 هي تشوفها مع بعض نظرية
+
+271
+00:29:26,280 --> 00:29:30,860
+1.6 لو أخدت b بيساوي صفر فبيطلع a أكبر من نصف a
+
+272
+00:29:30,860 --> 00:29:37,970
+أكبر من صفر إذا هذه نتيجة سريعة مظبوط Okay إذا يعني
+
+273
+00:29:37,970 --> 00:29:45,050
+هذه بعض الحاجات السهلة والبسيطة، هنا في نظرية كثير
+
+274
+00:29:45,050 --> 00:29:49,950
+مهمة، هذه برضه نظرية هنستخدمها بكره يعني في 
+
+275
+00:29:49,950 --> 00:29:55,770
+المستقبل، نظرية 1.8 نظرية كثير مهمة وأهميتها
+
+276
+00:29:55,770 --> 00:30:02,670
+هنشوفها في الشبات الرجاية إيش هذه النظرية بتقول؟ لو
+
+277
+00:30:02,670 --> 00:30:08,110
+في عندي عدد حقيقي غير سالب، غير سالب، وفي نفس
+
+278
+00:30:08,110 --> 00:30:14,050
+الوقت أصغر من إبسلون لكل عدد موجب إبسلون، فهذا
+
+279
+00:30:14,050 --> 00:30:19,350
+العدد لازم يكون هو الصفر، وهي برهان بالتناقض
+
+280
+00:30:22,990 --> 00:30:28,470
+كمان مرة العدد غير السالب اللي بيكون أي أصغر من أي 
+
+281
+00:30:28,470 --> 00:30:33,590
+عدد موجب هو الصفر مافيش غير الصفر اللي بيحقق
+
+282
+00:30:33,590 --> 00:30:40,250
+لخاصية هذه لبرهان ذلك نعمل برهان بالتناقض افرض أن 
+
+283
+00:30:40,250 --> 00:30:45,830
+الـ a أن الـ a هذا بيساوي الصفر و في نفس الوقت a
+
+284
+00:30:45,830 --> 00:30:48,850
+غير سالب إذا يعني a موجب صح؟
+
+285
+00:30:52,860 --> 00:30:57,980
+الآن حسب نظرية كورينة النتيجة 1.7 إذا a 
+
+286
+00:30:57,980 --> 00:31:05,260
+بيطلع أكبر من نصف a فأخذ epsilon zero هنا عدد موجب
+
+287
+00:31:05,260 --> 00:31:11,260
+بساوي a على اتنين نصف a هذا عدد موجب إذا أنا نجحت في
+
+288
+00:31:11,260 --> 00:31:18,670
+إيجاد عدد epsilon zero عدد موجب وال a أكبر منه هذا
+
+289
+00:31:18,670 --> 00:31:22,570
+يتناقض مع الفرض أن a أصغر من إبسلون لكل إبسلون
+
+290
+00:31:22,570 --> 00:31:29,190
+أكبر من الصفر أظبط؟ لأن هذا التناقض يثبت النظرية
+
+291
+00:31:29,190 --> 00:31:39,010
+واضح تمام؟ واضح البرهان؟ عيدها طيب أنا عندي a عدد
+
+292
+00:31:39,010 --> 00:31:43,190
+حقيقي غير سالب وفي نفس الوقت أصغر من كل الأعداد
+
+293
+00:31:43,190 --> 00:31:49,030
+الموجبة إبسلون بدي أثبت أن a بيساوي صفر برهان
+
+294
+00:31:49,030 --> 00:31:53,630
+بالتناقض prove by contradiction assume or suppose
+
+295
+00:31:53,630 --> 00:31:58,910
+the contrary النقيض أو النفي تبع النتيجة يعني a ما
+
+296
+00:31:58,910 --> 00:32:03,390
+بيساوي صفر نفي a بيساوي صفر a لا تساوي صفر طب أنا 
+
+297
+00:32:03,390 --> 00:32:07,470
+كاتب هنا الـ contrary a أكبر من صفر هذا صح بناء على 
+
+298
+00:32:07,470 --> 00:32:11,930
+أن الفرض a أكبر من أكبر من صفر وما بيساوي صفر إذن
+
+299
+00:32:11,930 --> 00:32:16,970
+أكبر من صفر صح طيب الآن
+
+300
+00:32:18,230 --> 00:32:23,270
+لو أخذت Epsilon Zero بيساوي نصف A وبما أنه A عدد
+
+301
+00:32:23,270 --> 00:32:27,650
+موجب فنتيجة 1.7 بتقول لو كان A عدد موجب
+
+302
+00:32:27,650 --> 00:32:35,520
+فنصف A بيطلع عدد موجب إذا أنا وفي نفس الوقت كمان الـ 
+
+303
+00:32:35,520 --> 00:32:40,820
+a أكبر من نصف a الـ a أكبر من نصف a وبالتالي إذا أنا 
+
+304
+00:32:40,820 --> 00:32:45,900
+نجحت في إيجاد epsilon zero عدد موجب و a أكبر منه
+
+305
+00:32:45,900 --> 00:32:54,700
+هذا بتناقض مع الفرض أنه بتناقض مع الفرض أنه a أصغر
+
+306
+00:32:54,700 --> 00:33:03,090
+من epsilon لكل epsilon موجبة صح؟ العبارة هذه هينفي
+
+307
+00:33:03,090 --> 00:33:07,710
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+308
+00:33:07,710 --> 00:33:10,950
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+309
+00:33:10,950 --> 00:33:11,090
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+310
+00:33:11,090 --> 00:33:11,430
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+311
+00:33:11,430 --> 00:33:12,810
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+312
+00:33:12,810 --> 00:33:13,730
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+313
+00:33:13,730 --> 00:33:21,570
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+314
+00:33:21,570 --> 00:33:25,250
+هذه نفي هذه نفي 
+
+315
+00:33:28,550 --> 00:33:32,050
+Okay، إذا احنا لحد الآن يعني كل شغلنا مبادئ
+
+316
+00:33:32,050 --> 00:33:37,270
+رياضيات، صح؟ طيب، طب ما هي مبادئ الرياضيات هي أساس
+
+317
+00:33:37,270 --> 00:33:46,390
+الـ .. اسمها أسس الرياضيات، فاسم على مسمى فبالتالي
+
+318
+00:33:46,390 --> 00:33:50,870
+فهم مادة هذه، جابت علينا منيح يعني، هترتاح في
+
+319
+00:33:50,870 --> 00:33:57,380
+المستقبل كتير Bernoulli inequality برنولي
+
+320
+00:33:57,380 --> 00:34:01,760
+الـ inequality هذه يعني في شوية متباينات طبعا مهمة في
+
+321
+00:34:01,760 --> 00:34:06,860
+الكتاب أنا اخترت واحدة منهم لكن في بعض المتباينات
+
+322
+00:34:06,860 --> 00:34:13,180
+الأخرى موجودة في الكتاب وأرجو أنكم تقرأوها فبرنولي 
+
+323
+00:34:13,180 --> 00:34:15,960
+الـ inequality هذه واحدة منهم متباينة برنولي يعني
+
+324
+00:34:15,960 --> 00:34:23,230
+بيقول لو كان X عدد حقيقي أكبر من سالب واحد فمجموعه
+
+325
+00:34:23,230 --> 00:34:28,750
+واحد و X to the power N دائما أكبر من أو يساوي واحد
+
+326
+00:34:28,750 --> 00:34:38,850
+زائد N ضرب X وهذا صحيح لكل الأعداد الطبيعية نعم
+
+327
+00:34:38,850 --> 00:34:46,290
+في نظرية جاب لها حجم ما خليناش نشوف
+
+328
+00:34:46,290 --> 00:34:46,890
+مع بعض
+
+329
+00:34:50,730 --> 00:35:07,610
+اه صحيح نشوف النظرية 1.9 نظرية
+
+330
+00:35:07,610 --> 00:35:10,870
+1.9 بتقول لو كان عندي عددين حقيقين حاصل
+
+331
+00:35:10,870 --> 00:35:15,610
+ضربهم موجب في إما إثنين موجبين يا إما إثنين 
+
+332
+00:35:15,610 --> 00:35:20,450
+سالبين صح؟ ممكن يكون الإثنين مختلفين في الإشارة و
+
+333
+00:35:20,450 --> 00:35:25,530
+حاصل ضربهم موجب إذا حصل ضرب عددين موجب بيقدر أنه
+
+334
+00:35:25,530 --> 00:35:33,670
+إما إثنين موجبين أو إثنين سالبين فالبرهان نشوف كيف
+
+335
+00:35:33,670 --> 00:35:39,640
+افرض الفرض تبعنا أن حاصل ضرب A وB موجب فهذا أكيد
+
+336
+00:35:39,640 --> 00:35:42,780
+بيؤدي أن لا الـ a بيساوي صفر ولا الـ b بيساوي صفر
+
+337
+00:35:42,780 --> 00:35:46,520
+لأن لو واحد منهم بيساوي صفر فحاصل الضرب هيطلع
+
+338
+00:35:46,520 --> 00:35:50,720
+بيساوي صفر contradiction تناقض صح؟ لأن هذا 
+
+339
+00:35:50,720 --> 00:36:02,020
+الاستنتاج منطقي طيب الآن احنا الـ a ناخذ ناخذ الجزء
+
+340
+00:36:02,020 --> 00:36:09,250
+هذا الآن أنا عند a لا يساوي صفر يبقى بتراي كاتومي
+
+341
+00:36:09,250 --> 00:36:14,650
+property حسب الخاصية التي هي إما a أكبر من صفر أو
+
+342
+00:36:14,650 --> 00:36:23,650
+a أصغر من صفر صح؟ يبقى في احتمالين طيب ناخذ الـ a لو
+
+343
+00:36:23,650 --> 00:36:30,370
+كان افرض أن a أكبر من صفر فهذا بيؤدي أن واحد على a
+
+344
+00:36:30,370 --> 00:36:42,620
+أكبر من صفر هذا يعني 1 على a أكبر من 0 بيؤدي
+
+345
+00:36:42,620 --> 00:36:48,700
+أيضًا إلى بي اللي هو بيساوي الـ 
+
+346
+00:36:48,700 --> 00:36:53,720
+بي ممكن أكتبها واحد في بي والواحد ممكن أبدله بواحد 
+
+347
+00:36:53,720 --> 00:36:57,800
+على a في a واستخدم الـ associative law واكتب هذا
+
+348
+00:36:57,800 --> 00:37:04,910
+على صورة واحد على a في a بي الآن هذا موجب وهذا موجب
+
+349
+00:37:04,910 --> 00:37:12,330
+إذا حصلت ضرب بيطلع موجب إذا هذه أثبتت أن الـ a أكبر
+
+350
+00:37:12,330 --> 00:37:19,950
+من الـ b أكبر من الصفر لأ احنا أخذنا الـ a أكبر من 
+
+351
+00:37:19,950 --> 00:37:24,290
+الصفر فأدت
+
+352
+00:37:24,290 --> 00:37:28,690
+إلى أن الـ b
+
+353
+00:37:28,690 --> 00:37:32,430
+أكبر من الصفر وبالتالي بيطلع الـ a والـ b موجبين
+
+354
+00:37:34,090 --> 00:37:40,810
+بالمثل لو افترضت .. أخذت لو افترضت أن a سالب فطبعا
+
+355
+00:37:40,810 --> 00:37:46,410
+مقلوب العدد السالب بيطلع سالب وبالتالي الـ b اللي
+
+356
+00:37:46,410 --> 00:37:52,830
+هي بتساوي واحد على a في a b زي ما عملنا هنا الـ b
+
+357
+00:37:52,830 --> 00:37:56,810
+بتطلع بتساوي واحد على a في a b ف ..
+
+358
+00:38:00,670 --> 00:38:05,850
+فهذا بيطلع الحاصل بضرب سالب لأن عندي أنا هي هذه
+
+359
+00:38:05,850 --> 00:38:12,290
+المتباينة هذه هي واحد على a سالب لو ضربت المتباينة
+
+360
+00:38:12,290 --> 00:38:18,370
+هذه في العدد الموجب a,b اللي هو عدد موجب فبيصير
+
+361
+00:38:18,370 --> 00:38:21,910
+المتباينة هذه عبارة عن واحد على a في a,b الطرف
+
+362
+00:38:21,910 --> 00:38:29,070
+الشمال وضربتها في عدد موجب فبيطلع أصغر من صفر في a
+
+363
+00:38:29,070 --> 00:38:30,030
+,b اللي هو صفر 
+
+364
+00:38:32,730 --> 00:38:38,730
+وبالتالي بيطلع عندي الـ B بيطلع عند الـ B التي هي
+
+365
+00:38:38,730 --> 00:38:46,390
+أصغر من الصفر إذا مرة ثانية لو فرضنا أن a,b أكبر من 0
+
+366
+00:38:46,390 --> 00:38:50,670
+فشفنا أن لا الـ a بيساوي 0 ولا الـ b بيساوي 0
+
+367
+00:38:50,670 --> 00:38:56,670
+وبالتالي إما بيطلع a أكبر من 0 أو a أصغر من 0 في
+
+368
+00:38:56,670 --> 00:39:01,010
+الحالة الأولى لو كان a أكبر من 0 بيطلع b أكبر من 0
+
+369
+00:39:01,010 --> 00:39:05,170
+وبالتالي a وb موجبين في الاحتمال الثاني أو الحالة
+
+370
+00:39:05,170 --> 00:39:09,940
+الثانية لو كان a سالب فشفنا أن بيطلع b سالب و
+
+371
+00:39:09,940 --> 00:39:18,480
+بالتالي إثنين سالبين okay تمام نشوف
+
+372
+00:39:18,480 --> 00:39:27,240
+الآن Bernoulli inequality اليوم
+
+373
+00:39:27,240 --> 00:39:32,620
+هناخد برهان by induction برضه مبادئ الرياضيات خلنا
+
+374
+00:39:32,620 --> 00:39:39,360
+برهان by contradiction و direct proof وهنشوف move
+
+375
+00:39:39,360 --> 00:39:44,880
+by induction نمسح 
+
+376
+00:39:44,880 --> 00:39:49,300
+اللوح برنولي 
+
+377
+00:39:49,300 --> 00:39:52,420
+الـ equality زي ما قلنا لو كان x عدد حقيقي أكبر من 
+
+378
+00:39:52,420 --> 00:39:58,000
+سالب واحد فلمّا أضيف عليه واحد وأرفع لقوة n هذا
+
+379
+00:39:58,000 --> 00:40:02,380
+بيطلع أكبر من أو يساوي واحد زائد n في x وهذا صحيح
+
+380
+00:40:02,380 --> 00:40:07,290
+لكل الأعداد الطبيعية البرهان by induction لو كانت n
+
+381
+00:40:07,290 --> 00:40:12,690
+بيساوي واحد بأثبت صحة العبارة عند n بيساوي واحد لأن
+
+382
+00:40:12,690 --> 00:40:20,350
+إبدأ من واحد فلو كان n بيساوي واحد فالطرف
+
+383
+00:40:20,350 --> 00:40:23,530
+الشمال 
+
+384
+00:40:23,530 --> 00:40:31,900
+بيطلع واحد زائد x صح؟ الطرف الشمال واحد زائد X والطرف
+
+385
+00:40:31,900 --> 00:40:37,240
+اليمين برضه واحد زائد X فبيطلع مساواة وطبعا
+
+386
+00:40:37,240 --> 00:40:42,720
+المساواة بقدر أبدلها بأكبر من أو يساوي مافي مشكلة
+
+387
+00:40:42,720 --> 00:40:46,020
+تمام؟
+
+388
+00:40:46,020 --> 00:40:52,700
+إذا العبارة هذه صحيحة عند N بيساوي واحد الآن نفترض
+
+389
+00:40:52,700 --> 00:40:57,780
+أن العبارة صحيحة عند N بيساوي K حيث K أكبر من
+
+390
+00:40:57,780 --> 00:41:05,800
+واحد هذا ما نسميه induction hypothesis الفرض تبع الـ
+
+391
+00:41:05,800 --> 00:41:12,740
+induction نفترض صحة العبارة عند N بيساوي K حيث K 
+
+392
+00:41:12,740 --> 00:41:18,080
+أكبر من 1 هذا معناه أن 1 زائد X to K bigger than or 
+
+393
+00:41:18,080 --> 00:41:24,510
+equal to 1 plus K X طيب الآن نريد نكمل الـ induction
+
+394
+00:41:24,510 --> 00:41:32,790
+عايزين نثبت صحة العبارة 1 اللي هي هذه العبارة
+
+395
+00:41:32,790 --> 00:41:38,730
+1 مش عارف من الواحد رايح العبارة 1 هذه نثبت 
+
+396
+00:41:38,730 --> 00:41:45,650
+الصحة عندنا بيساوي K زائد واحد طيب from 2 هذه
+
+397
+00:41:45,650 --> 00:41:47,870
+العبارة 2 اللي هي induction hypothesis
+
+398
+00:41:52,040 --> 00:41:59,640
+بتدفع في الـ type هاي العبارة هذه لما n بيساوي k
+
+399
+00:41:59,640 --> 00:42:05,880
+زائد واحد هتصير واحد زائد x الكل أس k زائد واحد
+
+400
+00:42:05,880 --> 00:42:12,410
+أكبر من أو يساوي واحد زائد k زائد واحد في X هذه
+
+401
+00:42:12,410 --> 00:42:18,390
+العبارة and N بيساوي K زائد 1 نبدأ بالطرف الشمال و
+
+402
+00:42:18,390 --> 00:42:22,670
+نثبت أنه أكبر من أو يساوي الطرف اليمين هاي الطرف
+
+403
+00:42:22,670 --> 00:42:27,950
+الشمال بقدر أجزئه حسب قوانين الأسس ل1 plus X to K
+
+404
+00:42:27,950 --> 00:42:34,410
+و1 زائد X to K ضرب 1 زائد X الآن من 2 من العبارة
+
+405
+00:42:34,410 --> 00:42:38,070
+الثانية one plus X to K اللي هو induction
+
+406
+00:42:38,070 --> 00:42:43,340
+hypothesis حسب 2 هذا أكبر من أو يساوي واحد زائد
+
+407
+00:42:43,340 --> 00:42:48,220
+K X مضروب في واحد زائد X بنضرب هدول في بعض و
+
+408
+00:42:48,220 --> 00:42:55,440
+بنرتب فبيطلع حاصل الضرب هذا هو واحد زائد K زائد
+
+409
+00:42:55,440 --> 00:43:01,340
+واحد في X زائد K في X تربيع الآن
+
+410
+00:43:01,340 --> 00:43:08,860
+هذا هذا عدد موجب هذا عدد موجب لأن K عدد طبيعي و X 
+
+411
+00:43:08,860 --> 00:43:16,060
+تربيع عدد موجب لما أشيل هذا أشطب فبيصغر المقدار
+
+412
+00:43:16,060 --> 00:43:20,580
+لما أشيل عدد موجب من عدد أو أنقص من عدد عدد موجب
+
+413
+00:43:20,580 --> 00:43:27,580
+بيصغر فبالتالي هذا أكبر من واحد زائد K زائد واحد في X
+
+414
+00:43:28,490 --> 00:43:33,310
+وهذا هو الطرف اليمين للمتباينة 1 اللي احنا عايزين 
+
+415
+00:43:33,310 --> 00:43:38,050
+نثبت صحتها عند m بيساوي k زائد واحد إذا this
+
+416
+00:43:38,050 --> 00:43:42,830
+completes the induction هذا بيكمل البرهان بال
+
+417
+00:43:42,830 --> 00:43:50,050
+induction مظبوط صح تمام واضح إذا هي صار في أن
+
+418
+00:43:50,050 --> 00:43:54,170
+متباينة
+
+419
+00:43:54,170 --> 00:44:01,080
+Bernoulli زي ما قلنا لكم في في الفي الـ section هذا
+
+420
+00:44:01,080 --> 00:44:08,240
+بعض المتباينات الأخرى فبإمكانكم تقرأوها الـ 
+
+421
+00:44:08,240 --> 00:44:12,260
+الآن خلصنا احنا section 2.1  أعتقد
+
+422
+00:44:12,260 --> 00:44:20,040
+فالمسائل المطلوب أنكم تحلوها اللي هي موجودة مرسومة
+
+423
+00:44:20,040 --> 00:44:23,280
+هنا وبرضه 
+
+424
+00:44:23,280 --> 00:44:26,500
+زي ما قلت لكم في الـ syllabus موجود على الصفحة تبعتي
+
+425
+00:44:27,940 --> 00:44:32,720
+فبرضه في الـ homework هذا موجود ل... مش للـ
+
+426
+00:44:32,720 --> 00:44:38,400
+section هذا لكل ال... المنهج إذا نبدأ نحل المسائل
+
+427
+00:44:38,400 --> 00:44:44,180
+هذه وإن شاء الله لأسبوع الجاي بنعمل مناقشة فأنا
+
+428
+00:44:44,180 --> 00:44:49,200
+هعمل مناقشة... أنا اللي هكون مناقشة لكم اليوم لأ
+
+429
+00:44:49,200 --> 00:44:52,960
+مافيش مناقشة لأنه لسه احنا يعني ماخدناش الـ material 
+
+430
+00:44:52,960 --> 00:45:00,110
+كافية أو اللي لسه يعني مش مهيئين أو مش محضرين
+
+431
+00:45:00,110 --> 00:45:06,170
+فهنواصل ونحاول إن شاء الله أسبوع الجاي ناخد كل
+
+432
+00:45:06,170 --> 00:45:10,450
+ساعة هذه الساعة
+
+433
+00:45:10,450 --> 00:45:12,930
+الأخيرة هذه المتأخرة نعملها مناقشة 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..c1cfe8622abc164369c3c94c30075d30aaa39f1c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Nztl0T85AIM_postprocess.srt
@@ -0,0 +1,1732 @@
+1
+00:00:21,210 --> 00:00:28,030
+انراجع مع بعض ال order properties of R او خواص
+
+2
+00:00:28,030 --> 00:00:33,790
+الترتيب للأعداد الحقيقية احنا من بداية ال chapter
+
+3
+00:00:33,790 --> 00:00:37,630
+قلنا انه ال real number system نظام الأعداد
+
+4
+00:00:37,630 --> 00:00:43,250
+الحقيقية نظام
+
+5
+00:00:43,250 --> 00:00:52,230
+الأعداد الحقيقية يتكون من مجموعة R boldface Rمع
+
+6
+00:00:52,230 --> 00:00:57,370
+عمليتين فنائيتين واحدة عملية الجامعة واحدة عملية
+
+7
+00:00:57,370 --> 00:01:04,390
+الضرب وافترضنا ان العمليات هذه بتحقق خمس خواص اللي
+
+8
+00:01:04,390 --> 00:01:08,910
+هي خواص ال field اللي هو ال commutative law,
+
+9
+00:01:09,050 --> 00:01:17,690
+associative law, distributive laws, existence of
+
+10
+00:01:17,690 --> 00:01:23,340
+identities, existence of inversesبعدين ضفنا على
+
+11
+00:01:23,340 --> 00:01:28,300
+ذلك انه افترضنا انه ال real number system R بتحقق
+
+12
+00:01:28,300 --> 00:01:33,020
+برضه خاصية الترتيب او خواص الترتيب اللي هي الخاصية
+
+13
+00:01:33,020 --> 00:01:38,440
+السادسة هذه الخاصية السادسة هذه تجزأت يعني تنص على
+
+14
+00:01:38,440 --> 00:01:43,560
+ما يليه نفترض انه يوجد مجموعة جزئية من R غير خالية
+
+15
+00:01:43,560 --> 00:01:49,360
+و المجموعة الجزئية هذه بنسميها P اللي هو اول حرف
+
+16
+00:01:49,360 --> 00:01:56,080
+في positiveعشان نسميها بعد هيك the set of positive
+
+17
+00:01:56,080 --> 00:02:02,020
+real numbers ف ال set P هذه closed under addition
+
+18
+00:02:02,020 --> 00:02:08,540
+and under multiplication كمان نفترض أن ال set P
+
+19
+00:02:08,540 --> 00:02:13,180
+هذه بتحقق الخاصية الثلاثية ال trichotomy property
+
+20
+00:02:14,240 --> 00:02:18,920
+which means that for any real number a exactly one
+
+21
+00:02:18,920 --> 00:02:27,060
+of the three possibilities holds either a belongs
+
+22
+00:02:27,060 --> 00:02:34,420
+to p or a equals zero or negative a belongs to p
+
+23
+00:02:34,420 --> 00:02:42,230
+بناء على هذه الخاصية شفنا أن الأعداد الحقيقيةgets
+
+24
+00:02:42,230 --> 00:02:47,350
+partitioned to three mutually disjoint sets يعني
+
+25
+00:02:47,350 --> 00:02:54,450
+الخاصية هذه بتجزق بتخليني أجزق العداد الحقيقية إلى
+
+26
+00:02:54,450 --> 00:03:00,110
+تلت مجموعات منفصلة مثنى مثنى pair-wise disjoint
+
+27
+00:03:00,110 --> 00:03:05,650
+يعني إن لو أخدت أي مجموعتين عشوائيتين من التلاتة
+
+28
+00:03:05,650 --> 00:03:09,310
+تقطعهم بساوي فايل مافيش بينهم عناصر و مش تلتين
+
+29
+00:03:10,410 --> 00:03:15,170
+واتحادهم بساوي ال R، لأن هذا بشكل تجزء على ال R،
+
+30
+00:03:15,170 --> 00:03:19,630
+تجزء على ال R أو بنسميها في الرياضيات partition of
+
+31
+00:03:19,630 --> 00:03:25,210
+R ال set P هذه سمنها set of positive real numbers
+
+32
+00:03:25,210 --> 00:03:35,210
+وعرفنا negative P على إنها negative عناصر ال set P
+
+33
+00:03:44,210 --> 00:03:49,430
+Okay فهي معرفة negative P هي كل ال elements
+
+34
+00:03:49,430 --> 00:03:56,850
+negative A such that A element in P بعدين
+
+35
+00:03:56,850 --> 00:04:02,430
+عرفنا علاقة الترتيب، الآن بنعرف اللي هو order
+
+36
+00:04:02,430 --> 00:04:08,340
+relation على Rما معنى أنه a لو في ending two real
+
+37
+00:04:08,340 --> 00:04:12,720
+numbers ما معنى a أصغر من b أو b أكبر من a قولنا
+
+38
+00:04:12,720 --> 00:04:19,820
+معناها أن الفرق بين b و a is positive real number
+
+39
+00:04:19,820 --> 00:04:24,480
+أو ينتمي لمجموعة الأعداد المجتمعةطب ما معناه a
+
+40
+00:04:24,480 --> 00:04:28,760
+أصغر من أو ساوي b أو b أكبر من أو ساوي a؟ معناته
+
+41
+00:04:28,760 --> 00:04:33,240
+الفرق بين b و a ينتمي للأعداد الموجبة، يعني الفرق
+
+42
+00:04:33,240 --> 00:04:40,120
+موجب أو يساوي سفر أو يساوي سفر، إذن معناه تاني طيب
+
+43
+00:04:40,120 --> 00:04:50,940
+و أعتقد إن احنا بعد هيك أثبتنا أه
+
+44
+00:04:50,940 --> 00:04:52,780
+وقفنا عند النظرية هذه
+
+45
+00:04:57,810 --> 00:05:02,310
+نظرية واحد خمسة قلنا إنه لأي لو أخدت أي تلت أعداد
+
+46
+00:05:02,310 --> 00:05:08,730
+حقيقية فعند الخواص التالية تتحقق نجموعة الخواص هذه
+
+47
+00:05:08,730 --> 00:05:17,210
+تتحقق فالخواص
+
+48
+00:05:17,210 --> 00:05:21,630
+هذه هذا
+
+49
+00:05:21,630 --> 00:05:27,330
+هي أمامكم transitivity خاصية التعدىايه يعني
+
+50
+00:05:27,330 --> 00:05:35,310
+التعدى؟ يعني اذا انا في عندي تلت أعداد حقه في A و
+
+51
+00:05:35,310 --> 00:05:38,770
+B و C
+
+52
+00:05:43,970 --> 00:05:52,530
+وكان B هنا أكبر .. B أكبر من A and C أكبر من B
+
+53
+00:05:52,530 --> 00:05:55,790
+فهذا
+
+54
+00:05:55,790 --> 00:06:09,770
+بيؤدي أنه C أكبر من A خليني
+
+55
+00:06:09,770 --> 00:06:15,290
+أنا أكسهم عشان .. كليهم زي ..مهمة موجودة هناك هي a
+
+56
+00:06:15,290 --> 00:06:27,090
+أكبر من b هي a أكبر من b و b أكبر من c فبطلع
+
+57
+00:06:27,090 --> 00:06:38,470
+c أو a بطلع أكبر من cهذه a أكبر من b و b أكبر من c
+
+58
+00:06:38,470 --> 00:06:44,750
+إذا نقدر نتعدى و نقول a أكبر من c فهذه بيسموها في
+
+59
+00:06:44,750 --> 00:06:50,730
+الرياضيات transitivity أو خاصية التعدى الخاصية
+
+60
+00:06:50,730 --> 00:06:53,990
+التانية
+
+61
+00:06:53,990 --> 00:06:59,930
+بنسميها tricotomy برضه خاصية ثلاثية جاية من
+
+62
+00:06:59,930 --> 00:07:05,770
+الخاصية الثلاثية اللى شفناها قبل شويةفبتقول
+
+63
+00:07:05,770 --> 00:07:08,650
+exactly one of the following holds واحد من تلات
+
+64
+00:07:08,650 --> 00:07:19,190
+احتمالات بتحصل اما a اكبر من b او a بتساوي b او a
+
+65
+00:07:19,190 --> 00:07:28,260
+اصغر من bلأي عددين حقيقيين A وB واحد فقط من
+
+66
+00:07:28,260 --> 00:07:32,800
+الاحتمالات التلاتة بيكون صحيح وهو اما A أكبر من B
+
+67
+00:07:32,800 --> 00:07:38,860
+أو A بساوي B أو A أصغر من B الـ Antisymmetry
+
+68
+00:07:38,860 --> 00:07:43,640
+property علاقة أكبر من أو ساويها دي بنسميها
+
+69
+00:07:43,640 --> 00:07:48,400
+Antisymmetric يعني ايه؟ بتحقق خاصية تضاد التماثل
+
+70
+00:07:49,760 --> 00:07:54,960
+إيه يعني؟ مع أن لو كانت A على علاقة مع B و B على
+
+71
+00:07:54,960 --> 00:08:00,940
+علاقة مع A فلازم يطلع A بساوي B، A أكبر من أو ساوي
+
+72
+00:08:00,940 --> 00:08:05,300
+B و B أكبر من أو ساوي A فلازم A ساوي B، هذي
+
+73
+00:08:05,300 --> 00:08:11,640
+بنسميها Anti-symmetry propertyهنا الخاصية هذه لو
+
+74
+00:08:11,640 --> 00:08:18,140
+كان a أكبر من b وضفت للطرفين أي عدد c فالمتباينة
+
+75
+00:08:18,140 --> 00:08:22,040
+تبقى زي ما هي شريتها زي ما هي طيب لو في عندي
+
+76
+00:08:22,040 --> 00:08:26,100
+متباينة a أكبر من b لان نتحدث عن متباينات
+
+77
+00:08:26,100 --> 00:08:31,960
+inequalitiesلو كان a أكبر من b و c عدد موجب وضربت
+
+78
+00:08:31,960 --> 00:08:35,960
+الطرفين في عدد الموجب c فإشارة المتباينة تبقى كما
+
+79
+00:08:35,960 --> 00:08:40,380
+هي لكن لو ضربت المتباينة في عدد سالب إشارة
+
+80
+00:08:40,380 --> 00:08:46,900
+المتباينة تناكز الخاصية f بتقول أنه لأي عدد حقيقي
+
+81
+00:08:46,900 --> 00:08:51,300
+لا يساوي سفر مربع أي عدد حقيقي لا يساوي سفر دائما
+
+82
+00:08:51,300 --> 00:08:55,400
+بيكون عدد موجب الواحد
+
+83
+00:08:55,900 --> 00:08:59,780
+الـ Distinguished elements في R أو في الـ real
+
+84
+00:08:59,780 --> 00:09:02,940
+number system اللي هم السفر والواحد اللي هو ال
+
+85
+00:09:02,940 --> 00:09:07,280
+identity elements سمناهم بيحققوا ان واحد دايما
+
+86
+00:09:07,280 --> 00:09:13,400
+اكبر من السفر و سالب واحد اصغر من السفر كمان لأي
+
+87
+00:09:13,400 --> 00:09:16,840
+عدد طبيعي هذي the set of natural numbers اي عدد
+
+88
+00:09:16,840 --> 00:09:23,210
+طبيعي بيكون دايما موجب اي عدد طبيعي بيطلع موجبلو
+
+89
+00:09:23,210 --> 00:09:27,450
+كان a عدد حقيقي موجب فمقلوبه موجب لو كان a عدد
+
+90
+00:09:27,450 --> 00:09:38,990
+حقيقي سالب مقلوبه بيطلع سالب الخاصية
+
+91
+00:09:38,990 --> 00:09:45,610
+الأخيرة ال لو كان a أصغر من b و اتنين موجبين
+
+92
+00:09:45,610 --> 00:09:53,160
+فمقلوب لصغير أكبر من مقلوبالكبير أو مقلوب الكبير
+
+93
+00:09:53,160 --> 00:09:56,680
+أصغر من مقلوب الصغير بصرت اتنين اللي هم نفس
+
+94
+00:09:56,680 --> 00:10:00,980
+الإشارة لكن لو كان واحد موجة بواحد سالب فالكلام
+
+95
+00:10:00,980 --> 00:10:08,060
+هذا مش صحيح خدوا بالك طيب نشوف نمر بسرعة على
+
+96
+00:10:08,060 --> 00:10:15,500
+البرهين قرأته البرهين انتوا؟ طيب
+
+97
+00:10:30,910 --> 00:10:38,550
+خاصية التعدى خاصية التعدى انا كان عندي a أكبر من b
+
+98
+00:10:38,550 --> 00:10:44,710
+and b أكبر من c بدنا نثبت ان هذا يعدي ان a أكبر من
+
+99
+00:10:44,710 --> 00:10:52,250
+c فالبرهان ذلك يكفي نثبت ان الفرق بين c و a موجب
+
+100
+00:10:52,990 --> 00:10:56,990
+يعني ينتمي لل set P of positive real numbers
+
+101
+00:10:56,990 --> 00:11:02,450
+فتعالوا نثبت الكلام هذا أنا عندي من المعطيات او من
+
+102
+00:11:02,450 --> 00:11:08,370
+الفرض الفرق هذا موجب والفرق هذا موجب من المعطيات
+
+103
+00:11:09,240 --> 00:11:13,680
+طيب set P closed under addition مغلقة تحت عملية
+
+104
+00:11:13,680 --> 00:11:18,820
+الجمع إذا مجموعة أنصرين في P بيطلع أنصر تالت في P
+
+105
+00:11:18,820 --> 00:11:22,660
+هذا الأنصر التالت اللي بيقول المجموعة طلع A سالب C
+
+106
+00:11:22,660 --> 00:11:28,560
+هذا معناه مادام الفرخ هذا تملى P معناته الفرخ هذا
+
+107
+00:11:28,560 --> 00:11:33,900
+موجب أو A أكبر من C as required كما هو مطلوب،
+
+108
+00:11:33,900 --> 00:11:38,040
+مظبوط؟ واضح؟ طيب
+
+109
+00:11:42,860 --> 00:11:49,940
+أي عدد حقيقي له واحد من تلت احتمالات اما موجب او
+
+110
+00:11:49,940 --> 00:11:56,720
+سفر او سالب الان بناء على هذه الخاصية ممكن نثبت
+
+111
+00:11:56,720 --> 00:12:01,940
+الخاصية الثلاثية الخاصية
+
+112
+00:12:01,940 --> 00:12:05,600
+بي
+
+113
+00:12:09,230 --> 00:12:15,030
+قلنا إن لو كان لأي عددين حقيقيين لأي عددين حقيقيين
+
+114
+00:12:15,030 --> 00:12:19,750
+a و b، a أكبر من b أو a بساوي b أو a أصغر من b
+
+115
+00:12:19,750 --> 00:12:22,910
+فالبرهان
+
+116
+00:12:22,910 --> 00:12:27,350
+ذلك بيعتمد على ال try-cutting property اللي شفناها
+
+117
+00:12:27,350 --> 00:12:33,470
+قبل شوية فأنا
+
+118
+00:12:33,470 --> 00:12:33,870
+عندي
+
+119
+00:12:37,890 --> 00:12:41,310
+حسب الـ trichotomy property، لو أخدت الفرق هذا،
+
+120
+00:12:41,310 --> 00:12:46,850
+هذا real number فأي real number إما positive أو
+
+121
+00:12:46,850 --> 00:12:54,390
+بساوي سفر أو negative، صح؟ وهذا بكافئ، الكلام هذا
+
+122
+00:12:54,390 --> 00:13:01,210
+بكافئ A سالب B ينتمي ل B بكافئ أنه الـ A أكبر من B
+
+123
+00:13:02,260 --> 00:13:07,220
+طب وهذا ينتمي لـ 0 بكافة أن a بساوي b أو الفرق
+
+124
+00:13:07,220 --> 00:13:11,660
+بساوي 0 وبالتالي a بساوي b و الفرق هذا ينتمي ل
+
+125
+00:13:11,660 --> 00:13:16,180
+negative b معناته الفرق هذا سالب يعني معناه أن a
+
+126
+00:13:16,180 --> 00:13:21,760
+أصغر من b وهذا اللي بدنا إياه هذا اللي بدنا إياه
+
+127
+00:13:21,760 --> 00:13:25,620
+طيب
+
+128
+00:13:25,620 --> 00:13:32,390
+الجزء C قلنا اللي هو ال antisymmetry propertyالـ
+
+129
+00:13:32,390 --> 00:13:37,990
+Anti-symmetry property نفكركم فيها بتقول لو كان a
+
+130
+00:13:37,990 --> 00:13:46,590
+أكبر من أو يساوي b and b أكبر من أو يساوي a فهذا
+
+131
+00:13:46,590 --> 00:13:52,210
+بيعدي أن a بساوي b، بظبط؟ طيب
+
+132
+00:13:57,150 --> 00:14:02,570
+أنا بدأ أثبت أن A بساوي B، هذه النتيجة، فبدأ أعمل
+
+133
+00:14:02,570 --> 00:14:07,750
+برهان بالتناقض، فبرهان بالتناقض دائما نفرض مافيه
+
+134
+00:14:07,750 --> 00:14:12,670
+النتيجة هو الصح، وبنفسها إلى التناقض، ف assume أن
+
+135
+00:14:12,670 --> 00:14:21,500
+A لا تساوي Bإذا حسب الخاصية الفلاثية هذا بيقدّي ان
+
+136
+00:14:21,500 --> 00:14:30,800
+اما a أصغر من b or b أصغر من a، مظبوط؟ طيب إذا هنا
+
+137
+00:14:30,800 --> 00:14:39,720
+.. الآن لو أخدت .. لو أخدت ال a أكبر من b اللي هو
+
+138
+00:14:39,720 --> 00:14:46,880
+الاحتمال هذالو أخدت .. لو قلت أن a أكبر من b فهذا
+
+139
+00:14:46,880 --> 00:14:54,140
+بتناقض مع الفرض .. بتناقض مع الفرض أن a أصغر من ..
+
+140
+00:14:54,140 --> 00:15:01,160
+a أصغر من أوسع من b هدول اتنين بيعطون التناقض طيب
+
+141
+00:15:01,160 --> 00:15:06,400
+لو افترضت الاحتمال التاني أن a أصغر من b فهذا
+
+142
+00:15:06,400 --> 00:15:15,210
+بتناقض مع الفرض أن a أكبر منأو يساوي الـ B إذا في
+
+143
+00:15:15,210 --> 00:15:20,790
+الحالتين لو فرضت هذا صح بتناقض مع هذا الجزء لو
+
+144
+00:15:20,790 --> 00:15:25,410
+فرضت هذا صح بتناقض مع هذا الجزء اللي هو جزء من
+
+145
+00:15:25,410 --> 00:15:29,970
+الفرض وبالتالي في كلتا الحالتين الفرض أن A لا
+
+146
+00:15:29,970 --> 00:15:35,050
+يساوي B أدى إلى تناقض إذا الصح أن A لازم تساوي B
+
+147
+00:15:35,050 --> 00:15:40,600
+كما هو مطلوب okay هذا برهان بالتناقضواضح تمام
+
+148
+00:15:40,600 --> 00:15:47,740
+مفهوم فاهمين ولا هيك يعني أمور
+
+149
+00:15:47,740 --> 00:15:53,040
+سهلة وبسيطة وكلها يعني مبادئ رياضيات احنا هنا يعني
+
+150
+00:15:53,040 --> 00:15:58,840
+مراجعة لمبادئ رياضيات أو طرق البرهان في مبادئ
+
+151
+00:15:58,840 --> 00:16:03,680
+رياضيات طيب
+
+152
+00:16:03,680 --> 00:16:07,220
+الآن بنثبت القصية F
+
+153
+00:16:13,430 --> 00:16:18,050
+لأي عدد حقيقي لا يساوي سفر دائما مربع و بيطلع موجب
+
+154
+00:16:18,050 --> 00:16:22,330
+فعشان أثبت مربع ال a موجب لازم أثبت ان مربع ال a
+
+155
+00:16:22,330 --> 00:16:30,890
+ينتمي لفئة او مجموعة العداد الموجبة طيب احنا فرضين
+
+156
+00:16:30,890 --> 00:16:34,950
+a لايساوي سفر اذا by tricotomy property بالخاصية
+
+157
+00:16:34,950 --> 00:16:40,470
+التلاتية اما a موجب او سالب يعني معناه هذا او هذا
+
+158
+00:16:40,470 --> 00:16:51,220
+الانلو كانت ال A موجبة فمربع و ال P مغلقة تحت
+
+159
+00:16:51,220 --> 00:16:56,440
+عملية الضرب فحاصل ضرب A في A اللي هو A تربية بيطلع
+
+160
+00:16:56,440 --> 00:17:03,580
+ينتمي يعني هذا بيساوي A تربية ال
+
+161
+00:17:03,580 --> 00:17:07,660
+A ينتمي ل P إذا حاصل الضرب ينتمي ل P وبالتالي A
+
+162
+00:17:07,660 --> 00:17:12,080
+تربية موجبة okay وهذا اللي احنا عايزينهالحالة
+
+163
+00:17:12,080 --> 00:17:17,600
+التانية طب افرض انه negative A تنتمي ل P او A
+
+164
+00:17:17,600 --> 00:17:23,200
+تنتمي ل negative P يعني A سالم ففي الحالة هذه لو
+
+165
+00:17:23,200 --> 00:17:29,220
+ضربت هذا العنصر في نفسه بطلع ينتمي إلى ال P بطلع
+
+166
+00:17:29,220 --> 00:17:33,480
+ينتمي إلى ال P وهذا بطلع بساوي من الخواص اللي
+
+167
+00:17:33,480 --> 00:17:37,510
+أخدناها قبل هيكيعني هذا عبارة عن هذا سالب إيه
+
+168
+00:17:37,510 --> 00:17:40,970
+بكتبه سالب واحد في إيه و سالب إيه التاني نفس
+
+169
+00:17:40,970 --> 00:17:45,650
+الحاجة سالب واحد في إيه فبطلع سالب واحد في سالب
+
+170
+00:17:45,650 --> 00:17:50,170
+واحد في إيه تربية و هذا واحد فبطلع إيه تربية تنتمي
+
+171
+00:17:50,170 --> 00:17:54,830
+لدي وبالتالي إيه تربية موجبة إذا هنا أثبتنا إن أي
+
+172
+00:17:54,830 --> 00:17:59,690
+عدد حقيقي مختلف عن السفر دائما مربع موجب
+
+173
+00:18:13,230 --> 00:18:23,990
+خاصية جي الخاصية
+
+174
+00:18:23,990 --> 00:18:24,510
+جي
+
+175
+00:18:30,830 --> 00:18:36,810
+احنا بنفبط أن الواحد أكبر من السفر فبكل بساطة واحد
+
+176
+00:18:36,810 --> 00:18:42,710
+بساوي واحد ضرب نفسه وهذا بيطلع واحد تربية و قبل
+
+177
+00:18:42,710 --> 00:18:46,710
+شوية شوفنا و الواحد مختلف عن السفر إذا المربع
+
+178
+00:18:46,710 --> 00:18:54,870
+بيطلع موجب حسب الخاصية السابقة، أثبت؟هذا معناه إذا
+
+179
+00:18:54,870 --> 00:18:59,770
+هيثبتنا واحد أكبر من السفر وبالتالي واحد ينتمي لل
+
+180
+00:18:59,770 --> 00:19:04,670
+positive real numbers إذا سالب واحد ينتمي ل
+
+181
+00:19:04,670 --> 00:19:07,610
+negative two يعني negative واحد أصغر من السفر
+
+182
+00:19:07,610 --> 00:19:20,050
+عملية بسيطة طيب احنا الآن بدنا نثبت ان كل
+
+183
+00:19:22,970 --> 00:19:31,370
+عدد حقيقي موجب مقلوبه موجب اه فبنعمل برهان
+
+184
+00:19:31,370 --> 00:19:36,190
+بالتناقض اذا هنا هندي اللي عايز اثبته هنا بس هنذكر
+
+185
+00:19:36,190 --> 00:19:41,430
+ال statement اللي بدنا نثبته يعني ال statement
+
+186
+00:19:41,430 --> 00:19:46,710
+اللي عايز اثبته لو كان a موجب ف reciprocal تبعه
+
+187
+00:19:46,710 --> 00:19:51,920
+بطلع موجب او مقلوبه بطلع موجبلبرهان ذلك نعمل برهان
+
+188
+00:19:51,920 --> 00:20:01,560
+بالتناقض نفرض أن واحد على a أقل من السفر وطبعا
+
+189
+00:20:01,560 --> 00:20:06,380
+عندي انا من الفرض هذا الفرض لازال قائم a أكبر من
+
+190
+00:20:06,380 --> 00:20:13,520
+السفر عندي الفرابين هدول فعندي a أكبر من السفر و
+
+191
+00:20:13,520 --> 00:20:17,640
+واحد على a أصغر من السفر فهذا بيقدي
+
+192
+00:20:20,870 --> 00:20:28,190
+لو ضربت المتباينة هذه في a اللي هو عدد موجب فهيصير
+
+193
+00:20:28,190 --> 00:20:32,450
+اندي واحد على a في a أصغر من سفر في a اللي هو
+
+194
+00:20:32,450 --> 00:20:37,290
+بيساوي سفر طب هدف بيساوي واحد ان هك بيطلع واحد
+
+195
+00:20:37,290 --> 00:20:42,530
+أصغر من سفر وبالتالي هدف يعطيني تناقض لأن الواحد
+
+196
+00:20:42,530 --> 00:20:47,530
+أكبر من سفر لسه مثبتينه قبل شوية ان هدف بيأدي إلى
+
+197
+00:20:47,530 --> 00:20:54,640
+تناقض وبالتاليمقلوب الـ A لازم يكون موجب بالمثل لو
+
+198
+00:20:54,640 --> 00:21:00,980
+كان مقلوب الـ A سالب فممكن نثبت انه مقلوب و ايضا
+
+199
+00:21:00,980 --> 00:21:05,900
+بيطلع سالب فالبرهان مشابه حسيبكم انتوا تكتبوه
+
+200
+00:21:05,900 --> 00:21:07,720
+تمام؟
+
+201
+00:21:21,570 --> 00:21:23,670
+أنا مش عارف لسه أنا هيك بعمل
+
+202
+00:21:50,000 --> 00:21:59,840
+طيب ال .. الجزء هذا الأخير إيش كان هذا؟ إيش كنا
+
+203
+00:21:59,840 --> 00:22:14,980
+بدنا نثبت هناك؟
+
+204
+00:22:17,920 --> 00:22:26,800
+اه إذا كان a عدد موجب و أصغر من b فهذا بيقدّي أن
+
+205
+00:22:26,800 --> 00:22:34,100
+مقلوب الكبير أصغر من مقلوب الصغير بظبط
+
+206
+00:22:34,100 --> 00:22:39,080
+و طبعا هذا موجب فلإثبات
+
+207
+00:22:39,080 --> 00:22:43,360
+أن واحد على بي أصغر من واحد على ايه بتثبت أن الفرق
+
+208
+00:22:43,360 --> 00:22:52,860
+بين واحد على ايه واحد على ايهو 1 على D ينتمي إلى P
+
+209
+00:22:52,860 --> 00:22:59,900
+أو موجة طيب الان هاي ناخد 1 على A سلب 1 على B
+
+210
+00:22:59,900 --> 00:23:05,140
+فاخدنا خاصية ناخد مقام مشترك A B و بعدين بيصير
+
+211
+00:23:05,140 --> 00:23:11,160
+عندى هذا بتحول لحاصل ضرب الان هذا positive number
+
+212
+00:23:11,160 --> 00:23:15,420
+لان احنا فرضين ان ال B أكبر من A فالفرق هذا
+
+213
+00:23:15,420 --> 00:23:23,780
+positiveو A B فبطلع
+
+214
+00:23:23,780 --> 00:23:29,200
+هذا مقلوب ال positive بطلع positive فهذا
+
+215
+00:23:29,200 --> 00:23:33,120
+positive و هذا positive و ال 6 دي closed under
+
+216
+00:23:33,120 --> 00:23:36,400
+multiplication إذن حاصر الضربة ده بطلع positive
+
+217
+00:23:36,400 --> 00:23:45,000
+لكون حاصر الضرب هنا العناصر فيه موجبة وبالتالي
+
+218
+00:23:46,170 --> 00:23:53,450
+إذا .. إذا هذا بيطلع أكبر من الصفر هذا بيطلع الفرق
+
+219
+00:23:53,450 --> 00:23:58,030
+أكبر من أوم وجب وبالتالي واحد على أيه أكبر من واحد
+
+220
+00:23:58,030 --> 00:24:03,770
+على بيه okay الأجزاء المتبقية D وE وH ممكن برهانة
+
+221
+00:24:03,770 --> 00:24:10,030
+بالمثل فاحنا دايما بنسيب للطالب شوية حاجات يثبتها
+
+222
+00:24:11,300 --> 00:24:15,740
+يعني عشان ان الطالب يشارك شوية و إلا بيصير عملية
+
+223
+00:24:15,740 --> 00:24:20,020
+التدريس مملة لو احنا بدنا نشرحلكم كل حاجة و مانخلش
+
+224
+00:24:20,020 --> 00:24:25,880
+ولا إيش للطالب فبصير عملية مملة و بعدين الفهم بكون
+
+225
+00:24:25,880 --> 00:24:31,860
+ماخص كل ما انت شاركت أكتر كل ما شعرتي أو حسيتي
+
+226
+00:24:31,860 --> 00:24:36,900
+بالمعلومة أكتر و كل ما فهمتيها أكتر فالحاجات هذه
+
+227
+00:24:36,900 --> 00:24:40,500
+بالإضافة للتمارين اللي في نهاية كل section في
+
+228
+00:24:40,500 --> 00:24:47,560
+الكتابحالها كتير بساعد في فهم المادة بدون ذلك بظل
+
+229
+00:24:47,560 --> 00:24:57,420
+فهمكم نقص ننتقل إلى نظرية أخرى نظرية واحد ستة
+
+230
+00:24:57,420 --> 00:25:02,680
+نظرية هذه نظرية يعني بسيطة ومهمة
+
+231
+00:25:04,730 --> 00:25:09,830
+رغم بساطتها لكن مهمة إيش بتقول النظرية هذه بتقول
+
+232
+00:25:09,830 --> 00:25:16,250
+لو أخدت أي عددين حقيقين و a أكبر من b فلازم يكون a
+
+233
+00:25:16,250 --> 00:25:26,310
+أكبر من متوسط a و b و أكبر من b البرهان بسيط هي
+
+234
+00:25:26,310 --> 00:25:32,330
+عند الفرض أنا فارض أن a أكبر من bبتثبت أن a أكبر
+
+235
+00:25:32,330 --> 00:25:39,330
+من نص مجموعة a و b و نص مجموعة a و b أكبر من b طيب
+
+236
+00:25:39,330 --> 00:25:44,490
+نثبت المتباينة الأولى هذه نثبت المتباينة الأولى
+
+237
+00:25:44,490 --> 00:25:49,230
+الأول بعدين نثبت التانية
+
+238
+00:25:52,810 --> 00:25:59,530
+فالإثبات الجزء الأول فهي عندي a أكبر من b إذا لو
+
+239
+00:25:59,530 --> 00:26:07,110
+جمعت a على نفسها ده اتنين a لو جمعت على الطرفين a
+
+240
+00:26:07,110 --> 00:26:11,630
+فبطلع عندي a زائد a أكبر من b زاد a هذه خاصية
+
+241
+00:26:11,630 --> 00:26:15,810
+أخدناها قبل a إذا اتنين a بيطلع أكبر من a زائد b
+
+242
+00:26:16,680 --> 00:26:22,900
+كذلك لو جمعت على الطرفين هنا B فبطلع A زائد B أكبر
+
+243
+00:26:22,900 --> 00:26:26,320
+من B زائد B A زائد B أكبر من B زائد B اللي هو
+
+244
+00:26:26,320 --> 00:26:32,320
+اتنين B إذا أنا في عندي الآن متباينتين اتنين A
+
+245
+00:26:32,320 --> 00:26:39,960
+أكبر من A زائد B هيا اتنين A أكبر من A زائد B وA
+
+246
+00:26:39,960 --> 00:26:46,700
+زائد B أكبر من اتنين B إذا by transitivityخاصية
+
+247
+00:26:46,700 --> 00:26:52,740
+التعدى ممكن استنتج ان اتنين a اكبر من a زايد b
+
+248
+00:26:52,740 --> 00:26:59,440
+اكبر من اتنين b الان العدد اتنين عدد طبيعي وشوفنا
+
+249
+00:26:59,440 --> 00:27:03,360
+في الخاصية بتقول اي عدد طبيعي هو عدد موجب فى
+
+250
+00:27:03,360 --> 00:27:09,520
+النظرية اللى فاتت كذلك اي عدد موجب مقلوبه موجب اذا
+
+251
+00:27:09,520 --> 00:27:14,520
+النص عدد موجب الان لو ضربت المتباينة هذه فى النص
+
+252
+00:27:14,520 --> 00:27:19,340
+اللى هو عدد موجبإشاراتها تبقى زي ما هي هذه خاصية
+
+253
+00:27:19,340 --> 00:27:27,740
+خلناها في النظرية هذه تمام؟ إذا أنا حضرب الفنص هي
+
+254
+00:27:27,740 --> 00:27:33,960
+ضربت طبعا هذا بيساوي a وهذا بيساوي b وبالتالي نحصل
+
+255
+00:27:33,960 --> 00:27:40,520
+على المطلوب إذا يعني براهين سهلة وبسيطة النظرية
+
+256
+00:27:40,520 --> 00:27:46,960
+هذه مهمة لأن نتيجة اللي بعدهاأو أهميتها تظهر في
+
+257
+00:27:46,960 --> 00:27:53,580
+النتيجة اللي بعدها اللي هي corollary 171 corollary
+
+258
+00:27:53,580 --> 00:28:04,620
+171 بيقول أن
+
+259
+00:28:04,620 --> 00:28:12,040
+أي عدد موجب بيكون أكبر من نصه اللي هو موجب أي عدد
+
+260
+00:28:12,040 --> 00:28:18,520
+حقيقي موجب دايما أكبر من نصهوبالتالي هذا معناه في
+
+261
+00:28:18,520 --> 00:28:23,200
+رياضيات أن الأعداد الحقيقية الموجبة مالهاش
+
+262
+00:28:23,200 --> 00:28:27,960
+smallest element مافيش .. لو أخدت الأعداد الحقيقية
+
+263
+00:28:27,960 --> 00:28:35,200
+الموجبة اللي هي set P فهذا ال set ماقدرش أحط أصبعي
+
+264
+00:28:35,200 --> 00:28:42,360
+على أصغر عنصر فيها مالهاش أصغر عنصرhas no smallest
+
+265
+00:28:42,360 --> 00:28:48,580
+element لأن لو أخدت أي عنصر موجب و سميته a فبقدر
+
+266
+00:28:48,580 --> 00:28:54,200
+ألاقي عدد موجب أخر أصغر منه اللي هو نصف فبالتالي
+
+267
+00:28:54,200 --> 00:28:59,800
+ال set of positive numbers has no strictly
+
+268
+00:28:59,800 --> 00:29:04,500
+positive element تمام؟ البرهان تبع الكرولري هذا
+
+269
+00:29:04,500 --> 00:29:09,360
+بينتج من نظريةيعني خد بي بساوة سفر في النظرية اللى
+
+270
+00:29:09,360 --> 00:29:26,280
+فاتت نظرية واحد ستة هي تشوفها مع بعض نظرية
+
+271
+00:29:26,280 --> 00:29:30,860
+واحد ستة لو أخدت بي بساوة سفر فبطلع ا اكبر من نص ا
+
+272
+00:29:30,860 --> 00:29:37,970
+اكبر من سفر اذا هذه نتيجة سريعة مظبوطOkay إذا يعني
+
+273
+00:29:37,970 --> 00:29:45,050
+هذه بعض الحاجات السهلة والبسيطة، هنا في نظرية كتير
+
+274
+00:29:45,050 --> 00:29:49,950
+مهمة، هذه برضه نظرية هنستخدمها بكرا يعني في
+
+275
+00:29:49,950 --> 00:29:55,770
+المستقبل، نظرية واحد تمنع، نظرية كتير مهمة وأهمتها
+
+276
+00:29:55,770 --> 00:30:02,670
+هنشوفها في الشبات الرجايةإيش هذه النظرية بتقول؟ لو
+
+277
+00:30:02,670 --> 00:30:08,110
+في عندي عدد حقيقي غير سالب، غير سالب، و في نفس
+
+278
+00:30:08,110 --> 00:30:14,050
+الوقت أصغر من إبسلون لكل عدد موجب إبسلون، فهذا
+
+279
+00:30:14,050 --> 00:30:19,350
+العدد لازم يكون هو السفر، وهي برهان بالتناقض
+
+280
+00:30:22,990 --> 00:30:28,470
+كمان مرة العدد غير السالب اللى بيكون اي اصغر من اي
+
+281
+00:30:28,470 --> 00:30:33,590
+عدد موجب هو السفر مافيش غير السفر اللى بيحقق
+
+282
+00:30:33,590 --> 00:30:40,250
+لخاصية هذه لبرهان ذلك نعمل برهان بالتناقض افرض ان
+
+283
+00:30:40,250 --> 00:30:45,830
+ال a ان ال a هذا بيسويش السفر و في نفس الوجهة a
+
+284
+00:30:45,830 --> 00:30:48,850
+غير سالب اذا يعني a موجب صح؟
+
+285
+00:30:52,860 --> 00:30:57,980
+الان حسب نظرية الكورينة النتيجة واحد سبعة اذا a
+
+286
+00:30:57,980 --> 00:31:05,260
+بطلع اكبر من نص a فاخد epsilon zero هنا عدد موجب
+
+287
+00:31:05,260 --> 00:31:11,260
+بساوي a ع اتنين نص a هذا عدد موجب اذا هاني نجحت في
+
+288
+00:31:11,260 --> 00:31:18,670
+ايجاد عدد epsilon zero عدد موجب وال a اكبر منههذا
+
+289
+00:31:18,670 --> 00:31:22,570
+يتناقض مع الفرض أن a أصغر من إبسلون لكل إبسلون
+
+290
+00:31:22,570 --> 00:31:29,190
+أكبر من السفر أظبط؟ لأن هذا التناقض يثبت النظرية
+
+291
+00:31:29,190 --> 00:31:39,010
+واضح تمام؟ واضح البرهن؟ عيده طيب أنا عندي a عدد
+
+292
+00:31:39,010 --> 00:31:43,190
+حقيقي غير سالم وفي نفس الوجهة أصغر من كل الأعداد
+
+293
+00:31:43,190 --> 00:31:49,030
+الموجبة إبسلون بدا أثبت أن a بساوي سفربرهان
+
+294
+00:31:49,030 --> 00:31:53,630
+بالتناقض prove by contradiction assume or suppose
+
+295
+00:31:53,630 --> 00:31:58,910
+the contrary النقيض أو النفي تبع النتيجة يعني a ما
+
+296
+00:31:58,910 --> 00:32:03,390
+بيستويش صفر نفي a بيستوي صفر a لا تستوي صفر طب أنا
+
+297
+00:32:03,390 --> 00:32:07,470
+كاتب هنا ال contrary a أكبر من صفر هذا صح بناء على
+
+298
+00:32:07,470 --> 00:32:11,930
+أن الفرض a أكبر من أكبر من صفر وما بيستويش صفر إذن
+
+299
+00:32:11,930 --> 00:32:16,970
+أكبر من صفر صح طيب الآن
+
+300
+00:32:18,230 --> 00:32:23,270
+لو أخدت Epsilon Zero بساوي نص A و بما أنه A عدد
+
+301
+00:32:23,270 --> 00:32:27,650
+موجب فنتيجة واحدة السابعة بتقول لو كان A عدد موجب
+
+302
+00:32:27,650 --> 00:32:35,520
+فنص A بطلع عدد موجبإذا هيني و في نفس الوجد كمان ال
+
+303
+00:32:35,520 --> 00:32:40,820
+a أكبر من نص a ال a أكبر من نص a وبالتالي إذا هيني
+
+304
+00:32:40,820 --> 00:32:45,900
+لجحت في إيجاد epsilon zero عدد موجب و a أكبر منه
+
+305
+00:32:45,900 --> 00:32:54,700
+هذا بتناقض مع الفرض أنه بتناقض مع الفرض أنه a أصغر
+
+306
+00:32:54,700 --> 00:33:03,090
+من epsilon لكل epsilon موجبة صح؟ الإبارة هذه هينفي
+
+307
+00:33:03,090 --> 00:33:07,710
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+308
+00:33:07,710 --> 00:33:10,950
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+309
+00:33:10,950 --> 00:33:11,090
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+310
+00:33:11,090 --> 00:33:11,430
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+311
+00:33:11,430 --> 00:33:12,810
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+312
+00:33:12,810 --> 00:33:13,730
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+313
+00:33:13,730 --> 00:33:21,570
+هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي هذه نفي
+
+314
+00:33:21,570 --> 00:33:25,250
+هذه نفي هذه نفي
+
+315
+00:33:28,550 --> 00:33:32,050
+Okay، إذا إحنا لحد الآن يعني كل شغلنا مبادئ
+
+316
+00:33:32,050 --> 00:33:37,270
+رياضيات، صح؟ طيب، طب ما هي مبادئ رياضيات هي أساس
+
+317
+00:33:37,270 --> 00:33:46,390
+ال .. اسمها أساسية الرياضيات، فاسم على مسمة فبختل
+
+318
+00:33:46,390 --> 00:33:50,870
+فهمة المادة هذه، جابت علينا منيحة يعني، هترتاح في
+
+319
+00:33:50,870 --> 00:33:57,380
+المستجبل كتير Bernoulli inequalityبرنول
+
+320
+00:33:57,380 --> 00:34:01,760
+الانيقوليتي هذه يعني في شوية متباينات طبعا مهمة في
+
+321
+00:34:01,760 --> 00:34:06,860
+الكتاب انا اختارت واحدة منهم لكن في بعض المتباينات
+
+322
+00:34:06,860 --> 00:34:13,180
+الأخرى موجودة في الكتاب وارجو انكم تقراوها فبرنول
+
+323
+00:34:13,180 --> 00:34:15,960
+الانيقوليتي هذه واحدة منهم متباينة برنول يعني
+
+324
+00:34:15,960 --> 00:34:23,230
+بيقول لو كان X عدد حقيقي أكبر من سالب واحدفمجموعة
+
+325
+00:34:23,230 --> 00:34:28,750
+واحد و X to the power N دايما أكبر من أو ساوي واحد
+
+326
+00:34:28,750 --> 00:34:38,850
+زائد N ضرب X وهذا صحيح لكل الأعداد الطبيعية نعم
+
+327
+00:34:38,850 --> 00:34:46,290
+في نظرية جاب الهاجم ماخلنهاش شوف
+
+328
+00:34:46,290 --> 00:34:46,890
+مع بعض
+
+329
+00:34:50,730 --> 00:35:07,610
+أه صحيح نشوف النظرية واحد تسعة نظرية
+
+330
+00:35:07,610 --> 00:35:10,870
+واحد تسعة بتقول لو كان أندي عددين حقيقين حاصل
+
+331
+00:35:10,870 --> 00:35:15,610
+ضربهم موجب فيا إما اتنين موجبين يا إما اتنين
+
+332
+00:35:15,610 --> 00:35:20,450
+سالبين صح؟ممكن يكون الاتنين مختلفين في الإشارة و
+
+333
+00:35:20,450 --> 00:35:25,530
+حصل ضربهم موجب إذا حصل ضرب عددين موجب بيقدر انه
+
+334
+00:35:25,530 --> 00:35:33,670
+اما اتنين موجبين او اتنين سالبين فالبرهان نشوف كيف
+
+335
+00:35:33,670 --> 00:35:39,640
+افرض الفرض تبعنا ان حصل ضرب A وB موجبفهذا أكيد
+
+336
+00:35:39,640 --> 00:35:42,780
+بيقدّي ان لا ال a بيساوي سفر ولا ال b بيساوي سفر
+
+337
+00:35:42,780 --> 00:35:46,520
+لأن لو واحد منهم بيساوي سفر فحاصل الدرب هيطلع
+
+338
+00:35:46,520 --> 00:35:50,720
+بيساوي سفر contradiction تناقض صح؟ لأن هذا
+
+339
+00:35:50,720 --> 00:36:02,020
+الاستنتاج منطقي طيب الان احنا ال a ناخد ناخد الجزء
+
+340
+00:36:02,020 --> 00:36:09,250
+هذا الان انا عند a لا يساوي سفربقى اتراي كاتومي
+
+341
+00:36:09,250 --> 00:36:14,650
+property حسب الخلصية التي هي اما a أكبر من سفر أو
+
+342
+00:36:14,650 --> 00:36:23,650
+a أصغر من سفر صح؟ بقى في احتمالين طيب ناخد ال a لو
+
+343
+00:36:23,650 --> 00:36:30,370
+كان افرض ان a أكبر من سفر فهذا بيدى ان واحد على a
+
+344
+00:36:30,370 --> 00:36:42,620
+أكبر من سفرهذا يعني 1 على a أكبر من 0 بيؤدي
+
+345
+00:36:42,620 --> 00:36:48,700
+أيضًا إلى بي اللي هو بساوي ال
+
+346
+00:36:48,700 --> 00:36:53,720
+بي ممكن اكتبها واحد في بي والواحد ممكن ابدله بواحد
+
+347
+00:36:53,720 --> 00:36:57,800
+على a في a واستخدم ال associative law واكتب هذا
+
+348
+00:36:57,800 --> 00:37:04,910
+على صورة واحد على a في a بي الان هذا موجبوهذا موجب
+
+349
+00:37:04,910 --> 00:37:12,330
+إذا حصلت ضرب بيطلع موجب إذا هذه أثبتت أن ال a أكبر
+
+350
+00:37:12,330 --> 00:37:19,950
+من ال b أكبر من السفر لأ احنا أخدنا ال a أكبر من
+
+351
+00:37:19,950 --> 00:37:24,290
+السفر فأدت
+
+352
+00:37:24,290 --> 00:37:28,690
+إلى أن ال b
+
+353
+00:37:28,690 --> 00:37:32,430
+أكبر من السفر وبالتالي بيطلع ال a و ال b موجبين
+
+354
+00:37:34,090 --> 00:37:40,810
+بالمثل لو افترضت .. اخدت لو افترضت ان a سالب فطبعا
+
+355
+00:37:40,810 --> 00:37:46,410
+مقلوب العدد السالب بيطلع سالب وبالتالي ال b اللي
+
+356
+00:37:46,410 --> 00:37:52,830
+هي بتساوي واحد على a في a b زي ما عملنا هنا ال b
+
+357
+00:37:52,830 --> 00:37:56,810
+بتطلع بتساوي واحد على a في a b ف ..
+
+358
+00:38:00,670 --> 00:38:05,850
+فهذا بيطلع الحاصل بضرب سالب لأن عندي انا هي هذه
+
+359
+00:38:05,850 --> 00:38:12,290
+المتباينة هذه هي واحد على ا سالب لو ضربت المتباينة
+
+360
+00:38:12,290 --> 00:38:18,370
+هذه في العدد الموجب a,b اللي هو عدد موجب فبصير
+
+361
+00:38:18,370 --> 00:38:21,910
+المتباينة هذه عبارة عن واحد على a في a,b الطرف
+
+362
+00:38:21,910 --> 00:38:29,070
+الشمال وضربتها في عدد موجب فبطلع أصغر من سفر في a
+
+363
+00:38:29,070 --> 00:38:30,030
+,b اللي هو سفر
+
+364
+00:38:32,730 --> 00:38:38,730
+وبالتالي بيطلع عندي الـ B بيطلع عند الـ B التي هي
+
+365
+00:38:38,730 --> 00:38:46,390
+أصغر للسفرإذا مرة تانية لو فرضنا أن a,b أكبر من 0
+
+366
+00:38:46,390 --> 00:38:50,670
+فشوفنا أن لا ال a بالساوية 0 ولا ال b بالساوية 0
+
+367
+00:38:50,670 --> 00:38:56,670
+وبالتالي أما بطلع a أكبر من 0 أو a أصغر من 0 في
+
+368
+00:38:56,670 --> 00:39:01,010
+الحالة الأولى لو كان a أكبر من 0 بطلع b أكبر من 0
+
+369
+00:39:01,010 --> 00:39:05,170
+وبالتالي a وb موجبين في الاحتمال التاني أو الحالة
+
+370
+00:39:05,170 --> 00:39:09,940
+التانية لو كان a سالب فشوفنا أن بطلع b سالبو
+
+371
+00:39:09,940 --> 00:39:18,480
+بالتالي اتنين سالبين okay تمام نشوف
+
+372
+00:39:18,480 --> 00:39:27,240
+الان Bernoulli inequality اليوم
+
+373
+00:39:27,240 --> 00:39:32,620
+هناخد برهان by induction برضه مبادئ الرياضيات خلنا
+
+374
+00:39:32,620 --> 00:39:39,360
+برهان by contradiction و direct proofوهنشوف move
+
+375
+00:39:39,360 --> 00:39:44,880
+by induction نمسح
+
+376
+00:39:44,880 --> 00:39:49,300
+اللوح بيرنول
+
+377
+00:39:49,300 --> 00:39:52,420
+ال equality زي ما قلنا لو كان x عدد حقيقي أكبر من
+
+378
+00:39:52,420 --> 00:39:58,000
+سالب واحد فلمّا أضيف عليه واحد وارفع لقوة n هذا
+
+379
+00:39:58,000 --> 00:40:02,380
+بيطلع أكبر من أو سالب واحد زائد n في x وهذا صحيح
+
+380
+00:40:02,380 --> 00:40:07,290
+لكل الأعداد الطبيعيةالبرغم by induction لو كانت n
+
+381
+00:40:07,290 --> 00:40:12,690
+بساوي واحد بثبت صحة العبارة عند n بساوي واحد لأن
+
+382
+00:40:12,690 --> 00:40:20,350
+ان تبدأ من واحد فلو كان n بساوي واحد فالطرف
+
+383
+00:40:20,350 --> 00:40:23,530
+الشمال
+
+384
+00:40:23,530 --> 00:40:31,900
+بطلع واحد زاد x صح؟الطرف الشمال واحد زائد X والطرف
+
+385
+00:40:31,900 --> 00:40:37,240
+اليمين برضه واحد زائد X فبطلع مساواة وطبعا
+
+386
+00:40:37,240 --> 00:40:42,720
+المساواة بقدر بدلها بأكبر من أوسعه مافي مشكلة
+
+387
+00:40:42,720 --> 00:40:46,020
+تمام؟
+
+388
+00:40:46,020 --> 00:40:52,700
+إذا العبارة هذه صحيحة عند N بالساوي واحد الآن نفرض
+
+389
+00:40:52,700 --> 00:40:57,780
+أن العبارة صحيحة عند N بالساوي K حيث K أكبر من
+
+390
+00:40:57,780 --> 00:41:05,800
+واحدهذا ما نسميه induction hypothesis الفرض تبع ال
+
+391
+00:41:05,800 --> 00:41:12,740
+induction نفرض صحة العبارة عند N بساوة K حيث K
+
+392
+00:41:12,740 --> 00:41:18,080
+أكبر من 1 هذا معناه أن 1 زاد X to K bigger than or
+
+393
+00:41:18,080 --> 00:41:24,510
+equal to 1 plus K Xطيب الان نريد نكمل ال induction
+
+394
+00:41:24,510 --> 00:41:32,790
+عايزين نثبت صحة العبارة واحد اللي هي هذه العبارة
+
+395
+00:41:32,790 --> 00:41:38,730
+واحد مش عارف من الواحد رايح العبارة واحد هذه نثب
+
+396
+00:41:38,730 --> 00:41:45,650
+الصحة عندنا بساوة K زياد واحد طيب from اتنين هذه
+
+397
+00:41:45,650 --> 00:41:47,870
+العبارة اتنين اللي هي induction hypothesis
+
+398
+00:41:52,040 --> 00:41:59,640
+بتدفع في الـ type هاي العبارة هذه لما n ساوي k
+
+399
+00:41:59,640 --> 00:42:05,880
+زائد واحد هصير واحد زائد x الكل أس k زائد واحد
+
+400
+00:42:05,880 --> 00:42:12,410
+أكبر من أو ساوي واحد زائد k زائد واحدفي X هذه
+
+401
+00:42:12,410 --> 00:42:18,390
+العبارة and N بساوي K زي 1 نبدأ بالطرف الشمال و
+
+402
+00:42:18,390 --> 00:42:22,670
+نثبت أنه أكبر من أو يساوي الطرف اليمين هاي الطرف
+
+403
+00:42:22,670 --> 00:42:27,950
+الشمال بقدر أجزئه حسب قوانين الأسس ل1 plus K to K
+
+404
+00:42:27,950 --> 00:42:34,410
+و 1 زي X to K ضرب 1 زي X الآن من اتنين من العبارة
+
+405
+00:42:34,410 --> 00:42:38,070
+التانية one plus X to K اللي هو induction
+
+406
+00:42:38,070 --> 00:42:43,340
+hypothesisحسب اتنين هذا اكبر من او يساوي واحد زياد
+
+407
+00:42:43,340 --> 00:42:48,220
+ك اكس مضروب في واحد زياد اكس بنضرب هدول في بعض و
+
+408
+00:42:48,220 --> 00:42:55,440
+بنرتب فبطلع حاصل الضرب هذا هو واحد زياد ك زياد
+
+409
+00:42:55,440 --> 00:43:01,340
+واحد في اكس زياد ك في اكس تربيه الآن
+
+410
+00:43:01,340 --> 00:43:08,860
+هذا هذا عدد موجب هذا عدد موجبلأن K عدد طبيعي و X
+
+411
+00:43:08,860 --> 00:43:16,060
+تربيه عدد موجب لما أشيل هذا أشطبه فبصغر المقدار
+
+412
+00:43:16,060 --> 00:43:20,580
+لما أشيل عدد موجب من عدد أو أنقص من عدد عدد موجب
+
+413
+00:43:20,580 --> 00:43:27,580
+بصغر فبالتالي هذا أكبر من واحد زاد K زاد واحد في X
+
+414
+00:43:28,490 --> 00:43:33,310
+وهذا هو الطرف اليمين للمتباينة 1 اللي احنا عايزين
+
+415
+00:43:33,310 --> 00:43:38,050
+نثبت صحتها عند m بساوي k زيادة واحدة اذا this
+
+416
+00:43:38,050 --> 00:43:42,830
+completes the induction هذا بيكمل البرهان بال
+
+417
+00:43:42,830 --> 00:43:50,050
+induction مظبوط صح تمام واضح اذا هاي صار في ان
+
+418
+00:43:50,050 --> 00:43:54,170
+متباينة
+
+419
+00:43:54,170 --> 00:44:01,080
+Bernoulli زي ما قلنا لكم في في الفي ال section هذا
+
+420
+00:44:01,080 --> 00:44:08,240
+بعض المتباينات الأخرى فبإمكانكم تقرؤوها ال
+
+421
+00:44:08,240 --> 00:44:12,260
+homework الآن خلصنا احنا section اتنين واحد اعتقد
+
+422
+00:44:12,260 --> 00:44:20,040
+فالمسائل المطلوب انكم تحلوها اللي هي موجودة مرسوصة
+
+423
+00:44:20,040 --> 00:44:23,280
+هنا وبرضه
+
+424
+00:44:23,280 --> 00:44:26,500
+زي ما قلتلكم في syllabus موجود على الصفحه تبعتي
+
+425
+00:44:27,940 --> 00:44:32,720
+فبارضه في ال homework هذا موجود ل .. مش ل ال
+
+426
+00:44:32,720 --> 00:44:38,400
+section هذا لكل ال .. المنهج اذا نبدأ نحل المسائل
+
+427
+00:44:38,400 --> 00:44:44,180
+هذه و ان شاء الله لسبوع الجاي بنعمل مناقشة فانا
+
+428
+00:44:44,180 --> 00:44:49,200
+هعمل مناقشة .. انا اللي هكون مناقشة لكم اليوم لأ
+
+429
+00:44:49,200 --> 00:44:52,960
+مافيش مناقشة لأنه لسه احنا يعني ماخدناش material
+
+430
+00:44:52,960 --> 00:45:00,110
+كافيةاو اللي لسه يعني مش مهيئين او مش محضرين
+
+431
+00:45:00,110 --> 00:45:06,170
+فهنواصل ونحاول ان شاء الله أسبوع الجاى ناخد كل
+
+432
+00:45:06,170 --> 00:45:10,450
+ساعة هذه الساعة
+
+433
+00:45:10,450 --> 00:45:12,930
+الأخيرة هذه المتأخرة نعملها مناقشة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/QCtISTGMQww_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/QCtISTGMQww_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..a9ea7dabfcccaf990cb54fe17ce79b8259f8fc69
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/QCtISTGMQww_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4869, "start": 20.77, "end": 48.69, "text": "السلام عليكم بسم الله الرحمن الرحيم اليوم طبعا فينا لقائن اللقاء الأول هناخد فيه مناقشة زي ما .. زي ما احنا متعودين هنناقش اليوم section تلاتة أربعة و تلاتة خمسة فإذا في عندكم أي أسئلة معينة في section تلاتة أربعة", "tokens": [6027, 3794, 37440, 25894, 24793, 4724, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 45595, 20498, 23032, 3555, 3615, 995, 8978, 8315, 5296, 4587, 16373, 1863, 13672, 4587, 16606, 16247, 12610, 8032, 1863, 47283, 3215, 8978, 3224, 3714, 8315, 4587, 8592, 3660, 30767, 1829, 19446, 4386, 30767, 1829, 19446, 1975, 5016, 8315, 44650, 3615, 23328, 9957, 8032, 1863, 8315, 4587, 8592, 45595, 20498, 3541, 6055, 1211, 9307, 3660, 5551, 25513, 27884, 4032, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 6156, 28814, 15730, 8978, 43242, 24793, 36632, 5551, 3794, 19986, 37977, 20449, 9957, 3660, 8978, 3541, 6055, 1211, 9307, 3660, 5551, 25513, 27884], "avg_logprob": -0.11741285842771713, 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13454, "start": 105.6, "end": 134.54, "text": "هذه السؤال تمانية D section تلاتة أربعة determine the following limits determine يعني حد دي أو أوج دي ال limit لل sequence اللي الحد العام تبعها", "tokens": [3224, 24192, 21136, 33604, 6027, 46811, 7649, 10632, 413, 3541, 6055, 1211, 9307, 3660, 5551, 25513, 27884, 6997, 264, 3480, 10406, 6997, 37495, 22653, 11331, 3215, 11778, 1829, 34051, 5551, 29245, 11778, 1829, 2423, 4948, 24976, 8310, 13672, 1829, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 11296], "avg_logprob": -0.26123047682146233, "compression_ratio": 1.3733333333333333, "no_speech_prob": 0.0, "words": [{"start": 105.6, "end": 106.08, "word": "هذه", "probability": 0.7587890625}, {"start": 106.08, "end": 106.86, "word": " السؤال", "probability": 0.9069010416666666}, {"start": 106.86, "end": 109.66, "word": " تمانية", "probability": 0.8841145833333334}, {"start": 109.66, "end": 110.78, "word": " D", "probability": 0.362548828125}, {"start": 110.78, "end": 112.36, "word": " 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"end": 161.68, "word": " ال", "probability": 0.86865234375}, {"start": 161.68, "end": 162.04, "word": " ..", "probability": 0.8466796875}], "temperature": 1.0}, {"id": 6, "seek": 19085, "start": 168.21, "end": 190.85, "text": "We know احنا فيه ان ال limit المعروفة limit واحد زايد X على M الكل أس M as M tends to infinity بساوي E أس X ال limit هذه معروفة موجودة هنا", "tokens": [4360, 458, 1975, 5016, 8315, 8978, 3224, 16472, 2423, 4948, 9673, 3615, 32887, 5172, 3660, 4948, 36764, 24401, 30767, 995, 25708, 1783, 15844, 376, 2423, 28820, 5551, 3794, 376, 382, 376, 12258, 281, 13202, 4724, 3794, 995, 45865, 462, 5551, 3794, 1783, 2423, 4948, 29538, 20449, 32887, 5172, 3660, 3714, 29245, 23328, 3660, 34105], "avg_logprob": -0.2659091006625782, "compression_ratio": 1.348993288590604, "no_speech_prob": 0.0, "words": [{"start": 168.21, "end": 168.51, "word": "We", "probability": 0.089599609375}, {"start": 168.51, "end": 168.99, "word": " know", "probability": 0.8623046875}, {"start": 168.99, 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"probability": 0.8193359375}, {"start": 178.83, "end": 179.13, "word": " أس", "probability": 0.430419921875}, {"start": 179.13, "end": 179.63, "word": " M", "probability": 0.89794921875}, {"start": 179.63, "end": 182.23, "word": " as", "probability": 0.70361328125}, {"start": 182.23, "end": 182.75, "word": " M", "probability": 0.97998046875}, {"start": 182.75, "end": 183.15, "word": " tends", "probability": 0.7841796875}, {"start": 183.15, "end": 183.37, "word": " to", "probability": 0.94140625}, {"start": 183.37, "end": 184.01, "word": " infinity", "probability": 0.83251953125}, {"start": 184.01, "end": 186.07, "word": " بساوي", "probability": 0.86181640625}, {"start": 186.07, "end": 186.45, "word": " E", "probability": 0.94189453125}, {"start": 186.45, "end": 186.73, "word": " أس", "probability": 0.8857421875}, {"start": 186.73, "end": 187.15, "word": " X", "probability": 0.9951171875}, {"start": 187.15, "end": 188.99, "word": " ال", "probability": 0.6611328125}, {"start": 188.99, "end": 189.21, "word": " limit", "probability": 0.94921875}, {"start": 189.21, "end": 189.49, "word": " هذه", "probability": 0.397216796875}, {"start": 189.49, "end": 190.13, "word": " معروفة", "probability": 0.966796875}, {"start": 190.13, "end": 190.61, "word": " موجودة", "probability": 0.8878173828125}, {"start": 190.61, "end": 190.85, "word": " هنا", "probability": 0.75634765625}], "temperature": 1.0}, {"id": 7, "seek": 21660, "start": 194.08, "end": 216.6, "text": "باستخدام ال limit هذه ممكن ان احنا وبالتالي تعالى نشوف واحد زائد واحد على اتنين in الكل اصلا تلاتة in", "tokens": [3555, 995, 14851, 9778, 3215, 10943, 2423, 4948, 29538, 3714, 43020, 16472, 1975, 5016, 8315, 46599, 6027, 2655, 6027, 1829, 37279, 6027, 7578, 8717, 8592, 38688, 36764, 24401, 30767, 16373, 3215, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 294, 2423, 28820, 1975, 9381, 15040, 6055, 1211, 9307, 3660, 294], "avg_logprob": -0.3187499940395355, "compression_ratio": 1.403225806451613, "no_speech_prob": 0.0, "words": [{"start": 194.08, "end": 195.3, "word": "باستخدام", "probability": 0.7371724446614584}, {"start": 195.3, "end": 195.42, "word": " ال", "probability": 0.262939453125}, {"start": 195.42, "end": 195.68, "word": " limit", "probability": 0.76708984375}, {"start": 195.68, "end": 196.1, "word": " هذه", "probability": 0.38037109375}, {"start": 196.1, "end": 197.8, "word": " ممكن", "probability": 0.651611328125}, {"start": 197.8, "end": 198.04, "word": " ان", "probability": 0.5439453125}, {"start": 198.04, "end": 200.22, "word": " احنا", "probability": 0.8746744791666666}, {"start": 200.22, "end": 206.66, "word": " وبالتالي", "probability": 0.857177734375}, {"start": 206.66, "end": 209.9, "word": " تعالى", "probability": 0.89013671875}, {"start": 209.9, "end": 210.46, "word": " نشوف", "probability": 0.9895833333333334}, {"start": 210.46, "end": 211.5, "word": " واحد", "probability": 0.783935546875}, {"start": 211.5, "end": 212.32, "word": " زائد", "probability": 0.7774251302083334}, {"start": 212.32, "end": 213.48, "word": " واحد", "probability": 0.889404296875}, {"start": 213.48, "end": 213.7, "word": " على", "probability": 0.4814453125}, {"start": 213.7, "end": 214.3, "word": " اتنين", "probability": 0.962646484375}, {"start": 214.3, "end": 214.7, "word": " in", "probability": 0.51171875}, {"start": 214.7, "end": 215.34, "word": " الكل", "probability": 0.783203125}, {"start": 215.34, "end": 215.76, "word": " اصلا", "probability": 0.5502115885416666}, {"start": 215.76, "end": 216.38, "word": " تلاتة", "probability": 0.89794921875}, {"start": 216.38, "end": 216.6, "word": " in", "probability": 0.916015625}], "temperature": 1.0}, {"id": 8, "seek": 24616, "start": 217.6, "end": 246.16, "text": "نحاول نكتب هذا على صورة واحد زائد X على M الكل قص M فهذا ممكن كتابته على صورة المقدار هذا على الصورة واحد زائد نص على M نص على M الكل قص M الكل تكيين", "tokens": [1863, 5016, 995, 12610, 8717, 4117, 2655, 3555, 23758, 15844, 20328, 13063, 3660, 36764, 24401, 30767, 16373, 3215, 1783, 15844, 376, 2423, 28820, 12174, 9381, 376, 6156, 3224, 15730, 3714, 43020, 9122, 2655, 16758, 47395, 15844, 20328, 13063, 3660, 9673, 28543, 9640, 23758, 15844, 31767, 13063, 3660, 36764, 24401, 30767, 16373, 3215, 8717, 9381, 15844, 376, 8717, 9381, 15844, 376, 2423, 28820, 12174, 9381, 376, 2423, 28820, 6055, 4117, 1829, 9957], "avg_logprob": -0.1280381963070896, "compression_ratio": 2.0, "no_speech_prob": 0.0, "words": [{"start": 217.6, "end": 218.16, "word": "نحاول", "probability": 0.93701171875}, {"start": 218.16, "end": 218.58, "word": " نكتب", "probability": 0.996826171875}, {"start": 218.58, "end": 218.84, "word": " هذا", "probability": 0.876953125}, {"start": 218.84, "end": 219.06, "word": " على", "probability": 0.94873046875}, {"start": 219.06, "end": 219.44, "word": " صورة", "probability": 0.9781901041666666}, {"start": 219.44, "end": 219.76, "word": " واحد", "probability": 0.9853515625}, {"start": 219.76, "end": 220.2, "word": " زائد", "probability": 0.9366861979166666}, {"start": 220.2, "end": 220.54, "word": " X", "probability": 0.484619140625}, {"start": 220.54, "end": 220.76, "word": " على", "probability": 0.8134765625}, {"start": 220.76, "end": 221.08, "word": " M", "probability": 0.900390625}, {"start": 221.08, "end": 222.06, "word": " الكل", "probability": 0.820556640625}, {"start": 222.06, "end": 222.3, "word": " قص", "probability": 0.6480712890625}, {"start": 222.3, "end": 222.58, "word": " M", "probability": 0.73828125}, {"start": 222.58, "end": 224.56, "word": " فهذا", "probability": 0.9466145833333334}, {"start": 224.56, "end": 224.94, "word": " ممكن", "probability": 0.879638671875}, {"start": 224.94, "end": 225.78, "word": " كتابته", "probability": 0.9547119140625}, {"start": 225.78, "end": 226.02, "word": " على", "probability": 0.96435546875}, {"start": 226.02, "end": 226.54, "word": " صورة", "probability": 0.96728515625}, {"start": 226.54, "end": 227.04, "word": " المقدار", "probability": 0.79443359375}, {"start": 227.04, "end": 227.3, "word": " هذا", "probability": 0.8681640625}, {"start": 227.3, "end": 228.28, "word": " على", "probability": 0.92578125}, {"start": 228.28, "end": 228.76, "word": " الصورة", "probability": 0.8767903645833334}, {"start": 228.76, "end": 229.2, "word": " واحد", "probability": 0.9912109375}, {"start": 229.2, "end": 229.94, "word": " زائد", "probability": 0.9876302083333334}, {"start": 229.94, "end": 232.02, "word": " نص", "probability": 0.990234375}, {"start": 232.02, "end": 232.92, "word": " على", "probability": 0.50537109375}, {"start": 232.92, "end": 233.28, "word": " M", "probability": 0.54541015625}, {"start": 233.28, "end": 237.12, "word": " نص", "probability": 0.934326171875}, {"start": 237.12, "end": 237.56, "word": " على", "probability": 0.8837890625}, {"start": 237.56, "end": 238.4, "word": " M", "probability": 0.9912109375}, {"start": 238.4, "end": 241.06, "word": " الكل", "probability": 0.920654296875}, {"start": 241.06, "end": 241.72, "word": " قص", "probability": 0.973876953125}, {"start": 241.72, "end": 242.3, "word": " M", "probability": 0.99267578125}, {"start": 242.3, "end": 245.6, "word": " الكل", "probability": 0.934326171875}, {"start": 245.6, "end": 246.16, "word": " تكيين", "probability": 0.718994140625}], "temperature": 1.0}, {"id": 9, "seek": 27402, "start": 249.9, "end": 274.02, "text": "صحيح وبالتالي therefore limit لما انتقل ل infinity ل ال sequence اللي لحد تبعها هذا بيساوي limit لما انتقل ل infinity ل الكلام هذا للمفضار اللي هنا", "tokens": [9381, 5016, 1829, 5016, 46599, 6027, 2655, 6027, 1829, 4412, 4948, 5296, 15042, 16472, 2655, 4587, 1211, 5296, 13202, 5296, 2423, 8310, 13672, 1829, 5296, 24401, 6055, 3555, 3615, 11296, 23758, 4724, 1829, 3794, 995, 45865, 4948, 5296, 15042, 16472, 2655, 4587, 1211, 5296, 13202, 5296, 2423, 28820, 10943, 23758, 5296, 19528, 5172, 11242, 9640, 13672, 1829, 34105], "avg_logprob": -0.3559321892463555, "compression_ratio": 1.5845070422535212, "no_speech_prob": 0.0, "words": 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0.62158203125}, {"start": 261.96, "end": 262.32, "word": " لحد", "probability": 0.642578125}, {"start": 262.32, "end": 263.1, "word": " تبعها", "probability": 0.77667236328125}, {"start": 263.1, "end": 263.5, "word": " هذا", "probability": 0.224365234375}, {"start": 263.5, "end": 265.56, "word": " بيساوي", "probability": 0.8357421875}, {"start": 265.56, "end": 266.1, "word": " limit", "probability": 0.90673828125}, {"start": 266.1, "end": 268.16, "word": " لما", "probability": 0.9404296875}, {"start": 268.16, "end": 268.98, "word": " انتقل", "probability": 0.960693359375}, {"start": 268.98, "end": 269.12, "word": " ل", "probability": 0.87548828125}, {"start": 269.12, "end": 269.72, "word": " infinity", "probability": 0.9013671875}, {"start": 269.72, "end": 270.92, "word": " ل", "probability": 0.94140625}, {"start": 270.92, "end": 271.6, "word": " الكلام", "probability": 0.8463541666666666}, {"start": 271.6, "end": 272.04, "word": " هذا", "probability": 0.8876953125}, {"start": 272.04, "end": 273.62, "word": " للمفضار", "probability": 0.625146484375}, {"start": 273.62, "end": 273.76, "word": " اللي", "probability": 0.95263671875}, {"start": 273.76, "end": 274.02, "word": " هنا", "probability": 0.98681640625}], "temperature": 1.0}, {"id": 10, "seek": 30167, "start": 274.99, "end": 301.67, "text": "و هي عندي واحد زي ال X على M او M الكل اص M ف limit المقدار اللي جوا بساوي E اص نص او E اص X الكل ده كئيب انا مقدر ادخل ال limit جوا القصير المربعين limit المقدار اللي جوا جوا القصير المربعين", "tokens": [2407, 39896, 18871, 16254, 36764, 24401, 30767, 1829, 2423, 1783, 15844, 376, 1975, 2407, 376, 2423, 28820, 1975, 9381, 376, 6156, 4948, 9673, 28543, 9640, 13672, 1829, 10874, 14407, 4724, 3794, 995, 45865, 462, 1975, 9381, 8717, 9381, 1975, 2407, 462, 1975, 9381, 1783, 2423, 28820, 11778, 3224, 9122, 19986, 1829, 3555, 1975, 8315, 3714, 28543, 2288, 1975, 3215, 9778, 1211, 2423, 4948, 10874, 14407, 25062, 9381, 13546, 9673, 25513, 3615, 9957, 4948, 9673, 28543, 9640, 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نص", "probability": 0.9765625}, {"start": 289.09, "end": 289.79, "word": " او", "probability": 0.971923828125}, {"start": 289.79, "end": 290.03, "word": " E", "probability": 0.9609375}, {"start": 290.03, "end": 290.25, "word": " اص", "probability": 0.923095703125}, {"start": 290.25, "end": 290.67, "word": " X", "probability": 0.99462890625}, {"start": 290.67, "end": 292.19, "word": " الكل", "probability": 0.8212890625}, {"start": 292.19, "end": 293.37, "word": " ده", "probability": 0.4921875}, {"start": 293.37, "end": 293.67, "word": " كئيب", "probability": 0.68048095703125}, {"start": 293.67, "end": 293.81, "word": " انا", "probability": 0.5694580078125}, {"start": 293.81, "end": 294.11, "word": " مقدر", "probability": 0.8816731770833334}, {"start": 294.11, "end": 294.53, "word": " ادخل", "probability": 0.9483642578125}, {"start": 294.53, "end": 294.65, "word": " ال", "probability": 0.53271484375}, {"start": 294.65, "end": 294.97, "word": " limit", "probability": 0.97900390625}, {"start": 294.97, "end": 295.87, "word": " جوا", "probability": 0.937744140625}, {"start": 295.87, "end": 296.37, "word": " القصير", "probability": 0.8899739583333334}, {"start": 296.37, "end": 297.15, "word": " المربعين", "probability": 0.8837890625}, {"start": 297.15, "end": 297.95, "word": " limit", "probability": 0.919921875}, {"start": 297.95, "end": 298.79, "word": " المقدار", "probability": 0.9921875}, {"start": 298.79, "end": 299.07, "word": " اللي", "probability": 0.98193359375}, {"start": 299.07, "end": 299.33, "word": " جوا", "probability": 0.967041015625}, {"start": 299.33, "end": 300.51, "word": " جوا", "probability": 0.6104736328125}, {"start": 300.51, "end": 300.93, "word": " القصير", "probability": 0.9889322916666666}, {"start": 300.93, "end": 301.67, "word": " المربعين", "probability": 0.9957275390625}], "temperature": 1.0}, {"id": 11, "seek": 32508, "start": 302.31, "end": 325.09, "text": "حسب القاعدة او القانون هذا E أُس X اللي هو نص وبعدين 2 بالكلتك A فبطلع E أُس 3 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"temperature": 1.0}, {"id": 15, "seek": 44755, "start": 421.91, "end": 447.55, "text": "إذا XIN و YIN هم سيكونات كوشي هم سيكونات كوشي هم سيكونات كوشي هم سيكونات كوشي", "tokens": [28814, 15730, 1783, 1464, 4032, 398, 1464, 8032, 2304, 8608, 1829, 4117, 2407, 8315, 2655, 9122, 2407, 8592, 1829, 8032, 2304, 8608, 1829, 4117, 2407, 8315, 2655, 9122, 2407, 8592, 1829, 8032, 2304, 8608, 1829, 4117, 2407, 8315, 2655, 9122, 2407, 8592, 1829, 8032, 2304, 8608, 1829, 4117, 2407, 8315, 2655, 9122, 2407, 8592, 1829], "avg_logprob": -0.37611606397799086, "compression_ratio": 2.462962962962963, "no_speech_prob": 0.0, "words": [{"start": 421.91, "end": 422.49, "word": "إذا", "probability": 0.5657958984375}, {"start": 422.49, "end": 424.71, "word": " XIN", "probability": 0.29736328125}, {"start": 424.71, "end": 426.51, "word": " و", "probability": 0.9482421875}, {"start": 426.51, "end": 427.73, "word": " YIN", "probability": 0.760498046875}, {"start": 427.73, "end": 430.15, "word": " هم", 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0.9737548828125}], "temperature": 1.0}, {"id": 33, "seek": 95511, "start": 927.01, "end": 955.11, "text": "و بعدين أخدت الأزواج هذه مع بعض و استخدمت ل triangle inequality ال absolute value ل مجموعة كبيرة أصغر من أوسع مجموعة absolute values فالان باستخدام لو سمينا المعادلة هاد ال star فباستخدام ال star", "tokens": [2407, 39182, 9957, 5551, 9778, 3215, 2655, 16247, 11622, 2407, 26108, 29538, 20449, 45030, 11242, 4032, 44713, 9778, 40448, 2655, 5296, 13369, 16970, 2423, 8236, 2158, 5296, 3714, 7435, 2304, 2407, 27884, 9122, 3555, 48923, 5551, 9381, 17082, 2288, 9154, 34051, 3794, 3615, 3714, 7435, 2304, 2407, 27884, 8236, 4190, 6156, 6027, 7649, 4724, 995, 14851, 9778, 3215, 10943, 45164, 8608, 2304, 1829, 8315, 9673, 3615, 18513, 37977, 8032, 18513, 2423, 3543, 6156, 3555, 995, 14851, 9778, 3215, 10943, 2423, 3543], "avg_logprob": -0.21684451910053812, "compression_ratio": 1.6117021276595744, "no_speech_prob": 0.0, "words": [{"start": 927.01, "end": 927.11, "word": "و", "probability": 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بساوي ال على two to n و هكذا و آخر حد هيكون ال over", "tokens": [6027, 39184, 8236, 2158, 29538, 4724, 3794, 995, 45865, 2423, 670, 732, 281, 297, 3175, 36764, 24401, 1392, 29538, 8717, 36178, 11296, 11778, 1829, 4032, 2423, 8236, 2158, 13672, 1829, 39182, 11296, 4724, 3794, 995, 45865, 2423, 15844, 732, 281, 297, 4032, 8032, 4117, 15730, 4032, 19753, 34740, 11331, 3215, 39896, 30544, 2423, 670], "avg_logprob": -0.2627840833230452, "compression_ratio": 1.4933333333333334, "no_speech_prob": 0.0, "words": [{"start": 957.42, "end": 957.74, "word": "الـ", "probability": 0.5125732421875}, {"start": 957.74, "end": 958.08, "word": " absolute", "probability": 0.62841796875}, {"start": 958.08, "end": 958.74, "word": " value", "probability": 0.99267578125}, {"start": 958.74, "end": 959.18, "word": " هذه", "probability": 0.4873046875}, {"start": 959.18, "end": 960.44, "word": " بساوي", "probability": 0.6380615234375}, {"start": 960.44, "end": 961.46, "word": " ال", "probability": 0.277099609375}, 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على اتنين قص ان نجاتف واحد والمجموع هذا اصغر من اتنين لان هذا جزء من الهارمونيك جيومتريكس سيريز اللي هي واحد على اتنين قص ان من ان equal zero to infinity هذه مجموعة اتنين", "tokens": [6027, 7649, 23758, 1975, 9381, 17082, 2288, 9154, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 12174, 9381, 16472, 8717, 7435, 9307, 5172, 36764, 24401, 16070, 2304, 7435, 2304, 45367, 23758, 1975, 9381, 17082, 2288, 9154, 1975, 2655, 1863, 9957, 5296, 7649, 23758, 10874, 11622, 38207, 9154, 2423, 3224, 9640, 2304, 11536, 1829, 4117, 10874, 1829, 20498, 2655, 16572, 4117, 3794, 8608, 13546, 1829, 11622, 13672, 1829, 39896, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 12174, 9381, 16472, 9154, 16472, 2681, 4018, 281, 13202, 29538, 3714, 7435, 2304, 2407, 27884, 1975, 2655, 1863, 9957], "avg_logprob": -0.28557180153562667, "compression_ratio": 1.8901734104046244, "no_speech_prob": 0.0, "words": [{"start": 1040.14, "end": 1041.1, "word": "الان", "probability": 0.64208984375}, {"start": 1041.1, "end": 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"end": 1067.7, "word": " مجموعة", "probability": 0.9703125}, {"start": 1067.7, "end": 1068.16, "word": " اتنين", "probability": 0.979736328125}], "temperature": 1.0}, {"id": 38, "seek": 109741, "start": 1069.35, "end": 1097.41, "text": "فهذا جزء منها وبالتالي المجموعة هذا أصغر من المجموعة الـ geometric series هذه المجموعة بساوة اتنين okay فإذا هذا المجموعة أصغر من اتنين إذا هذا أصغر من واحد على اتنين قص ان سالب واحد في اتنين اللي هو عبارة عن واحد على اتنين قص ان سالب اتنين", "tokens": [5172, 3224, 15730, 10874, 11622, 38207, 9154, 11296, 46599, 6027, 2655, 6027, 1829, 9673, 7435, 2304, 2407, 27884, 23758, 5551, 9381, 17082, 2288, 9154, 9673, 7435, 2304, 2407, 27884, 2423, 39184, 33246, 2638, 29538, 9673, 7435, 2304, 2407, 27884, 4724, 3794, 995, 2407, 3660, 1975, 2655, 1863, 9957, 1392, 6156, 28814, 15730, 23758, 9673, 7435, 2304, 2407, 27884, 5551, 9381, 17082, 2288, 9154, 1975, 2655, 1863, 9957, 11933, 15730, 23758, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 12174, 9381, 16472, 8608, 6027, 3555, 36764, 24401, 8978, 1975, 2655, 1863, 9957, 13672, 1829, 31439, 6225, 3555, 9640, 3660, 18871, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 12174, 9381, 16472, 8608, 6027, 3555, 1975, 2655, 1863, 9957], "avg_logprob": -0.1587035143670957, "compression_ratio": 2.2432432432432434, "no_speech_prob": 0.0, "words": [{"start": 1069.35, "end": 1069.81, "word": "فهذا", "probability": 0.75048828125}, {"start": 1069.81, "end": 1070.25, "word": " جزء", "probability": 0.97119140625}, {"start": 1070.25, "end": 1070.63, "word": " منها", "probability": 0.95947265625}, {"start": 1070.63, "end": 1071.07, "word": " وبالتالي", "probability": 0.8755859375}, {"start": 1071.07, "end": 1071.55, "word": " المجموعة", "probability": 0.8794921875}, {"start": 1071.55, "end": 1071.81, "word": " هذا", "probability": 0.45458984375}, {"start": 1071.81, "end": 1072.39, "word": " أصغر", "probability": 0.901611328125}, {"start": 1072.39, "end": 1072.53, "word": " من", 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"probability": 0.8978515625}, {"start": 1080.97, "end": 1081.49, "word": " أصغر", "probability": 0.9752197265625}, {"start": 1081.49, "end": 1081.63, "word": " من", "probability": 0.9951171875}, {"start": 1081.63, "end": 1082.15, "word": " اتنين", "probability": 0.961181640625}, {"start": 1082.15, "end": 1082.69, "word": " إذا", "probability": 0.591552734375}, {"start": 1082.69, "end": 1083.09, "word": " هذا", "probability": 0.767578125}, {"start": 1083.09, "end": 1083.59, "word": " أصغر", "probability": 0.975830078125}, {"start": 1083.59, "end": 1083.83, "word": " من", "probability": 0.99609375}, {"start": 1083.83, "end": 1084.25, "word": " واحد", "probability": 0.951416015625}, {"start": 1084.25, "end": 1084.41, "word": " على", "probability": 0.5703125}, {"start": 1084.41, "end": 1084.73, "word": " اتنين", "probability": 0.9697265625}, {"start": 1084.73, "end": 1085.07, "word": " قص", "probability": 0.65625}, {"start": 1085.07, "end": 1085.47, "word": " ان", "probability": 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0.929443359375}, {"start": 1094.99, "end": 1095.33, "word": " ان", "probability": 0.93896484375}, {"start": 1095.33, "end": 1096.09, "word": " سالب", "probability": 0.9674479166666666}, {"start": 1096.09, "end": 1097.41, "word": " اتنين", "probability": 0.987548828125}], "temperature": 1.0}, {"id": 39, "seek": 111443, "start": 1108.77, "end": 1114.43, "text": "الان هذا بيروح للسفر as n tends to infinity", "tokens": [6027, 7649, 23758, 4724, 13546, 2407, 5016, 24976, 3794, 5172, 2288, 382, 297, 12258, 281, 13202], "avg_logprob": -0.35041360294117646, "compression_ratio": 0.8450704225352113, "no_speech_prob": 0.0, "words": [{"start": 1108.77, "end": 1109.77, "word": "الان", "probability": 0.4915771484375}, {"start": 1109.77, "end": 1110.17, "word": " هذا", "probability": 0.5205078125}, {"start": 1110.17, "end": 1110.79, "word": " بيروح", "probability": 0.877685546875}, {"start": 1110.79, "end": 1111.71, "word": " للسفر", "probability": 0.7608642578125}, {"start": 1111.71, "end": 1112.93, "word": " as", "probability": 0.8427734375}, {"start": 1112.93, "end": 1113.43, "word": " n", "probability": 0.4462890625}, {"start": 1113.43, "end": 1113.77, "word": " tends", "probability": 0.55029296875}, {"start": 1113.77, "end": 1113.93, "word": " to", "probability": 0.9306640625}, {"start": 1113.93, "end": 1114.43, "word": " infinity", "probability": 0.87255859375}], "temperature": 1.0}, {"id": 40, "seek": 113727, "start": 1115.39, "end": 1137.27, "text": "هو بالتالي ده بتطلع عندي أنا ال limit ل absolute xn minus xm as m as n tenths of infinity و طبعاً m تقول infinity تساوي سفر therefore ال sequence xn is Cauchy", "tokens": [3224, 2407, 20666, 2655, 6027, 1829, 11778, 3224, 39894, 9566, 1211, 3615, 18871, 16254, 41850, 2423, 4948, 5296, 8236, 2031, 77, 3175, 2031, 76, 382, 275, 382, 297, 2064, 32184, 295, 13202, 4032, 23032, 3555, 3615, 995, 14111, 275, 6055, 39648, 13202, 6055, 3794, 995, 45865, 8608, 5172, 2288, 4412, 2423, 8310, 2031, 77, 307, 7544, 625, 88], "avg_logprob": -0.42108051554631376, "compression_ratio": 1.2830188679245282, "no_speech_prob": 0.0, "words": [{"start": 1115.39, "end": 1115.79, "word": "هو", "probability": 0.49945068359375}, {"start": 1115.79, "end": 1116.43, "word": " بالتالي", "probability": 0.826904296875}, {"start": 1116.43, "end": 1116.69, "word": " ده", "probability": 0.552734375}, {"start": 1116.69, "end": 1117.09, "word": " بتطلع", "probability": 0.74298095703125}, {"start": 1117.09, "end": 1117.39, "word": " عندي", "probability": 0.822021484375}, {"start": 1117.39, "end": 1117.57, "word": " أنا", "probability": 0.345703125}, {"start": 1117.57, "end": 1117.71, "word": " ال", "probability": 0.449462890625}, {"start": 1117.71, "end": 1118.09, "word": " limit", "probability": 0.8994140625}, {"start": 1118.09, "end": 1119.33, "word": " ل", "probability": 0.93310546875}, {"start": 1119.33, "end": 1119.89, "word": " absolute", "probability": 0.6845703125}, {"start": 1119.89, "end": 1120.77, "word": " xn", "probability": 0.583740234375}, {"start": 1120.77, "end": 1121.21, "word": " minus", "probability": 0.463134765625}, {"start": 1121.21, "end": 1122.41, "word": " xm", "probability": 0.872314453125}, {"start": 1122.41, "end": 1123.57, "word": " as", "probability": 0.81298828125}, {"start": 1123.57, "end": 1124.07, "word": " m", "probability": 0.321044921875}, {"start": 1124.07, "end": 1124.99, "word": " as", "probability": 0.87451171875}, {"start": 1124.99, "end": 1127.13, "word": " n", "probability": 0.56591796875}, {"start": 1127.13, "end": 1127.61, "word": " tenths", "probability": 0.320556640625}, {"start": 1127.61, "end": 1127.77, "word": " of", "probability": 0.45849609375}, {"start": 1127.77, "end": 1128.33, "word": " infinity", "probability": 0.88037109375}, {"start": 1128.33, "end": 1128.73, "word": " و", "probability": 0.82958984375}, {"start": 1128.73, "end": 1129.37, "word": " طبعاً", "probability": 0.80390625}, {"start": 1129.37, "end": 1129.67, "word": " m", "probability": 0.74658203125}, {"start": 1129.67, "end": 1130.17, "word": " تقول", "probability": 0.63818359375}, {"start": 1130.17, "end": 1130.89, "word": " infinity", "probability": 0.65185546875}, {"start": 1130.89, "end": 1131.79, "word": " تساوي", "probability": 0.638763427734375}, {"start": 1131.79, "end": 1132.37, "word": " سفر", "probability": 0.90283203125}, {"start": 1132.37, "end": 1133.31, "word": " therefore", "probability": 0.357421875}, {"start": 1133.31, "end": 1133.93, "word": " ال", "probability": 0.65576171875}, {"start": 1133.93, "end": 1134.49, "word": " sequence", "probability": 0.98681640625}, {"start": 1134.49, "end": 1135.41, "word": " xn", "probability": 0.913330078125}, {"start": 1135.41, "end": 1136.67, "word": " is", "probability": 0.900390625}, {"start": 1136.67, "end": 1137.27, "word": " Cauchy", "probability": 0.8834635416666666}], "temperature": 1.0}, {"id": 41, "seek": 115766, "start": 1142.5, "end": 1157.66, "text": "و بالتالي therefore by cauchy criterion by cauchy criterion x in convergence صح؟ say", "tokens": [2407, 20666, 2655, 6027, 1829, 4412, 538, 1335, 625, 88, 46691, 538, 1335, 625, 88, 46691, 2031, 294, 32181, 20328, 5016, 22807, 584], "avg_logprob": -0.3749999962747097, "compression_ratio": 1.1585365853658536, "no_speech_prob": 0.0, "words": [{"start": 1142.5, "end": 1142.76, "word": "و", "probability": 0.5078125}, {"start": 1142.76, "end": 1143.5, "word": " بالتالي", "probability": 0.877685546875}, {"start": 1143.5, "end": 1144.24, "word": " therefore", "probability": 0.2366943359375}, {"start": 1144.24, "end": 1145.56, "word": " by", "probability": 0.69287109375}, {"start": 1145.56, "end": 1146.14, "word": " cauchy", "probability": 0.7159016927083334}, {"start": 1146.14, "end": 1146.92, "word": " criterion", "probability": 0.93505859375}, {"start": 1146.92, "end": 1148.88, "word": " by", "probability": 0.66748046875}, {"start": 1148.88, "end": 1149.62, "word": " cauchy", "probability": 0.9259440104166666}, {"start": 1149.62, "end": 1150.4, "word": " criterion", "probability": 0.95068359375}, {"start": 1150.4, "end": 1152.72, "word": " x", "probability": 0.568359375}, {"start": 1152.72, "end": 1153.32, "word": " in", "probability": 0.425048828125}, {"start": 1153.32, "end": 1155.18, "word": " convergence", "probability": 0.37451171875}, {"start": 1155.18, "end": 1157.66, "word": " صح؟", "probability": 0.7509765625}, {"start": 1157.66, "end": 1157.66, "word": " say", "probability": 0.8076171875}], "temperature": 1.0}, {"id": 42, "seek": 118973, "start": 1166.77, "end": 1189.73, "text": "دعينا نسمي ال limit ل x in بساوي x حيث x is real number الان مطلوب ان احنا نجد قيمة ال x قيمة ال limit لل sequence اللى هى سمنها x فبنرجع للمعادلة هناك", "tokens": [3215, 3615, 1829, 8315, 8717, 38251, 1829, 2423, 4948, 5296, 2031, 294, 4724, 3794, 995, 45865, 2031, 11331, 1829, 12984, 2031, 307, 957, 1230, 2423, 7649, 3714, 9566, 1211, 37746, 16472, 1975, 5016, 8315, 8717, 7435, 3215, 12174, 32640, 3660, 2423, 2031, 12174, 32640, 3660, 2423, 4948, 24976, 8310, 13672, 7578, 8032, 7578, 8608, 27842, 11296, 2031, 6156, 3555, 1863, 47341, 3615, 5296, 19528, 3615, 18513, 37977, 34105, 4117], "avg_logprob": -0.28125000638621195, "compression_ratio": 1.45, "no_speech_prob": 0.0, "words": [{"start": 1166.77, "end": 1167.79, "word": "دعينا", "probability": 0.793701171875}, {"start": 1167.79, "end": 1168.47, "word": " نسمي", "probability": 0.92822265625}, {"start": 1168.47, "end": 1169.27, "word": " ال", "probability": 0.4404296875}, {"start": 1169.27, "end": 1169.67, "word": " limit", "probability": 0.783203125}, {"start": 1169.67, "end": 1170.83, "word": " ل", "probability": 0.7666015625}, {"start": 1170.83, "end": 1171.21, "word": " x", "probability": 0.296142578125}, {"start": 1171.21, "end": 1171.57, "word": " in", "probability": 0.513671875}, {"start": 1171.57, "end": 1172.55, "word": " بساوي", "probability": 0.8485107421875}, {"start": 1172.55, "end": 1173.09, "word": " x", "probability": 0.921875}, {"start": 1173.09, "end": 1174.23, "word": " حيث", "probability": 0.5959676106770834}, {"start": 1174.23, "end": 1174.55, "word": " x", "probability": 0.8955078125}, {"start": 1174.55, "end": 1174.83, "word": " is", "probability": 0.92578125}, {"start": 1174.83, "end": 1175.11, "word": " real", "probability": 0.71728515625}, {"start": 1175.11, "end": 1175.59, "word": " number", "probability": 0.974609375}, {"start": 1175.59, "end": 1178.01, "word": " الان", "probability": 0.61083984375}, {"start": 1178.01, "end": 1178.43, "word": " مطلوب", "probability": 0.853271484375}, {"start": 1178.43, "end": 1178.61, "word": " ان", "probability": 0.35302734375}, {"start": 1178.61, "end": 1178.87, "word": " احنا", "probability": 0.7608235677083334}, {"start": 1178.87, "end": 1179.49, "word": " نجد", "probability": 0.8047688802083334}, {"start": 1179.49, "end": 1180.15, "word": " قيمة", "probability": 0.9884440104166666}, {"start": 1180.15, "end": 1180.27, "word": " ال", "probability": 0.88623046875}, {"start": 1180.27, "end": 1180.51, "word": " x", "probability": 0.41552734375}, {"start": 1180.51, "end": 1180.97, "word": " قيمة", "probability": 0.9021809895833334}, {"start": 1180.97, "end": 1181.11, "word": " ال", "probability": 0.7685546875}, {"start": 1181.11, "end": 1181.45, "word": " limit", "probability": 0.97705078125}, {"start": 1181.45, "end": 1182.11, "word": " لل", "probability": 0.66064453125}, {"start": 1182.11, "end": 1182.69, "word": " sequence", "probability": 0.96240234375}, {"start": 1182.69, "end": 1184.07, "word": " اللى", "probability": 0.7127685546875}, {"start": 1184.07, "end": 1184.23, "word": " هى", "probability": 0.69677734375}, {"start": 1184.23, "end": 1184.79, "word": " سمنها", "probability": 0.74853515625}, {"start": 1184.79, "end": 1185.19, "word": " x", "probability": 0.92333984375}, {"start": 1185.19, "end": 1187.63, "word": " فبنرجع", "probability": 0.866796875}, {"start": 1187.63, "end": 1188.47, "word": " للمعادلة", "probability": 0.89853515625}, {"start": 1188.47, "end": 1189.73, "word": " هناك", "probability": 0.802734375}], "temperature": 1.0}, {"id": 43, "seek": 122694, "start": 1207.88, "end": 1226.94, "text": "to find x take limits of both sides", "tokens": [1353, 915, 2031, 747, 10406, 295, 1293, 4881], "avg_logprob": -0.2060546875, "compression_ratio": 0.813953488372093, "no_speech_prob": 0.0, "words": [{"start": 1207.8799999999999, "end": 1209.28, "word": "to", "probability": 0.64404296875}, {"start": 1209.28, "end": 1209.88, "word": " find", "probability": 0.947265625}, {"start": 1209.88, "end": 1213.02, "word": " x", "probability": 0.54296875}, {"start": 1213.02, "end": 1215.08, "word": " take", "probability": 0.61669921875}, {"start": 1215.08, "end": 1218.98, "word": " limits", "probability": 0.9794921875}, {"start": 1218.98, "end": 1223.96, "word": " of", "probability": 0.97900390625}, {"start": 1223.96, "end": 1226.3, "word": " both", "probability": 0.9208984375}, {"start": 1226.3, "end": 1226.94, 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directly that a bounded a bounded a bounded monotone a bounded monotone", "tokens": [2304, 8592, 8032, 2407, 3838, 300, 257, 37498, 257, 37498, 257, 37498, 1108, 310, 546, 257, 37498, 1108, 310, 546], "avg_logprob": -0.47805059523809523, "compression_ratio": 1.528301886792453, "no_speech_prob": 0.0, "words": [{"start": 1557.19, "end": 1557.29, "word": "مش", "probability": 0.2200927734375}, {"start": 1557.29, "end": 1557.59, "word": " هو", "probability": 0.35888671875}, {"start": 1557.59, "end": 1558.17, "word": " directly", "probability": 0.56298828125}, {"start": 1558.17, "end": 1565.59, "word": " that", "probability": 0.62060546875}, {"start": 1565.59, "end": 1568.45, "word": " a", "probability": 0.87548828125}, {"start": 1568.45, "end": 1569.01, "word": " bounded", "probability": 0.9697265625}, {"start": 1569.01, "end": 1572.55, "word": " a", "probability": 0.291015625}, {"start": 1572.55, "end": 1573.27, "word": " bounded", "probability": 0.9541015625}, {"start": 1573.27, "end": 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"text": "وهذا معناه ان absolute xn او xn minus xn اصغر من epsilon اصبت؟ صح؟ وهذا صحيح لكل M أكبر من أو ساوي M أكبر من أو ساوي capital M تمام؟ اذا بنقول هنا since", "tokens": [2407, 3224, 15730, 20449, 8315, 3224, 16472, 8236, 2031, 77, 1975, 2407, 2031, 77, 3175, 2031, 77, 1975, 9381, 17082, 2288, 9154, 17889, 1975, 9381, 3555, 2655, 22807, 20328, 5016, 22807, 37037, 15730, 20328, 5016, 1829, 5016, 5296, 28820, 376, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 376, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 376, 46811, 10943, 22807, 1975, 15730, 44945, 39648, 34105, 1670], "avg_logprob": -0.27377716700236004, "compression_ratio": 1.4743589743589745, "no_speech_prob": 0.0, "words": [{"start": 2168.37, "end": 2169.33, "word": "وهذا", "probability": 0.7887369791666666}, {"start": 2169.33, "end": 2170.15, "word": " معناه", "probability": 0.9124348958333334}, {"start": 2170.15, "end": 2170.41, "word": " ان", "probability": 0.57373046875}, {"start": 2170.41, "end": 2171.03, "word": " absolute", "probability": 0.49365234375}, {"start": 2171.03, "end": 2172.35, "word": " xn", "probability": 0.54638671875}, {"start": 2172.35, "end": 2173.33, "word": " او", "probability": 0.85595703125}, {"start": 2173.33, "end": 2174.03, "word": " xn", "probability": 0.7138671875}, {"start": 2174.03, "end": 2174.51, "word": " minus", "probability": 0.671875}, {"start": 2174.51, "end": 2177.15, "word": " xn", "probability": 0.906005859375}, {"start": 2177.15, "end": 2178.79, "word": " اصغر", "probability": 0.905029296875}, {"start": 2178.79, "end": 2179.01, "word": " من", "probability": 0.99267578125}, {"start": 2179.01, "end": 2179.57, "word": " epsilon", "probability": 0.4501953125}, {"start": 2179.57, "end": 2180.93, "word": " اصبت؟", "probability": 0.4868408203125}, {"start": 2180.93, "end": 2182.15, "word": " صح؟", "probability": 0.9073893229166666}, {"start": 2182.15, "end": 2184.45, "word": " وهذا", "probability": 0.76220703125}, {"start": 2184.45, "end": 2185.13, "word": " صحيح", "probability": 0.9959716796875}, {"start": 2185.13, "end": 2186.61, "word": " لكل", "probability": 0.965087890625}, {"start": 2186.61, "end": 2186.99, "word": " M", "probability": 0.55126953125}, {"start": 2186.99, "end": 2187.63, "word": " أكبر", "probability": 0.81005859375}, {"start": 2187.63, "end": 2187.87, "word": " من", "probability": 0.94873046875}, {"start": 2187.87, "end": 2188.11, "word": " أو", "probability": 0.8388671875}, {"start": 2188.11, "end": 2188.73, "word": " ساوي", "probability": 0.8790690104166666}, {"start": 2188.73, "end": 2189.05, "word": " M", "probability": 0.6103515625}, {"start": 2189.05, "end": 2189.61, "word": " أكبر", "probability": 0.9241536458333334}, {"start": 2189.61, "end": 2189.75, "word": " من", "probability": 0.99365234375}, {"start": 2189.75, "end": 2189.95, "word": " أو", "probability": 0.96337890625}, {"start": 2189.95, "end": 2190.43, "word": " ساوي", "probability": 0.97412109375}, {"start": 2190.43, "end": 2190.97, "word": " capital", "probability": 0.435546875}, {"start": 2190.97, "end": 2191.35, "word": " M", "probability": 0.94287109375}, {"start": 2191.35, "end": 2192.93, "word": " تمام؟", "probability": 0.73681640625}, {"start": 2192.93, "end": 2193.29, "word": " اذا", "probability": 0.6822509765625}, {"start": 2193.29, "end": 2193.77, "word": " بنقول", "probability": 0.6593017578125}, {"start": 2193.77, "end": 2194.03, "word": " هنا", "probability": 0.9892578125}, {"start": 2194.03, "end": 2194.55, "word": " since", "probability": 0.56494140625}], "temperature": 1.0}, {"id": 77, "seek": 221962, "start": 2196.14, "end": 2219.62, "text": "epsilon أكبر من السفر was arbitrary it follows بينتج من تعريف الكوشي sequence that sequence xm is cauchy", "tokens": [10653, 15754, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 390, 23211, 309, 10002, 4724, 9957, 2655, 7435, 9154, 37279, 16572, 5172, 33251, 2407, 8592, 1829, 8310, 300, 8310, 2031, 76, 307, 1335, 625, 88], "avg_logprob": -0.29575893197740827, "compression_ratio": 1.0991735537190082, "no_speech_prob": 0.0, "words": [{"start": 2196.14, "end": 2196.76, "word": "epsilon", "probability": 0.55010986328125}, {"start": 2196.76, "end": 2197.18, "word": " أكبر", "probability": 0.892578125}, {"start": 2197.18, "end": 2197.38, "word": " من", "probability": 0.98828125}, {"start": 2197.38, "end": 2197.96, "word": " السفر", "probability": 0.841796875}, {"start": 2197.96, "end": 2198.54, "word": " was", "probability": 0.438232421875}, {"start": 2198.54, "end": 2199.46, "word": " arbitrary", "probability": 0.76953125}, {"start": 2199.46, "end": 2206.34, "word": " it", "probability": 0.60986328125}, {"start": 2206.34, "end": 2207.0, "word": " follows", "probability": 0.8896484375}, {"start": 2207.0, "end": 2210.64, "word": " بينتج", "probability": 0.829833984375}, {"start": 2210.64, "end": 2210.86, "word": " من", "probability": 0.99169921875}, {"start": 2210.86, "end": 2211.48, "word": " تعريف", "probability": 0.99365234375}, {"start": 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8608, 33604, 6027, 6055, 7649, 10632, 4032, 46958, 5016, 2423, 26890, 3224, 7649, 8978, 36632, 8608, 5172, 9381, 9640], "avg_logprob": -0.26114003636218885, "compression_ratio": 1.4916666666666667, "no_speech_prob": 0.0, "words": [{"start": 2223.2, "end": 2224.02, "word": "وهنا", "probability": 0.4899495442708333}, {"start": 2224.02, "end": 2224.32, "word": " هيك", "probability": 0.681884765625}, {"start": 2224.32, "end": 2224.88, "word": " طبقنا", "probability": 0.9443359375}, {"start": 2224.88, "end": 2225.64, "word": " التعريف", "probability": 0.9493408203125}, {"start": 2225.64, "end": 2226.52, "word": " وهذا", "probability": 0.77685546875}, {"start": 2226.52, "end": 2227.04, "word": " برهان", "probability": 0.81494140625}, {"start": 2227.04, "end": 2227.76, "word": " مباشر", "probability": 0.9891357421875}, {"start": 2227.76, "end": 2228.86, "word": " okay", "probability": 0.468505859375}, {"start": 2228.86, "end": 2231.0, "word": " تمام", "probability": 0.96435546875}, {"start": 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2270.92, "text": "في أي سؤال تاني section تلاتة خمسة أو تلاتة ستة لبعده الناس اللي بدرسوا", "tokens": [41185, 36632, 8608, 33604, 6027, 6055, 7649, 1829, 3541, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 34051, 6055, 1211, 9307, 3660, 8608, 2655, 3660, 5296, 3555, 22488, 3224, 2423, 8315, 3794, 13672, 1829, 47525, 2288, 3794, 14407], "avg_logprob": -0.1219951915435302, "compression_ratio": 1.2323232323232323, "no_speech_prob": 0.0, "words": [{"start": 2252.98, "end": 2253.28, "word": "في", "probability": 0.83642578125}, {"start": 2253.28, "end": 2253.52, "word": " أي", "probability": 0.705078125}, {"start": 2253.52, "end": 2253.88, "word": " سؤال", "probability": 0.9925130208333334}, {"start": 2253.88, "end": 2254.3, "word": " تاني", "probability": 0.90673828125}, {"start": 2254.3, "end": 2254.62, "word": " section", "probability": 0.51513671875}, {"start": 2254.62, "end": 2255.14, "word": " تلاتة", "probability": 0.9010009765625}, {"start": 2255.14, "end": 2255.84, "word": " خمسة", "probability": 0.941162109375}, {"start": 2255.84, "end": 2257.24, "word": " أو", "probability": 0.7294921875}, {"start": 2257.24, "end": 2257.84, "word": " تلاتة", "probability": 0.9693603515625}, {"start": 2257.84, "end": 2258.32, "word": " ستة", "probability": 0.89892578125}, {"start": 2258.32, "end": 2261.6, "word": " لبعده", "probability": 0.85595703125}, {"start": 2261.6, "end": 2270.06, "word": " الناس", "probability": 0.9044596354166666}, {"start": 2270.06, "end": 2270.24, "word": " اللي", "probability": 0.95751953125}, {"start": 2270.24, "end": 2270.92, "word": " بدرسوا", "probability": 0.87353515625}], "temperature": 1.0}, {"id": 80, "seek": 230498, "start": 2283.44, "end": 2304.98, "text": "التلاتة ستة؟ سؤال أربعة في رياضينا. هذا ما حللناه إزاي؟", "tokens": [6027, 2655, 1211, 9307, 3660, 8608, 2655, 3660, 22807, 8608, 33604, 6027, 5551, 25513, 27884, 8978, 12602, 1829, 46958, 1829, 8315, 13, 23758, 19446, 11331, 1211, 1211, 8315, 3224, 11933, 11622, 47302, 22807], 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235294, "start": 2324.99, "end": 2352.95, "text": "السكتشن تلاتة ستة السؤال الرابع ترى بي establish establish the proper divergence of sequence", "tokens": [6027, 3794, 4117, 2655, 8592, 1863, 6055, 1211, 9307, 3660, 8608, 2655, 3660, 21136, 33604, 6027, 34892, 16758, 3615, 6055, 2288, 7578, 4724, 1829, 8327, 8327, 264, 2296, 47387, 295, 8310], "avg_logprob": -0.3659668015316129, "compression_ratio": 1.203883495145631, "no_speech_prob": 0.0, "words": [{"start": 2324.99, "end": 2325.57, "word": "السكتشن", "probability": 0.5915730794270834}, {"start": 2325.57, "end": 2326.13, "word": " تلاتة", "probability": 0.8104248046875}, {"start": 2326.13, "end": 2326.83, "word": " ستة", "probability": 0.90283203125}, {"start": 2326.83, "end": 2327.65, "word": " السؤال", "probability": 0.9694010416666666}, {"start": 2327.65, "end": 2328.43, "word": " الرابع", "probability": 0.9842122395833334}, {"start": 2328.43, "end": 2331.25, "word": " ترى", "probability": 0.43115234375}, {"start": 2331.25, 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new file mode 100644
index 0000000000000000000000000000000000000000..b992cfbf81081065dfda553bac25ddbcd3444f6c
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"word": " أكبر", "probability": 0.9658203125}, {"start": 105.82, "end": 106.04, "word": " من", "probability": 0.98046875}, {"start": 106.04, "end": 106.2, "word": " أو", "probability": 0.1339111328125}, {"start": 106.2, "end": 106.62, "word": " ساوي", "probability": 0.8408203125}, {"start": 106.62, "end": 107.1, "word": " capital", "probability": 0.254150390625}, {"start": 107.1, "end": 107.58, "word": " M", "probability": 0.93603515625}, {"start": 107.58, "end": 108.94, "word": " فهذا", "probability": 0.9462890625}, {"start": 108.94, "end": 109.46, "word": " بيقدي", "probability": 0.57745361328125}, {"start": 109.46, "end": 110.92, "word": " أنه", "probability": 0.623779296875}, {"start": 110.92, "end": 112.08, "word": " absolute", "probability": 0.7666015625}, {"start": 112.08, "end": 115.06, "word": " SM", "probability": 0.348876953125}, {"start": 115.06, "end": 115.74, "word": " minus", "probability": 0.6025390625}, {"start": 115.74, "end": 116.56, "word": " SN", "probability": 0.771484375}, {"start": 116.56, "end": 117.34, "word": " بساوي", "probability": 0.9495849609375}, {"start": 117.34, "end": 118.28, "word": " absolute", "probability": 0.8994140625}, {"start": 118.28, "end": 120.36, "word": " XN", "probability": 0.841796875}, {"start": 120.36, "end": 120.9, "word": " زاد", "probability": 0.5863037109375}, {"start": 120.9, "end": 121.42, "word": " واحد", "probability": 0.849609375}, {"start": 121.42, "end": 121.9, "word": " زاد", "probability": 0.94970703125}, {"start": 121.9, "end": 122.84, "word": " XN", "probability": 0.928955078125}, {"start": 122.84, "end": 124.56, "word": " زاد", "probability": 0.93505859375}, {"start": 124.56, "end": 125.02, "word": " اتنين", "probability": 0.964599609375}, {"start": 125.02, "end": 125.18, "word": " و", "probability": 0.7294921875}, {"start": 125.18, "end": 125.84, "word": " هكذا", "probability": 0.8797200520833334}, {"start": 125.84, "end": 126.18, "word": " إلى", "probability": 0.85205078125}, {"start": 126.18, "end": 127.3, "word": " XM", "probability": 0.94140625}], "temperature": 1.0}, {"id": 5, "seek": 16041, "start": 131.64, "end": 160.42, "text": "النظرية هذه تبعت كوشي أخدناها المرة اللي فاتت و برهناها مظبوط اليوم هناخد نظرية تانية و مهمة و النظرية هذه بتقول انه let x in be sequence of non-negative real numbers then series", "tokens": [6027, 1863, 19913, 2288, 10632, 29538, 6055, 3555, 34268, 9122, 2407, 8592, 1829, 5551, 9778, 3215, 8315, 11296, 9673, 25720, 13672, 1829, 6156, 9307, 2655, 4032, 4724, 2288, 3224, 8315, 11296, 3714, 19913, 3555, 2407, 9566, 45595, 20498, 8032, 1863, 47283, 3215, 8717, 19913, 2288, 10632, 6055, 7649, 10632, 4032, 3714, 16095, 3660, 4032, 28239, 19913, 2288, 10632, 29538, 39894, 39648, 16472, 3224, 718, 2031, 294, 312, 8310, 295, 2107, 12, 28561, 1166, 957, 3547, 550, 2638], "avg_logprob": -0.21814904476587588, "compression_ratio": 1.5251396648044693, "no_speech_prob": 0.0, "words": [{"start": 131.64, "end": 132.46, "word": "النظرية", "probability": 0.970703125}, {"start": 132.46, "end": 132.72, "word": " هذه", "probability": 0.54443359375}, {"start": 132.72, "end": 133.12, "word": " تبعت", "probability": 0.7574869791666666}, {"start": 133.12, "end": 133.62, "word": " كوشي", "probability": 0.772003173828125}, {"start": 133.62, "end": 134.18, "word": " أخدناها", "probability": 0.9205078125}, {"start": 134.18, "end": 134.46, "word": " المرة", "probability": 0.885009765625}, {"start": 134.46, "end": 134.62, "word": " اللي", "probability": 0.77587890625}, {"start": 134.62, "end": 135.04, "word": " فاتت", "probability": 0.9747721354166666}, {"start": 135.04, "end": 135.1, "word": " و", "probability": 0.9248046875}, {"start": 135.1, "end": 135.62, "word": " برهناها", "probability": 0.725830078125}, {"start": 135.62, "end": 136.12, "word": " مظبوط", "probability": 0.7244873046875}, {"start": 136.12, "end": 137.74, "word": " اليوم", "probability": 0.866943359375}, {"start": 137.74, "end": 138.38, "word": " هناخد", "probability": 0.900146484375}, {"start": 138.38, "end": 139.24, "word": " نظرية", "probability": 0.99267578125}, {"start": 139.24, "end": 139.86, "word": " تانية", "probability": 0.986328125}, {"start": 139.86, "end": 140.02, "word": " و", "probability": 0.9775390625}, {"start": 140.02, "end": 140.64, "word": " مهمة", "probability": 0.9365234375}, {"start": 140.64, "end": 142.08, "word": " و", "probability": 0.451171875}, {"start": 142.08, "end": 142.68, "word": " النظرية", "probability": 0.9708251953125}, {"start": 142.68, "end": 142.92, "word": " هذه", "probability": 0.89013671875}, {"start": 142.92, "end": 143.48, "word": " بتقول", "probability": 0.97900390625}, {"start": 143.48, "end": 146.0, "word": " انه", "probability": 0.444580078125}, {"start": 146.0, "end": 146.58, "word": " let", "probability": 0.8486328125}, {"start": 146.58, "end": 148.54, "word": " x", "probability": 0.69873046875}, {"start": 148.54, "end": 148.9, "word": " in", "probability": 0.505859375}, {"start": 148.9, "end": 149.12, "word": " be", "probability": 0.8056640625}, {"start": 149.12, "end": 149.96, "word": " sequence", "probability": 0.6181640625}, {"start": 149.96, "end": 151.36, "word": " of", "probability": 0.97998046875}, {"start": 151.36, "end": 151.66, "word": " non", "probability": 0.9306640625}, {"start": 151.66, "end": 152.1, "word": "-negative", "probability": 0.7692057291666666}, {"start": 152.1, "end": 152.38, "word": " real", "probability": 0.970703125}, {"start": 152.38, "end": 153.0, "word": " numbers", "probability": 0.89208984375}, {"start": 153.0, "end": 159.62, "word": " then", "probability": 0.69921875}, {"start": 159.62, "end": 160.42, "word": " series", "probability": 0.72216796875}], "temperature": 1.0}, {"id": 6, "seek": 18754, "start": 163.08, "end": 187.54, "text": "xn converges if and only if الsequence of partial sums its sequence of partial sums اللي هي sn is bounded", "tokens": [87, 77, 9652, 2880, 498, 293, 787, 498, 2423, 11834, 655, 295, 14641, 34499, 1080, 8310, 295, 14641, 34499, 13672, 1829, 39896, 2406, 307, 37498], "avg_logprob": -0.4134615453389975, "compression_ratio": 1.2696629213483146, "no_speech_prob": 0.0, "words": [{"start": 163.08, "end": 164.08, "word": "xn", "probability": 0.23931884765625}, {"start": 164.08, "end": 165.16, "word": " converges", "probability": 0.5767822265625}, {"start": 165.16, "end": 169.34, "word": " if", "probability": 0.87451171875}, {"start": 169.34, "end": 169.6, "word": " and", "probability": 0.92529296875}, {"start": 169.6, "end": 169.9, "word": " only", "probability": 0.908203125}, {"start": 169.9, "end": 170.32, "word": " if", "probability": 0.98486328125}, {"start": 170.32, "end": 172.26, "word": " الsequence", "probability": 0.6436360677083334}, {"start": 172.26, "end": 172.48, "word": " of", "probability": 0.96533203125}, {"start": 172.48, "end": 172.98, "word": " partial", "probability": 0.92333984375}, {"start": 172.98, "end": 173.5, "word": " sums", "probability": 0.9697265625}, {"start": 173.5, "end": 175.68, "word": " its", "probability": 0.2734375}, {"start": 175.68, "end": 176.46, "word": " sequence", "probability": 0.94921875}, {"start": 176.46, "end": 177.0, "word": " of", "probability": 0.9267578125}, {"start": 177.0, "end": 177.76, "word": " partial", "probability": 0.93701171875}, {"start": 177.76, "end": 179.0, "word": " sums", "probability": 0.958984375}, {"start": 179.0, "end": 185.6, "word": " اللي", "probability": 0.617431640625}, {"start": 185.6, "end": 185.88, "word": " هي", "probability": 0.681640625}, {"start": 185.88, "end": 186.6, "word": " sn", "probability": 0.5146484375}, {"start": 186.6, "end": 187.06, "word": " is", "probability": 0.912109375}, {"start": 187.06, "end": 187.54, "word": " bounded", "probability": 0.97998046875}], "temperature": 1.0}, {"id": 7, "seek": 21840, "start": 196.76, "end": 218.4, "text": "proof we have sn بساوي او sn زايد واحد بساوي sn زايد xn زايد واحد", "tokens": [15690, 321, 362, 2406, 4724, 3794, 995, 45865, 1975, 2407, 2406, 30767, 995, 25708, 36764, 24401, 4724, 3794, 995, 45865, 2406, 30767, 995, 25708, 2031, 77, 30767, 995, 25708, 36764, 24401], "avg_logprob": -0.16076660342514515, "compression_ratio": 1.5396825396825398, "no_speech_prob": 0.0, "words": [{"start": 196.76, "end": 198.16, "word": "proof", "probability": 0.455078125}, {"start": 198.16, "end": 199.56, "word": " we", "probability": 0.63037109375}, {"start": 199.56, "end": 201.04, "word": " have", "probability": 0.97900390625}, {"start": 201.04, "end": 204.08, "word": " sn", "probability": 0.662109375}, {"start": 204.08, "end": 205.26, "word": " بساوي", "probability": 0.7723388671875}, {"start": 205.26, "end": 206.14, "word": " او", "probability": 0.845947265625}, {"start": 206.14, "end": 206.86, "word": " sn", "probability": 0.9052734375}, {"start": 206.86, "end": 209.74, "word": " زايد", "probability": 0.7132975260416666}, {"start": 209.74, "end": 210.24, "word": " واحد", "probability": 0.963623046875}, {"start": 210.24, "end": 211.02, "word": " بساوي", "probability": 0.962890625}, {"start": 211.02, "end": 216.02, "word": " sn", "probability": 0.880859375}, {"start": 216.02, "end": 216.76, "word": " زايد", "probability": 0.93359375}, {"start": 216.76, "end": 217.54, "word": " xn", "probability": 0.87353515625}, {"start": 217.54, "end": 217.96, "word": " زايد", "probability": 0.98046875}, {"start": 217.96, "end": 218.4, "word": " واحد", "probability": 0.982177734375}], "temperature": 1.0}, {"id": 8, "seek": 22615, "start": 219.59, "end": 226.15, "text": "الان زاد فرص partial sum بيساوي الانف partial sum وبنضيف عليه لحد رقم ان زاد واحد", "tokens": [6027, 7649, 30767, 18513, 6156, 2288, 9381, 14641, 2408, 4724, 1829, 3794, 995, 45865, 2423, 7649, 5172, 14641, 2408, 46599, 1863, 11242, 33911, 47356, 5296, 24401, 12602, 4587, 2304, 16472, 30767, 18513, 36764, 24401], "avg_logprob": -0.4140624931880406, "compression_ratio": 1.2959183673469388, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 219.59, "end": 220.11, "word": "الان", "probability": 0.32806396484375}, {"start": 220.11, "end": 220.45, "word": " زاد", "probability": 0.56884765625}, {"start": 220.45, "end": 220.85, "word": " فرص", "probability": 0.4414876302083333}, {"start": 220.85, "end": 221.29, "word": " partial", "probability": 0.5634765625}, {"start": 221.29, "end": 221.61, "word": " sum", "probability": 0.81787109375}, {"start": 221.61, "end": 222.07, "word": " بيساوي", "probability": 0.6555419921875}, {"start": 222.07, "end": 222.47, "word": " الانف", "probability": 0.5662434895833334}, {"start": 222.47, "end": 222.79, "word": " partial", "probability": 0.76318359375}, {"start": 222.79, "end": 223.33, "word": " sum", "probability": 0.97119140625}, {"start": 223.33, "end": 224.37, "word": " وبنضيف", "probability": 0.81787109375}, {"start": 224.37, "end": 224.63, "word": " عليه", "probability": 0.9765625}, {"start": 224.63, "end": 224.95, "word": " لحد", "probability": 0.75244140625}, {"start": 224.95, "end": 225.37, "word": " رقم", "probability": 0.9801432291666666}, {"start": 225.37, "end": 225.57, "word": " ان", "probability": 0.5810546875}, {"start": 225.57, "end": 225.85, "word": " زاد", "probability": 0.888671875}, {"start": 225.85, "end": 226.15, "word": " واحد", "probability": 0.981201171875}], "temperature": 1.0}, {"id": 9, "seek": 26452, "start": 238.05, "end": 264.53, "text": "و طبعا ال Xn زاد واحد هذا احنا فرضين ان حدود ال sequence Xn كلها غير سالبة فهذا غير سالب وبالتالي المجموع هذا بالتأكيد اكبر من او سوى Sn لكل N في N فهذا معناه ان ال sequence Sn is increasing متزايدة", "tokens": [2407, 23032, 3555, 3615, 995, 2423, 1783, 77, 30767, 18513, 36764, 24401, 23758, 1975, 5016, 8315, 6156, 43042, 9957, 16472, 11331, 3215, 23328, 2423, 8310, 1783, 77, 28242, 11296, 32771, 13546, 8608, 6027, 49401, 6156, 3224, 15730, 32771, 13546, 8608, 6027, 3555, 46599, 6027, 2655, 6027, 1829, 9673, 7435, 2304, 45367, 23758, 20666, 2655, 10721, 4117, 25708, 1975, 4117, 26890, 9154, 1975, 2407, 8608, 2407, 7578, 9264, 5296, 28820, 426, 8978, 426, 6156, 3224, 15730, 20449, 8315, 3224, 16472, 2423, 8310, 9264, 307, 5662, 44650, 11622, 995, 25708, 3660], "avg_logprob": -0.15373263723320432, "compression_ratio": 1.6424870466321244, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 238.05, "end": 238.27, "word": "و", "probability": 0.44189453125}, {"start": 238.27, "end": 238.61, "word": " طبعا", "probability": 0.8326416015625}, {"start": 238.61, "end": 238.77, "word": " ال", "probability": 0.79541015625}, {"start": 238.77, "end": 239.41, "word": " Xn", "probability": 0.433837890625}, {"start": 239.41, "end": 239.89, "word": " زاد", "probability": 0.62060546875}, {"start": 239.89, "end": 240.31, "word": " واحد", "probability": 0.819091796875}, {"start": 240.31, "end": 240.65, "word": " هذا", "probability": 0.705078125}, {"start": 240.65, "end": 240.95, "word": " احنا", "probability": 0.7598470052083334}, {"start": 240.95, "end": 241.37, "word": " فرضين", "probability": 0.9708658854166666}, {"start": 241.37, "end": 241.51, "word": " ان", "probability": 0.7060546875}, {"start": 241.51, "end": 241.89, "word": " حدود", "probability": 0.98779296875}, {"start": 241.89, "end": 242.07, "word": " ال", "probability": 0.93603515625}, {"start": 242.07, "end": 242.63, "word": " sequence", "probability": 0.923828125}, {"start": 242.63, "end": 243.87, "word": " Xn", "probability": 0.755859375}, {"start": 243.87, "end": 244.27, "word": " كلها", "probability": 0.9560546875}, {"start": 244.27, "end": 244.55, "word": " غير", "probability": 0.9970703125}, {"start": 244.55, "end": 245.05, "word": " سالبة", "probability": 0.8736979166666666}, {"start": 245.05, "end": 245.55, "word": " فهذا", "probability": 0.9274088541666666}, {"start": 245.55, "end": 245.85, "word": " غير", "probability": 0.994873046875}, {"start": 245.85, "end": 246.51, "word": " سالب", "probability": 0.9697265625}, {"start": 246.51, "end": 248.53, "word": " وبالتالي", "probability": 0.89951171875}, {"start": 248.53, "end": 249.07, "word": " المجموع", "probability": 0.9173583984375}, {"start": 249.07, "end": 249.41, "word": " هذا", "probability": 0.9287109375}, {"start": 249.41, "end": 250.31, "word": " بالتأكيد", "probability": 0.96845703125}, {"start": 250.31, "end": 250.85, "word": " اكبر", "probability": 0.83154296875}, {"start": 250.85, "end": 251.05, "word": " من", "probability": 0.931640625}, {"start": 251.05, "end": 251.25, "word": " او", "probability": 0.739990234375}, {"start": 251.25, "end": 251.69, "word": " سوى", "probability": 0.8484700520833334}, {"start": 251.69, "end": 252.23, "word": " Sn", "probability": 0.83447265625}, {"start": 252.23, "end": 254.11, "word": " لكل", "probability": 0.97998046875}, {"start": 254.11, "end": 254.39, "word": " N", "probability": 0.397705078125}, {"start": 254.39, "end": 254.59, "word": " في", "probability": 0.890625}, {"start": 254.59, "end": 254.79, "word": " N", "probability": 0.8193359375}, {"start": 254.79, "end": 257.87, "word": " فهذا", "probability": 0.9786783854166666}, {"start": 257.87, "end": 258.47, "word": " معناه", "probability": 0.9742838541666666}, {"start": 258.47, "end": 258.77, "word": " ان", "probability": 0.875}, {"start": 258.77, "end": 258.99, "word": " ال", "probability": 0.83056640625}, {"start": 258.99, "end": 259.37, "word": " sequence", "probability": 0.9833984375}, {"start": 259.37, "end": 260.35, "word": " Sn", "probability": 0.89208984375}, {"start": 260.35, "end": 261.91, "word": " is", "probability": 0.8525390625}, {"start": 261.91, "end": 262.79, "word": " increasing", "probability": 0.96875}, {"start": 262.79, "end": 264.53, "word": " متزايدة", "probability": 0.89111328125}], "temperature": 1.0}, {"id": 10, "seek": 28856, "start": 266.96, "end": 288.56, "text": "بالتالي، من خلال الوضع الوضع الوضع الوضع الوضع الوضع الوضع الوضع الوضع الوضع", "tokens": [3555, 6027, 2655, 6027, 1829, 12399, 9154, 16490, 1211, 6027, 2423, 2407, 11242, 3615, 2423, 2407, 11242, 3615, 2423, 2407, 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"probability": 0.82431640625}, {"start": 1130.3, "end": 1130.7, "word": " اتنين", "probability": 0.8402099609375}, {"start": 1130.7, "end": 1134.92, "word": " لما", "probability": 0.727294921875}, {"start": 1134.92, "end": 1135.16, "word": " ج", "probability": 0.8369140625}, {"start": 1135.16, "end": 1135.6, "word": " بيساوي", "probability": 0.980859375}, {"start": 1135.6, "end": 1136.16, "word": " اتنين", "probability": 0.992919921875}], "temperature": 1.0}, {"id": 40, "seek": 116464, "start": 1137.02, "end": 1164.64, "text": "فطلع SK2 أصغر من واحد زائد R واحد زائد هاي اتنين سالب واحد الأسف لما J بساوية تلاتة عندي طلع ال SK تلاتة أصغر من واحد زائد R زائد R أس تلاتة سالب واحد إذا S sub KJ أصغر من واحد زائد R إلى R أس J minus واحد", "tokens": [5172, 9566, 1211, 3615, 21483, 17, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 30767, 16373, 3215, 497, 36764, 24401, 30767, 16373, 3215, 8032, 47302, 1975, 2655, 1863, 9957, 8608, 6027, 3555, 36764, 24401, 16247, 3794, 5172, 5296, 15042, 508, 4724, 3794, 995, 2407, 10632, 6055, 1211, 9307, 3660, 18871, 16254, 23032, 1211, 3615, 2423, 21483, 6055, 1211, 9307, 3660, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 30767, 16373, 3215, 497, 30767, 16373, 3215, 497, 5551, 3794, 6055, 1211, 9307, 3660, 8608, 6027, 3555, 36764, 24401, 11933, 15730, 318, 1422, 591, 41, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 30767, 16373, 3215, 497, 30731, 497, 5551, 3794, 508, 3175, 36764, 24401], "avg_logprob": -0.29645270055478756, "compression_ratio": 2.079754601226994, "no_speech_prob": 0.0, "words": [{"start": 1137.02, "end": 1137.64, "word": "فطلع", "probability": 0.58642578125}, {"start": 1137.64, "end": 1138.76, "word": " SK2", "probability": 0.5177001953125}, {"start": 1138.76, "end": 1139.5, "word": " أصغر", "probability": 0.9276123046875}, {"start": 1139.5, "end": 1139.9, "word": " من", "probability": 0.986328125}, {"start": 1139.9, "end": 1141.66, "word": " واحد", "probability": 0.56402587890625}, {"start": 1141.66, "end": 1142.1, "word": " زائد", "probability": 0.67236328125}, {"start": 1142.1, "end": 1142.4, "word": " R", "probability": 0.8486328125}, {"start": 1142.4, "end": 1142.98, "word": " واحد", "probability": 0.6427001953125}, {"start": 1142.98, "end": 1143.64, "word": " زائد", "probability": 0.9638671875}, {"start": 1143.64, "end": 1144.98, "word": " هاي", "probability": 0.3173828125}, {"start": 1144.98, "end": 1145.86, "word": " اتنين", "probability": 0.69921875}, {"start": 1145.86, "end": 1146.36, "word": " سالب", "probability": 0.9375}, {"start": 1146.36, "end": 1146.66, "word": " واحد", "probability": 0.985595703125}, {"start": 1146.66, "end": 1147.64, "word": " الأسف", "probability": 0.4894612630208333}, {"start": 1147.64, "end": 1148.26, "word": " لما", "probability": 0.741455078125}, {"start": 1148.26, "end": 1148.64, "word": " J", "probability": 0.40478515625}, {"start": 1148.64, "end": 1149.12, "word": " بساوية", "probability": 0.685498046875}, {"start": 1149.12, "end": 1149.66, "word": " تلاتة", "probability": 0.9317626953125}, {"start": 1149.66, "end": 1150.04, "word": " عندي", "probability": 0.6527099609375}, {"start": 1150.04, "end": 1150.34, "word": " طلع", "probability": 0.9680989583333334}, {"start": 1150.34, "end": 1150.46, "word": " ال", "probability": 0.34619140625}, {"start": 1150.46, "end": 1151.62, "word": " SK", "probability": 0.1412353515625}, {"start": 1151.62, "end": 1152.24, "word": " تلاتة", "probability": 0.820556640625}, {"start": 1152.24, "end": 1152.58, "word": " أصغر", "probability": 0.9691162109375}, {"start": 1152.58, "end": 1152.74, "word": " من", "probability": 0.9951171875}, {"start": 1152.74, "end": 1153.08, "word": " واحد", "probability": 0.99072265625}, {"start": 1153.08, "end": 1153.5, "word": " زائد", "probability": 0.8313802083333334}, {"start": 1153.5, "end": 1153.86, "word": " R", "probability": 0.97509765625}, {"start": 1153.86, "end": 1155.14, "word": " زائد", "probability": 0.8826497395833334}, {"start": 1155.14, "end": 1155.52, "word": " R", "probability": 0.98046875}, {"start": 1155.52, "end": 1155.8, "word": " أس", "probability": 0.5662841796875}, {"start": 1155.8, "end": 1156.38, "word": " تلاتة", "probability": 0.9530029296875}, {"start": 1156.38, "end": 1156.78, "word": " سالب", "probability": 0.9671223958333334}, {"start": 1156.78, "end": 1157.14, "word": " واحد", "probability": 0.994873046875}, {"start": 1157.14, "end": 1157.9, "word": " إذا", "probability": 0.66259765625}, {"start": 1157.9, "end": 1158.5, "word": " S", "probability": 0.462890625}, {"start": 1158.5, "end": 1159.08, "word": " sub", "probability": 0.6630859375}, {"start": 1159.08, "end": 1159.68, "word": " KJ", "probability": 0.812255859375}, {"start": 1159.68, "end": 1160.12, "word": " أصغر", "probability": 0.98779296875}, {"start": 1160.12, "end": 1160.26, "word": " من", "probability": 0.99609375}, {"start": 1160.26, "end": 1160.58, "word": " واحد", "probability": 0.9912109375}, {"start": 1160.58, "end": 1160.96, "word": " زائد", "probability": 0.9148763020833334}, {"start": 1160.96, "end": 1161.28, "word": " R", "probability": 0.99072265625}, {"start": 1161.28, "end": 1161.64, "word": " إلى", "probability": 0.8525390625}, {"start": 1161.64, "end": 1162.6, "word": " R", "probability": 0.9560546875}, {"start": 1162.6, "end": 1163.0, "word": " أس", "probability": 0.7587890625}, {"start": 1163.0, "end": 1163.74, "word": " J", "probability": 0.69482421875}, {"start": 1163.74, "end": 1164.16, "word": " minus", "probability": 0.51953125}, {"start": 1164.16, "end": 1164.64, "word": " واحد", "probability": 0.953857421875}], "temperature": 1.0}, {"id": 41, "seek": 118898, "start": 1166.26, "end": 1188.98, "text": "الان هذا المجموع أصغر من مجموع ال infinite series اللي هي summation من j بساوي سفر إلى ملا نهاية لR أس J صح؟", "tokens": [6027, 7649, 23758, 9673, 7435, 2304, 45367, 5551, 9381, 17082, 2288, 9154, 3714, 7435, 2304, 45367, 2423, 13785, 2638, 13672, 1829, 39896, 28811, 9154, 361, 4724, 3794, 995, 45865, 8608, 5172, 2288, 30731, 3714, 15040, 8717, 11296, 10632, 5296, 49, 5551, 3794, 508, 20328, 5016, 22807], "avg_logprob": -0.25033245822216604, "compression_ratio": 1.20863309352518, "no_speech_prob": 0.0, "words": [{"start": 1166.26, "end": 1166.76, "word": "الان", "probability": 0.772216796875}, {"start": 1166.76, "end": 1167.1, "word": " هذا", "probability": 0.72607421875}, {"start": 1167.1, "end": 1167.8, "word": " المجموع", "probability": 0.92431640625}, {"start": 1167.8, "end": 1170.1, "word": " أصغر", "probability": 0.83453369140625}, {"start": 1170.1, "end": 1170.54, "word": " من", "probability": 0.98388671875}, {"start": 1170.54, "end": 1172.52, "word": " مجموع", "probability": 0.936767578125}, {"start": 1172.52, "end": 1172.88, "word": " ال", "probability": 0.79833984375}, {"start": 1172.88, "end": 1173.32, "word": " infinite", "probability": 0.794921875}, {"start": 1173.32, "end": 1174.04, "word": " series", "probability": 0.9169921875}, {"start": 1174.04, "end": 1175.12, "word": " اللي", "probability": 0.5458984375}, {"start": 1175.12, "end": 1175.34, "word": " هي", "probability": 0.89892578125}, {"start": 1175.34, "end": 1175.98, "word": " summation", "probability": 0.8818359375}, {"start": 1175.98, "end": 1176.68, "word": " من", "probability": 0.95263671875}, {"start": 1176.68, "end": 1179.28, "word": " j", "probability": 0.5693359375}, {"start": 1179.28, "end": 1180.24, "word": " بساوي", "probability": 0.7890625}, {"start": 1180.24, "end": 1180.86, "word": " سفر", "probability": 0.8042805989583334}, {"start": 1180.86, "end": 1181.08, "word": " إلى", "probability": 0.92041015625}, {"start": 1181.08, "end": 1181.42, "word": " ملا", "probability": 0.6483154296875}, {"start": 1181.42, "end": 1182.12, "word": " نهاية", "probability": 0.9375}, {"start": 1182.12, "end": 1184.56, "word": " لR", "probability": 0.5948486328125}, {"start": 1184.56, "end": 1184.86, "word": " أس", "probability": 0.5025634765625}, {"start": 1184.86, "end": 1185.22, "word": " J", "probability": 0.595703125}, {"start": 1185.22, "end": 1188.98, "word": " صح؟", "probability": 0.8551432291666666}], "temperature": 1.0}, {"id": 42, "seek": 121682, "start": 1190.16, "end": 1216.82, "text": "هذا عبارة عن finite sum أصغر من مجموعة infinite series هذه عبارة عن geometric series with first term واحد وال ratio تبعها R والـ R أكبر من صفر أصغر من واحد", "tokens": [3224, 15730, 6225, 3555, 9640, 3660, 18871, 19362, 2408, 5551, 9381, 17082, 2288, 9154, 3714, 7435, 2304, 2407, 27884, 13785, 2638, 29538, 6225, 3555, 9640, 3660, 18871, 33246, 2638, 365, 700, 1433, 36764, 24401, 16070, 8509, 6055, 3555, 3615, 11296, 497, 16070, 39184, 497, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 5551, 9381, 17082, 2288, 9154, 36764, 24401], "avg_logprob": -0.3027012600737103, "compression_ratio": 1.4899328859060403, "no_speech_prob": 0.0, "words": [{"start": 1190.16, "end": 1190.62, "word": "هذا", "probability": 0.3240966796875}, {"start": 1190.62, "end": 1196.76, "word": " عبارة", "probability": 0.764556884765625}, {"start": 1196.76, "end": 1196.86, "word": " عن", "probability": 0.99462890625}, {"start": 1196.86, "end": 1197.16, "word": " finite", "probability": 0.427978515625}, {"start": 1197.16, "end": 1197.76, "word": " sum", "probability": 0.955078125}, {"start": 1197.76, "end": 1198.54, "word": " أصغر", "probability": 0.9439697265625}, {"start": 1198.54, "end": 1198.68, "word": " من", "probability": 0.9833984375}, {"start": 1198.68, "end": 1199.18, "word": " مجموعة", "probability": 0.869970703125}, {"start": 1199.18, "end": 1199.46, "word": " infinite", "probability": 0.266845703125}, {"start": 1199.46, "end": 1200.72, "word": " series", "probability": 0.8203125}, {"start": 1200.72, "end": 1201.44, "word": " هذه", "probability": 0.137939453125}, {"start": 1201.44, "end": 1201.8, "word": " عبارة", "probability": 0.9827880859375}, {"start": 1201.8, "end": 1201.98, "word": " عن", "probability": 0.99267578125}, {"start": 1201.98, "end": 1202.4, "word": " geometric", "probability": 0.39453125}, {"start": 1202.4, "end": 1203.18, "word": " series", "probability": 0.919921875}, {"start": 1203.18, "end": 1205.66, "word": " with", "probability": 0.7099609375}, {"start": 1205.66, "end": 1206.88, "word": " first", "probability": 0.74853515625}, {"start": 1206.88, "end": 1207.3, "word": " term", "probability": 0.97998046875}, {"start": 1207.3, "end": 1207.92, "word": " واحد", "probability": 0.780517578125}, {"start": 1207.92, "end": 1210.7, "word": " وال", "probability": 0.291259765625}, {"start": 1210.7, "end": 1211.84, "word": " ratio", "probability": 0.90673828125}, {"start": 1211.84, "end": 1212.48, "word": " تبعها", "probability": 0.798583984375}, {"start": 1212.48, "end": 1212.92, "word": " R", "probability": 0.52294921875}, {"start": 1212.92, "end": 1214.18, "word": " والـ", "probability": 0.4989013671875}, {"start": 1214.18, "end": 1214.5, "word": " R", "probability": 0.53564453125}, {"start": 1214.5, "end": 1214.98, "word": " أكبر", "probability": 0.97265625}, {"start": 1214.98, "end": 1215.22, "word": " من", "probability": 0.9951171875}, {"start": 1215.22, "end": 1215.62, "word": " صفر", "probability": 0.753173828125}, {"start": 1215.62, "end": 1216.08, "word": " أصغر", "probability": 0.958251953125}, {"start": 1216.08, "end": 1216.28, "word": " من", "probability": 0.9951171875}, {"start": 1216.28, "end": 1216.82, "word": " واحد", "probability": 0.984375}], "temperature": 1.0}, {"id": 43, "seek": 124456, "start": 1217.8, "end": 1244.56, "text": "إذا الـ geometric series هذه converges ومجموعة بساوي واحد على واحد minus R الكلام هذا صحيح for all j belong to M وطبعا ال SKJ هذا عبارة عن مجموعة partial sum partial sum لأعداد موجبة وبالتالي موجبة", "tokens": [28814, 15730, 2423, 39184, 33246, 2638, 29538, 9652, 2880, 4032, 2304, 7435, 2304, 2407, 27884, 4724, 3794, 995, 45865, 36764, 24401, 15844, 36764, 24401, 3175, 497, 2423, 28820, 10943, 23758, 20328, 5016, 1829, 5016, 337, 439, 361, 5784, 281, 376, 4032, 9566, 3555, 3615, 995, 2423, 21483, 41, 23758, 6225, 3555, 9640, 3660, 18871, 3714, 7435, 2304, 2407, 27884, 14641, 2408, 14641, 2408, 5296, 10721, 22488, 18513, 3714, 29245, 49401, 46599, 6027, 2655, 6027, 1829, 3714, 29245, 49401], "avg_logprob": -0.18601661731925193, "compression_ratio": 1.4522613065326633, "no_speech_prob": 0.0, "words": [{"start": 1217.8, "end": 1218.14, "word": "إذا", "probability": 0.468505859375}, {"start": 1218.14, "end": 1218.32, "word": " الـ", "probability": 0.7841796875}, {"start": 1218.32, "end": 1218.6, "word": " geometric", "probability": 0.60498046875}, {"start": 1218.6, "end": 1219.12, "word": " series", "probability": 0.89892578125}, {"start": 1219.12, "end": 1219.52, "word": " هذه", "probability": 0.453857421875}, {"start": 1219.52, "end": 1221.1, "word": " converges", "probability": 0.640869140625}, {"start": 1221.1, "end": 1222.52, "word": " ومجموعة", "probability": 0.9151204427083334}, {"start": 1222.52, "end": 1223.08, "word": " بساوي", "probability": 0.8055419921875}, {"start": 1223.08, "end": 1223.72, "word": " واحد", "probability": 0.7763671875}, {"start": 1223.72, "end": 1225.72, "word": " على", "probability": 0.732421875}, {"start": 1225.72, "end": 1226.32, "word": " واحد", "probability": 0.983154296875}, {"start": 1226.32, "end": 1226.94, "word": " minus", "probability": 0.91064453125}, {"start": 1226.94, "end": 1227.5, "word": " R", "probability": 0.716796875}, {"start": 1227.5, "end": 1230.82, "word": " الكلام", "probability": 0.9049479166666666}, {"start": 1230.82, "end": 1231.1, "word": " هذا", "probability": 0.9345703125}, {"start": 1231.1, "end": 1231.66, "word": " صحيح", "probability": 0.9974365234375}, {"start": 1231.66, "end": 1231.98, "word": " for", "probability": 0.76953125}, {"start": 1231.98, "end": 1232.44, "word": " all", "probability": 0.951171875}, {"start": 1232.44, "end": 1232.8, "word": " j", "probability": 0.72021484375}, {"start": 1232.8, "end": 1233.32, "word": " belong", "probability": 0.462158203125}, {"start": 1233.32, "end": 1233.6, "word": " to", "probability": 0.97900390625}, {"start": 1233.6, "end": 1233.9, "word": " M", "probability": 0.52001953125}, {"start": 1233.9, "end": 1235.96, "word": " وطبعا", "probability": 0.941015625}, {"start": 1235.96, "end": 1236.12, "word": " ال", "probability": 0.9599609375}, {"start": 1236.12, "end": 1236.92, "word": " SKJ", "probability": 0.42626953125}, {"start": 1236.92, "end": 1237.32, "word": " هذا", "probability": 0.7333984375}, {"start": 1237.32, "end": 1237.68, "word": " عبارة", "probability": 0.985107421875}, {"start": 1237.68, "end": 1237.9, "word": " عن", "probability": 0.9921875}, {"start": 1237.9, "end": 1238.4, "word": " مجموعة", "probability": 0.98154296875}, {"start": 1238.4, "end": 1238.76, "word": " partial", "probability": 0.94287109375}, {"start": 1238.76, "end": 1239.22, "word": " sum", "probability": 0.94873046875}, {"start": 1239.22, "end": 1240.32, "word": " partial", "probability": 0.83349609375}, {"start": 1240.32, "end": 1240.92, "word": " sum", "probability": 0.97119140625}, {"start": 1240.92, "end": 1242.76, "word": " لأعداد", "probability": 0.905517578125}, {"start": 1242.76, "end": 1243.32, "word": " موجبة", "probability": 0.9677734375}, {"start": 1243.32, "end": 1244.02, "word": " وبالتالي", "probability": 0.86787109375}, {"start": 1244.02, "end": 1244.56, "word": " موجبة", "probability": 0.93603515625}], "temperature": 1.0}, {"id": 44, "seek": 127016, "start": 1247.7, "end": 1270.16, "text": "إذا أنا هيك أثبتت إن ال sub sequence SKJ bounded below by 0 و bounded above by العدد الموجب 1 على 1 minus R وبالتالي، إذا هيك نستنتج، therefore the subsequence", "tokens": [28814, 15730, 41850, 39896, 4117, 5551, 12984, 3555, 2655, 2655, 36145, 2423, 1422, 8310, 21483, 41, 37498, 2507, 538, 1958, 4032, 37498, 3673, 538, 2423, 22488, 3215, 9673, 29245, 3555, 502, 15844, 502, 3175, 497, 46599, 6027, 2655, 6027, 1829, 12399, 11933, 15730, 39896, 4117, 8717, 14851, 29399, 7435, 12399, 4412, 264, 13924, 655], "avg_logprob": -0.3531249989162792, "compression_ratio": 1.2891566265060241, "no_speech_prob": 0.0, "words": [{"start": 1247.7, "end": 1248.1, "word": "إذا", "probability": 0.46453857421875}, {"start": 1248.1, "end": 1248.44, "word": " أنا", "probability": 0.55322265625}, {"start": 1248.44, "end": 1248.68, "word": " هيك", "probability": 0.710205078125}, {"start": 1248.68, "end": 1249.22, "word": " أثبتت", "probability": 0.98251953125}, {"start": 1249.22, "end": 1249.44, "word": " إن", "probability": 0.385986328125}, {"start": 1249.44, "end": 1249.66, "word": " ال", "probability": 0.4716796875}, {"start": 1249.66, "end": 1252.72, "word": " sub", "probability": 0.2113037109375}, {"start": 1252.72, "end": 1253.36, "word": " sequence", "probability": 0.58203125}, {"start": 1253.36, "end": 1254.26, "word": " SKJ", "probability": 0.605712890625}, {"start": 1254.26, "end": 1254.68, "word": " bounded", "probability": 0.68896484375}, {"start": 1254.68, "end": 1255.16, "word": " below", "probability": 0.78466796875}, {"start": 1255.16, "end": 1255.48, "word": " by", "probability": 0.837890625}, {"start": 1255.48, "end": 1255.9, "word": " 0", "probability": 0.69384765625}, {"start": 1255.9, "end": 1256.94, "word": " و", "probability": 0.76904296875}, {"start": 1256.94, "end": 1257.38, "word": " bounded", "probability": 0.859375}, {"start": 1257.38, "end": 1258.06, "word": " above", "probability": 0.95068359375}, {"start": 1258.06, "end": 1258.7, "word": " by", "probability": 0.87646484375}, {"start": 1258.7, "end": 1259.26, "word": " العدد", "probability": 0.813720703125}, {"start": 1259.26, "end": 1259.82, "word": " الموجب", "probability": 0.8727213541666666}, {"start": 1259.82, "end": 1260.18, "word": " 1", "probability": 0.82470703125}, {"start": 1260.18, "end": 1260.52, "word": " على", "probability": 0.56298828125}, {"start": 1260.52, "end": 1260.86, "word": " 1", "probability": 0.8837890625}, {"start": 1260.86, "end": 1261.34, "word": " minus", "probability": 0.748046875}, {"start": 1261.34, "end": 1261.72, "word": " R", "probability": 0.6943359375}, {"start": 1261.72, "end": 1265.42, "word": " وبالتالي،", "probability": 0.7632649739583334}, {"start": 1265.42, "end": 1265.92, "word": " إذا", "probability": 0.774169921875}, {"start": 1265.92, "end": 1266.26, "word": " هيك", "probability": 0.948486328125}, {"start": 1266.26, "end": 1267.08, "word": " نستنتج،", "probability": 0.7013427734375}, {"start": 1267.08, "end": 1267.48, "word": " therefore", "probability": 0.7001953125}, {"start": 1267.48, "end": 1269.04, "word": " the", "probability": 0.5390625}, {"start": 1269.04, "end": 1270.16, "word": " subsequence", "probability": 0.793701171875}], "temperature": 1.0}, {"id": 45, "seek": 129369, "start": 1273.37, "end": 1293.69, "text": "ده sub-sequence اللي هي SKJ هاد ال sub-sequence من مين؟ من ال sequence of partial sums SN اشملها is bounded أثبتنا أنها ايه؟ bounded، مصدقوط؟ طلعنا احنا عملنا construction ل sub-sequence", "tokens": [3215, 3224, 1422, 12, 11834, 655, 13672, 1829, 39896, 21483, 41, 8032, 18513, 2423, 1422, 12, 11834, 655, 9154, 3714, 9957, 22807, 9154, 2423, 8310, 295, 14641, 34499, 13955, 1975, 8592, 2304, 1211, 11296, 307, 37498, 5551, 12984, 3555, 2655, 8315, 14739, 11296, 1975, 1829, 3224, 22807, 37498, 12399, 3714, 9381, 3215, 4587, 2407, 9566, 22807, 23032, 1211, 3615, 8315, 1975, 5016, 8315, 6225, 42213, 8315, 6435, 5296, 1422, 12, 11834, 655], "avg_logprob": -0.38570206785855227, "compression_ratio": 1.44, "no_speech_prob": 0.0, "words": [{"start": 1273.37, "end": 1273.71, "word": "ده", "probability": 0.483123779296875}, {"start": 1273.71, "end": 1273.93, "word": " sub", "probability": 0.250732421875}, {"start": 1273.93, "end": 1274.47, "word": "-sequence", "probability": 0.75390625}, {"start": 1274.47, "end": 1274.61, "word": " اللي", "probability": 0.56494140625}, {"start": 1274.61, "end": 1274.87, "word": " هي", "probability": 0.83154296875}, {"start": 1274.87, "end": 1276.49, "word": " SKJ", "probability": 0.72802734375}, {"start": 1276.49, "end": 1277.31, "word": " هاد", "probability": 0.3509521484375}, {"start": 1277.31, "end": 1277.41, "word": " ال", "probability": 0.7763671875}, {"start": 1277.41, "end": 1277.57, "word": " sub", "probability": 0.466796875}, {"start": 1277.57, "end": 1277.99, "word": "-sequence", "probability": 0.9461263020833334}, {"start": 1277.99, "end": 1278.21, "word": " من", "probability": 0.98681640625}, {"start": 1278.21, "end": 1279.63, "word": " مين؟", "probability": 0.8304036458333334}, {"start": 1279.63, "end": 1280.01, "word": " من", "probability": 0.92236328125}, {"start": 1280.01, "end": 1280.23, "word": " ال", "probability": 0.78759765625}, {"start": 1280.23, "end": 1280.73, "word": " sequence", "probability": 0.93115234375}, {"start": 1280.73, "end": 1280.99, "word": " of", "probability": 0.828125}, {"start": 1280.99, "end": 1281.35, "word": " partial", "probability": 0.9013671875}, {"start": 1281.35, "end": 1281.65, "word": " sums", "probability": 0.9482421875}, {"start": 1281.65, "end": 1282.29, "word": " SN", "probability": 0.48388671875}, {"start": 1282.29, "end": 1283.07, "word": " اشملها", "probability": 0.742578125}, {"start": 1283.07, "end": 1284.37, "word": " is", "probability": 0.58349609375}, {"start": 1284.37, "end": 1284.99, "word": " bounded", "probability": 0.9267578125}, {"start": 1284.99, "end": 1286.19, "word": " أثبتنا", "probability": 0.9390625}, {"start": 1286.19, "end": 1286.93, "word": " أنها", "probability": 0.705322265625}, {"start": 1286.93, "end": 1287.35, "word": " ايه؟", "probability": 0.63299560546875}, {"start": 1287.35, "end": 1288.07, "word": " bounded،", "probability": 0.3812255859375}, {"start": 1288.07, "end": 1290.41, "word": " مصدقوط؟", "probability": 0.6548549107142857}, {"start": 1290.41, "end": 1291.49, "word": " طلعنا", "probability": 0.75616455078125}, {"start": 1291.49, "end": 1291.69, "word": " احنا", "probability": 0.7977701822916666}, {"start": 1291.69, "end": 1292.03, "word": " عملنا", "probability": 0.9708658854166666}, {"start": 1292.03, "end": 1292.69, "word": " construction", "probability": 0.92041015625}, {"start": 1292.69, "end": 1292.91, "word": " ل", "probability": 0.94970703125}, {"start": 1292.91, "end": 1293.15, "word": " sub", "probability": 0.72900390625}, {"start": 1293.15, "end": 1293.69, "word": "-sequence", "probability": 0.9558919270833334}], "temperature": 1.0}, {"id": 46, "seek": 131673, "start": 1294.67, "end": 1316.73, "text": "من الـ sequence of partial sums وهي طلعت bounded هي bounded below by 0 bounded above by 1 over 1 minus r لأن حسب اللمّة اللي فاتت so by above لمّة ال sequence of partial sums نفسها is bounded", "tokens": [27842, 2423, 39184, 8310, 295, 14641, 34499, 37037, 1829, 23032, 1211, 34268, 37498, 39896, 37498, 2507, 538, 1958, 37498, 3673, 538, 502, 670, 502, 3175, 367, 5296, 33456, 11331, 35457, 13672, 2304, 11703, 3660, 13672, 1829, 6156, 9307, 2655, 370, 538, 3673, 32767, 11703, 3660, 2423, 8310, 295, 14641, 34499, 8717, 36178, 11296, 307, 37498], "avg_logprob": -0.24818638871823037, "compression_ratio": 1.4842767295597483, "no_speech_prob": 0.0, "words": [{"start": 1294.67, "end": 1294.93, "word": "من", "probability": 0.494140625}, {"start": 1294.93, "end": 1295.09, "word": " الـ", "probability": 0.51806640625}, {"start": 1295.09, "end": 1295.35, "word": " sequence", "probability": 0.67236328125}, {"start": 1295.35, "end": 1295.65, "word": " of", "probability": 0.95654296875}, {"start": 1295.65, "end": 1296.01, "word": " partial", "probability": 0.94677734375}, {"start": 1296.01, "end": 1296.35, "word": " sums", "probability": 0.9599609375}, {"start": 1296.35, "end": 1296.57, "word": " وهي", "probability": 0.697021484375}, {"start": 1296.57, "end": 1296.91, "word": " طلعت", "probability": 0.9353841145833334}, {"start": 1296.91, "end": 1297.19, "word": " bounded", "probability": 0.447021484375}, {"start": 1297.19, "end": 1297.37, "word": " هي", "probability": 0.2115478515625}, {"start": 1297.37, "end": 1297.77, "word": " bounded", "probability": 0.90771484375}, {"start": 1297.77, "end": 1298.13, "word": " below", "probability": 0.85595703125}, {"start": 1298.13, "end": 1298.45, "word": " by", "probability": 0.759765625}, {"start": 1298.45, "end": 1298.87, "word": " 0", "probability": 0.449462890625}, {"start": 1298.87, "end": 1299.63, "word": " bounded", "probability": 0.67138671875}, {"start": 1299.63, "end": 1300.09, "word": " above", "probability": 0.955078125}, {"start": 1300.09, "end": 1300.41, "word": " by", "probability": 0.96875}, {"start": 1300.41, "end": 1300.69, "word": " 1", "probability": 0.876953125}, {"start": 1300.69, "end": 1301.05, "word": " over", "probability": 0.8671875}, {"start": 1301.05, "end": 1301.27, "word": " 1", "probability": 0.84716796875}, {"start": 1301.27, "end": 1301.67, "word": " minus", "probability": 0.7119140625}, {"start": 1301.67, "end": 1302.03, "word": " r", "probability": 0.465576171875}, {"start": 1302.03, "end": 1302.97, "word": " لأن", "probability": 0.65771484375}, {"start": 1302.97, "end": 1303.33, "word": " حسب", "probability": 0.983154296875}, {"start": 1303.33, "end": 1303.75, "word": " اللمّة", "probability": 0.7086181640625}, {"start": 1303.75, "end": 1303.91, "word": " اللي", "probability": 0.9169921875}, {"start": 1303.91, "end": 1304.57, "word": " فاتت", "probability": 0.994140625}, {"start": 1304.57, "end": 1306.39, "word": " so", "probability": 0.287841796875}, {"start": 1306.39, "end": 1306.93, "word": " by", "probability": 0.96484375}, {"start": 1306.93, "end": 1307.91, "word": " above", "probability": 0.8837890625}, {"start": 1307.91, "end": 1308.53, "word": " لمّة", "probability": 0.7906901041666666}, {"start": 1308.53, "end": 1313.61, "word": " ال", "probability": 0.92236328125}, {"start": 1313.61, "end": 1314.19, "word": " sequence", "probability": 0.95361328125}, {"start": 1314.19, "end": 1314.57, "word": " of", "probability": 0.9775390625}, {"start": 1314.57, "end": 1314.97, "word": " partial", "probability": 0.947265625}, {"start": 1314.97, "end": 1315.35, "word": " sums", "probability": 0.95263671875}, {"start": 1315.35, "end": 1315.95, "word": " نفسها", "probability": 0.974609375}, {"start": 1315.95, "end": 1316.27, "word": " is", "probability": 0.95654296875}, {"start": 1316.27, "end": 1316.73, "word": " bounded", "probability": 0.95458984375}], "temperature": 1.0}, {"id": 47, "seek": 135091, "start": 1322.95, "end": 1350.91, "text": "وبالتالي إذا by above theorem مدام ال sequence of partial sums is bounded إذا ال series converges okay إذا بعديكم نقول so by above theorem ال series sigma", "tokens": [37746, 6027, 2655, 6027, 1829, 11933, 15730, 538, 3673, 20904, 3714, 3215, 10943, 2423, 8310, 295, 14641, 34499, 307, 37498, 11933, 15730, 2423, 2638, 9652, 2880, 1392, 11933, 15730, 45030, 16254, 24793, 8717, 39648, 370, 538, 3673, 20904, 2423, 2638, 12771], "avg_logprob": -0.3158482185431889, "compression_ratio": 1.3741007194244603, "no_speech_prob": 0.0, "words": [{"start": 1322.95, "end": 1323.95, "word": "وبالتالي", "probability": 0.92822265625}, {"start": 1323.95, "end": 1324.57, "word": " إذا", "probability": 0.51885986328125}, {"start": 1324.57, "end": 1325.03, "word": " by", "probability": 0.58935546875}, {"start": 1325.03, "end": 1325.61, "word": " above", "probability": 0.9365234375}, {"start": 1325.61, "end": 1326.17, "word": " theorem", "probability": 0.580078125}, {"start": 1326.17, "end": 1327.77, "word": " مدام", "probability": 0.602294921875}, {"start": 1327.77, "end": 1327.93, "word": " ال", "probability": 0.78466796875}, {"start": 1327.93, "end": 1328.21, "word": " sequence", "probability": 0.87353515625}, {"start": 1328.21, "end": 1328.57, "word": " of", "probability": 0.83837890625}, {"start": 1328.57, "end": 1328.93, "word": " partial", "probability": 0.93701171875}, {"start": 1328.93, "end": 1329.29, "word": " sums", "probability": 0.97265625}, {"start": 1329.29, "end": 1329.51, "word": " is", "probability": 0.92236328125}, {"start": 1329.51, "end": 1329.95, "word": " bounded", "probability": 0.93994140625}, {"start": 1329.95, "end": 1330.67, "word": " إذا", "probability": 0.7294921875}, {"start": 1330.67, "end": 1330.89, "word": " ال", "probability": 0.80078125}, {"start": 1330.89, "end": 1331.09, "word": " series", "probability": 0.82275390625}, {"start": 1331.09, "end": 1331.91, "word": " converges", "probability": 0.852783203125}, {"start": 1331.91, "end": 1335.71, "word": " okay", "probability": 0.52587890625}, {"start": 1335.71, "end": 1336.23, "word": " إذا", "probability": 0.879638671875}, {"start": 1336.23, "end": 1338.61, "word": " بعديكم", "probability": 0.6291910807291666}, {"start": 1338.61, "end": 1339.07, "word": " نقول", "probability": 0.97509765625}, {"start": 1339.07, "end": 1343.79, "word": " so", "probability": 0.55810546875}, {"start": 1343.79, "end": 1344.39, "word": " by", "probability": 0.98046875}, {"start": 1344.39, "end": 1344.91, "word": " above", "probability": 0.97216796875}, {"start": 1344.91, "end": 1345.55, "word": " theorem", "probability": 0.7861328125}, {"start": 1345.55, "end": 1349.93, "word": " ال", "probability": 0.65380859375}, {"start": 1349.93, "end": 1350.27, "word": " series", "probability": 0.953125}, {"start": 1350.27, "end": 1350.91, "word": " sigma", "probability": 0.48388671875}], "temperature": 1.0}, {"id": 48, "seek": 137464, "start": 1351.94, "end": 1374.64, "text": "1 على N أكبر من 1 فهذا يثبت الجزء الأول من النظرية خلّينا نثبت الجزء التاني", "tokens": [16, 15844, 426, 5551, 4117, 26890, 9154, 502, 6156, 3224, 15730, 7251, 12984, 3555, 2655, 25724, 11622, 38207, 16247, 12610, 9154, 28239, 19913, 2288, 10632, 16490, 1211, 11703, 9957, 995, 8717, 12984, 3555, 2655, 25724, 11622, 38207, 16712, 7649, 1829], "avg_logprob": -0.39977134727850194, "compression_ratio": 1.375, "no_speech_prob": 0.0, "words": [{"start": 1351.94, "end": 1352.44, "word": "1", "probability": 0.1978759765625}, {"start": 1352.44, "end": 1352.84, "word": " على", "probability": 0.263427734375}, {"start": 1352.84, "end": 1354.02, "word": " N", "probability": 0.110595703125}, {"start": 1354.02, "end": 1354.34, "word": " أكبر", "probability": 0.5881754557291666}, {"start": 1354.34, "end": 1354.76, "word": " من", "probability": 0.96240234375}, {"start": 1354.76, "end": 1354.76, "word": " 1", "probability": 0.81103515625}, {"start": 1354.76, "end": 1366.08, "word": " فهذا", "probability": 0.4403483072916667}, {"start": 1366.08, "end": 1366.78, "word": " يثبت", "probability": 0.78546142578125}, {"start": 1366.78, "end": 1367.22, "word": " الجزء", "probability": 0.9425455729166666}, {"start": 1367.22, "end": 1367.76, "word": " الأول", "probability": 0.9765625}, {"start": 1367.76, "end": 1368.76, "word": " من", "probability": 0.86962890625}, {"start": 1368.76, "end": 1369.54, "word": " النظرية", "probability": 0.83013916015625}, {"start": 1369.54, "end": 1373.3, "word": " خلّينا", "probability": 0.6406494140625}, {"start": 1373.3, "end": 1373.7, "word": " نثبت", "probability": 0.97900390625}, {"start": 1373.7, "end": 1374.08, "word": " الجزء", "probability": 0.9903971354166666}, {"start": 1374.08, "end": 1374.64, "word": " التاني", "probability": 0.8937174479166666}], "temperature": 1.0}, {"id": 49, "seek": 140914, "start": 1390.08, "end": 1409.14, "text": "using induction you can show أنه for P أكبر من صفر أصغر من واحد", "tokens": [7981, 33371, 291, 393, 855, 14739, 3224, 337, 430, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 5551, 9381, 17082, 2288, 9154, 36764, 24401], "avg_logprob": -0.24755858381589255, "compression_ratio": 0.9883720930232558, "no_speech_prob": 0.0, "words": [{"start": 1390.08, "end": 1391.26, "word": "using", "probability": 0.268310546875}, {"start": 1391.26, "end": 1392.32, "word": " induction", "probability": 0.96533203125}, {"start": 1392.32, "end": 1397.2, "word": " you", "probability": 0.7138671875}, {"start": 1397.2, "end": 1397.78, "word": " can", "probability": 0.9638671875}, {"start": 1397.78, "end": 1398.42, "word": " show", "probability": 0.96142578125}, {"start": 1398.42, "end": 1401.66, "word": " أنه", "probability": 0.572509765625}, {"start": 1401.66, "end": 1402.26, "word": " for", "probability": 0.47119140625}, {"start": 1402.26, "end": 1406.8, "word": " P", "probability": 0.359619140625}, {"start": 1406.8, "end": 1407.3, "word": " أكبر", "probability": 0.9405924479166666}, {"start": 1407.3, "end": 1407.54, "word": " من", "probability": 0.98828125}, {"start": 1407.54, "end": 1407.92, "word": " صفر", "probability": 0.7914225260416666}, {"start": 1407.92, "end": 1408.4, "word": " أصغر", "probability": 0.962890625}, {"start": 1408.4, "end": 1408.58, "word": " من", "probability": 0.99462890625}, {"start": 1408.58, "end": 1409.14, "word": " واحد", "probability": 0.905029296875}], "temperature": 1.0}, {"id": 50, "seek": 143123, "start": 1412.37, "end": 1431.23, "text": "لو كانت ال P طبعا أكبر من صفر وأصغر من أو ساوي الواحد ف N أوس P بطلع أصغر من أو ساوي N لكل N في N صح؟ انا ممكن اثباته by induction", "tokens": [1211, 2407, 25961, 2655, 2423, 430, 23032, 3555, 3615, 995, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 36725, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 14407, 24401, 6156, 426, 34051, 3794, 430, 4724, 9566, 1211, 3615, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 426, 5296, 28820, 426, 8978, 426, 20328, 5016, 22807, 1975, 8315, 3714, 43020, 1975, 12984, 3555, 9307, 3224, 538, 33371], "avg_logprob": -0.2101332738119013, "compression_ratio": 1.4452054794520548, "no_speech_prob": 0.0, "words": [{"start": 1412.37, "end": 1412.77, "word": "لو", "probability": 0.7763671875}, {"start": 1412.77, "end": 1413.17, "word": " كانت", "probability": 0.939208984375}, {"start": 1413.17, "end": 1413.31, "word": " ال", "probability": 0.88134765625}, {"start": 1413.31, "end": 1413.63, "word": " P", "probability": 0.59765625}, {"start": 1413.63, "end": 1415.73, "word": " طبعا", "probability": 0.915283203125}, {"start": 1415.73, "end": 1416.11, "word": " أكبر", "probability": 0.8846028645833334}, {"start": 1416.11, "end": 1416.31, "word": " من", "probability": 0.99365234375}, {"start": 1416.31, "end": 1416.61, "word": " صفر", "probability": 0.8126627604166666}, {"start": 1416.61, "end": 1417.15, "word": " وأصغر", "probability": 0.8927001953125}, {"start": 1417.15, "end": 1417.35, "word": " من", "probability": 0.984375}, {"start": 1417.35, "end": 1417.53, "word": " أو", "probability": 0.89794921875}, {"start": 1417.53, "end": 1417.97, "word": " ساوي", "probability": 0.8776041666666666}, {"start": 1417.97, "end": 1418.53, "word": " الواحد", "probability": 0.7989908854166666}, {"start": 1418.53, "end": 1420.07, "word": " ف", "probability": 0.87841796875}, {"start": 1420.07, "end": 1420.53, "word": " N", "probability": 0.41455078125}, {"start": 1420.53, "end": 1420.93, "word": " أوس", "probability": 0.4659423828125}, {"start": 1420.93, "end": 1421.25, "word": " P", "probability": 0.97509765625}, {"start": 1421.25, "end": 1422.91, "word": " بطلع", "probability": 0.7083740234375}, {"start": 1422.91, "end": 1423.37, "word": " أصغر", "probability": 0.9786376953125}, {"start": 1423.37, "end": 1423.55, "word": " من", "probability": 0.99072265625}, {"start": 1423.55, "end": 1423.73, "word": " أو", "probability": 0.9814453125}, {"start": 1423.73, "end": 1424.35, "word": " ساوي", "probability": 0.9777018229166666}, {"start": 1424.35, "end": 1424.75, "word": " N", "probability": 0.9462890625}, {"start": 1424.75, "end": 1425.29, "word": " لكل", "probability": 0.935791015625}, {"start": 1425.29, "end": 1425.65, "word": " N", "probability": 0.66796875}, {"start": 1425.65, "end": 1425.85, "word": " في", "probability": 0.85400390625}, {"start": 1425.85, "end": 1426.11, "word": " N", "probability": 0.89013671875}, {"start": 1426.11, "end": 1429.67, "word": " صح؟", "probability": 0.8048502604166666}, {"start": 1429.67, "end": 1429.91, "word": " انا", "probability": 0.596435546875}, {"start": 1429.91, "end": 1430.15, "word": " ممكن", "probability": 0.939697265625}, {"start": 1430.15, "end": 1430.61, "word": " اثباته", "probability": 0.7634765625}, {"start": 1430.61, "end": 1430.79, "word": " by", "probability": 0.880859375}, {"start": 1430.79, "end": 1431.23, "word": " induction", "probability": 0.98583984375}], "temperature": 1.0}, {"id": 51, "seek": 145271, "start": 1432.53, "end": 1452.71, "text": "So, 1 على n أصغر لو ساقى 1 على n أُس في for all n ينتمي إلى n فإن هذا بيقدّي هذا بيقدّي", "tokens": [6455, 11, 502, 15844, 297, 5551, 9381, 17082, 2288, 45164, 8608, 995, 4587, 7578, 502, 15844, 297, 5551, 10859, 3794, 8978, 337, 439, 297, 7251, 29399, 2304, 1829, 30731, 297, 6156, 28814, 1863, 23758, 4724, 1829, 28543, 11703, 1829, 23758, 4724, 1829, 28543, 11703, 1829], "avg_logprob": -0.4194972858480785, "compression_ratio": 1.3173076923076923, "no_speech_prob": 0.0, "words": [{"start": 1432.53, "end": 1433.05, "word": "So,", "probability": 0.066650390625}, {"start": 1434.39, "end": 1434.99, "word": " 1", "probability": 0.5009765625}, {"start": 1434.99, "end": 1435.45, "word": " على", "probability": 0.426513671875}, {"start": 1435.45, "end": 1436.63, "word": " n", "probability": 0.64990234375}, {"start": 1436.63, "end": 1437.19, "word": " أصغر", "probability": 0.9366455078125}, {"start": 1437.19, "end": 1437.43, "word": " لو", "probability": 0.55712890625}, {"start": 1437.43, "end": 1437.87, "word": " ساقى", "probability": 0.638336181640625}, {"start": 1437.87, "end": 1438.19, "word": " 1", "probability": 0.482421875}, {"start": 1438.19, "end": 1438.47, "word": " على", "probability": 0.84716796875}, {"start": 1438.47, "end": 1438.85, "word": " n", "probability": 0.90673828125}, {"start": 1438.85, "end": 1439.23, "word": " أُس", "probability": 0.6078287760416666}, {"start": 1439.23, "end": 1439.47, "word": " في", "probability": 0.30029296875}, {"start": 1439.47, "end": 1440.61, "word": " for", "probability": 0.320556640625}, {"start": 1440.61, "end": 1441.09, "word": " all", "probability": 0.95361328125}, {"start": 1441.09, "end": 1441.47, "word": " n", "probability": 0.8623046875}, {"start": 1441.47, "end": 1442.25, "word": " ينتمي", "probability": 0.953125}, {"start": 1442.25, "end": 1442.53, "word": " إلى", "probability": 0.96826171875}, {"start": 1442.53, "end": 1442.87, "word": " n", "probability": 0.38818359375}, {"start": 1442.87, "end": 1448.47, "word": " فإن", "probability": 0.7810872395833334}, {"start": 1448.47, "end": 1448.79, "word": " هذا", "probability": 0.72216796875}, {"start": 1448.79, "end": 1449.43, "word": " بيقدّي", "probability": 0.67021484375}, {"start": 1449.43, "end": 1452.03, "word": " هذا", "probability": 0.75146484375}, {"start": 1452.03, "end": 1452.71, "word": " بيقدّي", "probability": 0.9640625}], "temperature": 1.0}, {"id": 52, "seek": 149263, "start": 1470.45, "end": 1492.63, "text": "this implies أن ال summation from k بساوي واحد to n لواحد على k أصغر لو ساوي summation من k بساوي واحد إلى n لواحد على n على k أوس P", "tokens": [11176, 18779, 14739, 2423, 28811, 490, 350, 4724, 3794, 995, 45865, 36764, 24401, 281, 297, 5296, 14407, 24401, 15844, 350, 5551, 9381, 17082, 2288, 45164, 8608, 995, 45865, 28811, 9154, 350, 4724, 3794, 995, 45865, 36764, 24401, 30731, 297, 5296, 14407, 24401, 15844, 297, 15844, 350, 34051, 3794, 430], "avg_logprob": -0.2499999976158142, "compression_ratio": 1.578512396694215, "no_speech_prob": 0.0, "words": [{"start": 1470.45, "end": 1470.89, "word": "this", "probability": 0.29736328125}, {"start": 1470.89, "end": 1471.59, "word": " implies", "probability": 0.9541015625}, {"start": 1471.59, "end": 1471.89, "word": " أن", "probability": 0.568359375}, {"start": 1471.89, "end": 1472.11, "word": " ال", "probability": 0.5810546875}, {"start": 1472.11, "end": 1472.73, "word": " summation", "probability": 0.70361328125}, {"start": 1472.73, "end": 1473.53, "word": " from", "probability": 0.66552734375}, {"start": 1473.53, "end": 1474.23, "word": " k", "probability": 0.71826171875}, {"start": 1474.23, "end": 1475.59, "word": " بساوي", "probability": 0.6666259765625}, {"start": 1475.59, "end": 1476.23, "word": " واحد", "probability": 0.916748046875}, {"start": 1476.23, "end": 1476.59, "word": " to", "probability": 0.7099609375}, {"start": 1476.59, "end": 1477.23, "word": " n", "probability": 0.75390625}, {"start": 1477.23, "end": 1479.23, "word": " لواحد", "probability": 0.82421875}, {"start": 1479.23, "end": 1479.47, "word": " على", "probability": 0.71533203125}, {"start": 1479.47, "end": 1479.89, "word": " k", "probability": 0.5068359375}, {"start": 1479.89, "end": 1480.45, "word": " أصغر", "probability": 0.945068359375}, {"start": 1480.45, "end": 1480.69, "word": " لو", "probability": 0.88671875}, {"start": 1480.69, "end": 1481.13, "word": " ساوي", "probability": 0.9703776041666666}, {"start": 1481.13, "end": 1482.01, "word": " summation", "probability": 0.4853515625}, {"start": 1482.01, "end": 1483.33, "word": " من", "probability": 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"temperature": 1.0}, {"id": 53, "seek": 152185, "start": 1494.65, "end": 1521.85, "text": "طب ما هذا عبارة عن ال partial sum لسمي S N لل harmonic series .. لل .. لل harmonic series و هذا عبارة عن ال partial sum سمي S N star لل P series صح؟ إن أنا أصبح عندي أنا S N أصغر من أو ساوي S N star", "tokens": [9566, 3555, 19446, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 14641, 2408, 5296, 38251, 1829, 318, 426, 24976, 32270, 2638, 4386, 24976, 4386, 24976, 32270, 2638, 4032, 23758, 6225, 3555, 9640, 3660, 18871, 2423, 14641, 2408, 8608, 2304, 1829, 318, 426, 3543, 24976, 430, 2638, 20328, 5016, 22807, 36145, 41850, 5551, 9381, 49628, 18871, 16254, 41850, 318, 426, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 318, 426, 3543], "avg_logprob": -0.23041373197461518, "compression_ratio": 1.70625, "no_speech_prob": 0.0, "words": [{"start": 1494.65, "end": 1494.97, "word": "طب", "probability": 0.865966796875}, {"start": 1494.97, "end": 1495.11, "word": " ما", "probability": 0.46826171875}, {"start": 1495.11, "end": 1495.41, "word": " هذا", "probability": 0.54638671875}, {"start": 1495.41, "end": 1495.85, "word": " عبارة", "probability": 0.9945068359375}, {"start": 1495.85, "end": 1496.05, "word": " عن", "probability": 0.998046875}, {"start": 1496.05, "end": 1496.21, "word": " ال", "probability": 0.91796875}, {"start": 1496.21, "end": 1496.65, "word": " partial", "probability": 0.87451171875}, {"start": 1496.65, "end": 1497.27, "word": " sum", "probability": 0.96484375}, {"start": 1497.27, "end": 1497.95, "word": " لسمي", "probability": 0.7083333333333334}, {"start": 1497.95, "end": 1498.39, "word": " S", "probability": 0.2342529296875}, {"start": 1498.39, "end": 1499.79, "word": " N", "probability": 0.6337890625}, {"start": 1499.79, "end": 1501.21, "word": " لل", "probability": 0.45068359375}, {"start": 1501.21, "end": 1501.69, "word": " harmonic", "probability": 0.82958984375}, {"start": 1501.69, "end": 1502.25, "word": " series", "probability": 0.76708984375}, {"start": 1502.25, "end": 1502.27, "word": " ..", "probability": 0.419189453125}, {"start": 1502.27, "end": 1502.59, "word": " لل", "probability": 0.88037109375}, {"start": 1502.59, "end": 1503.99, "word": " ..", "probability": 0.90576171875}, {"start": 1503.99, "end": 1505.93, "word": " لل", "probability": 0.95751953125}, {"start": 1505.93, "end": 1506.31, "word": " harmonic", "probability": 0.974609375}, {"start": 1506.31, "end": 1506.95, "word": " series", "probability": 0.94677734375}, {"start": 1506.95, "end": 1508.03, "word": " و", "probability": 0.56640625}, {"start": 1508.03, "end": 1508.27, "word": " هذا", "probability": 0.85888671875}, {"start": 1508.27, "end": 1508.69, "word": " عبارة", "probability": 0.9986572265625}, {"start": 1508.69, "end": 1508.85, "word": " عن", "probability": 0.99755859375}, {"start": 1508.85, "end": 1509.03, "word": " ال", "probability": 0.94482421875}, {"start": 1509.03, "end": 1509.41, "word": " partial", "probability": 0.97314453125}, {"start": 1509.41, "end": 1509.93, "word": " sum", "probability": 0.98193359375}, {"start": 1509.93, "end": 1510.45, "word": " سمي", "probability": 0.8631184895833334}, {"start": 1510.45, "end": 1510.77, "word": " S", "probability": 0.8525390625}, {"start": 1510.77, "end": 1511.07, "word": " N", "probability": 0.8984375}, {"start": 1511.07, "end": 1511.61, "word": " star", "probability": 0.15625}, {"start": 1511.61, "end": 1513.19, "word": " لل", "probability": 0.6796875}, {"start": 1513.19, "end": 1513.43, "word": " P", "probability": 0.95849609375}, {"start": 1513.43, "end": 1513.89, "word": " series", "probability": 0.70654296875}, {"start": 1513.89, "end": 1516.81, "word": " صح؟", "probability": 0.86669921875}, {"start": 1516.81, "end": 1516.97, "word": " إن", "probability": 0.183837890625}, {"start": 1516.97, "end": 1517.11, "word": " أنا", "probability": 0.62744140625}, {"start": 1517.11, "end": 1517.51, "word": " أصبح", "probability": 0.8504231770833334}, {"start": 1517.51, "end": 1517.87, "word": " عندي", "probability": 0.777099609375}, {"start": 1517.87, "end": 1518.19, "word": " أنا", "probability": 0.70947265625}, {"start": 1518.19, "end": 1518.61, "word": " S", "probability": 0.9208984375}, {"start": 1518.61, "end": 1519.11, "word": " N", "probability": 0.96923828125}, {"start": 1519.11, "end": 1519.73, "word": " أصغر", "probability": 0.9942626953125}, {"start": 1519.73, "end": 1519.89, "word": " من", "probability": 0.98193359375}, {"start": 1519.89, "end": 1520.05, "word": " أو", "probability": 0.943359375}, {"start": 1520.05, "end": 1520.57, "word": " ساوي", "probability": 0.8898111979166666}, {"start": 1520.57, "end": 1520.93, "word": " S", "probability": 0.364013671875}, {"start": 1520.93, "end": 1521.31, "word": " N", "probability": 0.978515625}, {"start": 1521.31, "end": 1521.85, "word": " star", "probability": 0.755859375}], "temperature": 1.0}, {"id": 54, "seek": 155323, "start": 1524.55, "end": 1553.23, "text": "where لكل n where sn بساوي sigma واحد على k من k بساوي واحد إلى n و sn star بساوي sigma من k بساوي واحد إلى n لواحد على k أُس P طيب since أثبتنا احنا قبل هيك انه ال sequence of partial sums", "tokens": [1992, 5296, 28820, 297, 689, 2406, 4724, 3794, 995, 45865, 12771, 36764, 24401, 15844, 350, 9154, 350, 4724, 3794, 995, 45865, 36764, 24401, 30731, 297, 4032, 2406, 3543, 4724, 3794, 995, 45865, 12771, 9154, 350, 4724, 3794, 995, 45865, 36764, 24401, 30731, 297, 5296, 14407, 24401, 15844, 350, 5551, 10859, 3794, 430, 23032, 1829, 3555, 1670, 5551, 12984, 3555, 2655, 8315, 1975, 5016, 8315, 12174, 36150, 39896, 4117, 16472, 3224, 2423, 8310, 295, 14641, 34499], "avg_logprob": -0.22923519932910016, "compression_ratio": 1.6646341463414633, "no_speech_prob": 0.0, "words": [{"start": 1524.55, "end": 1525.07, "word": "where", "probability": 0.308837890625}, {"start": 1525.07, "end": 1525.95, "word": " لكل", "probability": 0.928466796875}, {"start": 1525.95, "end": 1526.41, "word": " n", "probability": 0.6806640625}, {"start": 1526.41, "end": 1529.43, "word": " where", "probability": 0.61865234375}, {"start": 1529.43, "end": 1530.05, "word": " sn", "probability": 0.377197265625}, {"start": 1530.05, "end": 1530.79, "word": " بساوي", "probability": 0.53076171875}, {"start": 1530.79, "end": 1531.33, "word": " sigma", "probability": 0.40869140625}, {"start": 1531.33, "end": 1532.15, "word": " واحد", "probability": 0.791748046875}, {"start": 1532.15, "end": 1532.37, "word": " على", "probability": 0.54638671875}, {"start": 1532.37, "end": 1532.79, "word": " k", "probability": 0.69482421875}, {"start": 1532.79, "end": 1532.91, "word": " من", "probability": 0.9443359375}, {"start": 1532.91, "end": 1533.21, "word": " k", "probability": 0.8046875}, {"start": 1533.21, "end": 1533.69, "word": " بساوي", "probability": 0.936279296875}, {"start": 1533.69, "end": 1534.17, "word": " واحد", "probability": 0.980712890625}, {"start": 1534.17, "end": 1535.81, "word": " إلى", "probability": 0.65625}, {"start": 1535.81, "end": 1536.29, "word": " n", "probability": 0.83544921875}, {"start": 1536.29, "end": 1536.53, "word": " و", "probability": 0.904296875}, {"start": 1536.53, "end": 1537.11, "word": " sn", "probability": 0.6943359375}, {"start": 1537.11, "end": 1537.93, "word": " star", "probability": 0.149169921875}, {"start": 1537.93, "end": 1539.57, "word": " بساوي", "probability": 0.9462890625}, {"start": 1539.57, "end": 1539.91, "word": " sigma", "probability": 0.81494140625}, {"start": 1539.91, "end": 1540.15, "word": " من", "probability": 0.98779296875}, {"start": 1540.15, "end": 1540.41, "word": " k", "probability": 0.94482421875}, {"start": 1540.41, "end": 1540.87, "word": " بساوي", "probability": 0.9727783203125}, {"start": 1540.87, "end": 1541.27, "word": " واحد", "probability": 0.989990234375}, {"start": 1541.27, "end": 1541.45, "word": " إلى", "probability": 0.85595703125}, {"start": 1541.45, "end": 1541.67, "word": " n", "probability": 0.9541015625}, {"start": 1541.67, "end": 1542.23, "word": " لواحد", "probability": 0.7486165364583334}, {"start": 1542.23, "end": 1542.51, "word": " على", "probability": 0.90087890625}, {"start": 1542.51, "end": 1543.11, "word": " k", "probability": 0.9560546875}, {"start": 1543.11, "end": 1543.61, "word": " أُس", "probability": 0.6221516927083334}, {"start": 1543.61, "end": 1543.91, "word": " P", "probability": 0.490478515625}, {"start": 1543.91, "end": 1546.89, "word": " طيب", "probability": 0.9303385416666666}, {"start": 1546.89, "end": 1547.43, "word": " since", "probability": 0.7392578125}, {"start": 1547.43, "end": 1549.23, "word": " أثبتنا", "probability": 0.98671875}, {"start": 1549.23, "end": 1549.39, "word": " احنا", "probability": 0.82373046875}, {"start": 1549.39, "end": 1549.69, "word": " قبل", "probability": 0.66259765625}, {"start": 1549.69, "end": 1550.17, "word": " هيك", "probability": 0.938232421875}, {"start": 1550.17, "end": 1551.35, "word": " انه", "probability": 0.525634765625}, {"start": 1551.35, "end": 1551.63, "word": " ال", "probability": 0.92333984375}, {"start": 1551.63, "end": 1552.01, "word": " sequence", "probability": 0.95849609375}, {"start": 1552.01, "end": 1552.35, "word": " of", "probability": 0.7509765625}, {"start": 1552.35, "end": 1552.77, "word": " partial", "probability": 0.951171875}, {"start": 1552.77, "end": 1553.23, "word": " sums", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 55, "seek": 156229, "start": 1556.31, "end": 1562.29, "text": "السيكوانس SN هذا عبارة عن السيكوانس of partial sums", "tokens": [6027, 3794, 1829, 4117, 2407, 7649, 3794, 13955, 23758, 6225, 3555, 9640, 3660, 18871, 21136, 1829, 4117, 2407, 7649, 3794, 295, 14641, 34499], "avg_logprob": -0.358398428807656, "compression_ratio": 1.082191780821918, "no_speech_prob": 0.0, "words": [{"start": 1556.31, "end": 1557.27, "word": "السيكوانس", "probability": 0.703125}, {"start": 1557.27, "end": 1558.01, "word": " SN", "probability": 0.329833984375}, {"start": 1558.01, "end": 1560.27, "word": " هذا", "probability": 0.17529296875}, {"start": 1560.27, "end": 1560.61, "word": " عبارة", "probability": 0.978271484375}, {"start": 1560.61, "end": 1560.73, "word": " عن", "probability": 0.99658203125}, {"start": 1560.73, "end": 1561.29, "word": " السيكوانس", "probability": 0.8376668294270834}, {"start": 1561.29, "end": 1561.41, "word": " of", "probability": 0.69677734375}, {"start": 1561.41, "end": 1561.79, "word": " partial", "probability": 0.95751953125}, {"start": 1561.79, "end": 1562.29, "word": " sums", "probability": 0.81591796875}], "temperature": 1.0}, {"id": 56, "seek": 160037, "start": 1572.51, "end": 1600.37, "text": "of the harmonic series sigma 1 على n ايش مالهم؟ اثبتنا انهم unbounded ال sequence هذه is unbounded بنالنا ذلك في مثال سابق unbounded فلو سمينا المتباين هذا star then it follows", "tokens": [2670, 264, 32270, 2638, 12771, 502, 15844, 297, 1975, 1829, 8592, 3714, 6027, 16095, 22807, 1975, 12984, 3555, 2655, 8315, 16472, 16095, 517, 18767, 292, 2423, 8310, 29538, 307, 517, 18767, 292, 44945, 6027, 8315, 29910, 23275, 8978, 50113, 6027, 8608, 16758, 4587, 517, 18767, 292, 6156, 1211, 2407, 8608, 2304, 1829, 8315, 9673, 2655, 3555, 995, 9957, 23758, 3543, 550, 309, 10002], "avg_logprob": -0.3371582077816129, "compression_ratio": 1.393063583815029, "no_speech_prob": 0.0, "words": [{"start": 1572.51, "end": 1573.03, "word": "of", "probability": 0.057220458984375}, {"start": 1573.03, "end": 1573.41, "word": " the", "probability": 0.259033203125}, {"start": 1573.41, "end": 1573.91, "word": " harmonic", "probability": 0.8720703125}, {"start": 1573.91, "end": 1574.45, "word": " series", "probability": 0.8916015625}, {"start": 1574.45, "end": 1574.77, "word": " sigma", "probability": 0.3330078125}, {"start": 1574.77, "end": 1575.13, "word": " 1", "probability": 0.487548828125}, {"start": 1575.13, "end": 1575.47, "word": " على", "probability": 0.466552734375}, {"start": 1575.47, "end": 1575.85, "word": " n", "probability": 0.51123046875}, {"start": 1575.85, "end": 1576.71, "word": " ايش", "probability": 0.5757649739583334}, {"start": 1576.71, "end": 1577.51, "word": " مالهم؟", "probability": 0.80859375}, {"start": 1577.51, "end": 1578.33, "word": " اثبتنا", "probability": 0.87099609375}, {"start": 1578.33, "end": 1578.71, "word": " انهم", "probability": 0.830078125}, {"start": 1578.71, "end": 1579.53, "word": " unbounded", "probability": 0.8230794270833334}, {"start": 1579.53, "end": 1579.75, "word": " ال", "probability": 0.6611328125}, {"start": 1579.75, "end": 1580.05, "word": " sequence", "probability": 0.456787109375}, {"start": 1580.05, "end": 1580.57, "word": " هذه", "probability": 0.4912109375}, {"start": 1580.57, "end": 1580.83, "word": " is", "probability": 0.59619140625}, {"start": 1580.83, "end": 1583.07, "word": " unbounded", "probability": 0.8782552083333334}, {"start": 1583.07, "end": 1583.57, "word": " بنالنا", "probability": 0.68212890625}, {"start": 1583.57, "end": 1583.89, "word": " ذلك", "probability": 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follows", "probability": 0.9091796875}], "temperature": 1.0}, {"id": 57, "seek": 163360, "start": 1604.9, "end": 1633.6, "text": "from star ينتج من المتباينة star إذا كانت ال sequence هذه unbounded لصغيرة unbounded فالحدود أكبر is unbounded ال sequence SM star is unbounded وبالتالي حسب النظرية أعلى so by", "tokens": [20579, 3543, 7251, 29399, 7435, 9154, 9673, 2655, 3555, 995, 9957, 3660, 3543, 11933, 15730, 25961, 2655, 2423, 8310, 29538, 517, 18767, 292, 5296, 9381, 17082, 48923, 517, 18767, 292, 6156, 6027, 24401, 23328, 5551, 4117, 26890, 307, 517, 18767, 292, 2423, 8310, 13115, 3543, 307, 517, 18767, 292, 46599, 6027, 2655, 6027, 1829, 11331, 35457, 28239, 19913, 2288, 10632, 5551, 3615, 23942, 370, 538], "avg_logprob": -0.23615057382619742, "compression_ratio": 1.4727272727272727, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1604.9, "end": 1605.3, "word": "from", "probability": 0.1429443359375}, {"start": 1605.3, "end": 1605.9, "word": " star", "probability": 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"probability": 0.91650390625}, {"start": 1617.06, "end": 1617.84, "word": " فالحدود", "probability": 0.931396484375}, {"start": 1617.84, "end": 1618.46, "word": " أكبر", "probability": 0.9013671875}, {"start": 1618.46, "end": 1619.7, "word": " is", "probability": 0.71923828125}, {"start": 1619.7, "end": 1620.4, "word": " unbounded", "probability": 0.9611002604166666}, {"start": 1620.4, "end": 1620.62, "word": " ال", "probability": 0.29248046875}, {"start": 1620.62, "end": 1621.24, "word": " sequence", "probability": 0.95361328125}, {"start": 1621.24, "end": 1622.0, "word": " SM", "probability": 0.29052734375}, {"start": 1622.0, "end": 1622.62, "word": " star", "probability": 0.176025390625}, {"start": 1622.62, "end": 1624.36, "word": " is", "probability": 0.85205078125}, {"start": 1624.36, "end": 1627.24, "word": " unbounded", "probability": 0.9669596354166666}, {"start": 1627.24, "end": 1631.14, "word": " وبالتالي", "probability": 0.9517578125}, {"start": 1631.14, "end": 1631.56, 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"end": 1648.85, "word": " بساوة", "probability": 0.484130859375}, {"start": 1648.85, "end": 1649.37, "word": " واحد", "probability": 0.911865234375}, {"start": 1649.37, "end": 1650.09, "word": " وإنفينيتي", "probability": 0.7238595145089286}, {"start": 1650.09, "end": 1651.55, "word": " لواحد", "probability": 0.7743326822916666}, {"start": 1651.55, "end": 1651.79, "word": " على", "probability": 0.595703125}, {"start": 1651.79, "end": 1652.23, "word": " K", "probability": 0.82470703125}, {"start": 1652.23, "end": 1652.61, "word": " أوس", "probability": 0.50933837890625}, {"start": 1652.61, "end": 1652.93, "word": " P", "probability": 0.8251953125}, {"start": 1652.93, "end": 1655.39, "word": " is", "probability": 0.318359375}, {"start": 1655.39, "end": 1656.29, "word": " divergent", "probability": 0.739990234375}], "temperature": 1.0}, {"id": 59, "seek": 168229, "start": 1661.61, "end": 1682.29, "text": "حسب النظرية اللى ذكرناها سابقًا، طيب، في عندي هاي series of positive numbers أو non-negative real numbers و ال sequence of partial sums أبعدتها unbounded، إذا حسب النظرية ال series is not convergent أو divergent، صح؟", "tokens": [5016, 35457, 28239, 19913, 2288, 10632, 13672, 7578, 29910, 37983, 8315, 11296, 8608, 16758, 4587, 14111, 995, 12399, 23032, 1829, 3555, 12399, 8978, 18871, 16254, 8032, 47302, 2638, 295, 3353, 3547, 34051, 2107, 12, 28561, 1166, 957, 3547, 4032, 2423, 8310, 295, 14641, 34499, 5551, 3555, 22488, 2655, 11296, 517, 18767, 292, 12399, 11933, 15730, 11331, 35457, 28239, 19913, 2288, 10632, 2423, 2638, 307, 406, 9652, 6930, 34051, 18558, 6930, 12399, 20328, 5016, 22807], "avg_logprob": -0.2945833396911621, "compression_ratio": 1.4846938775510203, "no_speech_prob": 0.0, "words": [{"start": 1661.61, "end": 1662.09, "word": "حسب", "probability": 0.7376708984375}, {"start": 1662.09, "end": 1662.85, "word": " النظرية", "probability": 0.944580078125}, {"start": 1662.85, "end": 1664.41, "word": " اللى", "probability": 0.444091796875}, {"start": 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"probability": 0.64111328125}, {"start": 1676.97, "end": 1677.31, "word": " حسب", "probability": 0.9345703125}, {"start": 1677.31, "end": 1677.95, "word": " النظرية", "probability": 0.992431640625}, {"start": 1677.95, "end": 1679.09, "word": " ال", "probability": 0.6923828125}, {"start": 1679.09, "end": 1679.49, "word": " series", "probability": 0.92919921875}, {"start": 1679.49, "end": 1679.91, "word": " is", "probability": 0.92724609375}, {"start": 1679.91, "end": 1680.25, "word": " not", "probability": 0.96533203125}, {"start": 1680.25, "end": 1680.97, "word": " convergent", "probability": 0.959228515625}, {"start": 1680.97, "end": 1681.19, "word": " أو", "probability": 0.8916015625}, {"start": 1681.19, "end": 1681.91, "word": " divergent،", "probability": 0.8289388020833334}, {"start": 1681.91, "end": 1682.29, "word": " صح؟", "probability": 0.98681640625}], "temperature": 1.0}, {"id": 60, "seek": 170622, "start": 1683.92, "end": 1706.22, "text": "إذن هذا بتثبت الجزء التاني وبالتاني هيك بيكون بارهاننا اختبار ال P-series test او ال P-series test إذن أي P-series بتكون convergent إذا ال P أكبر من واحد و divergent إذا P أصغر من أول ساول واحد في أي سؤال؟", "tokens": [28814, 8848, 1863, 23758, 39894, 12984, 3555, 2655, 25724, 11622, 38207, 16712, 7649, 1829, 46599, 6027, 2655, 7649, 1829, 39896, 4117, 4724, 1829, 30544, 4724, 9640, 3224, 7649, 8315, 1975, 46456, 3555, 9640, 2423, 430, 12, 12484, 530, 1500, 1975, 2407, 2423, 430, 12, 12484, 530, 1500, 11933, 8848, 1863, 36632, 430, 12, 12484, 530, 39894, 30544, 9652, 6930, 11933, 15730, 2423, 430, 5551, 4117, 26890, 9154, 36764, 24401, 4032, 18558, 6930, 11933, 15730, 430, 5551, 9381, 17082, 2288, 9154, 5551, 12610, 8608, 995, 12610, 36764, 24401, 8978, 36632, 8608, 33604, 6027, 22807], "avg_logprob": -0.23454122086788745, "compression_ratio": 1.7472527472527473, "no_speech_prob": 0.0, "words": [{"start": 1683.92, "end": 1684.24, "word": "إذن", "probability": 0.4517415364583333}, {"start": 1684.24, "end": 1684.46, 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"probability": 0.78173828125}, {"start": 1740.4, "end": 1740.92, "word": " series", "probability": 0.88232421875}, {"start": 1740.92, "end": 1748.9, "word": " comparison", "probability": 0.8408203125}, {"start": 1748.9, "end": 1753.96, "word": " test", "probability": 0.9365234375}, {"start": 1753.96, "end": 1756.34, "word": " for", "probability": 0.91845703125}, {"start": 1756.34, "end": 1757.02, "word": " series", "probability": 0.92138671875}], "temperature": 1.0}, {"id": 62, "seek": 178138, "start": 1773.68, "end": 1781.38, "text": "لت XIN و YIN بـ sequences of non-negative real numbers", "tokens": [1211, 2655, 1783, 1464, 4032, 398, 1464, 4724, 39184, 22978, 295, 2107, 12, 28561, 1166, 957, 3547], "avg_logprob": -0.5377604299121432, "compression_ratio": 0.8939393939393939, "no_speech_prob": 0.0, "words": [{"start": 1773.68, "end": 1774.24, "word": "لت", "probability": 0.398284912109375}, {"start": 1774.24, "end": 1775.2, "word": " XIN", "probability": 0.38916015625}, {"start": 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series الأكبر بالتأكيد diverge وهي البرهان البرهن الجزء الأول", "tokens": [1671, 331, 6930, 39894, 4587, 16254, 16472, 2423, 2638, 16247, 9381, 17082, 2288, 41881, 293, 45164, 25961, 2655, 2423, 2638, 16247, 4117, 26890, 16247, 9381, 17082, 2288, 18558, 432, 6156, 3555, 2655, 4587, 16254, 14739, 2423, 2638, 16247, 4117, 26890, 20666, 2655, 10721, 4117, 25708, 18558, 432, 37037, 1829, 2423, 26890, 3224, 7649, 2423, 26890, 3224, 1863, 25724, 11622, 38207, 16247, 12610], "avg_logprob": -0.21775793414267283, "compression_ratio": 1.8043478260869565, "no_speech_prob": 0.0, "words": [{"start": 1869.92, "end": 1870.8, "word": "convergent", "probability": 0.8185221354166666}, {"start": 1870.8, "end": 1871.42, "word": " بتقدي", "probability": 0.6357421875}, {"start": 1871.42, "end": 1871.64, "word": " ان", "probability": 0.378173828125}, {"start": 1871.64, "end": 1871.82, "word": " ال", "probability": 0.5556640625}, {"start": 1871.82, "end": 1872.14, "word": " series", "probability": 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cauchy criterion for series given", "tokens": [640, 2540, 14739, 3224, 2423, 2638, 12771, 17861, 9652, 2880, 550, 538, 1335, 625, 88, 46691, 337, 2638, 2212], "avg_logprob": -0.2908203214406967, "compression_ratio": 1.036144578313253, "no_speech_prob": 0.0, "words": [{"start": 1901.02, "end": 1901.88, "word": "assume", "probability": 0.798828125}, {"start": 1901.88, "end": 1904.76, "word": " أنه", "probability": 0.5260009765625}, {"start": 1904.76, "end": 1905.02, "word": " ال", "probability": 0.8671875}, {"start": 1905.02, "end": 1905.38, "word": " series", "probability": 0.61669921875}, {"start": 1905.38, "end": 1905.9, "word": " sigma", "probability": 0.67724609375}, {"start": 1905.9, "end": 1906.5, "word": " yn", "probability": 0.59033203125}, {"start": 1906.5, "end": 1908.96, "word": " converges", "probability": 0.776611328125}, {"start": 1908.96, "end": 1916.18, "word": " then", "probability": 0.7568359375}, {"start": 1916.18, "end": 1916.78, "word": " by", "probability": 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{"start": 2362.67, "end": 2363.97, "word": " فهذا", "probability": 0.9635416666666666}, {"start": 2363.97, "end": 2364.57, "word": " بكافئ", "probability": 0.9910888671875}, {"start": 2364.57, "end": 2365.03, "word": " not", "probability": 0.93994140625}, {"start": 2365.03, "end": 2365.57, "word": " Q", "probability": 0.88525390625}, {"start": 2365.57, "end": 2366.13, "word": " implies", "probability": 0.9521484375}, {"start": 2366.13, "end": 2367.09, "word": " not", "probability": 0.95263671875}, {"start": 2367.09, "end": 2367.51, "word": " P", "probability": 0.94580078125}, {"start": 2367.51, "end": 2370.55, "word": " فذا", "probability": 0.67626953125}, {"start": 2370.55, "end": 2370.99, "word": " أثبتنا", "probability": 0.94296875}, {"start": 2370.99, "end": 2371.13, "word": " إن", "probability": 0.484375}, {"start": 2371.13, "end": 2371.43, "word": " هذا", "probability": 0.91650390625}, {"start": 2371.43, "end": 2371.79, "word": " true", "probability": 0.91357421875}, {"start": 2371.79, "end": 2373.73, "word": " فهذا", "probability": 0.9842122395833334}, {"start": 2373.73, "end": 2374.09, "word": " بيكون", "probability": 0.90966796875}, {"start": 2374.09, "end": 2374.39, "word": " true", "probability": 0.955078125}], "temperature": 1.0}, {"id": 84, "seek": 238359, "start": 2376.15, "end": 2383.59, "text": "طب تعالى نشوف هذا هذا اللى انا اثبتنا انه true اللى هو الجزء ا او الجزء واحد الجزء التانى هو ال counter positive", "tokens": [9566, 3555, 37279, 6027, 7578, 8717, 8592, 38688, 23758, 23758, 13672, 7578, 1975, 8315, 1975, 12984, 3555, 2655, 8315, 16472, 3224, 2074, 13672, 7578, 31439, 25724, 11622, 38207, 1975, 1975, 2407, 25724, 11622, 38207, 36764, 24401, 25724, 11622, 38207, 16712, 7649, 7578, 31439, 2423, 5682, 3353], "avg_logprob": -0.38297871959970353, "compression_ratio": 1.564102564102564, "no_speech_prob": 0.0, "words": [{"start": 2376.15, "end": 2376.41, "word": "طب", "probability": 0.4805145263671875}, {"start": 2376.41, "end": 2376.75, "word": " 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contrapositive للعبارة في الجزء الأول نفي هذا اللي هو ال series sigma x in by dirge بيقدي الى نفي هذا اللي هو ال series y in by dirge", "tokens": [3224, 1211, 23758, 20328, 5016, 1829, 5016, 37279, 6027, 14407, 8717, 4587, 2288, 10721, 25724, 11622, 38207, 16712, 7649, 1829, 1975, 2407, 37279, 6027, 14407, 8717, 7435, 1829, 3555, 2423, 660, 4007, 329, 2187, 24976, 3615, 3555, 9640, 3660, 8978, 25724, 11622, 38207, 16247, 12610, 2423, 660, 4007, 329, 2187, 24976, 3615, 3555, 9640, 3660, 8978, 25724, 11622, 38207, 16247, 12610, 8717, 41185, 23758, 13672, 1829, 31439, 2423, 2638, 12771, 2031, 294, 538, 4746, 432, 4724, 1829, 4587, 16254, 2423, 7578, 8717, 41185, 23758, 13672, 1829, 31439, 2423, 2638, 288, 294, 538, 4746, 432], "avg_logprob": -0.23240131296609579, "compression_ratio": 2.0290697674418605, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2384.96, "end": 2385.24, "word": "هل", "probability": 0.9716796875}, {"start": 2385.24, "end": 2385.44, "word": " هذا", 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{"start": 2399.8, "end": 2400.06, "word": " هو", "probability": 0.986328125}, {"start": 2400.06, "end": 2400.26, "word": " ال", "probability": 0.363525390625}, {"start": 2400.26, "end": 2400.52, "word": " series", "probability": 0.9052734375}, {"start": 2400.52, "end": 2400.98, "word": " sigma", "probability": 0.202880859375}, {"start": 2400.98, "end": 2401.22, "word": " x", "probability": 0.8876953125}, {"start": 2401.22, "end": 2401.42, "word": " in", "probability": 0.14697265625}, {"start": 2401.42, "end": 2401.56, "word": " by", "probability": 0.55322265625}, {"start": 2401.56, "end": 2402.22, "word": " dirge", "probability": 0.174560546875}, {"start": 2402.22, "end": 2403.1, "word": " بيقدي", "probability": 0.65673828125}, {"start": 2403.1, "end": 2403.56, "word": " الى", "probability": 0.4683837890625}, {"start": 2403.56, "end": 2403.86, "word": " نفي", "probability": 0.94384765625}, {"start": 2403.86, "end": 2404.16, "word": " هذا", "probability": 0.96435546875}, {"start": 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سهل نعمل مقارنة زي هذه وبالتالي بنلجأ إلى اختبار اخر بنسميه limit comparison test", "tokens": [2304, 4587, 9640, 1863, 3660, 49374, 11331, 3215, 23328, 2423, 2638, 12771, 2031, 77, 4032, 11331, 3215, 23328, 2423, 2638, 12771, 17861, 20666, 9566, 16572, 28671, 9673, 3555, 33599, 25720, 30767, 1829, 19446, 8978, 2423, 3543, 5551, 5016, 1829, 7649, 995, 37893, 8608, 3224, 1211, 8717, 25957, 1211, 3714, 4587, 9640, 1863, 3660, 30767, 1829, 29538, 46599, 6027, 2655, 6027, 1829, 44945, 1211, 7435, 10721, 30731, 1975, 46456, 3555, 9640, 1975, 34740, 44945, 38251, 1829, 3224, 4948, 9660, 1500], "avg_logprob": -0.13496094159781932, "compression_ratio": 1.5675675675675675, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2423.08, "end": 2424.0, "word": "مقارنة", "probability": 0.93310546875}, {"start": 2424.0, "end": 2424.46, "word": " بين", "probability": 0.89892578125}, {"start": 2424.46, "end": 2424.96, "word": " حدود", "probability": 0.98828125}, {"start": 2424.96, "end": 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يعني لو كانت ال series ال series مثلا هذه ال sigma yn diverse", "tokens": [5172, 1863, 4117, 2655, 3555, 2423, 1500, 23758, 11778, 4117, 2655, 13063, 22807, 8717, 25957, 8032, 1211, 10721, 6225, 3555, 3794, 28239, 19913, 2288, 10632, 4724, 43042, 3224, 16490, 9566, 10721, 37495, 22653, 8717, 36178, 8717, 5016, 2407, 13672, 1829, 9122, 8315, 8717, 25957, 43761, 20666, 8310, 8717, 25957, 43761, 20666, 2638, 8032, 7578, 27188, 2655, 25720, 16712, 6027, 10632, 22807, 1975, 3224, 23032, 3555, 3615, 995, 37495, 22653, 16472, 2655, 45164, 25961, 50113, 15040, 29538, 31767, 5016, 10632, 28242, 10943, 4117, 20666, 19913, 3555, 9566, 37495, 22653, 45164, 25961, 2655, 2423, 2638, 2423, 2638, 50113, 15040, 29538, 2423, 12771, 17861, 9521], "avg_logprob": -0.266535199961616, "compression_ratio": 1.817351598173516, "no_speech_prob": 0.0, "words": [{"start": 2445.54, "end": 2446.32, "word": "فنكتب", "probability": 0.9609375}, {"start": 2446.32, "end": 2446.46, "word": " ال", "probability": 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"probability": 0.99169921875}, {"start": 2464.4, "end": 2464.72, "word": " كانت", "probability": 0.99462890625}, {"start": 2464.72, "end": 2464.88, "word": " ال", "probability": 0.96826171875}, {"start": 2464.88, "end": 2465.34, "word": " series", "probability": 0.97509765625}, {"start": 2465.34, "end": 2467.34, "word": " ال", "probability": 0.58544921875}, {"start": 2467.34, "end": 2467.74, "word": " series", "probability": 0.9658203125}, {"start": 2467.74, "end": 2468.16, "word": " مثلا", "probability": 0.997802734375}, {"start": 2468.16, "end": 2468.76, "word": " هذه", "probability": 0.763671875}, {"start": 2468.76, "end": 2469.52, "word": " ال", "probability": 0.41552734375}, {"start": 2469.52, "end": 2469.86, "word": " sigma", "probability": 0.93310546875}, {"start": 2469.86, "end": 2470.42, "word": " yn", "probability": 0.52587890625}, {"start": 2470.42, "end": 2470.92, "word": " diverse", "probability": 0.268798828125}], "temperature": 1.0}, {"id": 89, "seek": 249685, "start": 2472.07, "end": 2496.85, "text": "هل هذا بيقعد ان sigma xn diverge؟ مش بالضرورة ممكن diverge و ممكن converge نفس الحاجة لو كانت ال series sigma xn converge هل sigma yn converge؟ ممكن او ممكن لأ اكيد و ممكن نجيب counter examples لازم تفكري في إيجاد counter examples و انت بتدرسيه", "tokens": [3224, 1211, 23758, 4724, 1829, 4587, 22488, 16472, 12771, 2031, 77, 18558, 432, 22807, 37893, 20666, 11242, 2288, 13063, 3660, 3714, 43020, 18558, 432, 4032, 3714, 43020, 41881, 8717, 36178, 21542, 26108, 3660, 45164, 25961, 2655, 2423, 2638, 12771, 2031, 77, 41881, 8032, 1211, 12771, 17861, 41881, 22807, 3714, 43020, 1975, 2407, 3714, 43020, 5296, 10721, 1975, 4117, 25708, 4032, 3714, 43020, 8717, 7435, 1829, 3555, 5682, 5110, 5296, 31377, 2304, 6055, 5172, 4117, 16572, 8978, 11933, 1829, 7435, 18513, 5682, 5110, 4032, 16472, 2655, 39894, 3215, 2288, 3794, 1829, 3224], "avg_logprob": -0.1946331534696662, "compression_ratio": 1.7058823529411764, "no_speech_prob": 0.0, "words": [{"start": 2472.07, 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2491.81, "end": 2492.43, "word": " examples", "probability": 0.853515625}, {"start": 2492.43, "end": 2493.25, "word": " لازم", "probability": 0.8301595052083334}, {"start": 2493.25, "end": 2493.75, "word": " تفكري", "probability": 0.81756591796875}, {"start": 2493.75, "end": 2493.91, "word": " في", "probability": 0.97265625}, {"start": 2493.91, "end": 2494.33, "word": " إيجاد", "probability": 0.9393310546875}, {"start": 2494.33, "end": 2494.69, "word": " counter", "probability": 0.79150390625}, {"start": 2494.69, "end": 2495.27, "word": " examples", "probability": 0.88427734375}, {"start": 2495.27, "end": 2495.89, "word": " و", "probability": 0.9697265625}, {"start": 2495.89, "end": 2496.07, "word": " انت", "probability": 0.76123046875}, {"start": 2496.07, "end": 2496.85, "word": " بتدرسيه", "probability": 0.9192708333333334}], "temperature": 1.0}, {"id": 90, "seek": 252203, "start": 2500.81, "end": 2522.03, "text": "نشوف ال limit comparison test limit comparison test", "tokens": [1863, 8592, 38688, 2423, 4948, 9660, 1500, 4948, 9660, 1500], "avg_logprob": -0.297762778672305, "compression_ratio": 1.2391304347826086, "no_speech_prob": 0.0, "words": [{"start": 2500.81, "end": 2501.31, "word": "نشوف", "probability": 0.7620442708333334}, {"start": 2501.31, "end": 2501.47, "word": " ال", "probability": 0.352294921875}, {"start": 2501.47, "end": 2501.65, "word": " limit", "probability": 0.65576171875}, {"start": 2501.65, "end": 2502.23, "word": " comparison", "probability": 0.92626953125}, {"start": 2502.23, "end": 2503.15, "word": " test", "probability": 0.93115234375}, {"start": 2503.15, "end": 2513.63, "word": " limit", "probability": 0.818359375}, {"start": 2513.63, "end": 2515.09, "word": " comparison", "probability": 0.94482421875}, {"start": 2515.09, "end": 2522.03, "word": " test", "probability": 0.9296875}], "temperature": 1.0}, {"id": 91, "seek": 256048, "start": 2538.3, "end": 2560.48, "text": "لت Xn وYn بيكونوا سيكوانس من حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة بيكون حدود حقيقية مفتوحة به", "tokens": [1211, 2655, 1783, 77, 4032, 56, 77, 4724, 1829, 30544, 14407, 8608, 1829, 4117, 2407, 7649, 3794, 9154, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 4724, 1829, 30544, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 4724, 1829, 30544, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 4724, 1829, 30544, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 4724, 1829, 30544, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 4724, 1829, 30544, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 4724, 1829, 30544, 11331, 3215, 23328, 11331, 38436, 4587, 10632, 3714, 5172, 2655, 2407, 5016, 3660, 39627], "avg_logprob": -0.35901162790697677, "compression_ratio": 4.22093023255814, "no_speech_prob": 0.0, "words": [{"start": 2538.2999999999997, "end": 2539.7, "word": "لت", "probability": 0.52850341796875}, {"start": 2539.7, "end": 2541.1, "word": " Xn", "probability": 0.2882080078125}, {"start": 2541.1, "end": 2543.14, "word": " وYn", "probability": 0.8056640625}, {"start": 2543.14, "end": 2543.8, "word": " بيكونوا", "probability": 0.65423583984375}, {"start": 2543.8, "end": 2544.62, "word": " سيكوانس", "probability": 0.634033203125}, {"start": 2544.62, "end": 2545.82, "word": " من", "probability": 0.212158203125}, {"start": 2545.82, "end": 2546.42, "word": " حدود", "probability": 0.2880452473958333}, {"start": 2546.42, "end": 2547.12, "word": " حقيقية", "probability": 0.8104248046875}, {"start": 2547.12, "end": 2547.12, "word": " مفتوحة", "probability": 0.4933675130208333}, {"start": 2547.12, "end": 2550.12, "word": " بيكون", "probability": 0.7083333333333334}, {"start": 2550.12, "end": 2550.42, "word": " حدود", "probability": 0.34197998046875}, {"start": 2550.42, "end": 2550.42, "word": " حقيقية", "probability": 0.617431640625}, {"start": 2550.42, "end": 2550.42, "word": " مفتوحة", "probability": 0.8872477213541666}, {"start": 2550.42, "end": 2553.22, "word": " بيكون", "probability": 0.5984700520833334}, {"start": 2553.22, "end": 2553.44, "word": " حدود", "probability": 0.828125}, {"start": 2553.44, "end": 2555.3, "word": " حقيقية", "probability": 0.9539794921875}, {"start": 2555.3, "end": 2556.56, "word": " مفتوحة", "probability": 0.9469401041666666}, {"start": 2556.56, "end": 2556.96, "word": " بيكون", "probability": 0.7079264322916666}, {"start": 2556.96, "end": 2557.02, "word": " حدود", "probability": 0.8787434895833334}, {"start": 2557.02, "end": 2558.02, "word": " حقيقية", "probability": 0.9732666015625}, {"start": 2558.02, "end": 2558.02, "word": " مفتوحة", "probability": 0.96533203125}, {"start": 2558.02, "end": 2558.34, "word": " بيكون", "probability": 0.80908203125}, {"start": 2558.34, "end": 2558.34, "word": " حدود", "probability": 0.9016927083333334}, {"start": 2558.34, "end": 2558.34, "word": " حقيقية", "probability": 0.9791259765625}, {"start": 2558.34, "end": 2558.34, "word": " مفتوحة", "probability": 0.974853515625}, {"start": 2558.34, "end": 2558.46, "word": " بيكون", "probability": 0.9065755208333334}, {"start": 2558.46, "end": 2558.46, "word": " حدود", "probability": 0.9314778645833334}, {"start": 2558.46, "end": 2558.52, "word": " حقيقية", "probability": 0.982177734375}, {"start": 2558.52, "end": 2558.52, "word": " مفتوحة", "probability": 0.981201171875}, {"start": 2558.52, "end": 2558.52, "word": " بيكون", "probability": 0.953125}, {"start": 2558.52, "end": 2558.52, "word": " حدود", "probability": 0.9482421875}, {"start": 2558.52, "end": 2560.48, "word": " حقيقية", "probability": 0.9822998046875}, {"start": 2560.48, "end": 2560.48, "word": " مفتوحة", "probability": 0.9853515625}, {"start": 2560.48, "end": 2560.48, "word": " به", "probability": 0.007335662841796875}], "temperature": 1.0}, {"id": 92, "seek": 259533, "start": 2567.73, "end": 2595.33, "text": "و ال R هذا يعني عدد حقيقي طبعا ينتمي إلى R ففي عندي برضه نتيجتين النتيجة الأولى إذا كان ال R بسويش سفر then ال series sigma X M converges if and only if", "tokens": [2407, 2423, 497, 23758, 37495, 22653, 6225, 3215, 3215, 11331, 38436, 38436, 23032, 3555, 3615, 995, 7251, 29399, 2304, 1829, 30731, 497, 6156, 41185, 18871, 16254, 4724, 43042, 3224, 8717, 31371, 7435, 2655, 9957, 28239, 31371, 7435, 3660, 16247, 12610, 7578, 11933, 15730, 25961, 2423, 497, 4724, 3794, 45865, 8592, 8608, 5172, 2288, 550, 2423, 2638, 12771, 1783, 376, 9652, 2880, 498, 293, 787, 498], "avg_logprob": -0.21425188942389053, "compression_ratio": 1.3668639053254439, "no_speech_prob": 0.0, "words": [{"start": 2567.73, "end": 2568.01, "word": "و", "probability": 0.77294921875}, {"start": 2568.01, "end": 2568.17, "word": " ال", "probability": 0.65234375}, {"start": 2568.17, "end": 2568.45, "word": " R", "probability": 0.51611328125}, {"start": 2568.45, "end": 2569.03, "word": " هذا", "probability": 0.6962890625}, {"start": 2569.03, "end": 2569.47, "word": " يعني", "probability": 0.93359375}, {"start": 2569.47, "end": 2569.85, "word": " عدد", "probability": 0.9833984375}, {"start": 2569.85, "end": 2570.33, "word": " حقيقي", "probability": 0.9856770833333334}, {"start": 2570.33, "end": 2570.81, "word": " طبعا", "probability": 0.986328125}, {"start": 2570.81, "end": 2572.11, "word": " ينتمي", "probability": 0.75250244140625}, {"start": 2572.11, "end": 2572.33, "word": " إلى", "probability": 0.446044921875}, {"start": 2572.33, "end": 2572.81, "word": " R", "probability": 0.82421875}, {"start": 2572.81, "end": 2576.85, "word": " ففي", "probability": 0.85888671875}, {"start": 2576.85, "end": 2577.39, "word": " عندي", "probability": 0.91650390625}, {"start": 2577.39, "end": 2579.01, "word": " برضه", "probability": 0.9423828125}, {"start": 2579.01, "end": 2580.05, "word": " نتيجتين", "probability": 0.95703125}, {"start": 2580.05, "end": 2581.51, "word": " النتيجة", "probability": 0.9342041015625}, {"start": 2581.51, "end": 2581.93, "word": " الأولى", "probability": 0.8997395833333334}, {"start": 2581.93, "end": 2582.17, "word": " إذا", "probability": 0.796875}, {"start": 2582.17, "end": 2582.71, "word": " كان", "probability": 0.98681640625}, {"start": 2582.71, "end": 2584.35, "word": " ال", "probability": 0.89404296875}, {"start": 2584.35, "end": 2584.67, "word": " R", "probability": 0.96142578125}, {"start": 2584.67, "end": 2585.29, "word": " بسويش", "probability": 0.747314453125}, {"start": 2585.29, "end": 2585.79, "word": " سفر", "probability": 0.7730305989583334}, {"start": 2585.79, "end": 2589.63, "word": " then", "probability": 0.5302734375}, {"start": 2589.63, "end": 2589.97, "word": " ال", "probability": 0.37890625}, {"start": 2589.97, "end": 2590.37, "word": " series", "probability": 0.9150390625}, {"start": 2590.37, "end": 2590.83, "word": " sigma", "probability": 0.52392578125}, {"start": 2590.83, "end": 2591.15, "word": " X", "probability": 0.52783203125}, {"start": 2591.15, "end": 2591.47, "word": " M", "probability": 0.40673828125}, {"start": 2591.47, "end": 2592.45, "word": " converges", "probability": 0.8447265625}, {"start": 2592.45, "end": 2594.45, "word": " if", "probability": 0.814453125}, {"start": 2594.45, "end": 2594.67, "word": " and", "probability": 0.89501953125}, {"start": 2594.67, "end": 2594.95, "word": " only", "probability": 0.68017578125}, {"start": 2594.95, "end": 2595.33, "word": " if", "probability": 0.978515625}], "temperature": 1.0}, {"id": 93, "seek": 262583, "start": 2604.15, "end": 2625.83, "text": "الجزء التاني من النظرية لو كان R بساوي سفر", "tokens": [6027, 7435, 11622, 38207, 16712, 7649, 1829, 9154, 28239, 19913, 2288, 10632, 45164, 25961, 497, 4724, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.26307743528614874, "compression_ratio": 1.056338028169014, "no_speech_prob": 0.0, "words": [{"start": 2604.15, "end": 2605.35, "word": "الجزء", "probability": 0.7430419921875}, {"start": 2605.35, "end": 2605.95, "word": " التاني", "probability": 0.8592122395833334}, {"start": 2605.95, "end": 2606.29, "word": " من", "probability": 0.9365234375}, {"start": 2606.29, "end": 2607.53, "word": " النظرية", "probability": 0.96630859375}, {"start": 2607.53, "end": 2623.65, "word": " لو", "probability": 0.387451171875}, {"start": 2623.65, "end": 2624.15, "word": " كان", "probability": 0.97021484375}, {"start": 2624.15, "end": 2624.67, "word": " R", "probability": 0.37255859375}, {"start": 2624.67, "end": 2625.29, "word": " بساوي", "probability": 0.5796051025390625}, {"start": 2625.29, "end": 2625.83, "word": " سفر", "probability": 0.82763671875}], "temperature": 1.0}, {"id": 94, "seek": 264784, "start": 2628.2, "end": 2647.84, "text": "بعد ذلك لو الار بيسوي سفر يعني ان الار بيسويش سفر then ال series sigma y in دي درجز بيقدي ان ال series sigma x in دي درجز", "tokens": [3555, 22488, 29910, 23275, 45164, 2423, 9640, 4724, 1829, 3794, 45865, 8608, 5172, 2288, 37495, 22653, 16472, 2423, 9640, 4724, 1829, 3794, 45865, 8592, 8608, 5172, 2288, 550, 2423, 2638, 12771, 288, 294, 11778, 1829, 11778, 47341, 11622, 4724, 1829, 4587, 16254, 16472, 2423, 2638, 12771, 2031, 294, 11778, 1829, 11778, 47341, 11622], "avg_logprob": -0.527199058621018, "compression_ratio": 1.564102564102564, "no_speech_prob": 0.0, "words": [{"start": 2628.2, "end": 2628.78, "word": "بعد", "probability": 0.37554931640625}, {"start": 2628.78, "end": 2628.86, "word": " ذلك", "probability": 0.75341796875}, {"start": 2628.86, "end": 2630.38, "word": " لو", "probability": 0.1248779296875}, {"start": 2630.38, "end": 2630.94, "word": " الار", "probability": 0.36572265625}, {"start": 2630.94, "end": 2631.4, "word": " بيسوي", "probability": 0.5701904296875}, {"start": 2631.4, "end": 2631.88, "word": " سفر", "probability": 0.7125651041666666}, {"start": 2631.88, "end": 2632.06, "word": " يعني", "probability": 0.6029052734375}, {"start": 2632.06, "end": 2632.22, "word": " ان", "probability": 0.1578369140625}, {"start": 2632.22, "end": 2632.54, "word": " الار", "probability": 0.8720703125}, {"start": 2632.54, "end": 2633.18, "word": " بيسويش", "probability": 0.80361328125}, {"start": 2633.18, "end": 2633.66, "word": " سفر", "probability": 0.970703125}, {"start": 2633.66, "end": 2636.66, "word": " then", "probability": 0.1434326171875}, {"start": 2636.66, "end": 2638.12, "word": " ال", "probability": 0.359375}, {"start": 2638.12, "end": 2638.5, "word": " series", "probability": 0.583984375}, {"start": 2638.5, "end": 2639.08, "word": " sigma", "probability": 0.57958984375}, {"start": 2639.08, "end": 2639.56, "word": " y", "probability": 0.71875}, {"start": 2639.56, "end": 2639.94, "word": " in", "probability": 0.5966796875}, {"start": 2639.94, "end": 2640.4, "word": " دي", "probability": 0.279052734375}, {"start": 2640.4, "end": 2641.22, "word": " درجز", "probability": 0.64794921875}, {"start": 2641.22, "end": 2643.16, "word": " بيقدي", "probability": 0.713134765625}, {"start": 2643.16, "end": 2643.38, "word": " ان", "probability": 0.77978515625}, {"start": 2643.38, "end": 2643.58, "word": " ال", "probability": 0.89453125}, {"start": 2643.58, "end": 2643.86, "word": " series", "probability": 0.8876953125}, {"start": 2643.86, "end": 2644.34, "word": " sigma", "probability": 0.91064453125}, {"start": 2644.34, "end": 2644.7, "word": " x", "probability": 0.93798828125}, {"start": 2644.7, "end": 2645.36, "word": " in", "probability": 0.91845703125}, {"start": 2645.36, "end": 2647.18, "word": " دي", "probability": 0.750244140625}, {"start": 2647.18, "end": 2647.84, "word": " درجز", "probability": 0.9884440104166666}], "temperature": 1.0}, {"id": 95, "seek": 267852, "start": 2651.38, "end": 2678.52, "text": "او لأ convergence افضل لو كانت series sigma yn convergence بيقدي ان series sigma xn ايبان convergence فقط اتجاه واحد لكن الاكس مش شرط تكون صحيح okay تمام هو البرهان يعني كتير سهل", "tokens": [995, 2407, 5296, 10721, 32181, 1975, 5172, 11242, 1211, 45164, 25961, 2655, 2638, 12771, 17861, 32181, 4724, 1829, 4587, 16254, 16472, 2638, 12771, 2031, 77, 1975, 1829, 3555, 7649, 32181, 6156, 47432, 1975, 2655, 7435, 40294, 36764, 24401, 44381, 42963, 4117, 3794, 37893, 13412, 2288, 9566, 6055, 30544, 20328, 5016, 1829, 5016, 1392, 46811, 10943, 31439, 2423, 26890, 3224, 7649, 37495, 22653, 9122, 2655, 13546, 8608, 3224, 1211], "avg_logprob": -0.3034420393515324, "compression_ratio": 1.5086705202312138, "no_speech_prob": 0.0, "words": [{"start": 2651.38, "end": 2651.8, "word": "او", "probability": 0.5211944580078125}, {"start": 2651.8, "end": 2652.04, "word": " لأ", "probability": 0.6624755859375}, {"start": 2652.04, "end": 2652.58, "word": " convergence", "probability": 0.210205078125}, {"start": 2652.58, "end": 2653.22, "word": " افضل", "probability": 0.7657470703125}, {"start": 2653.22, "end": 2654.66, "word": " لو", "probability": 0.7451171875}, {"start": 2654.66, "end": 2655.2, "word": " كانت", "probability": 0.955810546875}, {"start": 2655.2, "end": 2657.06, "word": " series", "probability": 0.59228515625}, {"start": 2657.06, "end": 2657.74, "word": " sigma", "probability": 0.6318359375}, {"start": 2657.74, "end": 2659.3, "word": " yn", "probability": 0.67431640625}, {"start": 2659.3, "end": 2660.12, "word": " convergence", "probability": 0.83544921875}, {"start": 2660.12, "end": 2660.66, "word": " بيقدي", "probability": 0.6197509765625}, {"start": 2660.66, "end": 2660.84, "word": " ان", "probability": 0.75}, {"start": 2660.84, "end": 2661.74, "word": " series", "probability": 0.46240234375}, {"start": 2661.74, "end": 2662.32, "word": " sigma", "probability": 0.880859375}, {"start": 2662.32, "end": 2663.24, "word": " xn", "probability": 0.953369140625}, {"start": 2663.24, "end": 2664.48, "word": " ايبان", "probability": 0.639984130859375}, {"start": 2664.48, "end": 2665.16, "word": " convergence", "probability": 0.7158203125}, {"start": 2665.16, "end": 2665.6, "word": " فقط", "probability": 0.9814453125}, {"start": 2665.6, "end": 2666.06, "word": " اتجاه", "probability": 0.9781494140625}, {"start": 2666.06, "end": 2666.38, "word": " واحد", "probability": 0.994140625}, {"start": 2666.38, "end": 2666.62, "word": " لكن", "probability": 0.77294921875}, {"start": 2666.62, "end": 2666.98, "word": " الاكس", "probability": 0.7740071614583334}, {"start": 2666.98, "end": 2667.16, "word": " مش", "probability": 0.96337890625}, {"start": 2667.16, "end": 2667.4, "word": " شرط", "probability": 0.98779296875}, {"start": 2667.4, "end": 2667.62, "word": " تكون", "probability": 0.673828125}, {"start": 2667.62, "end": 2669.74, "word": " صحيح", "probability": 0.9835205078125}, {"start": 2669.74, "end": 2670.5, "word": " okay", "probability": 0.426025390625}, {"start": 2670.5, "end": 2672.44, "word": " تمام", "probability": 0.97998046875}, {"start": 2672.44, "end": 2674.44, "word": " هو", "probability": 0.464111328125}, {"start": 2674.44, "end": 2677.46, "word": " البرهان", "probability": 0.8734130859375}, {"start": 2677.46, "end": 2677.72, "word": " يعني", "probability": 0.970458984375}, {"start": 2677.72, "end": 2678.08, "word": " كتير", "probability": 0.96728515625}, {"start": 2678.08, "end": 2678.52, "word": " سهل", "probability": 0.9973958333333334}], "temperature": 1.0}, {"id": 96, "seek": 271770, "start": 2689.52, "end": 2717.7, "text": "بنسمي الشرط هذا star الجزء الأول assume ان R لا يساوي سفر طبعا في الحالة هذه ال R بطلع موجد", "tokens": [3555, 1863, 38251, 1829, 25124, 2288, 9566, 23758, 3543, 25724, 11622, 38207, 16247, 12610, 6552, 16472, 497, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 23032, 3555, 3615, 995, 8978, 21542, 6027, 3660, 29538, 2423, 497, 4724, 9566, 1211, 3615, 3714, 29245, 3215], "avg_logprob": -0.27876419912685046, "compression_ratio": 1.2276422764227641, "no_speech_prob": 0.0, "words": [{"start": 2689.52, "end": 2690.34, "word": "بنسمي", "probability": 0.73187255859375}, {"start": 2690.34, "end": 2690.9, "word": " الشرط", "probability": 0.9147135416666666}, {"start": 2690.9, "end": 2691.28, "word": " هذا", "probability": 0.5517578125}, {"start": 2691.28, "end": 2691.96, "word": " star", "probability": 0.60546875}, {"start": 2691.96, "end": 2702.3, "word": " الجزء", "probability": 0.8512369791666666}, {"start": 2702.3, "end": 2702.88, "word": " الأول", "probability": 0.9375}, {"start": 2702.88, "end": 2704.26, "word": " assume", "probability": 0.78662109375}, {"start": 2704.26, "end": 2706.6, "word": " ان", "probability": 0.477783203125}, {"start": 2706.6, "end": 2707.26, "word": " R", "probability": 0.2010498046875}, {"start": 2707.26, "end": 2707.52, "word": " لا", "probability": 0.81201171875}, {"start": 2707.52, "end": 2708.1, "word": " يساوي", "probability": 0.7548828125}, {"start": 2708.1, "end": 2708.62, "word": " سفر", "probability": 0.7556966145833334}, {"start": 2708.62, "end": 2712.94, "word": " طبعا", "probability": 0.938720703125}, {"start": 2712.94, "end": 2713.14, "word": " في", "probability": 0.81982421875}, {"start": 2713.14, "end": 2713.56, "word": " الحالة", "probability": 0.9547526041666666}, {"start": 2713.56, "end": 2713.96, "word": " هذه", "probability": 0.84033203125}, {"start": 2713.96, "end": 2714.38, "word": " ال", "probability": 0.78466796875}, {"start": 2714.38, "end": 2715.32, "word": " R", "probability": 0.66455078125}, {"start": 2715.32, "end": 2717.18, "word": " بطلع", "probability": 0.7569580078125}, {"start": 2717.18, "end": 2717.7, "word": " موجد", "probability": 0.771484375}], "temperature": 1.0}, {"id": 97, "seek": 274795, "start": 2719.57, "end": 2747.95, "text": "لأن لحظة انتوا ان xn على yn هذه كلها أعداد موجبة و ال limit ل sequence أعداء كل حدودها موجبة بيطلع المفروض يطلع ال limit تبعتها موجبة إذا كانت ال limit موجودة واضح؟ إذن هذا من نظرية سابقة لما أن حدود ال sequence xn على yn هذه ال sequence حدودها كلها موجبة إذا نهايتها تطلع أيضا موجبة", "tokens": [1211, 33456, 5296, 5016, 19913, 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2720.01, "end": 2720.43, "word": " لحظة", "probability": 0.6595611572265625}, {"start": 2720.43, "end": 2720.73, "word": " انتوا", "probability": 0.7945963541666666}, {"start": 2720.73, "end": 2720.83, "word": " ان", "probability": 0.6064453125}, {"start": 2720.83, "end": 2721.67, "word": " xn", "probability": 0.4541015625}, {"start": 2721.67, "end": 2721.93, "word": " على", "probability": 0.6572265625}, {"start": 2721.93, "end": 2722.41, "word": " yn", "probability": 0.9140625}, {"start": 2722.41, "end": 2723.89, "word": " هذه", "probability": 0.3779296875}, {"start": 2723.89, "end": 2724.21, "word": " كلها", "probability": 0.930908203125}, {"start": 2724.21, "end": 2724.61, "word": " أعداد", "probability": 0.8990885416666666}, {"start": 2724.61, "end": 2725.17, "word": " موجبة", "probability": 0.9695638020833334}, {"start": 2725.17, "end": 2726.55, "word": " و", "probability": 0.57568359375}, {"start": 2726.55, "end": 2726.69, "word": " ال", "probability": 0.1544189453125}, {"start": 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0.83544921875}, {"start": 2850.59, "end": 2851.33, "word": " صحيح", "probability": 0.9951171875}, {"start": 2851.33, "end": 2851.89, "word": " لكل", "probability": 0.981201171875}, {"start": 2851.89, "end": 2852.27, "word": " N", "probability": 0.8818359375}, {"start": 2852.27, "end": 2852.79, "word": " أكبر", "probability": 0.9554036458333334}, {"start": 2852.79, "end": 2853.01, "word": " من", "probability": 0.99462890625}, {"start": 2853.01, "end": 2853.19, "word": " أو", "probability": 0.9638671875}, {"start": 2853.19, "end": 2853.51, "word": " ساوي", "probability": 0.9742838541666666}, {"start": 2853.51, "end": 2853.91, "word": " capital", "probability": 0.52734375}, {"start": 2853.91, "end": 2854.49, "word": " N", "probability": 0.66064453125}, {"start": 2854.49, "end": 2855.11, "word": " تمام", "probability": 0.677734375}, {"start": 2855.11, "end": 2857.13, "word": " الان", "probability": 0.6884765625}, {"start": 2857.13, "end": 2857.33, "word": " now", "probability": 0.476318359375}], "temperature": 1.0}, {"id": 102, "seek": 289318, "start": 2865.12, "end": 2893.18, "text": "نسمي هذه double star فالان if sigma x and converge then by double star and comparison test and comparison test", "tokens": [1863, 38251, 1829, 29538, 3834, 3543, 6156, 6027, 7649, 498, 12771, 2031, 293, 41881, 550, 538, 3834, 3543, 293, 9660, 1500, 293, 9660, 1500], "avg_logprob": -0.2326562547683716, "compression_ratio": 1.2708333333333333, "no_speech_prob": 0.0, "words": [{"start": 2865.12, "end": 2865.82, "word": "نسمي", "probability": 0.7162272135416666}, {"start": 2865.82, "end": 2866.2, "word": " هذه", "probability": 0.8173828125}, {"start": 2866.2, "end": 2866.56, "word": " double", "probability": 0.75341796875}, {"start": 2866.56, "end": 2867.08, "word": " star", "probability": 0.89990234375}, {"start": 2867.08, "end": 2873.94, "word": " فالان", "probability": 0.8653971354166666}, {"start": 2873.94, "end": 2874.66, "word": " if", "probability": 0.58447265625}, {"start": 2874.66, "end": 2877.16, "word": " sigma", "probability": 0.734375}, {"start": 2877.16, "end": 2878.86, "word": " x", "probability": 0.83740234375}, {"start": 2878.86, "end": 2879.28, "word": " and", "probability": 0.491943359375}, {"start": 2879.28, "end": 2880.22, "word": " converge", "probability": 0.8359375}, {"start": 2880.22, "end": 2883.82, "word": " then", "probability": 0.77587890625}, {"start": 2883.82, "end": 2886.2, "word": " by", "probability": 0.94677734375}, {"start": 2886.2, "end": 2886.58, "word": " double", "probability": 0.93798828125}, {"start": 2886.58, "end": 2887.16, "word": " star", "probability": 0.94970703125}, {"start": 2887.16, "end": 2889.52, "word": " and", "probability": 0.9296875}, {"start": 2889.52, "end": 2890.2, "word": " comparison", "probability": 0.93896484375}, {"start": 2890.2, "end": 2890.78, "word": " test", "probability": 0.93359375}, {"start": 2890.78, "end": 2891.72, "word": " and", "probability": 0.73828125}, {"start": 2891.72, "end": 2892.54, "word": " comparison", "probability": 0.912109375}, {"start": 2892.54, "end": 2893.18, "word": " test", "probability": 0.9072265625}], "temperature": 1.0}, {"id": 103, "seek": 292249, "start": 2895.01, "end": 2922.49, "text": "واختبار المقارنة إذا كانت هذه convergent فبطلع هذه ال series convergent صح وبالتالي بطلع sigma yn convergence هذا ثابت موجة ممكن نضرب في مقلوب و نتخلص منه إذا كانت هذه convergent فهذه convergent", "tokens": [2407, 47283, 2655, 3555, 9640, 9673, 4587, 9640, 1863, 3660, 11933, 15730, 25961, 2655, 29538, 9652, 6930, 6156, 3555, 9566, 1211, 3615, 29538, 2423, 2638, 9652, 6930, 20328, 5016, 46599, 6027, 2655, 6027, 1829, 4724, 9566, 1211, 3615, 12771, 17861, 32181, 23758, 38637, 16758, 2655, 3714, 29245, 3660, 3714, 43020, 8717, 11242, 25513, 8978, 3714, 4587, 1211, 37746, 4032, 8717, 2655, 9778, 1211, 9381, 9154, 3224, 11933, 15730, 25961, 2655, 29538, 9652, 6930, 6156, 3224, 24192, 9652, 6930], "avg_logprob": -0.15822785338269005, "compression_ratio": 1.697674418604651, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2895.01, "end": 2895.69, "word": "واختبار", "probability": 0.8736328125}, {"start": 2895.69, "end": 2896.47, "word": " المقارنة", "probability": 0.990234375}, {"start": 2896.47, "end": 2897.07, "word": " إذا", "probability": 0.7047119140625}, {"start": 2897.07, "end": 2897.53, "word": " كانت", "probability": 0.92822265625}, {"start": 2897.53, "end": 2897.81, "word": " هذه", "probability": 0.69921875}, {"start": 2897.81, "end": 2898.77, "word": " convergent", "probability": 0.66064453125}, {"start": 2898.77, "end": 2902.45, "word": " فبطلع", "probability": 0.84521484375}, {"start": 2902.45, "end": 2902.95, "word": " هذه", "probability": 0.87548828125}, {"start": 2902.95, "end": 2903.19, "word": " ال", "probability": 0.77685546875}, {"start": 2903.19, "end": 2903.67, "word": " series", "probability": 0.80615234375}, {"start": 2903.67, "end": 2905.11, "word": " convergent", "probability": 0.900390625}, {"start": 2905.11, "end": 2906.01, "word": " صح", "probability": 0.792236328125}, {"start": 2906.01, "end": 2908.01, "word": " وبالتالي", "probability": 0.851318359375}, {"start": 2908.01, "end": 2909.95, "word": " بطلع", "probability": 0.9327392578125}, {"start": 2909.95, "end": 2911.29, "word": " sigma", "probability": 0.6416015625}, {"start": 2911.29, "end": 2911.81, "word": " yn", "probability": 0.59375}, {"start": 2911.81, "end": 2913.05, "word": " convergence", "probability": 0.56640625}, {"start": 2913.05, "end": 2915.79, "word": " هذا", "probability": 0.411865234375}, {"start": 2915.79, "end": 2916.19, "word": " ثابت", "probability": 0.9812825520833334}, {"start": 2916.19, "end": 2916.57, "word": " موجة", "probability": 0.96044921875}, {"start": 2916.57, "end": 2916.81, "word": " ممكن", "probability": 0.89208984375}, {"start": 2916.81, "end": 2917.29, "word": " نضرب", "probability": 0.9783528645833334}, {"start": 2917.29, "end": 2917.83, "word": " في", "probability": 0.7724609375}, {"start": 2917.83, "end": 2918.27, "word": " مقلوب", "probability": 0.9317626953125}, {"start": 2918.27, "end": 2918.37, "word": " و", "probability": 0.70654296875}, {"start": 2918.37, "end": 2918.87, "word": " نتخلص", "probability": 0.8955078125}, {"start": 2918.87, "end": 2919.27, "word": " منه", "probability": 0.984619140625}, {"start": 2919.27, "end": 2920.13, "word": " إذا", "probability": 0.939697265625}, {"start": 2920.13, "end": 2920.49, "word": " كانت", "probability": 0.9794921875}, {"start": 2920.49, "end": 2920.65, "word": " هذه", "probability": 0.8701171875}, {"start": 2920.65, "end": 2921.37, "word": " convergent", "probability": 0.951904296875}, {"start": 2921.37, "end": 2921.79, "word": " فهذه", "probability": 0.9554036458333334}, {"start": 2921.79, "end": 2922.49, "word": " convergent", "probability": 0.881103515625}], "temperature": 1.0}, {"id": 104, "seek": 294607, "start": 2926.95, "end": 2946.07, "text": "Also إذا كانت ال series sigma y in converge then برضه by المتباينة double star and ال comparison test إذا ناخد الجزء هذا", "tokens": [9171, 539, 11933, 15730, 25961, 2655, 2423, 2638, 12771, 288, 294, 41881, 550, 4724, 43042, 3224, 538, 9673, 2655, 3555, 995, 9957, 3660, 3834, 3543, 293, 2423, 9660, 1500, 11933, 15730, 8717, 47283, 3215, 25724, 11622, 38207, 23758], "avg_logprob": -0.26161857904532015, "compression_ratio": 1.1691176470588236, "no_speech_prob": 0.0, "words": [{"start": 2926.9500000000003, "end": 2927.9900000000002, "word": "Also", "probability": 0.6490478515625}, {"start": 2927.9900000000002, "end": 2929.03, "word": " إذا", "probability": 0.7064208984375}, {"start": 2929.03, "end": 2929.75, "word": " كانت", "probability": 0.97314453125}, {"start": 2929.75, "end": 2930.81, "word": " ال", "probability": 0.68798828125}, {"start": 2930.81, "end": 2931.09, "word": " series", "probability": 0.62451171875}, {"start": 2931.09, "end": 2931.53, "word": " sigma", "probability": 0.54833984375}, {"start": 2931.53, "end": 2931.89, "word": " y", "probability": 0.71240234375}, {"start": 2931.89, "end": 2932.21, "word": " in", "probability": 0.5517578125}, {"start": 2932.21, "end": 2933.07, "word": " converge", "probability": 0.85107421875}, {"start": 2933.07, "end": 2935.33, "word": " then", "probability": 0.3291015625}, {"start": 2935.33, "end": 2936.41, "word": " برضه", "probability": 0.9619140625}, {"start": 2936.41, "end": 2936.91, "word": " by", "probability": 0.5634765625}, {"start": 2936.91, "end": 2938.73, "word": " المتباينة", "probability": 0.8550211588541666}, {"start": 2938.73, "end": 2938.99, "word": " double", "probability": 0.75830078125}, {"start": 2938.99, "end": 2939.57, "word": " star", "probability": 0.87255859375}, {"start": 2939.57, "end": 2940.09, "word": " and", "probability": 0.8759765625}, {"start": 2940.09, "end": 2940.31, "word": " ال", "probability": 0.54833984375}, {"start": 2940.31, "end": 2940.81, "word": " comparison", "probability": 0.9560546875}, {"start": 2940.81, "end": 2941.53, "word": " test", "probability": 0.93798828125}, {"start": 2941.53, "end": 2944.89, "word": " إذا", "probability": 0.885009765625}, {"start": 2944.89, "end": 2945.33, "word": " ناخد", "probability": 0.9388020833333334}, {"start": 2945.33, "end": 2945.71, "word": " الجزء", "probability": 0.984375}, {"start": 2945.71, "end": 2946.07, "word": " هذا", "probability": 0.962890625}], "temperature": 1.0}, {"id": 105, "seek": 295784, "start": 2949.44, "end": 2957.84, "text": "إذا كانت ال series هذي convergent و الكبيرة هذي convergent ففي ثابت موجب convergent فهذه تطلع", "tokens": [28814, 15730, 25961, 2655, 2423, 2638, 8032, 8848, 1829, 9652, 6930, 4032, 33251, 3555, 48923, 8032, 8848, 1829, 9652, 6930, 6156, 41185, 38637, 16758, 2655, 3714, 29245, 3555, 9652, 6930, 6156, 3224, 24192, 6055, 9566, 1211, 3615], "avg_logprob": -0.19212581530997627, "compression_ratio": 1.3366336633663367, "no_speech_prob": 0.0, "words": [{"start": 2949.44, "end": 2949.8, "word": "إذا", "probability": 0.850341796875}, {"start": 2949.8, "end": 2950.18, "word": " كانت", "probability": 0.98876953125}, {"start": 2950.18, "end": 2950.28, "word": " ال", "probability": 0.8505859375}, {"start": 2950.28, "end": 2950.56, "word": " series", "probability": 0.92529296875}, {"start": 2950.56, "end": 2950.86, "word": " هذي", "probability": 0.5774739583333334}, {"start": 2950.86, "end": 2951.56, "word": " convergent", "probability": 0.876953125}, {"start": 2951.56, "end": 2951.68, "word": " و", "probability": 0.1373291015625}, {"start": 2951.68, "end": 2952.0, "word": " الكبيرة", "probability": 0.88037109375}, {"start": 2952.0, "end": 2952.72, "word": " هذي", "probability": 0.9148763020833334}, {"start": 2952.72, "end": 2953.46, "word": " convergent", "probability": 0.857421875}, {"start": 2953.46, "end": 2954.36, "word": " ففي", "probability": 0.91552734375}, {"start": 2954.36, "end": 2954.82, "word": " ثابت", "probability": 0.9783528645833334}, {"start": 2954.82, "end": 2955.2, "word": " موجب", "probability": 0.943359375}, {"start": 2955.2, "end": 2955.94, "word": " convergent", "probability": 0.867919921875}, {"start": 2955.94, "end": 2956.6, "word": " فهذه", "probability": 0.9013671875}, {"start": 2956.6, "end": 2957.84, "word": " تطلع", "probability": 0.941650390625}], "temperature": 1.0}, {"id": 106, "seek": 298373, "start": 2959.77, "end": 2983.73, "text": "بطلع sigma xn conversion وبالتالي هذا بكمل برهان الجزء الأول طبعا برهان الجزء التاني هيكون يعني مشابه فلأن الواجهة انتهى هنوقف و هسيبكم تقرؤ برهان الجزء الأول من الكتاب فنكتفي بهذا القدر و ان شاء الله نكمل المحاضرة الجاية", "tokens": [3555, 9566, 1211, 3615, 12771, 2031, 77, 14298, 46599, 6027, 2655, 6027, 1829, 23758, 4724, 24793, 1211, 4724, 2288, 3224, 7649, 25724, 11622, 38207, 16247, 12610, 23032, 3555, 3615, 995, 4724, 2288, 3224, 7649, 25724, 11622, 38207, 16712, 7649, 1829, 39896, 30544, 37495, 22653, 37893, 16758, 3224, 6156, 1211, 33456, 2423, 2407, 26108, 3224, 3660, 16472, 47395, 7578, 8032, 1863, 30543, 5172, 4032, 8032, 3794, 1829, 3555, 24793, 6055, 4587, 2288, 33604, 4724, 2288, 3224, 7649, 25724, 11622, 38207, 16247, 12610, 9154, 33251, 2655, 16758, 6156, 1863, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288, 4032, 16472, 13412, 16606, 21984, 8717, 24793, 1211, 9673, 5016, 46958, 25720, 25724, 995, 10632], "avg_logprob": -0.14569256891001453, "compression_ratio": 1.8341232227488151, "no_speech_prob": 1.7881393432617188e-06, "words": [{"start": 2959.77, "end": 2960.33, "word": "بطلع", "probability": 0.628082275390625}, {"start": 2960.33, "end": 2960.71, "word": " sigma", "probability": 0.509765625}, {"start": 2960.71, "end": 2961.43, "word": " xn", "probability": 0.5712890625}, {"start": 2961.43, "end": 2962.39, "word": " conversion", "probability": 0.87841796875}, {"start": 2962.39, "end": 2964.19, "word": " وبالتالي", "probability": 0.8990234375}, {"start": 2964.19, "end": 2964.43, "word": " هذا", "probability": 0.88330078125}, {"start": 2964.43, "end": 2965.03, "word": " بكمل", "probability": 0.9182942708333334}, {"start": 2965.03, "end": 2965.65, "word": " برهان", "probability": 0.958984375}, {"start": 2965.65, "end": 2966.35, "word": " الجزء", "probability": 0.9778645833333334}, {"start": 2966.35, "end": 2967.15, "word": " الأول", "probability": 0.87060546875}, {"start": 2967.15, "end": 2968.03, "word": " طبعا", "probability": 0.9661865234375}, {"start": 2968.03, "end": 2968.39, "word": " برهان", "probability": 0.98388671875}, {"start": 2968.39, "end": 2968.73, "word": " الجزء", "probability": 0.9881184895833334}, {"start": 2968.73, "end": 2969.15, "word": " التاني", "probability": 0.8543294270833334}, {"start": 2969.15, "end": 2969.71, "word": " هيكون", "probability": 0.81298828125}, {"start": 2969.71, "end": 2970.11, "word": " يعني", "probability": 0.71923828125}, {"start": 2970.11, "end": 2971.15, "word": " مشابه", "probability": 0.9698893229166666}, {"start": 2971.15, "end": 2972.27, "word": " فلأن", "probability": 0.7566731770833334}, {"start": 2972.27, "end": 2972.67, "word": " الواجهة", "probability": 0.84970703125}, {"start": 2972.67, "end": 2973.27, "word": " انتهى", "probability": 0.7482096354166666}, {"start": 2973.27, "end": 2974.05, "word": " هنوقف", "probability": 0.9151611328125}, {"start": 2974.05, "end": 2974.29, "word": " و", "probability": 0.943359375}, {"start": 2974.29, "end": 2974.75, "word": " هسيبكم", "probability": 0.80166015625}, {"start": 2974.75, "end": 2975.31, "word": " تقرؤ", "probability": 0.8404541015625}, {"start": 2975.31, "end": 2976.53, "word": " برهان", "probability": 0.9798583984375}, {"start": 2976.53, "end": 2976.95, "word": " الجزء", "probability": 0.9912109375}, {"start": 2976.95, "end": 2977.43, "word": " الأول", "probability": 0.96337890625}, {"start": 2977.43, "end": 2977.69, "word": " من", "probability": 0.98486328125}, {"start": 2977.69, "end": 2978.25, "word": " الكتاب", "probability": 0.9869791666666666}, {"start": 2978.25, "end": 2980.17, "word": " فنكتفي", "probability": 0.95380859375}, {"start": 2980.17, "end": 2980.47, "word": " بهذا", "probability": 0.94775390625}, {"start": 2980.47, "end": 2980.99, "word": " القدر", "probability": 0.99365234375}, {"start": 2980.99, "end": 2981.47, "word": " و", "probability": 0.99072265625}, {"start": 2981.47, "end": 2981.65, "word": " ان", "probability": 0.7685546875}, {"start": 2981.65, "end": 2981.89, "word": " شاء", "probability": 0.9736328125}, {"start": 2981.89, "end": 2981.95, "word": " الله", "probability": 0.94921875}, {"start": 2981.95, "end": 2982.63, "word": " نكمل", "probability": 0.8016764322916666}, {"start": 2982.63, "end": 2983.23, "word": " المحاضرة", "probability": 0.93310546875}, {"start": 2983.23, "end": 2983.73, "word": " الجاية", "probability": 0.8427734375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2985.018, "duration_after_vad": 2607.0068749999873}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Sym_17KvBqE_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Sym_17KvBqE_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6325ef36ac3969e87ae74bdc1e77d3b658051e9e
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/Sym_17KvBqE_postprocess.srt
@@ -0,0 +1,1724 @@
+1
+00:00:20,830 --> 00:00:26,410
+بسم الله الرحمن الرحيم احنا في المحاضرة اللى فاتت
+
+2
+00:00:26,410 --> 00:00:32,690
+اتحدثنا عن ال limit comparison test وبرهننا
+
+3
+00:00:32,690 --> 00:00:37,470
+الجزء الاول منه فنرجع مع بعض ال limit comparison
+
+4
+00:00:37,470 --> 00:00:43,190
+test for infinite series طبعا طبعا في limit
+
+5
+00:00:43,190 --> 00:00:47,450
+comparison test for sequences الان هذا الافتبار
+
+6
+00:00:47,450 --> 00:00:52,340
+قصد ال infinite seriesلو في عندي two sequences of
+
+7
+00:00:52,340 --> 00:00:57,760
+positive real numbers بحيث ان limit ال quotient
+
+8
+00:00:57,760 --> 00:01:05,700
+تبعهم exist بساوي عدد R ففي عندي نتيجتين، لو كان
+
+9
+00:01:05,700 --> 00:01:12,170
+العدد R أو limit R هذه لا تساوي 0ففي الحالة هذه
+
+10
+00:01:12,170 --> 00:01:18,130
+sigma x in series sigma x in convergence if and
+
+11
+00:01:18,130 --> 00:01:21,550
+only if ال series sigma y in convergence يعني
+
+12
+00:01:21,550 --> 00:01:24,910
+اتنين اما اتنين بيكونوا convergence زي بعض او
+
+13
+00:01:24,910 --> 00:01:28,630
+اتنين بيكونوا divergence زي بعض الجزء التاني بيقول
+
+14
+00:01:28,630 --> 00:01:32,010
+لو كانت ال R اللي هي limit لل quotient مساوة سفر
+
+15
+00:01:32,010 --> 00:01:37,110
+وإذا كانت ال series اللي الحد العام تبع Y in
+
+16
+00:01:37,110 --> 00:01:41,770
+convergenceفال series هذا بيقدر ال series اللي هي
+
+17
+00:01:41,770 --> 00:01:48,510
+sigma xn كلها يعني اعتقد ان احنا برهن الجزء الأول
+
+18
+00:01:48,510 --> 00:01:55,750
+برا اللي فاتت بظبط و خلينا نبرهن الجزء التاني طبعا
+
+19
+00:01:55,750 --> 00:02:06,370
+since اذا هنا let assume r
+
+20
+00:02:06,370 --> 00:02:07,650
+بساوي سفر
+
+21
+00:02:18,190 --> 00:02:24,490
+أما لو أخدت إبسلون أنا بساوي العدد واحد فهذا
+
+22
+00:02:24,490 --> 00:02:29,910
+إبسلون موجبة إحنا
+
+23
+00:02:29,910 --> 00:02:38,070
+لدينا من الفرض sense limit xn over yn as n tends
+
+24
+00:02:38,070 --> 00:02:45,640
+to infinityبساوي R اللي هو سفر الآن فمن تعريف by
+
+25
+00:02:45,640 --> 00:02:51,260
+definition of limit for epsilon positive زي هذه
+
+26
+00:02:51,260 --> 00:02:57,420
+يوجد capital N يعتمد على epsilon اللي هو الواحد
+
+27
+00:02:57,420 --> 00:03:03,900
+natural number بحيث انه لكل N أكبر من أو ساوي
+
+28
+00:03:03,900 --> 00:03:11,260
+capital Nهذا بيدّي أن ال absolute value ل xn على
+
+29
+00:03:11,260 --> 00:03:18,120
+yn minus zero بيطلع أصغر من ال epsilon اللي احنا
+
+30
+00:03:18,120 --> 00:03:26,660
+ماخدينها واحد طب xn عدد موجب و yn عدد موجبفال
+
+31
+00:03:26,660 --> 00:03:33,760
+quotient هذا كسر هذا موجب سالد سفر فهذا بيقدي ان
+
+32
+00:03:33,760 --> 00:03:42,880
+xn over yn أصغر من واحد لو ضربنا الطرفين العدد
+
+33
+00:03:42,880 --> 00:03:58,110
+الموجب yn فهذا هيقدي ان xn أصغر من ynوهذا صحيح لكل
+
+34
+00:03:58,110 --> 00:04:05,550
+N أكبر من أو يستوي capital N now
+
+35
+00:04:05,550 --> 00:04:09,090
+if
+
+36
+00:04:09,090 --> 00:04:20,550
+sigma yn converges then
+
+37
+00:04:21,920 --> 00:04:26,140
+by direct comparison test اللي أخدناها المرة اللي
+
+38
+00:04:26,140 --> 00:04:30,420
+فاتت إذا ال series الحد اللي عام تبعها أكبر
+
+39
+00:04:30,420 --> 00:04:34,480
+convergent فالأصغر
+
+40
+00:04:34,480 --> 00:04:42,920
+ال series الأصغر converges وهذا هو المطلوب هذا
+
+41
+00:04:42,920 --> 00:04:46,340
+اللي احنا عايزين نتبته إنه لو كانت ال series yn
+
+42
+00:04:46,340 --> 00:04:50,690
+convergent فلازم هذا يطلع convergent هذا صحيحby
+
+43
+00:04:50,690 --> 00:04:55,110
+direct comparison test لذلك هذا يكمل برهان الجزء
+
+44
+00:04:55,110 --> 00:05:02,230
+التالي نرجع الأن ناخد أمثلة على تطبيقات على ال
+
+45
+00:05:02,230 --> 00:05:08,590
+direct comparison test و على limit comparison test
+
+46
+00:05:11,980 --> 00:05:15,680
+كيف نستخدم ال comparison tests الاختبارين هدول
+
+47
+00:05:15,680 --> 00:05:27,780
+فيثبات ان ال series معينة is convergent discuss
+
+48
+00:05:27,780 --> 00:05:38,840
+.. discuss the convergence of
+
+49
+00:05:38,840 --> 00:05:40,360
+the following series
+
+50
+00:06:00,990 --> 00:06:07,110
+فناخد series sigma from n equals one to infinity ل
+
+51
+00:06:07,110 --> 00:06:17,370
+one over n squared plus n بالمناسبة
+
+52
+00:06:17,370 --> 00:06:18,450
+ال series هذه
+
+53
+00:06:23,110 --> 00:06:29,010
+ممكن نقارنها، الحد العام تبعها هذا، لما N تكون
+
+54
+00:06:29,010 --> 00:06:36,410
+large فممكن نهمل ال N بالنسبة ل N تربية و نعتبر أن
+
+55
+00:06:36,410 --> 00:06:42,730
+هذه ال series شبيهة أو behaves like تتصرف زي ال
+
+56
+00:06:42,730 --> 00:06:45,650
+series sigma 1 على N تربية
+
+57
+00:06:50,030 --> 00:06:54,610
+الان بنشوف إذا ممكن نطبق اختبار المقارنة المباشرة
+
+58
+00:06:54,610 --> 00:06:58,670
+ال direct comparison test بنطبقه وإذا ما اقدرناش
+
+59
+00:06:58,670 --> 00:07:06,950
+بنلجأ لاختبار تبع ال limit comparison test
+
+60
+00:07:24,400 --> 00:07:37,040
+فهنا ممكن يعني من السهل أن احنا نستخدم ال
+
+61
+00:07:37,040 --> 00:07:43,040
+direct comparison test لأنه انا عندي ال N تربيع
+
+62
+00:07:43,040 --> 00:07:50,400
+زائد N أكبر من أو يساوي Nأكبر من أو ساوي N تربية
+
+63
+00:07:50,400 --> 00:08:00,220
+لكل N ينتمي ل N هذا بيقدي أنه مقلوب N تربية زايد N
+
+64
+00:08:00,220 --> 00:08:08,680
+أصغر من أو ساوي مقلوب N تربية لكل N ك N الان
+
+65
+00:08:08,680 --> 00:08:13,020
+ال series
+
+66
+00:08:13,020 --> 00:08:15,360
+sigma واحد على N تربية
+
+67
+00:08:18,710 --> 00:08:29,730
+a P series is P series صح؟ with P
+
+68
+00:08:29,730 --> 00:08:40,270
+بيساوي اتنين اكبر من واحد so
+
+69
+00:08:40,270 --> 00:08:49,230
+it convergesby .. it is convergent by P series
+
+70
+00:08:49,230 --> 00:08:56,550
+test في ال P series test بيقوللي إذا كان أي P
+
+71
+00:08:56,550 --> 00:09:02,890
+series زي هذه بتكون convergent إذا كان P أكبر من
+
+72
+00:09:02,890 --> 00:09:08,530
+واحد و divergent إذا كان P أصغر من أوسع و أعلى و
+
+73
+00:09:08,530 --> 00:09:14,510
+برهننا الكلام هذا في المحاضرة السابقة أو الجبلةإذا
+
+74
+00:09:14,510 --> 00:09:20,250
+أنا في عندى two series واحدة الحد العام تبعها واحد
+
+75
+00:09:20,250 --> 00:09:23,790
+على انتر بيه وهذا الconversion وواحدة الحد العام
+
+76
+00:09:23,790 --> 00:09:28,090
+تبعها واحد على انتر بيه الزادة وهذا الحد العام
+
+77
+00:09:28,090 --> 00:09:31,250
+أصغر من أو ساوي الحد العام لهذه الconversion إذا
+
+78
+00:09:31,250 --> 00:09:35,630
+ممكن استخدم so
+
+79
+00:09:35,630 --> 00:09:38,550
+by direct comparison test
+
+80
+00:09:42,520 --> 00:09:46,740
+السيريز اللي هي sigma من n equals one to infinity
+
+81
+00:09:46,740 --> 00:09:56,860
+لواحد على n squared plus n converges
+
+82
+00:09:56,860 --> 00:10:03,340
+إذا السيريز هذه أتباعنا هي انها convergence by
+
+83
+00:10:03,340 --> 00:10:07,140
+direct comparison استخدمنا ال direct comparison
+
+84
+00:10:07,140 --> 00:10:09,160
+test مفهوم واضح؟
+
+85
+00:10:12,050 --> 00:10:13,950
+ناخد مثال تاني
+
+86
+00:10:36,080 --> 00:10:39,580
+بتاعة اتنين لو أخدنا series sigma from n equals
+
+87
+00:10:39,580 --> 00:10:47,780
+one to infinity لواحد على n تربية سالف n زائد
+
+88
+00:10:47,780 --> 00:10:54,180
+واحد بما نفحص هل ال series هذي convergent ولا
+
+89
+00:10:54,180 --> 00:10:57,520
+divergent طبعا
+
+90
+00:10:59,200 --> 00:11:04,280
+أول شيء بنفكر فيه، بنشوف كيف ال series هذه بتتصرف،
+
+91
+00:11:04,280 --> 00:11:07,740
+ما هي ال series القريبة منها، و اللي احنا عارفين
+
+92
+00:11:07,740 --> 00:11:12,600
+أنها أو ممكن نحكم عليها بسهولة، ن be convergent أو
+
+93
+00:11:12,600 --> 00:11:15,980
+divergent، يعني بدي أقارن ال series هذه ب series
+
+94
+00:11:15,980 --> 00:11:20,520
+تانيةمن السهل اني احكم عليها هل هي convergent او
+
+95
+00:11:20,520 --> 00:11:27,160
+divergent فلما N تكون كبيرة و ان N is sufficiently
+
+96
+00:11:27,160 --> 00:11:32,900
+large لما N تقول infinity ممكن اهمل N و اهمل 1
+
+97
+00:11:32,900 --> 00:11:41,080
+وبالتالي ال series هذه behaves تتصرف زي ال series
+
+98
+00:11:41,080 --> 00:11:42,880
+1 على N ترمية
+
+99
+00:11:45,470 --> 00:11:55,230
+اللي هي احنا عارفين which is كل بيت واحد طبعا by P
+
+100
+00:11:55,230 --> 00:12:01,270
+seriousness زي ما شرحنا في المثال الأول الآن
+
+101
+00:12:01,270 --> 00:12:10,120
+السؤال اللي بيطرح نفسه is it true هل واحد علىإن
+
+102
+00:12:10,120 --> 00:12:15,000
+تربية سالف إن زاد واحد أصغر من أو يساوي واحد على
+
+103
+00:12:15,000 --> 00:12:20,640
+إن تربية عشان نستخدم .. هل هذا الكلام صحيح لكل إن؟
+
+104
+00:12:20,640 --> 00:12:25,920
+لأ مش فاكرش أنا فللأسف هذا مش صحيح وبالتالي
+
+105
+00:12:25,920 --> 00:12:29,940
+مابقدرش أستخدم إن هذا not true
+
+106
+00:12:34,430 --> 00:12:41,410
+for example على سبيل المثال take m بساوي اتنين
+
+107
+00:12:41,410 --> 00:12:50,310
+هنجد المتباين هذه مش صح اذا مقدرش انا استخدم ال
+
+108
+00:12:50,310 --> 00:12:54,310
+direct comparison test اذا في الحالة هذه لازم
+
+109
+00:12:54,310 --> 00:12:59,190
+استخدم ال limit comparison test او ابحث عن مقارنة
+
+110
+00:12:59,190 --> 00:13:01,310
+تانية however
+
+111
+00:13:06,140 --> 00:13:17,200
+you can show بإمكانكم تخبطه أنه الواحد على n تربية
+
+112
+00:13:17,200 --> 00:13:23,500
+negative n زائد واحد هذا أصغر من أو ساوي اتنين على
+
+113
+00:13:23,500 --> 00:13:31,280
+n تربية وهذا صحيح لكل n في n إذن هذه المتباينة
+
+114
+00:13:31,280 --> 00:13:34,420
+صحيحة وبالتالي ممكن الآن
+
+115
+00:13:39,990 --> 00:13:46,330
+الان بإمكانك استخدام
+
+116
+00:13:46,330 --> 00:13:53,030
+تجارة مقارنة مباشرة للتأكيد
+
+117
+00:13:53,030 --> 00:14:02,650
+عشان تستنتجوا ان سيريز سيجما واحد على إنتر بيه
+
+118
+00:14:02,650 --> 00:14:08,980
+نيجاتيب ن بلس واحدconvergent لأنه ال series هذه
+
+119
+00:14:08,980 --> 00:14:15,920
+لأنه since ال series اللي الحد العام تبعها اتنين
+
+120
+00:14:15,920 --> 00:14:20,740
+على انتر بيها هي نفسها اتنين ضارب ال series sigma
+
+121
+00:14:20,740 --> 00:14:26,600
+واحد على انتر بيها و ال series هذه قلنا convergent
+
+122
+00:14:26,600 --> 00:14:29,660
+لأنها في series نضربها في عدد موجب بتضلها
+
+123
+00:14:29,660 --> 00:14:31,700
+convergent
+
+124
+00:14:34,390 --> 00:14:38,990
+لازم نثبت على ذلك الكلام هذا الكلام لازم تثبتيه صح
+
+125
+00:14:38,990 --> 00:14:45,970
+المشكلة في الحل هذا ان انا او انتوا كيف نبيه يخطر
+
+126
+00:14:45,970 --> 00:14:53,170
+على بالكم ان المتباين هذا صح اه it is not easy to
+
+127
+00:14:53,170 --> 00:14:57,030
+figure out this inequality مش سهل ان يختر على
+
+128
+00:14:57,030 --> 00:15:04,110
+بالنا او نستنتج ال .. او يعني ..بنعرف إنه في
+
+129
+00:15:04,110 --> 00:15:09,870
+متباينة زي هذه صحيحة هذا مش سهل وبالتالي ممكن
+
+130
+00:15:09,870 --> 00:15:14,090
+نستخدم ال limit comparison test ونرايح رأسنا ال
+
+131
+00:15:14,090 --> 00:15:16,690
+limit comparison test في الحالة هذه أسهل من إن أنا
+
+132
+00:15:16,690 --> 00:15:21,550
+يعني أخمن
+
+133
+00:15:21,550 --> 00:15:25,950
+.. أخمن يعني حاجة زي هذه okay فتعالوا نشوف كيف
+
+134
+00:15:25,950 --> 00:15:28,070
+نستخدم ال limit comparison test
+
+135
+00:15:31,920 --> 00:15:40,220
+أذا هنا we use limit
+
+136
+00:15:40,220 --> 00:15:45,160
+comparison test with
+
+137
+00:15:45,160 --> 00:15:54,640
+a n بساوي واحد على n تربيع minus n زايد واحد أو xn
+
+138
+00:15:54,640 --> 00:15:55,720
+فالبسامينات
+
+139
+00:15:57,840 --> 00:16:07,600
+و Yn بساوية واحد على M تربية فاني
+
+140
+00:16:07,600 --> 00:16:13,340
+ايجي نحسب ال limit ل Xn over Yn as N tenths of
+
+141
+00:16:13,340 --> 00:16:21,720
+infinity بساوية limit هاي Xn تقسيم Yn بتطلع M
+
+142
+00:16:21,720 --> 00:16:28,990
+تربية على M تربية negative M plus oneو ال limit
+
+143
+00:16:28,990 --> 00:16:36,930
+هذا عشان نحسبها بالجسم bust مقام على n تربية ففي
+
+144
+00:16:36,930 --> 00:16:41,750
+ال bust واحد واحد سالب واحد على n موجب واحد على n
+
+145
+00:16:41,750 --> 00:16:47,210
+تربية لإن تقول ال infinity وهذا بطلع واحد على واحد
+
+146
+00:16:47,210 --> 00:16:54,410
+سالب صفر موجب صفر ويساوي واحد لايساوي صفر إذن ال R
+
+147
+00:16:55,770 --> 00:16:59,910
+الـ R في ال limit comparison test طلعت بالساوي
+
+148
+00:16:59,910 --> 00:17:07,630
+واحد لا يساوي سفر وانا عندى اذا since وانا عندى ال
+
+149
+00:17:07,630 --> 00:17:13,050
+series sigma yn اللى هى sigma واحد على انتر بيان
+
+150
+00:17:13,050 --> 00:17:17,830
+is convergent then
+
+151
+00:17:17,830 --> 00:17:26,700
+by limit comparison test ال series sigma xnاللي هو
+
+152
+00:17:26,700 --> 00:17:32,960
+الحد اللي عم تبعها واحد على انتر بيه minus ان زاد
+
+153
+00:17:32,960 --> 00:17:41,560
+واحد كون بيعجز وهو مطلوب okay إذا هنا استخدمنا ال
+
+154
+00:17:41,560 --> 00:17:46,020
+limit كون .. لما يعجز أو يفشل ال comparison أو ال
+
+155
+00:17:46,020 --> 00:17:49,920
+direct comparison test بنرجع إلى limit comparison
+
+156
+00:17:49,920 --> 00:17:57,220
+testهنا لازم يجب ملاحظة انه اي سؤال بنحل بال
+
+157
+00:17:57,220 --> 00:18:02,660
+comparison test ممكن حله او نطبق عليه ال limit
+
+158
+00:18:02,660 --> 00:18:07,800
+comparison test لكن العكس ليس صحيح وبالتالي ال
+
+159
+00:18:07,800 --> 00:18:12,240
+limit comparison test اشمل و اعام من ال direct
+
+160
+00:18:12,240 --> 00:18:17,460
+comparison test ناخد مثال تالت واضح الحل في اي
+
+161
+00:18:17,460 --> 00:18:22,470
+سؤال او استفسار؟إذا دائما في مخرج يعني إذا انت مش
+
+162
+00:18:22,470 --> 00:18:26,390
+عارف تعمل direct comparison فاستخدم ال limit
+
+163
+00:18:26,390 --> 00:18:30,670
+comparison test وهذا مش صعب تشوفي دائما ال series
+
+164
+00:18:30,670 --> 00:18:35,730
+اللي قدامك behaves like some familiar series تتصرف
+
+165
+00:18:35,730 --> 00:18:41,130
+زي series معروفة لدينا و احنا عارف نقدر من السهل
+
+166
+00:18:41,130 --> 00:18:43,530
+نحكم عليها هل convergent او divergent
+
+167
+00:18:49,830 --> 00:18:57,370
+فلو أخدنا مثلا ال series هذه summation from
+
+168
+00:18:57,370 --> 00:19:03,630
+n equals one to infinity ل one over square root of
+
+169
+00:19:03,630 --> 00:19:08,530
+n plus one ف
+
+170
+00:19:08,530 --> 00:19:11,910
+ال series .. this series behaves طبعا لما n ..
+
+171
+00:19:11,910 --> 00:19:19,330
+when n gets large we neglect الواحد نهم الواحدوهذه
+
+172
+00:19:19,330 --> 00:19:29,190
+السيريز تتصرف من حيث التقارب والتباعد مثل سيجما
+
+173
+00:19:29,190 --> 00:19:31,210
+واحد على جذر الان
+
+174
+00:19:38,390 --> 00:19:46,390
+طيب can we السؤال يتفرج نفسه can we use direct
+
+175
+00:19:46,390 --> 00:19:51,670
+comparison test للإجابة
+
+176
+00:19:51,670 --> 00:19:57,770
+على السؤال هذا بنلاحظ أن n زائد 1 أكبر منها ويساوي
+
+177
+00:19:57,770 --> 00:20:06,310
+n لكل n هذا بيقدر أن واحدوبالتالي الجدر التربيعي ل
+
+178
+00:20:06,310 --> 00:20:10,110
+N زائد واحد أكبر من أو ساوي جدر ال N لكل N
+
+179
+00:20:10,110 --> 00:20:17,910
+وبالتالي هذا بيقدي أن واحد على الجدر التربيعي ل N
+
+180
+00:20:17,910 --> 00:20:26,950
+زائد واحد أقل من أو ساوي واحد على جدر ال N لكل Nو
+
+181
+00:20:26,950 --> 00:20:31,850
+احنا عارفين ان ال series هذه divergent لأنها P
+
+182
+00:20:31,850 --> 00:20:36,850
+series و ال P بساوي نص أصغر من واحد و هاد ال
+
+183
+00:20:36,850 --> 00:20:42,770
+series أصغر منها أو أصغر منها و يساويهافال direct
+
+184
+00:20:42,770 --> 00:20:46,970
+comparison test بيعطينيش نتيجة، بيعطينيش نتيجة إذا
+
+185
+00:20:46,970 --> 00:20:50,550
+الكبيرة divergent فالصغيرة ممكن تكون convergent
+
+186
+00:20:50,550 --> 00:20:57,150
+وممكن تكون divergent إذا هنا ال direct comparison
+
+187
+00:20:57,150 --> 00:21:01,050
+test fails،
+
+188
+00:21:01,050 --> 00:21:09,290
+fails يعني يفشل، يفشل وبالتالي مافيش أمامنا خيار
+
+189
+00:21:09,290 --> 00:21:13,350
+اللي احنا .. اللي .. اللي هالنا نستعملأو نستخدم
+
+190
+00:21:13,350 --> 00:21:27,350
+limit comparison test نستخدم
+
+191
+00:21:27,350 --> 00:21:29,970
+limit comparison test
+
+192
+00:21:34,290 --> 00:21:39,970
+with xn بيساوي واحد على ال square root of n plus
+
+193
+00:21:39,970 --> 00:21:48,230
+one و yn بيساوي one over square root of n نحسم ال
+
+194
+00:21:48,230 --> 00:21:53,990
+limit ل xn over yn as n tends to infinity بيساوي
+
+195
+00:21:53,990 --> 00:21:57,770
+ال limit هاي
+
+196
+00:21:57,770 --> 00:22:06,100
+جسم xn على yn بيطلع الجدر التربيعيلان على ان plus
+
+197
+00:22:06,100 --> 00:22:11,500
+one لما ان تقول ال infinity دخل ال limit تحت الجدر
+
+198
+00:22:11,500 --> 00:22:15,740
+لأن ال square root function is continuous فاندخل
+
+199
+00:22:15,740 --> 00:22:21,660
+ال limit و limit المقدار تحت الجدر بطلع واحد
+
+200
+00:22:21,660 --> 00:22:28,920
+وبالتالي واحد لا يساوي سوى إذا ال R في limit
+
+201
+00:22:28,920 --> 00:22:34,540
+comparison first طلعتdifferent from zero لأ تساوي
+
+202
+00:22:34,540 --> 00:22:44,020
+سفر و since ال series sigma من n equals one to
+
+203
+00:22:44,020 --> 00:22:52,860
+infinity لواحد على جدر ال n يعبر عن sigma واحد على
+
+204
+00:22:52,860 --> 00:22:58,160
+n أصمص is a p-series with
+
+205
+00:23:03,240 --> 00:23:11,940
+P بساوي نص أصغر من واحد it diverges
+
+206
+00:23:11,940 --> 00:23:24,780
+يعني بتطلع divergent by P series test ال series
+
+207
+00:23:24,780 --> 00:23:28,560
+يعني divergent وبالتالي
+
+208
+00:23:31,020 --> 00:23:34,760
+by limit comparison test حسب ال limit comparison
+
+209
+00:23:34,760 --> 00:23:44,020
+test هيعندي sigma x in و sigma y in sigma y in ده
+
+210
+00:23:44,020 --> 00:23:51,200
+هي طلعت divergent و ال R limit لرئيسه لا يساوي سفر
+
+211
+00:23:51,200 --> 00:23:56,100
+لان التانية زيها divergent ده is sigma x in اللي
+
+212
+00:23:56,100 --> 00:24:02,990
+هو واحد على الجذر التربيهي ال N زي واحدby agents
+
+213
+00:24:02,990 --> 00:24:17,470
+حسب ال limit comparison test okay تمام واضح طيب
+
+214
+00:24:17,470 --> 00:24:18,790
+ناخد كمان مثال
+
+215
+00:24:30,910 --> 00:24:37,470
+مثال رقم أربعة خلّينا نفحص ال series اللي هي
+
+216
+00:24:37,470 --> 00:24:44,770
+summation from n equals one to infinity ل one over
+
+217
+00:24:44,770 --> 00:24:52,070
+n factorial طبعا
+
+218
+00:24:52,070 --> 00:25:00,050
+هذه مش واضحممكن تقارنها لأن N factorial N
+
+219
+00:25:00,050 --> 00:25:04,810
+factorial بالساوي N نقش واحد N negative واحد N
+
+220
+00:25:04,810 --> 00:25:11,890
+negative اتنين إلى تلاتة في اتنين في واحد فمش
+
+221
+00:25:11,890 --> 00:25:21,350
+عارفين ايش نقارنها اه فهذا مش واضح لكن by trial
+
+222
+00:25:21,350 --> 00:25:31,990
+انا بتقوله بالتجريبنقدر احنا نحاول يعني نقرر او
+
+223
+00:25:31,990 --> 00:25:37,330
+يعني نشوف ان هنا عند عشان n في n سالب واحد في n
+
+224
+00:25:37,330 --> 00:25:43,510
+سالب اتنين فممكن نقارن ال series هذه بواحد على n
+
+225
+00:25:43,510 --> 00:25:51,750
+ترمية نشوف كيف ممكن نعمل المقارنة اذا هنا في حالين
+
+226
+00:25:51,750 --> 00:26:01,200
+هناSolution واحد نحن نحاول نقارن بالإيه فال
+
+227
+00:26:01,200 --> 00:26:08,740
+solution الأول أو الحل الأول بيعتمد use
+
+228
+00:26:08,740 --> 00:26:17,460
+induction to show that ممكن
+
+229
+00:26:17,460 --> 00:26:24,210
+نثبت بال induction أنهN تربية أصغر من N factorial
+
+230
+00:26:24,210 --> 00:26:30,290
+لكل N أكبر من أو ساوي أربعة المتباينة هذه صحيحة
+
+231
+00:26:30,290 --> 00:26:34,050
+لكل الأعداد الطبيعية أكبر من أو ساوي أربعة هذا
+
+232
+00:26:34,050 --> 00:26:38,390
+ممكن نثبته by induction زي ما اتعلمته هذا سؤال في
+
+233
+00:26:38,390 --> 00:26:44,670
+مبادئ رياضياتنشوف مع بعض الهدى صح نشوف أول حالة
+
+234
+00:26:44,670 --> 00:26:48,990
+لحظة ال N بتبدأ من أربعة مش من واحد ف N بساوي واحد
+
+235
+00:26:48,990 --> 00:26:52,390
+هنا هصير N بساوي أربعة و الباقى ال induction زي ما
+
+236
+00:26:52,390 --> 00:26:57,210
+اتعلمنا فلو N بساوي أربعة أربعة تربيه ستة عشر أصغر
+
+237
+00:26:57,210 --> 00:27:01,070
+من أربعة فاكتوريا الأربعة و عشرين ستة عشر أصغر من
+
+238
+00:27:01,070 --> 00:27:05,050
+أربعة و عشرين صحيحإذا العبارة صحيحة عند n بالساوية
+
+239
+00:27:05,050 --> 00:27:09,410
+أربعة افرض صحيتها عند n بالساوية k حيث k أي عدد
+
+240
+00:27:09,410 --> 00:27:13,570
+طبيعي أكبر من أربعة وثبت صحيتها عند n بالساوية k
+
+241
+00:27:13,570 --> 00:27:18,830
+زادة، أعتقد هذه مثلة في أخدت زيها في مبادئ رياضية،
+
+242
+00:27:18,830 --> 00:27:23,050
+رح نسيب .. سيبقى لكم .. ليه؟ ايه شو بتهارفنا مثلا
+
+243
+00:27:23,050 --> 00:27:25,610
+نختار الأربعة؟ ليش ما هو مثلا تلاتة أو واحد، سيبقى
+
+244
+00:27:25,610 --> 00:27:29,150
+احنا متعودين في ال induction؟أه لأنه انت ال ..
+
+245
+00:27:29,150 --> 00:27:34,030
+يعني نضل نجرب لحد ما نصر نصر بره صح اه من أربعة و
+
+246
+00:27:34,030 --> 00:27:37,690
+انت طالع تصير صحيحة أما قبل أربعة بتكون خطأ
+
+247
+00:27:37,690 --> 00:27:42,210
+وبالتالي مالهاش معناه أما من أربعة و أنت طالع
+
+248
+00:27:42,210 --> 00:27:49,830
+هتكون صحيحة فبنهم الأول تلت قيم لهم okay اذا و
+
+249
+00:27:49,830 --> 00:27:58,220
+بالتاليهذا بيقدي ان واحد على n factorial أصغر من
+
+250
+00:27:58,220 --> 00:28:04,040
+واحد على n تردية لكل n أكبر من أو ساوية أربعة
+
+251
+00:28:04,040 --> 00:28:11,960
+وبالتالي و ال series طبعا وبالتالي ممكن نستخدم ال
+
+252
+00:28:11,960 --> 00:28:15,800
+direct comparison test يعني الحالة هذه
+
+253
+00:28:24,200 --> 00:28:28,020
+و نستخدم الاختصار الوحيد الوحيد الوحيد الوحيد
+
+254
+00:28:28,020 --> 00:28:43,880
+الوحيد الوحيد الوحيد الوحيد
+
+255
+00:28:45,400 --> 00:28:50,520
+هذه الـ series هي ال key series بس بتبدأ من أربعة
+
+256
+00:28:50,520 --> 00:28:55,240
+فكأني حدث يتأول تلات حدود منها فهذا بيأثرش على ال
+
+257
+00:28:55,240 --> 00:28:59,420
+divergence أو ال convergence لل series إذا حدث
+
+258
+00:28:59,420 --> 00:29:04,980
+omitting أو deleting finite number of terms from
+
+259
+00:29:04,980 --> 00:29:09,000
+an infinite series does not affect the convergence
+
+260
+00:29:09,000 --> 00:29:13,240
+or the divergence of the series حدث عدد منتهي من
+
+261
+00:29:13,240 --> 00:29:19,540
+حدود ال seriesأو إضافة عدد منتهي كمان إلى حدود ال
+
+262
+00:29:19,540 --> 00:29:24,180
+series لا يؤثر لا على التقارب ولا على التباعد تبع
+
+263
+00:29:24,180 --> 00:29:34,900
+ال series هذا حقيقة سهل لو يعني و بدهاش برهان لأن
+
+264
+00:29:34,900 --> 00:29:40,300
+الحدود المنتهية هذه مجموعة بيطلع عدد منتهي فما
+
+265
+00:29:40,300 --> 00:29:47,230
+بأثرش على التقاربمن series بفرش على التقارب او
+
+266
+00:29:47,230 --> 00:29:52,310
+التباعد او اضافة عدد لان بما ان ال series
+
+267
+00:29:52,310 --> 00:29:58,110
+converges then ال series sigma واحد على n
+
+268
+00:29:58,110 --> 00:30:03,990
+factorial converges
+
+269
+00:30:03,990 --> 00:30:11,160
+من n بالساوية اربعة الى ملامية طبعا هذا بقدرإن أنا
+
+270
+00:30:11,160 --> 00:30:15,360
+لو ضفت لل series الحدود المتبقية من n بالساعة واحد
+
+271
+00:30:15,360 --> 00:30:22,160
+إلى تلاتة وبتصير من infinity هنا لواحد
+
+272
+00:30:22,160 --> 00:30:28,160
+على n factorial تطلع
+
+273
+00:30:28,160 --> 00:30:33,280
+conversion وهذا اللي بدنا يعني، إذن هذا أحد
+
+274
+00:30:33,280 --> 00:30:38,100
+الحلولة، okay؟ زي ما زملتكم يعني اخترحت، بتقول طب
+
+275
+00:30:38,100 --> 00:30:44,050
+و أنا إيش بدي أختار على بالي؟إن هذا المتباينة
+
+276
+00:30:44,050 --> 00:30:47,890
+الصحيحة اللي اعتمد عليها الحل أو اعتمدت عليها
+
+277
+00:30:47,890 --> 00:30:53,430
+المقارنة فمعاكم حاجة ممكن أنك .. يعني ماحدش يقدر
+
+278
+00:30:53,430 --> 00:30:57,970
+يعني يصل إلى ال .. أو ال percentage المتباينة هذه
+
+279
+00:30:57,970 --> 00:31:03,710
+اللي عليها بيرتكز الحل ففي حل تاني آخر نشوف الحل
+
+280
+00:31:03,710 --> 00:31:05,930
+التاني ال direct limit
+
+281
+00:31:09,330 --> 00:31:14,150
+الحل التاني solution
+
+282
+00:31:14,150 --> 00:31:18,430
+2 احنا
+
+283
+00:31:18,430 --> 00:31:26,430
+عارفين انه لو جسمت ناخد
+
+284
+00:31:26,430 --> 00:31:35,130
+xn بسعر واحد على n factorialبساوي واحد على ال
+
+285
+00:31:35,130 --> 00:31:41,770
+تربية كويس؟ زي ما عملناه في الحل الأول و بده قارن
+
+286
+00:31:41,770 --> 00:31:47,320
+التنتين هدول بس المقارنة المرة هذه هتكونبطريقة
+
+287
+00:31:47,320 --> 00:31:53,760
+مختلفة فلو أخدت xn و جسمتها على yn فطبعا هذا أكبر
+
+288
+00:31:53,760 --> 00:32:00,140
+من السبب لأن xn عدد موجب دايما لكل n و yn عدد موجب
+
+289
+00:32:00,140 --> 00:32:06,640
+فقسمت على دين موجبين بطلعة موجب وهذا بساوي n تربية
+
+290
+00:32:06,640 --> 00:32:13,720
+على yn اللي هو n factorial على n factorial
+
+291
+00:32:18,080 --> 00:32:26,220
+و هدا بساوي تاي n تربية على n factorial عبارة عن
+
+292
+00:32:26,220 --> 00:32:34,160
+واحد في اتنين في تلاتة الى n سالب اتنين في n سالب
+
+293
+00:32:34,160 --> 00:32:45,700
+واحد في n مظبوط؟ ممكن اختصر n مع n و هيبقى عندي
+
+294
+00:32:54,210 --> 00:33:05,130
+فهيبقى عندي n على واحد في اتنين الى n سالب اتنين
+
+295
+00:33:05,130 --> 00:33:14,650
+في n سالب واحد الان ممكن اثبات ان المقام هذا اكبر
+
+296
+00:33:14,650 --> 00:33:17,570
+من اتنين
+
+297
+00:33:20,110 --> 00:33:28,310
+إثنين في N سالب إثنين في N سالب واحد وهذا أكبر من
+
+298
+00:33:28,310 --> 00:33:38,410
+إثنين في N سالب واحد في N وهذا صحيح ليس لكل الـ N
+
+299
+00:33:38,410 --> 00:33:48,270
+مش لكل الأعداد الطبيعية N هذا أكبر من N سالب إثنين
+
+300
+00:33:48,270 --> 00:33:56,730
+في Nو هذا صحيح فقط لكل n أكبر من أو يساوي خمسة
+
+301
+00:33:56,730 --> 00:34:01,990
+يعني عند الأربعة مش صح و عند التلاتة و اتنين و
+
+302
+00:34:01,990 --> 00:34:08,580
+الواحد مش صحOkay؟ إذن N تربية على N factorial
+
+303
+00:34:08,580 --> 00:34:14,940
+بتطلع .. الآن هذا المقام أكبر من العدد هذا
+
+304
+00:34:14,940 --> 00:34:23,280
+وبالتالي المقلوب بتطلع أصغر من N على N في N سالب 2
+
+305
+00:34:23,280 --> 00:34:32,100
+طبعا N بتروح مع Nبيبقى عندي واحد على n ساوى اتنين
+
+306
+00:34:32,100 --> 00:34:37,140
+ويقول الكلام هذا صحيح لكل n أكبر من أو ساوى خمسة
+
+307
+00:34:48,530 --> 00:34:55,770
+xn على yn أصغر من واحد على n ثالث اتنين طبعا أكبر
+
+308
+00:34:55,770 --> 00:35:01,690
+من سفر أو أكبر من أو يساوي سفر وهذا صحيح لكل n
+
+309
+00:35:01,690 --> 00:35:06,890
+أكبر من أو يساوي خمسة الان هذا لما انتقل ل
+
+310
+00:35:06,890 --> 00:35:11,950
+infinity هذا بيروح لسفر لما انتقل ل infinity هذا
+
+311
+00:35:11,950 --> 00:35:16,610
+بيروح لسفر اذا by sandwich theorem
+
+312
+00:35:23,770 --> 00:35:30,910
+بطل عند ال limit ل xn over yn as n tends to
+
+313
+00:35:30,910 --> 00:35:36,570
+infinity بساوي سفر هاد هى ال R في ال limit
+
+314
+00:35:36,570 --> 00:35:42,150
+comparison test طيب since
+
+315
+00:35:44,540 --> 00:35:49,640
+سيجما واي ان اللي هي سيجما واحد على ان تربيعي
+
+316
+00:35:49,640 --> 00:35:58,740
+converges حسب الجزء الثاني من limit comparison
+
+317
+00:35:58,740 --> 00:36:02,160
+test limit comparison test بيقول إذا كان limit ال
+
+318
+00:36:02,160 --> 00:36:07,600
+ratio بساوي سفر وكانت سيجما واي ان convergent إذا
+
+319
+00:36:07,600 --> 00:36:10,400
+هذا بيقدر
+
+320
+00:36:13,430 --> 00:36:19,310
+سيجما اكس ام اللي هي سيجما وان اوبر ام فاكتوريال
+
+321
+00:36:19,310 --> 00:36:23,490
+convergence رغم المفهوم
+
+322
+00:36:26,800 --> 00:36:29,680
+واحد استخدم ال direct comparison test، التاني
+
+323
+00:36:29,680 --> 00:36:33,660
+استخدم ال limit comparison test، اتنين كان فيهم
+
+324
+00:36:33,660 --> 00:36:39,940
+شوية شغل مش سهل، لكن هذا هو الموجود، مفيش أسهل من
+
+325
+00:36:39,940 --> 00:36:46,400
+هذا فعلى أي حال يعني ال .. الأسئلة في الكتاب هتكون
+
+326
+00:36:46,400 --> 00:36:50,780
+معظمها سهلة إما في الحل بال limit comparison test
+
+327
+00:36:50,780 --> 00:36:55,520
+أو بال direct comparison test، في أي سؤال أو
+
+328
+00:36:55,520 --> 00:37:00,800
+استفسار؟الامور واضحة الحل واضح انا عارف انه كيف
+
+329
+00:37:00,800 --> 00:37:05,580
+يخطر على بالنا نعمل المقارنات هذه وهذا كلامكم صحيح
+
+330
+00:37:05,580 --> 00:37:12,980
+هذا يعني شيء مش سهل لكن في بعض المسائل ال .. يعني
+
+331
+00:37:12,980 --> 00:37:21,740
+ال .. مش سهل ان احنا نعمل المقارنة لكن بنحاول ..
+
+332
+00:37:21,740 --> 00:37:33,030
+بنحاول اللي بيحاولبيصل إلى حل خليني يعني احنا مش
+
+333
+00:37:33,030 --> 00:37:37,710
+عايزين نبدأ section جديد الصحيح ان هيك يعني ال
+
+334
+00:37:37,710 --> 00:37:42,550
+chapter خلص فعشان مابداش يعني نبدأ المرة الجاية
+
+335
+00:37:42,550 --> 00:37:49,210
+chapter جديد فخليني اخد احل سؤال من ال homework
+
+336
+00:37:49,210 --> 00:37:53,590
+problems السؤال هنا question
+
+337
+00:37:57,590 --> 00:38:06,030
+exercise رقم خمسة section تلاتة سبعة لأن هذا تمرين
+
+338
+00:38:06,030 --> 00:38:10,190
+خمسة في section تلاتة سبعة اللي هو آخر section في
+
+339
+00:38:10,190 --> 00:38:17,010
+chapter تلاتة السؤال بيقول can
+
+340
+00:38:17,010 --> 00:38:24,030
+you السؤال كتير يعني مهم و interesting can you
+
+341
+00:38:24,030 --> 00:38:31,770
+giveيعني كتاب بخاطب الطالب بيقوله can you give an
+
+342
+00:38:31,770 --> 00:38:44,810
+example هل بإمكانك تعطي مثال of a convergent of a
+
+343
+00:38:44,810 --> 00:38:53,550
+convergent series sigma xn and a divergent
+
+344
+00:39:03,070 --> 00:39:11,470
+بحيث ان المجموعة تبع ال two series يكون
+
+345
+00:39:11,470 --> 00:39:20,010
+convergent is convergent explain
+
+346
+00:39:20,010 --> 00:39:29,830
+وضحي الإجابةهتكون يا yes يا no و في كل تلحالتين بن
+
+347
+00:39:29,830 --> 00:39:37,710
+.. نعطيك تفسر ال yes او انه تبعتك فانا بقول انه
+
+348
+00:39:37,710 --> 00:39:44,450
+خلينا نعطيلكم يعني تشوفكم تفكروا نعطيكم دقيقة
+
+349
+00:39:44,450 --> 00:39:53,230
+تفكروا و تحاولوا تجيبوا مثال زي ما هو مطلوب إذا
+
+350
+00:39:53,230 --> 00:39:53,990
+كده إذا أمكن
+
+351
+00:39:57,650 --> 00:40:04,570
+فمين عندها مثال؟ كمان مرة بنجيب مثال ل two series
+
+352
+00:40:04,570 --> 00:40:10,370
+واحدة convergent اللي هي هذه الأولى والتانية
+
+353
+00:40:10,370 --> 00:40:16,190
+divergent بحيث أن مجموعهم يكون convergent هل هذا
+
+354
+00:40:16,190 --> 00:40:23,190
+ممكن؟ إذا ممكن طيب ممكن تعطيني مثال على ذلك يعني
+
+355
+00:40:23,190 --> 00:40:27,890
+اعطيني مثاليوضح صحة ال .. الكلام هذه ال example
+
+356
+00:40:27,890 --> 00:40:30,930
+مثلا نخدها هي أسهل إيش الواحد على الأن أو الأول و
+
+357
+00:40:30,930 --> 00:40:36,130
+أنت الرابعين خليني لحظة شوية لو سمحت هاي أخبرتكم
+
+358
+00:40:36,130 --> 00:40:37,350
+طرح example
+
+359
+00:40:41,280 --> 00:40:47,500
+أيه وقتك؟ ال XN قبل عن الواحد على الان تربية واحد
+
+360
+00:40:47,500 --> 00:40:55,520
+على ان تربية فطبعا هذا بقدر سيجما XN كل ذات يسار
+
+361
+00:40:55,520 --> 00:41:02,740
+سيجما واحد على ان تربية كل بيرزلأن هذه P series و
+
+362
+00:41:02,740 --> 00:41:07,380
+ال P بيساوي اتنين اكبر من واحد، صح؟ والتانية
+
+363
+00:41:07,380 --> 00:41:13,120
+الواحدة الجدر الأن نخدها YM بيساوي واحد على الجدر
+
+364
+00:41:13,120 --> 00:41:19,860
+الأن بتصير أص نص، طيب، بتصير سماشة للواحدالان
+
+365
+00:41:19,860 --> 00:41:25,920
+sigma yn بيساوي sigma 1 على n اصلا اصلا بي سيريز
+
+366
+00:41:25,920 --> 00:41:30,100
+هادي divergent بي بي سيريز هادي بي بي بي بي بي بي
+
+367
+00:41:30,100 --> 00:41:30,400
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+368
+00:41:30,400 --> 00:41:34,760
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+369
+00:41:34,760 --> 00:41:34,800
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+370
+00:41:34,800 --> 00:41:35,040
+بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي بي
+
+371
+00:41:35,040 --> 00:41:44,900
+بي
+
+372
+00:41:44,900 --> 00:41:53,900
+بيهي sigma واحد على N تربية زائد واحد على N أص نص
+
+373
+00:41:53,900 --> 00:42:02,420
+صح؟ وهذا بيساوي summation ناخد مقام مشترك N تربية
+
+374
+00:42:02,420 --> 00:42:10,140
+فبطلع واحد زائد N أص .. أص تلاتة عشان .. أص تلاتة
+
+375
+00:42:10,140 --> 00:42:14,040
+عشان .. مظبوط؟
+
+376
+00:42:21,390 --> 00:42:30,430
+هل هذه convergent؟ لما n تكون كبيرة .. اه لما n
+
+377
+00:42:30,430 --> 00:42:37,790
+تكون كبيرة هذه بتكون behaves like sigma
+
+378
+00:42:39,040 --> 00:42:45,440
+واحد لأ مش واحد ع انتر بياني مهم للواحد وفضل
+
+379
+00:42:45,440 --> 00:42:50,420
+عندي N أس ثلاثة ع اتنين ع انتر بيان اللي بيساوي
+
+380
+00:42:50,420 --> 00:42:54,740
+سيجما واحد ع ن أس نص
+
+381
+00:42:58,550 --> 00:43:03,570
+و ممكن الأن نستخدم ال limit comparison test نثبت
+
+382
+00:43:03,570 --> 00:43:07,430
+أن هذه divergent لأن هذه divergent باستخدام ال
+
+383
+00:43:07,430 --> 00:43:11,630
+limit comparison testزيادة .. زيادة .. زيادة ..
+
+384
+00:43:11,630 --> 00:43:14,670
+زيادة .. زيادة .. زيادة .. زيادة .. زيادة .. زيادة
+
+385
+00:43:14,670 --> 00:43:19,910
+.. زيادة
+
+386
+00:43:19,910 --> 00:43:30,990
+.. زيادة .. زيادة .. زيادة .. زيادة
+
+387
+00:43:30,990 --> 00:43:31,310
+..
+
+388
+00:43:34,540 --> 00:43:43,240
+another example طيب xn بساوي سالب واحد و سالب ن
+
+389
+00:43:43,240 --> 00:43:51,160
+مثلا yn بساوي واحد yn بساوي واحداه ف ال series
+
+390
+00:43:51,160 --> 00:43:58,520
+sigma x n diverge و sigma y n diverge فتنتهي ال
+
+391
+00:43:58,520 --> 00:44:01,540
+diverge، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+392
+00:44:01,540 --> 00:44:01,880
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+393
+00:44:01,880 --> 00:44:02,160
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+394
+00:44:02,160 --> 00:44:03,340
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+395
+00:44:03,340 --> 00:44:08,340
+مانفعش، مانفعش، مانفعش، مانفعش، مانفعش، مانفعش،
+
+396
+00:44:08,340 --> 00:44:14,220
+مانفعش، مانفعش، مانفعش، مانفعش، مانعلى مدرسة الأرض
+
+397
+00:44:14,220 --> 00:44:18,380
+ان هو من من أنتوا ساوي أربع على مالة نهاية حكينا
+
+398
+00:44:18,380 --> 00:44:20,820
+انه من أنتوا ساوي أربع على مالة نهاية هذا converge
+
+399
+00:44:20,820 --> 00:44:25,300
+بس اللي جابل حكينا انه converge انه diverge اللي
+
+400
+00:44:25,300 --> 00:44:28,280
+جابلنا طيب احنا عشان بسم ان احنا بنحكي على السؤال
+
+401
+00:44:28,280 --> 00:44:33,860
+هذا خليني احنا في هذا المثال في عندك مثال؟ خلاص
+
+402
+00:44:33,860 --> 00:44:38,440
+طبعا ال .. ال .. السابق هذا بعدين بنتناقش فيه
+
+403
+00:44:38,440 --> 00:44:43,270
+خليني أجرب عشان أنا مافيش وجهة على السؤال هذالو
+
+404
+00:44:43,270 --> 00:44:46,950
+كلكم حاولتوا .. كل واحدة حاولت تجيب مثال، كل أمثلة
+
+405
+00:44:46,950 --> 00:44:51,370
+أبقاتكم هتكون غلطة أو هتفشل، ليه؟ لأن مافيش ولا
+
+406
+00:44:51,370 --> 00:44:56,890
+مثال، لأن مافيش مثال، فانت قاعدين بتجيبوا .. تعطوا
+
+407
+00:44:56,890 --> 00:45:01,790
+حاجة مستحيلة، مش موجودة، إذا الإجابة على هذا
+
+408
+00:45:01,790 --> 00:45:02,330
+السؤال
+
+409
+00:45:08,820 --> 00:45:19,880
+إن ال answer ال answer is no لا يمكن يعطى مثال على
+
+410
+00:45:19,880 --> 00:45:22,700
+two series واحدة convergent والتانية divergent
+
+411
+00:45:22,700 --> 00:45:26,020
+مجموعة بتطلع convergent مستحيل this is impossible
+
+412
+00:45:26,020 --> 00:45:34,660
+لبرهان أو لثبات ذلك if
+
+413
+00:45:34,660 --> 00:45:47,190
+if thisإذا كان هذا صحيح أو إذا كان هذا صحيح يعني
+
+414
+00:45:47,190 --> 00:45:52,230
+لو اقدرت النجيب series convergent و series
+
+415
+00:45:52,230 --> 00:45:57,710
+divergent و مجموعة convergent then
+
+416
+00:45:57,710 --> 00:46:01,610
+we would have
+
+417
+00:46:03,890 --> 00:46:08,590
+إنه الـ series sigma yn اللي احنا فرضين انها
+
+418
+00:46:08,590 --> 00:46:15,650
+divergent اللي هي بساوي sigma xn زائد yn minus
+
+419
+00:46:15,650 --> 00:46:23,290
+sigma xn احنا قلنا لو هذا كان true معناته ال
+
+420
+00:46:23,290 --> 00:46:27,690
+series هذه convergent معناته هذه convergent ومن
+
+421
+00:46:27,690 --> 00:46:32,610
+الفرض هذه convergentوالفرق بين two convergent
+
+422
+00:46:32,610 --> 00:46:38,330
+series is convergent، إذن هذا هتطلع .. إذن الفرق
+
+423
+00:46:38,330 --> 00:46:43,830
+هيكون convergent وبالتالي إذن ال series sigma yn
+
+424
+00:46:43,830 --> 00:46:48,930
+is convergent، وهذا contradiction لإن احنا فرضين
+
+425
+00:46:48,930 --> 00:46:56,040
+أنها divergentهذا مش ممكن يكون true عشان هي كانت
+
+426
+00:46:56,040 --> 00:47:01,600
+كتبت if it were true مستحيل ..مستحيل مش if it was
+
+427
+00:47:01,600 --> 00:47:08,280
+true okay طبعا اذا انا ساطيع اعطاء مثال يعطيني
+
+428
+00:47:08,280 --> 00:47:13,800
+المواصفات هذه بالمرةتمام؟ إذا بنوقف هنا و هيك
+
+429
+00:47:13,800 --> 00:47:17,120
+بنكون خلصنا ال chapter تلاتة المرة الجاية ان شاء
+
+430
+00:47:17,120 --> 00:47:22,620
+الله هنبقى في chapter أربعة فشكرا لكم و نشوفكم ان
+
+431
+00:47:22,620 --> 00:47:23,640
+شاء الله يوم السبت
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/WVOztu-xKaw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/WVOztu-xKaw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3946e391728efc2857ff704e161d3c25e8817842
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/WVOztu-xKaw.srt
@@ -0,0 +1,1051 @@
+1
+00:00:20,670 --> 00:00:27,110
+Okay اليوم هنراجع بس نظرية ال-divergence theorem
+
+2
+00:00:27,110 --> 00:00:33,870
+اللي أخدناها آخر مرة بس نسترجعها بسرعة ال-
+
+3
+00:00:33,870 --> 00:00:40,370
+divergence theorem
+
+4
+00:00:40,370 --> 00:00:47,370
+عطلناها رقم 2.17 في النظرية
+
+5
+00:00:47,370 --> 00:00:48,150
+هذه بتقول
+
+6
+00:00:51,110 --> 00:01:04,590
+Let X and contained in R be a sequence then
+
+7
+00:01:04,590 --> 00:01:09,410
+the following statements are equivalent
+
+8
+00:01:11,820 --> 00:01:18,400
+فأول statement sequence x in does not converge to x
+
+9
+00:01:18,400 --> 00:01:28,200
+ينتمي لـ R for any x ينتمي لـ R وفي شرط الثاني وفي
+
+10
+00:01:28,200 --> 00:01:35,420
+شرط الثالث أنا بهمش شرط الثالث أو العبارة الثالثة
+
+11
+00:01:35,420 --> 00:01:39,840
+there exist ε>0 and a
+
+12
+00:01:39,840 --> 00:01:48,390
+subsequence أو subsequence X<sub>n</sub>
+
+13
+00:01:48,390 --> 00:01:53,410
+of
+
+14
+00:01:53,410 --> 00:02:05,750
+sequence X<sub>n</sub> such that |X<sub>n</sub> - X| >
+
+15
+00:02:05,750 --> 00:02:08,550
+أو يساوي ε<sub>0</sub> لكل M
+
+16
+00:02:13,970 --> 00:02:18,330
+و بدنا ناخد مثال على النظرية هذه أخدناها المرة
+
+17
+00:02:18,330 --> 00:02:22,290
+الماضية مثال
+
+18
+00:02:22,290 --> 00:02:29,470
+مثال
+
+19
+00:02:29,470 --> 00:02:39,490
+الثاني example consider
+
+20
+00:02:41,810 --> 00:02:49,510
+ناخد ال-sequence consider x<sub>n</sub> where sequence x<sub>n</sub> لحد
+
+21
+00:02:49,510 --> 00:03:04,610
+الآخر تبعها x<sub>n</sub> معرف على أنه بساوي n if n is odd و
+
+22
+00:03:04,610 --> 00:03:06,250
+بتساوي 1 على n
+
+23
+00:03:20,350 --> 00:03:28,690
+مطلوب أن أثبت أن ال-sequence x<sub>n</sub> is divergent
+
+24
+00:03:41,550 --> 00:03:47,630
+solution ولاحظوا أن sequence X<sub>n</sub> إنّها بيباري حدودها
+
+25
+00:03:47,630 --> 00:03:53,350
+واحد، نص، و
+
+26
+00:03:53,350 --> 00:03:59,850
+واحد، نص، ثلاثة، رابع، و
+
+27
+00:03:59,850 --> 00:04:00,390
+هكذا
+
+28
+00:04:08,080 --> 00:04:12,340
+فالـ sequence هذه بالنسبة لها does not converge
+
+29
+00:04:12,340 --> 00:04:23,040
+لأي x ينتمي لـ R to show أن x<sub>n</sub> does not converge لـ
+
+30
+00:04:23,040 --> 00:04:29,720
+x for any x
+
+31
+00:04:29,720 --> 00:04:31,440
+ينتمي إلى R
+
+32
+00:04:35,590 --> 00:04:44,030
+fix x ينتمي لـ r خلّينا ناخد arbitrary الـ
+
+33
+00:04:44,030 --> 00:04:47,470
+real number x ينتمي لـ r ونثبت أن x<sub>n</sub> does not
+
+34
+00:04:47,470 --> 00:04:54,730
+converge enough فعندي then
+
+35
+00:04:54,730 --> 00:04:59,670
+|x| طبعًا أكبر عدد حقيقي أكبر من الصفر
+
+36
+00:04:59,670 --> 00:05:11,460
+so by Archimedean property يوجد
+
+37
+00:05:11,460 --> 00:05:19,660
+n عدد طبيعي يعتمد على x عدد طبيعي بحيث
+
+38
+00:05:19,660 --> 00:05:38,420
+أن n x هذا is even and |x| < n x هذا
+
+39
+00:05:38,420 --> 00:05:42,990
+ممكن نحصل عليه من ال-Archimedean property الـ
+
+40
+00:05:42,990 --> 00:05:46,990
+argument property بتقول أي عدد حقيقي زي |x|
+
+41
+00:05:46,990 --> 00:05:53,450
+نقدر نلاقي عدد طبيعي يعتمد على الـ x بحيث أنه العدد
+
+42
+00:05:53,450 --> 00:06:02,110
+الطبيعي أكبر من العدد الحقيقي طيب الآن لو أخدت
+
+43
+00:06:02,110 --> 00:06:05,150
+ε
+
+44
+00:06:05,150 --> 00:06:11,530
+=0 معرفة على أنها n x - |x|
+
+45
+00:06:14,250 --> 00:06:24,170
+فهذا بيطلع عدد موجب هذا بيطلع عدد موجب و
+
+46
+00:06:24,170 --> 00:06:30,190
+الـ sequence الـ
+
+47
+00:06:30,190 --> 00:06:39,390
+sequence الحد العام تبعها n x + 2 m -
+
+48
+00:06:39,390 --> 00:06:39,930
+1
+
+49
+00:06:43,110 --> 00:06:51,010
+من m = 1 to infinity هذه بالمناسبة الحدود
+
+50
+00:06:51,010 --> 00:07:00,770
+تبعها هذا عدد فردي عدد طبيعي فردي وهذا
+
+51
+00:07:00,770 --> 00:07:09,190
+n x عدد طبيعي زوجي هذا odd وهذا
+
+52
+00:07:09,190 --> 00:07:09,890
+even
+
+53
+00:07:13,300 --> 00:07:22,960
+ف odd + even بيطلع odd إذاً الحدود
+
+54
+00:07:22,960 --> 00:07:33,340
+العامة للـ sequence هذه بتكون odd وبالتالي
+
+55
+00:07:33,340 --> 00:07:39,600
+الحدود هذه كلها حسب التعريف بتطلع بيساوي
+
+56
+00:07:41,460 --> 00:07:48,180
+هي طبعًا sub-sequence يعني
+
+57
+00:07:48,180 --> 00:07:54,300
+هذا هتكون لو m = 1 هيطلع n x + 1 n x
+
+58
+00:07:54,300 --> 00:08:03,820
++ 3 وهكذا صح؟ حسب التعريف هذا عبارة عن sub-
+
+59
+00:08:03,820 --> 00:08:11,700
+sequence subsequence من ال-sequence x<sub>m</sub> لأن هذه
+
+60
+00:08:11,700 --> 00:08:18,300
+جزء من الحدود الفردية، صح؟ وبالتالي هذه
+
+61
+00:08:18,300 --> 00:08:23,640
+subsequence من x<sub>n</sub> ومش
+
+62
+00:08:23,640 --> 00:08:35,320
+هيكوا بس and ال-absolute value لـ n x + 2 m - 1
+
+63
+00:08:35,320 --> 00:08:41,640
+الحد العام لل-subsequence هذه المسافة بينه وبين الـ
+
+64
+00:08:41,640 --> 00:08:46,220
+x باستخدام الـ triangle inequality في واحدة من الـ
+
+65
+00:08:46,220 --> 00:08:50,940
+triangle inequalities بتقول |a - b|
+
+66
+00:08:50,940 --> 00:08:56,200
+أكبر من أو يساوي |a| - |b| فهذا
+
+67
+00:08:56,200 --> 00:09:02,540
+أكبر من أو يساوي |n x + 2 m - 1|
+
+68
+00:09:02,540 --> 00:09:09,720
+- |x| هذا عدد موجب هذا عدد طبيعي وهذا
+
+69
+00:09:09,720 --> 00:09:24,860
+عدد طبيعي فردي فممكن نشيل absolute value وهذا
+
+70
+00:09:24,860 --> 00:09:35,880
+العدد هذا العدد أكبر من n x - |x| هذا
+
+71
+00:09:35,880 --> 00:09:45,580
+العدد هنا أكبر من n x لأن هذا عدد موجب صح ف n x +
+
+72
+00:09:45,580 --> 00:09:50,440
+عدد موجب أكبر من n x - |x| طب ما هذا
+
+73
+00:09:50,440 --> 00:09:56,000
+بيساوي عرفناه على n ε<sub>0</sub> صح؟ الآن الكلام هذا
+
+74
+00:09:56,000 --> 00:09:59,460
+صحيح لكل m ينتمي إلى N
+
+75
+00:10:09,010 --> 00:10:15,890
+أنا أثبتت أن يوجد ε<sub>0</sub> > 0 و sub sequence
+
+76
+00:10:15,890 --> 00:10:21,850
+من ال-sequence x<sub>n</sub> و المسافة بين الحد العام للـ
+
+77
+00:10:21,850 --> 00:10:28,430
+sequence هذه والـ x أكبر من ε<sub>0</sub> لكل m في n
+
+78
+00:10:34,110 --> 00:10:44,630
+الـ divergence theorem الجزء الثالث هيتحقق شروطه
+
+79
+00:10:44,630 --> 00:10:52,330
+وبالتالي ال-sequence x<sub>n</sub> does not converge to x
+
+80
+00:10:56,670 --> 00:10:59,930
+بما أن x was arbitrary إذاً ال-sequence x<sub>n</sub> does
+
+81
+00:10:59,930 --> 00:11:04,030
+not converge لأي عدد حقيقي x وبالتالي divergent
+
+82
+00:11:04,030 --> 00:11:10,130
+تمام؟ أنا هنا استخدمت ال-divergence theorem في
+
+83
+00:11:10,130 --> 00:11:16,150
+إثبات أن ال-sequence هذه divergent لاحظوا أن الـ
+
+84
+00:11:16,150 --> 00:11:23,710
+sequence 1 على n هذه subsequence من x<sub>n</sub> وهذه
+
+85
+00:11:23,710 --> 00:11:24,890
+convergent لـ 0
+
+86
+00:11:28,580 --> 00:11:32,700
+لكن الـ sequence نفسها does not converge هذه برضه
+
+87
+00:11:32,700 --> 00:11:37,220
+subsequence n subsequence .. لما تكون الـ n فردية
+
+88
+00:11:37,220 --> 00:11:42,620
+subsequence من X<sub>n</sub> وهذه is not convergent هذه الـ
+
+89
+00:11:42,620 --> 00:11:49,340
+limit بتاعها infinity طبعًا؟ و
+
+90
+00:11:49,340 --> 00:11:50,500
+هذا؟ في عايز سؤال؟
+
+91
+00:12:05,310 --> 00:12:12,050
+monotone subsequence theorem النظرية
+
+92
+00:12:12,050 --> 00:12:23,330
+2.18 أقامها هنا النظرية
+
+93
+00:12:23,330 --> 00:12:32,230
+هذه بتقول every sequence in R has
+
+94
+00:12:36,910 --> 00:12:44,930
+a monotone subsequence كل sequence of real numbers
+
+95
+00:12:44,930 --> 00:12:53,770
+ممكن نستخلص منها monotone subsequence البرهان تبع
+
+96
+00:12:53,770 --> 00:12:58,050
+النظرية هذه مش صعب موجود في الكتاب هخليكم تقرأوه
+
+97
+00:13:00,180 --> 00:13:09,360
+هخليكم تقرأوا البرهان من الكتاب المقرر C theorem
+
+98
+00:13:09,360 --> 00:13:20,360
+رقم 3.4.7 page 8
+
+99
+00:13:20,360 --> 00:13:22,020
+و70 in textbook
+
+100
+00:13:27,060 --> 00:13:32,200
+إذن حدؤوكم لقراءة البرهان، البرهان سهل مش صعب و
+
+101
+00:13:32,200 --> 00:13:36,380
+حاولوا
+
+102
+00:13:36,380 --> 00:13:40,280
+تقرأوه و تفهموه وده في أي صعوبة تسألوني أو
+
+103
+00:13:40,280 --> 00:13:46,280
+تتناقشوا معي في البرهان okay
+
+104
+00:13:46,280 --> 00:13:49,680
+في
+
+105
+00:13:49,680 --> 00:13:51,320
+نظرية ثانية
+
+106
+00:14:00,880 --> 00:14:06,900
+Bolzano-Weierstrass theorem
+
+107
+00:14:06,900 --> 00:14:17,880
+رقم 2.19 النظرية
+
+108
+00:14:17,880 --> 00:14:23,980
+هذه بتقول every bounded sequence every
+
+109
+00:14:23,980 --> 00:14:26,540
+bounded
+
+110
+00:14:28,920 --> 00:14:34,800
+sequence of real
+
+111
+00:14:34,800 --> 00:14:41,980
+numbers in R has a
+
+112
+00:14:41,980 --> 00:14:48,500
+convergent subsequence
+
+113
+00:14:53,720 --> 00:14:58,080
+لو في عندي bounded sequence of real numbers فبقدر
+
+114
+00:14:58,080 --> 00:15:02,820
+ألاقي جواتها convergent subsequence البرهان تبع
+
+115
+00:15:02,820 --> 00:15:07,720
+النظرية دي سهل باستخدام النظريات السابقة في لها
+
+116
+00:15:07,720 --> 00:15:14,300
+برهانين واحد short يعني قصير يعتمد على النظريات
+
+117
+00:15:14,300 --> 00:15:17,840
+الكبيرة اللي برهناها سابقًا وفي لها برهان
+
+118
+00:15:21,560 --> 00:15:26,680
+طويل نوعًا ما وهذا البرهان constructive proof يعني
+
+119
+00:15:26,680 --> 00:15:30,860
+بوريكم كيف تبنوا ال-subsequence اللي هتكون
+
+120
+00:15:30,860 --> 00:15:36,920
+convergent هناخد ال-short proof ونخليكم تقرأوا الـ
+
+121
+00:15:36,920 --> 00:15:40,980
+long proof رقم
+
+122
+00:15:40,980 --> 00:15:46,700
+1 let
+
+123
+00:15:46,700 --> 00:15:47,560
+x<sub>n</sub>
+
+124
+00:15:50,470 --> 00:15:58,910
+be a bounded sequence
+
+125
+00:15:58,910 --> 00:16:05,110
+of real numbers by
+
+126
+00:16:05,110 --> 00:16:17,490
+حسب النظرية الأخيرة رقم 2.18 by theorem 2.18
+
+127
+00:16:17,490 --> 00:16:19,410
+الـ monotone
+
+128
+00:16:22,580 --> 00:16:30,100
+بتقول أي sequence in R سواء bounded أو unbounded
+
+129
+00:16:30,100 --> 00:16:35,540
+every sequence of real numbers has a monotone
+
+130
+00:16:35,540 --> 00:16:44,360
+subsequence إذاً الـ X<sub>n</sub> has a monotone
+
+131
+00:16:44,360 --> 00:16:46,980
+subsequence
+
+132
+00:16:51,810 --> 00:17:02,790
+خليني أسميها x<sub>n</sub> k إذاً
+
+133
+00:17:02,790 --> 00:17:12,110
+هذه عبارة عن monotone subsequence طيب since x<sub>n</sub>
+
+134
+00:17:12,110 --> 00:17:18,150
+is bounded الـ
+
+135
+00:17:18,150 --> 00:17:21,090
+subsequence .. the subsequence
+
+136
+00:17:23,040 --> 00:17:32,960
+x<sub>n</sub> k is also bounded أي
+
+137
+00:17:32,960 --> 00:17:36,540
+sub sequence من bounded sequence is bounded مظبوط
+
+138
+00:17:36,540 --> 00:17:41,520
+صح لأن x
+
+139
+00:17:41,520 --> 00:17:42,880
+<sub>n</sub> bounded
+
+140
+00:17:46,080 --> 00:17:52,660
+بقدر أن يوجد M عدد موجب بحيث أن |x<sub>n</sub>| < 
+
+141
+00:17:52,660 --> 00:18:01,440
+من أو يساوي M لكل N طب ما هذا بقدّي أن absolute xnk
+
+142
+00:18:01,440 --> 00:18:10,380
+أيضا أصغر من أو يساوي M لكل N لأن ال subsequence
+
+143
+00:18:10,380 --> 00:18:13,500
+xnk هي subset من xn
+
+144
+00:18:16,250 --> 00:18:20,130
+مظبوط؟ كل حد في ال subsequence هو حد في ال
+
+145
+00:18:20,130 --> 00:18:26,370
+sequence وبالتالي تحقق نفس الشرط لأن هذا بيقدّي أن
+
+146
+00:18:26,370 --> 00:18:34,190
+كل .. هذا صحيح لكل K لأن هذه .. هذا الشرط منه
+
+147
+00:18:34,190 --> 00:18:37,830
+بيطلع X ان K is bounded
+
+148
+00:18:40,520 --> 00:18:44,940
+Okay إذا أنا كتبت برهان أنّ أي subsequence من
+
+149
+00:18:44,940 --> 00:18:54,900
+bounded sequence is bounded تمام Now ال
+
+150
+00:18:54,900 --> 00:19:01,980
+subsequence xnk is monotone
+
+151
+00:19:01,980 --> 00:19:05,420
+and
+
+152
+00:19:05,420 --> 00:19:05,980
+bounded
+
+153
+00:19:10,700 --> 00:19:15,920
+So by monotone convergence theorem حسب الـ
+
+154
+00:19:15,920 --> 00:19:20,960
+monotone convergence theorem it is convergent
+
+155
+00:19:26,310 --> 00:19:34,310
+ما هو المطلوب؟ لأن هنا أثبتنا أنّ أي sequence أي
+
+156
+00:19:34,310 --> 00:19:37,510
+sequence which is bounded أي sequence of real
+
+157
+00:19:37,510 --> 00:19:41,690
+numbers which is bounded has a convergent
+
+158
+00:19:41,690 --> 00:19:50,490
+subsequence تمام؟ لأن هذا برهان النظرية تمام؟ واضح؟
+
+159
+00:19:50,490 --> 00:20:03,750
+في طبعاً برهان ثاني، هذا البرهان موجود في الكتاب أنّ
+
+160
+00:20:03,750 --> 00:20:09,610
+البرهان رقم اثنين proof رقم اثنين موجود في الكتاب
+
+161
+00:20:09,610 --> 00:20:16,130
+see page 79
+
+162
+00:20:16,130 --> 00:20:17,630
+in textbook
+
+163
+00:20:23,250 --> 00:20:27,150
+إذا أنا هنسيبكم تقرأوا البرهان الثاني من الكتاب
+
+164
+00:20:27,150 --> 00:20:30,190
+طبعاً البرهان هذا هيكون constructive proof بيورجيكم
+
+165
+00:20:30,190 --> 00:20:36,090
+كيف بيبني ال subsequence خطوة خطوة وبحيث أنّها
+
+166
+00:20:36,090 --> 00:20:43,030
+تطلع convergent طيب في كمان نظرية في هذا السياق
+
+167
+00:20:50,570 --> 00:21:03,690
+theorem اثنين وعشرين let
+
+168
+00:21:03,690 --> 00:21:07,910
+x n content
+
+169
+00:21:07,910 --> 00:21:23,610
+in R be bounded bounded sequence and let X ينتمي
+
+170
+00:21:23,610 --> 00:21:29,370
+إلى R بـ
+
+171
+00:21:29,370 --> 00:21:36,870
+such that every … every convergent … every
+
+172
+00:21:36,870 --> 00:21:41,610
+convergent subsequence
+
+173
+00:21:41,610 --> 00:21:46,310
+… every convergent subsequence
+
+174
+00:21:50,710 --> 00:21:59,550
+of سمّيها xn أو
+
+175
+00:21:59,550 --> 00:22:08,450
+every convergent subsequence of xn converges to
+
+176
+00:22:08,450 --> 00:22:12,430
+x then
+
+177
+00:22:12,430 --> 00:22:16,930
+النتيجة أنّ ال sequence xn
+
+178
+00:22:19,750 --> 00:22:27,190
+converges to x إذا
+
+179
+00:22:27,190 --> 00:22:31,730
+أنا في عندي bounded sequence of real numbers وفي
+
+180
+00:22:31,730 --> 00:22:36,910
+عندي عدد حقيقي هذا العدد بيحقق الخاصية أنّ كل
+
+181
+00:22:36,910 --> 00:22:41,070
+convergent subsequence من ال sequence xn بتكون
+
+182
+00:22:41,070 --> 00:22:45,950
+convergent ل x فالحالة هذه ال sequence الأصلية
+
+183
+00:22:45,950 --> 00:22:51,730
+بتكون convergent وال limit تبعتها هي نفس العدد x
+
+184
+00:22:51,730 --> 00:22:55,430
+برهان نظرية هذه سهل مش صعب
+
+185
+00:22:59,850 --> 00:23:04,850
+هنستخدم البولزانو Weierstrass theorem اللي هي
+
+186
+00:23:04,850 --> 00:23:22,730
+نظرية تسعة عشر في البرهان هنشوف مع بعض prove assume
+
+187
+00:23:29,790 --> 00:23:37,770
+assume on contrary that
+
+188
+00:23:37,770 --> 00:23:42,890
+احنا عايزين نثبت أنّ ال sequence xn converge ل x
+
+189
+00:23:42,890 --> 00:23:51,550
+فنفرض أنّ xn does not converge to any x then
+
+190
+00:23:51,550 --> 00:23:55,330
+by divergence
+
+191
+00:23:57,210 --> 00:24:03,170
+by divergence theorem اللي
+
+192
+00:24:03,170 --> 00:24:10,530
+هي اثنين وثمانين كذبوت
+
+193
+00:24:10,530 --> 00:24:16,390
+أو اثنين وسبعين أعتقد هيك صح اه ال divergence
+
+194
+00:24:16,390 --> 00:24:22,140
+theorem بتقول لي xn does not converge ل xمعناته
+
+195
+00:24:22,140 --> 00:24:26,460
+there exist ε0 عدد موجب and there exist
+
+196
+00:24:26,460 --> 00:24:32,980
+a subsequence a subsequence
+
+197
+00:24:32,980 --> 00:24:36,180
+xrn
+
+198
+00:24:36,180 --> 00:24:40,200
+of xn
+
+199
+00:24:42,720 --> 00:24:51,320
+such that absolute xrn minus x أكبر من أو يساوي
+
+200
+00:24:51,320 --> 00:25:01,280
+ε0 هذا الكلام صحيح لكل n في N نسمي
+
+201
+00:25:01,280 --> 00:25:07,560
+المتباينة هذه star تمام؟ هذا من نظرية ال
+
+202
+00:25:07,560 --> 00:25:11,740
+divergence اثنين وسبعين ممكن نحصل على كل هذا
+
+203
+00:25:17,200 --> 00:25:22,140
+طيب since xn
+
+204
+00:25:22,140 --> 00:25:36,260
+is bounded لما أنّ ال sequence xn is bounded then
+
+205
+00:25:36,260 --> 00:25:45,160
+the subsequence the subsequence xrn is bounded
+
+206
+00:25:49,260 --> 00:25:53,140
+أي subsequence من bounded sequence تطلع bounded جب
+
+207
+00:25:53,140 --> 00:25:59,600
+شوية أثبتنا الكلام هذا so
+
+208
+00:25:59,600 --> 00:26:04,780
+by Bolzano
+
+209
+00:26:04,780 --> 00:26:11,680
+Weierstrass
+
+210
+00:26:11,680 --> 00:26:19,730
+theorem نظرية Bolzano Weierstrass هنطبقها على الـ
+
+211
+00:26:19,730 --> 00:26:28,190
+sequence xrn اللي هي bounded فبتقول
+
+212
+00:26:28,190 --> 00:26:33,010
+Bolzano Weierstrass theorem إذا في عندي sequence و
+
+213
+00:26:33,010 --> 00:26:39,970
+bounded ففي إلها convergent subsequence إذن ال
+
+214
+00:26:39,970 --> 00:26:45,070
+sequence ال
+
+215
+00:26:45,070 --> 00:26:47,410
+sequence xrn
+
+216
+00:26:48,810 --> 00:27:01,550
+has a convergent has a convergent subsequence
+
+217
+00:27:01,550 --> 00:27:04,570
+say
+
+218
+00:27:04,570 --> 00:27:13,270
+خلينا نسميها xkxkn
+
+219
+00:27:13,270 --> 00:27:16,170
+تمام؟
+
+220
+00:27:45,490 --> 00:27:50,890
+فإنّ ال …
+
+221
+00:27:50,890 --> 00:27:56,170
+إذا أنا في عندي convergent subsequence من ال
+
+222
+00:27:56,170 --> 00:28:03,390
+subsequence هذه وطبعاً هذه subsequence من xn وهذه
+
+223
+00:28:03,390 --> 00:28:07,070
+subsequence من هذه، إذا هذه subsequence من xn
+
+224
+00:28:11,210 --> 00:28:15,570
+بما أنّ xkn
+
+225
+00:28:15,570 --> 00:28:22,950
+هو أيضاً subsequence لسيكوينس
+
+226
+00:28:22,950 --> 00:28:28,430
+الأصلية xn ثم
+
+227
+00:28:28,430 --> 00:28:32,330
+بي وهو مرتبط
+
+228
+00:28:41,880 --> 00:28:48,020
+وهو مرتبط ثم
+
+229
+00:28:48,020 --> 00:28:54,650
+من حيث ال hypothesis من الفرض احنا فرضين في النظرية
+
+230
+00:28:54,650 --> 00:28:59,310
+هذه أنّ every convergent subsequence of xn لازم
+
+231
+00:28:59,310 --> 00:29:04,890
+تكون convergent للعدد x فهي في عندي subsequence من
+
+232
+00:29:04,890 --> 00:29:09,530
+ال sequence xn وconvergent إذا لازم تكون ال limit
+
+233
+00:29:09,530 --> 00:29:13,430
+تبعتها x إذا من الفرض limit
+
+234
+00:29:15,760 --> 00:29:28,480
+ل xkn as n tends to infinity بساوي x تمام طب
+
+235
+00:29:28,480 --> 00:29:37,720
+احنا عايزين نثبت أنّ ال xn نفسك converge لل x طيب
+
+236
+00:29:37,720 --> 00:29:44,400
+الآن من تعريف أنا في عندي ε0 موجود هاي في
+
+237
+00:29:44,400 --> 00:29:48,900
+عندي ε0 أنا في عندي ε0 هاد عدد
+
+238
+00:29:48,900 --> 00:29:56,460
+موجود من ال divergence في الفيلم hence وعندي ال
+
+239
+00:29:56,460 --> 00:29:59,520
+subsequence هاد ال converge ل x لذا لما أنت عارف
+
+240
+00:29:59,520 --> 00:30:06,660
+ال convergence there exist capital N يعتمد على
+
+241
+00:30:06,660 --> 00:30:08,000
+ε0
+
+242
+00:30:11,600 --> 00:30:19,000
+بحيث أنّ ال absolute value للحد العام لل sequence
+
+243
+00:30:19,000 --> 00:30:27,320
+xkn المسافة بينه بين ال x أصغر من ε0
+
+244
+00:30:27,320 --> 00:30:35,740
+وهذا الكلام صحيح لكل N في N تمام؟
+
+245
+00:30:37,500 --> 00:30:43,940
+بنسمي المتباينة هذي double star الآن
+
+246
+00:30:43,940 --> 00:30:57,360
+now star and double star بيقدّوا أنّ ال … هاي عندي
+
+247
+00:30:57,360 --> 00:31:06,910
+absolute xkn minus x هذا أصغر من ε0 هذا
+
+248
+00:31:06,910 --> 00:31:18,330
+من double star من هنا طب ومن ال star أنا
+
+249
+00:31:18,330 --> 00:31:23,250
+عندي xrn المسافة بين xrn و x أكبر من أو يساوي
+
+250
+00:31:23,250 --> 00:31:24,630
+ε0
+
+251
+00:31:27,280 --> 00:31:34,640
+وهذه subsequence من ال subsequence هذه يعني كل
+
+252
+00:31:34,640 --> 00:31:39,520
+حد في ال sequence هذه هو حد في هذه وبالتالي ال
+
+253
+00:31:39,520 --> 00:31:47,860
+subsequence هذه بتحقق المتباينة star إذن هذا صحيح
+
+254
+00:31:47,860 --> 00:31:52,920
+من star ال absolute الفرقة ده بيطلع أكبر من أو يساوي
+
+255
+00:31:52,920 --> 00:31:59,240
+ε0 الكلام هذا صحيح لكل n في N إذا أنا
+
+256
+00:31:59,240 --> 00:32:05,280
+عندي طلع هيك ε0 أصغر من ε0 و
+
+257
+00:32:05,280 --> 00:32:09,960
+هذا بديني تناقض contradiction إذا التناقض هذا
+
+258
+00:32:09,960 --> 00:32:19,040
+السبب تبعه أنّ إحنا فرضنا أنّ xn does not converge ل
+
+259
+00:32:19,040 --> 00:32:23,660
+x إذا ال contradiction هذه بتقول أنّ xn لازم
+
+260
+00:32:23,660 --> 00:32:30,900
+converge ل x وهذا اللي بدنا إيّاه وهو المطلوب هو
+
+261
+00:32:30,900 --> 00:32:35,700
+المطلوب okay إذا هيك بنكون برهنا النظرية هذه
+
+262
+00:32:35,700 --> 00:32:43,200
+خلينا ناخد إحنا break عشان فينا لقاء ثاني لمدة خمس
+
+263
+00:32:43,200 --> 00:32:46,120
+دقائق وبعدين يعني نرجع، ماشي الحال؟ 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0a1097ab4d6beab4aa79e4e70641be4882ca0e2b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XhLWrN2SkOQ_raw.srt
@@ -0,0 +1,1388 @@
+1
+00:00:20,890 --> 00:00:26,630
+أحنا هنكمل الموضوع لـ Properly Divergent Sequences
+
+2
+00:00:26,630 --> 00:00:31,690
+اللي بدأناه المحاضرة السابقة فشوفنا في المحاضرة
+
+3
+00:00:31,690 --> 00:00:36,530
+السابقة تعريف ما معنى أنه limit ل sequence xn
+
+4
+00:00:36,530 --> 00:00:41,450
+بساوي infinity وما معنى أنه limit ل sequence xn
+
+5
+00:00:41,450 --> 00:00:46,490
+بساوي negative infinityطبعاً الـ sequence بتكون
+
+6
+00:00:46,490 --> 00:00:49,470
+properly divergent إذا كانت ال limit تبعتها
+
+7
+00:00:49,470 --> 00:00:54,030
+بالساوي infinity أو سالب infinity في عندي
+
+8
+00:00:54,030 --> 00:00:58,550
+comparison test ل .. ل properly divergent
+
+9
+00:00:58,550 --> 00:01:01,790
+sequences هذا ال test بيقول لي لو في عندي two
+
+10
+00:01:01,790 --> 00:01:06,330
+sequences xn و yn two sequences of real numbers
+
+11
+00:01:06,330 --> 00:01:10,370
+بيحققوا الشرط star satisfy the condition star وهو
+
+12
+00:01:10,370 --> 00:01:15,400
+أن كل حدفي xn أصغر من أو ساوي الحد اللي بنظره في
+
+13
+00:01:15,400 --> 00:01:22,080
+ال sequence التانية yn هذا صحيح لكل n فإذا كانت ال
+
+14
+00:01:22,080 --> 00:01:26,140
+limit of the bigger sequence of the smaller
+
+15
+00:01:26,140 --> 00:01:30,240
+sequence is infinity then the limit of the bigger
+
+16
+00:01:30,240 --> 00:01:36,040
+sequence is infinity and if the limit of the big
+
+17
+00:01:36,040 --> 00:01:39,580
+the bigger sequence is negative infinity then the
+
+18
+00:01:39,580 --> 00:01:40,720
+limit of the smaller
+
+19
+00:01:50,070 --> 00:01:55,680
+الجزء بي هو نقاط نقاط نقاط نقاط نقاط نقاطan
+
+20
+00:01:55,680 --> 00:01:58,620
+application of the definition طبقنا التعريف
+
+21
+00:01:58,620 --> 00:02:03,740
+بالبرهان زي ما انتوا شايفينه برهان الجزء A similar
+
+22
+00:02:03,740 --> 00:02:09,840
+مشابه لجزء B فحنسيبوا تمرين لكم اتحاولوا يعني
+
+23
+00:02:09,840 --> 00:02:15,020
+اتبرهنوا بنفس الطريقة okay تمام فلو سمحتوا حاولوا
+
+24
+00:02:15,020 --> 00:02:21,720
+انكم اتبرهنوا الجزء A بنفس الطريقة في عندنا شوية
+
+25
+00:02:21,720 --> 00:02:24,300
+ملاحظات على النظرية
+
+26
+00:02:31,050 --> 00:02:36,390
+نعطيلها رقم تسعة و عشرين فالملاحظات
+
+27
+00:02:36,390 --> 00:02:44,850
+في عندي تلت ملاحظات الملاحظة الأولى انه theorem
+
+28
+00:02:44,850 --> 00:02:47,910
+النظرية
+
+29
+00:02:47,910 --> 00:02:53,270
+السابقة اعتقد ان هذا الرقم المفروض يكون تلاتين
+
+30
+00:02:59,480 --> 00:03:15,440
+theorem 29 remains true تبقى صحيحة if condition if
+
+31
+00:03:15,440 --> 00:03:22,720
+condition star is replaced إذا بدلنا الشرط star by
+
+32
+00:03:22,720 --> 00:03:34,060
+the weaker conditionby the weaker condition
+
+33
+00:03:34,060 --> 00:03:39,440
+اللي
+
+34
+00:03:39,440 --> 00:03:46,080
+هو xn less than or equal yn لكل
+
+35
+00:03:46,080 --> 00:03:55,540
+n أكبر من أو ساوي m for some n natural number
+
+36
+00:03:58,530 --> 00:04:05,110
+يعني بدل ما Xn أصغر من أو ساوي Yn لكل الان لكل
+
+37
+00:04:05,110 --> 00:04:09,910
+الأعداد الطبيعية N فلأ نفرض أن يوجد M عدد طبيعي
+
+38
+00:04:09,910 --> 00:04:15,190
+نفرض أن يوجد some M عدد طبيعي بحيث أن المتباين هذه
+
+39
+00:04:15,190 --> 00:04:21,590
+تتحقق لكل N أكبر من أو ساوي M يعني مش شرط تتحقق
+
+40
+00:04:21,590 --> 00:04:26,270
+للأعداد الطبيعية اللي أصغر من M فالنظرية برضه تبقى
+
+41
+00:04:26,270 --> 00:04:32,390
+صحيحةولو بدنا نبرهن النظرية اللى فاتت تحت الشرط
+
+42
+00:04:32,390 --> 00:04:37,150
+الأضعف هذا الشرط أضعف من الشرط ال star لكن برضه
+
+43
+00:04:37,150 --> 00:04:44,970
+بيعطينى نفس النظرية فال ..
+
+44
+00:04:44,970 --> 00:04:55,390
+ففي الحالة هذه in fact في حقيقة القمر in fact in
+
+45
+00:04:55,390 --> 00:05:07,220
+the proofsin the proofs of النظرية السابقة take
+
+46
+00:05:07,220 --> 00:05:19,460
+the required the required in to be that
+
+47
+00:05:19,460 --> 00:05:26,660
+corresponds that corresponds
+
+48
+00:05:28,960 --> 00:05:34,420
+that corresponds to the given to
+
+49
+00:05:34,420 --> 00:05:38,880
+the given alpha or
+
+50
+00:05:38,880 --> 00:05:45,360
+given beta to
+
+51
+00:05:45,360 --> 00:05:59,160
+be in ابارة عن ال maximum the m و n of alphaأو n
+
+52
+00:05:59,160 --> 00:06:07,920
+بالساوي ال maximum الأكبر بين العدد الطبيعي m و n
+
+53
+00:06:07,920 --> 00:06:17,920
+of beta إذن
+
+54
+00:06:17,920 --> 00:06:22,480
+في البرهان مثلا هذه الجزء التالي برهاننا عادي هي
+
+55
+00:06:22,480 --> 00:06:29,810
+نقلت الكلام هذا صحيح و يوجد capital Nهيعتمد على
+
+56
+00:06:29,810 --> 00:06:34,910
+beta بحيث ان الكلام هذا يتحقق الان ال star ماقدرش
+
+57
+00:06:34,910 --> 00:06:38,850
+ا say bye star هذه هتكون double star بدل ال star
+
+58
+00:06:38,850 --> 00:06:45,070
+فانا ساميها double star فالان
+
+59
+00:06:45,070 --> 00:06:51,590
+باخد بعرف n ال n هذه بعرفها على انها الأكبر بين m
+
+60
+00:06:51,590 --> 00:06:59,720
+و n of beta وبالتالي ال n هذهأكبر من أو ساوي M و
+
+61
+00:06:59,720 --> 00:07:05,920
+أكبر من أو ساوي N of beta وبالتالي لما أجي أخد N
+
+62
+00:07:05,920 --> 00:07:10,640
+أكبر من أو ساوي capital N بأضمن أن ال N تبعتي هذه
+
+63
+00:07:10,640 --> 00:07:17,820
+أكبر من أو ساوي M وبالتالي XN أصغر من أو ساوي YN
+
+64
+00:07:17,820 --> 00:07:22,940
+وكذلك
+
+65
+00:07:22,940 --> 00:07:28,120
+ال N لما تكون ال N أكبر من أو ساوي Nالـ N هذه
+
+66
+00:07:28,120 --> 00:07:33,020
+فبطمن أنها أكبر من أو ساوي N of beta وبالتالي
+
+67
+00:07:33,020 --> 00:07:40,000
+الكلام هذا بتحقق ويعني هيك بيكون برهن الجزء B بال
+
+68
+00:07:40,000 --> 00:07:44,960
+.. باستخدام الشرط الأضعف double star بالمثل طبعا
+
+69
+00:07:44,960 --> 00:07:48,960
+ممكن نعطيه في حالة لما يكون بدي أبرهن الجزء A
+
+70
+00:07:48,960 --> 00:07:53,540
+فباخد ال N المطلوبة هي ال maximum ل M وN of alpha
+
+71
+00:07:53,540 --> 00:08:00,680
+في برهان A تحت شرط starهذه أول ملاحظة الملاحظة
+
+72
+00:08:00,680 --> 00:08:15,360
+التانية الملاحظة
+
+73
+00:08:15,360 --> 00:08:25,480
+التانية if condition star holds إذا كان الشرط star
+
+74
+00:08:25,480 --> 00:08:27,660
+holds then
+
+75
+00:08:37,130 --> 00:08:42,070
+النتيجة ان y in tends to infinity does not
+
+76
+00:08:42,070 --> 00:08:46,650
+necessarily implies ان x in tends to infinity
+
+77
+00:08:53,430 --> 00:09:00,270
+وكان limit ال yn بساوي infinity فليس من الضروري ان
+
+78
+00:09:00,270 --> 00:09:06,090
+يكون limit xn بساوي infinity وهي مثال يوضح ذلك for
+
+79
+00:09:06,090 --> 00:09:13,550
+example على سبيل المثال consider consider
+
+80
+00:09:13,550 --> 00:09:20,230
+ال sequence 1 على n أصغر من أو بساوي n لكل n في n
+
+81
+00:09:21,990 --> 00:09:27,890
+إذن هى انا عندي xn وهى عندي yn وهى xn أصغر من
+
+82
+00:09:27,890 --> 00:09:34,090
+يساوي yn الشرط الصغير متحقق لكن أنا عندي ال limit
+
+83
+00:09:34,090 --> 00:09:39,330
+ل sequence yn اللى الحد العام تبعها n هدى بساوي
+
+84
+00:09:39,330 --> 00:09:51,440
+infinity but ال limit ل xn اللى هى واحد على nبساوي
+
+85
+00:09:51,440 --> 00:09:59,860
+سفر لا تساوي infinity السفر لا يساوي infinity okay
+
+86
+00:09:59,860 --> 00:10:06,440
+تمام إذا الشرط star ما بيخلنيش أستنتج أنه limit x
+
+87
+00:10:06,440 --> 00:10:09,200
+in بالساوية infinity عندما limit y in بالساوية
+
+88
+00:10:09,200 --> 00:10:19,280
+infinity بالمثل if condition star holdsإذا كان
+
+89
+00:10:19,280 --> 00:10:24,660
+الشرط الـ start متحقق then x
+
+90
+00:10:24,660 --> 00:10:30,840
+in تقول إلى negative infinity ليس بالضرورة بيؤدي
+
+91
+00:10:30,840 --> 00:10:36,620
+مش شرط يؤدي ان ال sequence y in تقول ل negative
+
+92
+00:10:36,620 --> 00:10:44,360
+infinity هذا مش شرط يكون صحيح بنا مثال على ذلك
+
+93
+00:10:44,360 --> 00:10:48,020
+ممكن نفس المثال بس
+
+94
+00:10:51,510 --> 00:10:57,370
+for example بس نضرب في سالب هي عندي negative n
+
+95
+00:10:57,370 --> 00:11:04,900
+أصغر من أو ساوي negative واحد على n لكل n في nهل
+
+96
+00:11:04,900 --> 00:11:09,780
+هذا كلام صح؟ انا عندي واحد على n أصغر من أو ساوي n
+
+97
+00:11:09,780 --> 00:11:17,040
+لكل n هذا صح اضرب في سالب واحد تناقص هاه؟ هاي عندك
+
+98
+00:11:17,040 --> 00:11:24,990
+xn بساوي سالب n وهي عندنا ynبساوي negative واحد
+
+99
+00:11:24,990 --> 00:11:31,590
+على n الان انا عندي limit xn اللي هو سالب n لما
+
+100
+00:11:31,590 --> 00:11:37,170
+طبعا n تقول infinity بساوي negative infinity but
+
+101
+00:11:37,170 --> 00:11:44,910
+لكن limit ال yn اللي هو واحد على n ايش بتساوي؟
+
+102
+00:11:44,910 --> 00:11:49,950
+ساوي سفر سالب واحد عفوا سالب واحد على n limit سالب
+
+103
+00:11:49,950 --> 00:11:56,630
+واحد على n بساوي سفرو ليست سالب infinity okay
+
+104
+00:11:56,630 --> 00:12:02,430
+تمام؟ إذا النظرية ال comparison test لا يقبل .. لا
+
+105
+00:12:02,430 --> 00:12:07,010
+يقبل التأويل زي ما بيقولوا بس النتائج تبعتها كما
+
+106
+00:12:07,010 --> 00:12:13,310
+هي في a و b أي شيء آخر مش مظبوطهو أمثلة بتوضح
+
+107
+00:12:13,310 --> 00:12:20,970
+الأشياء الأخرى تمام؟ في كمان اختبار أخر زي هذا
+
+108
+00:12:20,970 --> 00:12:25,550
+بنسميه limit comparison
+
+109
+00:12:25,550 --> 00:12:34,250
+test فال
+
+110
+00:12:34,250 --> 00:12:35,350
+.. نمسح
+
+111
+00:12:56,530 --> 00:13:11,090
+limit comparison test خلّيني
+
+112
+00:13:11,090 --> 00:13:19,550
+أاخد two sequences x in و y in بsequences of
+
+113
+00:13:19,550 --> 00:13:21,030
+positive real numbers
+
+114
+00:13:24,560 --> 00:13:28,760
+بالتالي سيكون الحدود
+
+115
+00:13:28,760 --> 00:13:33,660
+الموجبة لكي يكونوا سالبة وبعضهم سالبة وبعضهم موجبة
+
+116
+00:13:33,660 --> 00:13:36,820
+بي
+
+117
+00:13:36,820 --> 00:13:41,240
+such that limit
+
+118
+00:13:41,240 --> 00:13:49,480
+ل xn over yn as n tends to infinity بساوي L عدد
+
+119
+00:13:49,480 --> 00:13:50,200
+موجبة
+
+120
+00:13:55,720 --> 00:14:02,320
+بنسمي المعادلة add star then
+
+121
+00:14:02,320 --> 00:14:12,380
+limit xn بساوي infinity if and only if limit yn
+
+122
+00:14:12,380 --> 00:14:22,160
+بساوي infinity إذا
+
+123
+00:14:22,160 --> 00:14:26,190
+هنا في عندي limit comparison testالذي يتم استخدامه
+
+124
+00:14:26,190 --> 00:14:28,110
+لسيقونسات واحدة فقط من حدوث واحدة واحدة فقط من
+
+125
+00:14:28,110 --> 00:14:29,250
+حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط
+
+126
+00:14:29,250 --> 00:14:33,210
+من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة
+
+127
+00:14:33,210 --> 00:14:35,150
+فقط من حدوث واحدة فقط من حدوث واحدة فقط من حدوث
+
+128
+00:14:35,150 --> 00:14:35,310
+واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط من
+
+129
+00:14:35,310 --> 00:14:37,070
+حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة فقط
+
+130
+00:14:37,070 --> 00:14:43,510
+من حدوث واحدة فقط من حدوث واحدة فقط من حدوث واحدة
+
+131
+00:14:43,510 --> 00:14:51,010
+فقط من حدوث واحدة فقط من
+
+132
+00:14:51,010 --> 00:14:52,090
+حدوث واحدة ف
+
+133
+00:14:59,920 --> 00:15:10,860
+let assume ال أكبر من السفر satisfies
+
+134
+00:15:10,860 --> 00:15:15,300
+المعادلة
+
+135
+00:15:15,300 --> 00:15:21,390
+لسه نفرض ان في عدد حقيقي الوهو بحقق star يعني هو
+
+136
+00:15:21,390 --> 00:15:30,890
+limit ل ratio ل xn على yn تمام؟
+
+137
+00:15:30,890 --> 00:15:34,650
+take epsilon
+
+138
+00:15:34,650 --> 00:15:38,930
+بساوي
+
+139
+00:15:38,930 --> 00:15:46,920
+L على 2Since L is positive L over 2 is positive
+
+140
+00:15:46,920 --> 00:15:57,360
+لأن أنا جبت إبسلون which is positive طيب since من
+
+141
+00:15:57,360 --> 00:16:08,300
+الفرض since by star XN over YN converges to Lوهي
+
+142
+00:16:08,300 --> 00:16:10,980
+عندي إبسلون أكبر من الصفر is given إذا by
+
+143
+00:16:10,980 --> 00:16:15,400
+definition of convergence by epsilon capital N
+
+144
+00:16:15,400 --> 00:16:22,400
+definition لإبسلون هذه for this إبسلونthere exist
+
+145
+00:16:22,400 --> 00:16:29,600
+capital N يعتمد على إبسلون يعتمد
+
+146
+00:16:29,600 --> 00:16:34,780
+على الإبسلون اللي هي بتعتمد على العدد L natural
+
+147
+00:16:34,780 --> 00:16:41,620
+number بحيث أنه لكل N أكبر منه سوى capital N بيطلع
+
+148
+00:16:41,620 --> 00:16:48,220
+عندي absolute xn over yn negative L less than
+
+149
+00:16:48,220 --> 00:16:57,240
+إبسلونهزبوت هيك؟ طيب الـ Y بسوي L over 2 خلّينا
+
+150
+00:16:57,240 --> 00:17:02,560
+نشيل ال absolute value فبصير اندي Xn over Yn minus
+
+151
+00:17:02,560 --> 00:17:10,960
+L less than L two bigger than negative L ع اتنين و
+
+152
+00:17:10,960 --> 00:17:16,900
+هذا صحيح لكل N bigger than or equal N اجمع L على
+
+153
+00:17:16,900 --> 00:17:24,960
+كل الأطرافإن أنا بطلع عندي xn over yn less than
+
+154
+00:17:24,960 --> 00:17:31,920
+three over two L bigger than L over two and this
+
+155
+00:17:31,920 --> 00:17:36,620
+is true for every n bigger than or equal n نسمي
+
+156
+00:17:36,620 --> 00:17:46,680
+المتباينة هذه double star now
+
+157
+00:17:52,640 --> 00:18:02,400
+by double star انا عندي xn على yn اصغر من تلاتة ع
+
+158
+00:18:02,400 --> 00:18:09,020
+اتنين ال اللي هو الجزء هذا وهذا صحيح لكل n bigger
+
+159
+00:18:09,020 --> 00:18:15,820
+than or equal to n بيقدي انه xn
+
+160
+00:18:24,520 --> 00:18:34,680
+في اتنين على التلاتة L less than Y N وهذا صحيح لكل
+
+161
+00:18:34,680 --> 00:18:45,260
+N bigger than or equal N تصبوت هيك صح؟ هذه
+
+162
+00:18:45,260 --> 00:18:49,720
+المتباينة بتقدر هذه هي نفس هذه
+
+163
+00:18:53,610 --> 00:19:05,290
+so as limit احنا فرض .. now now
+
+164
+00:19:05,290 --> 00:19:10,690
+او .. او so if
+
+165
+00:19:12,930 --> 00:19:19,250
+limit xn بساوي infinity إذا كان limit xn بساوي
+
+166
+00:19:19,250 --> 00:19:23,490
+infinity و هذا ثابت موجب this is positive constant
+
+167
+00:19:23,490 --> 00:19:31,650
+ف limit كل هذا برضه بساوي plus infinity و
+
+168
+00:19:31,650 --> 00:19:39,450
+بالتالي by comparison test then by comparison by
+
+169
+00:19:39,450 --> 00:19:40,170
+comparison
+
+170
+00:19:42,940 --> 00:19:47,480
+by comparison test النظرية اللى فاتت مع الشرط
+
+171
+00:19:47,480 --> 00:19:51,640
+المخفف مع الشرط المخفف لأن فى النظرية اللى فاتت
+
+172
+00:19:51,640 --> 00:19:56,940
+كان عندي xn أصغر من أو يسوى yn لكل n بعدين قلنا أن
+
+173
+00:19:56,940 --> 00:20:01,200
+هذا الشرط لو خففناه لكل n أكبر من أو سوى عدد طبيعى
+
+174
+00:20:01,200 --> 00:20:06,440
+ما وليكن capital N هنا برضه بتظل صحيحةف by
+
+175
+00:20:06,440 --> 00:20:12,620
+comparison test and limit ال sequence هذه بساوي
+
+176
+00:20:12,620 --> 00:20:20,700
+infinity إذا limit الأكبر limit yn بساوي infinity
+
+177
+00:20:20,700 --> 00:20:31,220
+تمام؟ الآن بنثبت العكس نثبت الآن العكس طيب
+
+178
+00:20:31,220 --> 00:20:32,380
+conversely
+
+179
+00:20:40,740 --> 00:20:51,080
+Conversely Assume Assume المرهد أنه limit yn بساوي
+
+180
+00:20:51,080 --> 00:20:59,700
+infinity من double star من double star لو أخدت
+
+181
+00:20:59,700 --> 00:21:06,520
+النص .. النص هذا من المتباينة النص الآخرفعندي انا
+
+182
+00:21:06,520 --> 00:21:13,120
+L على 2 أصغر من XN على YN هذا صحيح for every N
+
+183
+00:21:13,120 --> 00:21:22,120
+أكبر من أو ساوية capital N طيب هذا بيقدي ان ال L
+
+184
+00:21:22,120 --> 00:21:29,400
+over 2 في YN أصغر من XN لكل N bigger than or equal
+
+185
+00:21:29,400 --> 00:21:33,380
+to capital N طيب
+
+186
+00:21:33,380 --> 00:21:34,900
+since
+
+187
+00:21:37,140 --> 00:21:44,200
+limit yn بساوي infinity بيقدي انه limit ثابت موجب
+
+188
+00:21:44,200 --> 00:21:51,060
+في yn لأن هذا بيقدي انه limit ثابت الا اتنين في yn
+
+189
+00:21:51,060 --> 00:21:57,760
+بساوي infinity so by comparison
+
+190
+00:21:57,760 --> 00:22:01,700
+by comparison test
+
+191
+00:22:07,170 --> 00:22:11,210
+أنا عندي limit ال sequence لصغيرة infinity، إذا
+
+192
+00:22:11,210 --> 00:22:15,550
+limit ال sequence الأكبر بطلع infinity، إذا limit
+
+193
+00:22:15,550 --> 00:22:28,070
+xn equals infinity وهذا بكمل البرهان، okay؟
+
+194
+00:22:28,070 --> 00:22:32,390
+تمام؟ إذا هذا بكمل ال limit comparison .. برهان ال
+
+195
+00:22:32,390 --> 00:22:36,780
+limit comparison testطبعا ال test هذا و ال test
+
+196
+00:22:36,780 --> 00:22:40,100
+اللي جابله ال comparison test في عليهم هتجدوا فيه
+
+197
+00:22:40,100 --> 00:22:44,160
+بعض التمرين ممكن
+
+198
+00:22:44,160 --> 00:22:47,660
+تطبيقهم على بعض ال sequences موجودة في التمرين
+
+199
+00:22:47,660 --> 00:22:53,560
+فهنسيبكم طبعا تحلوا التمرين عشان تشوفوا كيف ممكن
+
+200
+00:22:53,560 --> 00:22:54,280
+تطبيقهم
+
+201
+00:22:58,500 --> 00:23:05,220
+باقي section واحد في ال chapter تلاتة
+
+202
+00:23:32,720 --> 00:23:37,660
+السيكشن الأخير سيكشن
+
+203
+00:23:37,660 --> 00:23:43,820
+تلاتة سبعة في شبكر 3 هذا هيكون أبراعا مقدمة
+
+204
+00:23:43,820 --> 00:23:47,860
+introduction to
+
+205
+00:23:47,860 --> 00:23:52,920
+infinite series
+
+206
+00:23:57,560 --> 00:24:02,380
+introduction to infinite series مقدمة في
+
+207
+00:24:02,380 --> 00:24:06,940
+المتسلسلات اللانهائية
+
+208
+00:24:06,940 --> 00:24:19,460
+نعرف شو معناه متسلسلة لانهائية let xn
+
+209
+00:24:19,460 --> 00:24:27,330
+contained in R be a sequencesequence of real
+
+210
+00:24:27,330 --> 00:24:43,210
+numbers sum
+
+211
+00:24:43,210 --> 00:24:47,010
+x1
+
+212
+00:24:47,010 --> 00:24:56,200
+plus x2 plus ..x3 plus و هكذا plus xn plus و هكذا
+
+213
+00:24:56,200 --> 00:25:03,400
+و ممكن نكتبه بالصورة using sigma notation نستخدم
+
+214
+00:25:03,400 --> 00:25:09,920
+رمز sigma ممكن هذا نسميه summation from n equals
+
+215
+00:25:09,920 --> 00:25:17,280
+one to infinity إلى xn فالصن
+
+216
+00:25:17,280 --> 00:25:25,190
+المجموع هذاهذا expanded هذا compact form of
+
+217
+00:25:25,190 --> 00:25:39,010
+summation is called an infinite series generated
+
+218
+00:25:39,010 --> 00:25:42,970
+by
+
+219
+00:25:46,320 --> 00:25:54,100
+متولدة من .. by الـ sequence x in إذن
+
+220
+00:25:54,100 --> 00:26:00,280
+infinite series generated by الـ sequence x in إذا
+
+221
+00:26:00,280 --> 00:26:04,680
+هذه عبارة عن infinite series بتسميها متولدة من الـ
+
+222
+00:26:04,680 --> 00:26:09,340
+sequence x in طيب for every
+
+223
+00:26:12,430 --> 00:26:23,290
+for each n belong to N define خلّيني أعرف S1 على
+
+224
+00:26:23,290 --> 00:26:37,950
+أنه X1 S2 على أنه S1 زاد X2 بساوي X1 زاد X2 S3
+
+225
+00:26:37,950 --> 00:26:50,000
+بساوي S2 زاد X3Y ساوي X1 زايد X2 زايد X3 and
+
+226
+00:26:50,000 --> 00:26:58,380
+so on و هكذا نعرف SN على انه SN negative one زايد
+
+227
+00:26:58,380 --> 00:27:07,060
+XN و طبعا ال SN negative one هيكون عبارة عن
+
+228
+00:27:07,060 --> 00:27:07,700
+summation
+
+229
+00:27:10,600 --> 00:27:19,140
+x1 زائد x2 زائد و هكذا إلى أخر حد xn-1 هذا عبارة
+
+230
+00:27:19,140 --> 00:27:25,620
+عن ايه هذا عبارة عن s in negative one بنضيف لها xn
+
+231
+00:27:25,620 --> 00:27:34,340
+فهذا بيطلع بيساوي summation من k equals one to n
+
+232
+00:27:38,080 --> 00:27:43,700
+to for xk اذا
+
+233
+00:27:43,700 --> 00:27:49,840
+sn هو مجموع الحدود من اول حد الى حد رقم n وهكذا
+
+234
+00:27:49,840 --> 00:27:55,960
+ممكن نستمر الى ملا نهاية and so on الان انا كوّنت
+
+235
+00:27:55,960 --> 00:28:00,360
+sequence لاحظوا s1, s2, s3, sn هذا عبارة عن
+
+236
+00:28:00,360 --> 00:28:06,970
+sequence ال sequence الجديدة هذه لها اسمو sequence
+
+237
+00:28:06,970 --> 00:28:12,210
+مهمة of partial sums مظبوط قعدت نسميها اذا طرست
+
+238
+00:28:12,210 --> 00:28:18,790
+تفاضل الف وفهمته الموضوع هذا هناك الموضوع ال
+
+239
+00:28:18,790 --> 00:28:23,110
+series قعدت نسميها the sequence of partial sums
+
+240
+00:28:23,110 --> 00:28:29,210
+اذا the sequence
+
+241
+00:28:30,940 --> 00:28:37,980
+SN from N equals one to infinity is called بنسميها
+
+242
+00:28:37,980 --> 00:28:51,180
+the sequence the sequence of partial sums
+
+243
+00:28:51,180 --> 00:29:03,040
+sequence of partial sums of the seriesاللي هي
+
+244
+00:29:03,040 --> 00:29:11,080
+sigma xn او sigma من n بساعة واحد لانفينيتي okay
+
+245
+00:29:11,080 --> 00:29:18,660
+الان now if
+
+246
+00:29:18,660 --> 00:29:29,280
+ال sequence sn converges say
+
+247
+00:29:31,110 --> 00:29:42,090
+limit sn بالساوي عدد s ينتمي إلى r طبعا then
+
+248
+00:29:42,090 --> 00:29:51,390
+we say في الحالة هذه بنقول أنه the series اللي
+
+249
+00:29:51,390 --> 00:29:58,630
+هي summation xn from n equals one to infinity
+
+250
+00:29:58,630 --> 00:30:00,270
+converges
+
+251
+00:30:09,070 --> 00:30:18,290
+and its sum is summation from n equals one to
+
+252
+00:30:18,290 --> 00:30:23,530
+infinity ل x in ال summation تبعها أو المجموعة
+
+253
+00:30:23,530 --> 00:30:28,450
+تبعها عبارة عن limit لل sequence of partial sums
+
+254
+00:30:28,450 --> 00:30:32,730
+اللي هو العدد S
+
+255
+00:30:37,160 --> 00:30:43,180
+لو كانت الـ sequence divergent
+
+256
+00:30:43,180 --> 00:30:50,740
+if الـ sequence is in diverges we
+
+257
+00:30:50,740 --> 00:31:00,120
+say أنه الـ series sigma
+
+258
+00:31:00,120 --> 00:31:05,880
+x in diverges
+
+259
+00:31:09,090 --> 00:31:13,410
+إذا ال convergence و ال divergence depends on the
+
+260
+00:31:13,410 --> 00:31:18,630
+divergence أو convergence of the infinite series
+
+261
+00:31:18,630 --> 00:31:23,910
+depends on the convergence or divergence of the
+
+262
+00:31:23,910 --> 00:31:30,690
+sequence of partial sums مرتبط بيها ال sequence of
+
+263
+00:31:30,690 --> 00:31:34,350
+partial sums convergent السيريز اللي تابع إليها
+
+264
+00:31:34,350 --> 00:31:38,360
+convergentو العكس إذا كانت ال sequence of partial
+
+265
+00:31:38,360 --> 00:31:40,780
+sums divergent ال series ال infinite series
+
+266
+00:31:40,780 --> 00:31:51,840
+التابعة إلى divergent طيب
+
+267
+00:31:51,840 --> 00:31:58,180
+ناخد بعض الأمثلة طبعا
+
+268
+00:31:58,180 --> 00:32:01,720
+ال Sn هذا ال Sn
+
+269
+00:32:04,610 --> 00:32:15,810
+هذا بنسميه الانث partial sum الانث partial sum
+
+270
+00:32:15,810 --> 00:32:25,690
+انث partial sum المجموع الجزئي أنوني okay هو
+
+271
+00:32:25,690 --> 00:32:30,760
+الحد العام لل sequence و partial sumsأذا لما بدى
+
+272
+00:32:30,760 --> 00:32:34,760
+نخبر هل ال series convergent ولا divergent بجيب ال
+
+273
+00:32:34,760 --> 00:32:38,380
+sequence of partial sums وبجيب الحد العام لل
+
+274
+00:32:38,380 --> 00:32:41,780
+sequence of partial sums وبأفحص هل ال sequence هذي
+
+275
+00:32:41,780 --> 00:32:47,680
+convergent ولا divergent ناخد
+
+276
+00:32:47,680 --> 00:32:48,620
+بعض الأمثلة
+
+277
+00:33:02,760 --> 00:33:14,560
+المثال الأول consider
+
+278
+00:33:14,560 --> 00:33:17,780
+sequence
+
+279
+00:33:17,780 --> 00:33:25,480
+R to N from N equals 0 to infinity طبعا هذه
+
+280
+00:33:25,480 --> 00:33:33,650
+sequence of real numbersWhere R is a real number
+
+281
+00:33:33,650 --> 00:33:38,210
+which
+
+282
+00:33:38,210 --> 00:33:49,150
+generates الsequence هذه generates the geometric
+
+283
+00:33:49,150 --> 00:33:53,010
+.. the so-called geometric series .. geometric
+
+284
+00:33:53,010 --> 00:33:54,330
+series
+
+285
+00:33:57,050 --> 00:34:02,610
+اللي هي summation from n equals zero to infinity
+
+286
+00:34:02,610 --> 00:34:10,210
+from r to n okay اذا هي ال sequence هذه of real
+
+287
+00:34:10,210 --> 00:34:15,650
+numbers بتولد infinite series او generates this
+
+288
+00:34:15,650 --> 00:34:21,210
+infinite series اللي هي حدودها اول حد لما n بساوي
+
+289
+00:34:21,210 --> 00:34:33,620
+سفر واحد بعدين r بعدين r تربيهو R أس N و
+
+290
+00:34:33,620 --> 00:34:41,120
+هكذا ف such series is called geometric series هذه
+
+291
+00:34:41,120 --> 00:34:44,300
+ال series اللي على الصورة هذه بنسميها geometric
+
+292
+00:34:44,300 --> 00:34:49,820
+series الآن ال series هذه
+
+293
+00:34:58,170 --> 00:35:08,530
+this series واحد converges and
+
+294
+00:35:08,530 --> 00:35:15,910
+its sum اللي هو sigma from n equals zero to
+
+295
+00:35:15,910 --> 00:35:22,470
+infinity لRn بساوي واحد على واحد minus R إذا كان
+
+296
+00:35:22,470 --> 00:35:35,210
+absolute R أصغر من واحدand diverges and
+
+297
+00:35:35,210 --> 00:35:41,350
+اتنين diverges if
+
+298
+00:35:41,350 --> 00:35:48,830
+absolute are أكبر من أو يساوي واحد خلّينا
+
+299
+00:35:48,830 --> 00:35:50,050
+نثبت الجزء الأول
+
+300
+00:35:58,010 --> 00:36:04,110
+to prove one أنا
+
+301
+00:36:04,110 --> 00:36:12,410
+عندي ال S N بساوي سيجما
+
+302
+00:36:12,410 --> 00:36:21,050
+من K بساوي سفر إلى N ل R أس K اللي هو واحد زائد R
+
+303
+00:36:21,050 --> 00:36:34,640
+زائد R تلبية زائدR Sn وفي عندي .. في عندي ..
+
+304
+00:36:34,640 --> 00:36:37,780
+لو
+
+305
+00:36:37,780 --> 00:36:48,440
+ضربت Sn في Rفادرب الطرف اليمين في R فبطلع R زاد R
+
+306
+00:36:48,440 --> 00:36:56,380
+تربيه زاد و هكذا زاد R أس N و آخر حد هيكون R أس N
+
+307
+00:36:56,380 --> 00:37:00,940
+زاد 1 تمام؟ الآن خلّينا نطرح ال subtract
+
+308
+00:37:05,370 --> 00:37:10,850
+subtract نطرح المعادلة لتحت من اللي فوق فبطلع عندي
+
+309
+00:37:10,850 --> 00:37:18,590
+sn في واحد minus r أخدت عامل مشترك sn ولمّا أطرح
+
+310
+00:37:18,590 --> 00:37:23,330
+هذا بروح مع هذا كل الهدوء بتروح مع بعضها بظل عندي
+
+311
+00:37:23,330 --> 00:37:33,130
+واحد سالب r to n زاد واحد تمام ومن هنا اذا sn
+
+312
+00:37:36,050 --> 00:37:44,310
+بساوي واحد على واحد سالب R سالب R to N زايد واحد
+
+313
+00:37:44,310 --> 00:37:52,990
+على واحد سالب R ممكن
+
+314
+00:37:52,990 --> 00:37:59,070
+هذا نوديه على ناحية التانية فبصير عندى هذا سالب
+
+315
+00:37:59,070 --> 00:38:01,350
+هذا بساوي
+
+316
+00:38:03,070 --> 00:38:08,930
+سالب R to N زياد واحد على واحد سالب R الآن إذا
+
+317
+00:38:08,930 --> 00:38:13,590
+ناخد ال absolute value للطرفين Sn سالب واحد على
+
+318
+00:38:13,590 --> 00:38:21,030
+واحد سالب R بيطلع بيساوي الكلام هذا وهذا أصغر من
+
+319
+00:38:21,030 --> 00:38:27,830
+أو ساوي absolute R أسن زياد واحد على absolute واحد
+
+320
+00:38:27,830 --> 00:38:30,870
+minus R تمام؟
+
+321
+00:38:34,940 --> 00:38:41,200
+أذا عندي أنا هاي واحد على absolute واحد سالب R ضرب
+
+322
+00:38:41,200 --> 00:38:51,360
+absolute R أسن زائد واحد الان if absolute R أصغر
+
+323
+00:38:51,360 --> 00:39:00,870
+من واحدفهذا بيؤدي ان ال limit ل absolute R to N زي
+
+324
+00:39:00,870 --> 00:39:05,790
+1 لما N تقول ل infinity هذا بيساوي سفر أخدناها قبل
+
+325
+00:39:05,790 --> 00:39:10,430
+هيك وبالتالي
+
+326
+00:39:10,430 --> 00:39:14,950
+اذا ال ..
+
+327
+00:39:14,950 --> 00:39:18,290
+اذا انا عند ال absolute value هذه أكبر من أو ساوي
+
+328
+00:39:18,290 --> 00:39:24,270
+سفر و أصغر من أو ساوي ثابت موجب في sequenceالـ
+
+329
+00:39:24,270 --> 00:39:28,610
+sequence هذه تقول لـ 0 وهذه الـ sequence ثابتة
+
+330
+00:39:28,610 --> 00:39:32,330
+تقول لـ 0 اذا by sandwich theorem
+
+331
+00:39:40,720 --> 00:39:47,760
+بتطلع عند ال limit ل absolute sn minus 1 على 1
+
+332
+00:39:47,760 --> 00:39:52,820
+minus r لما n تقول ل infinity بساوي سفر و ممكن
+
+333
+00:39:52,820 --> 00:39:58,600
+ندخل ال limit جوا فهذا بقدر انه limit 1 على sn
+
+334
+00:39:58,600 --> 00:40:05,040
+عفوا limit sn لما n تقول ل infinity بساوي 1 على 1
+
+335
+00:40:05,040 --> 00:40:11,560
+سالب rوبالتالي إذا الـ series sigma from N equal 0
+
+336
+00:40:11,560 --> 00:40:17,240
+to infinity لR to N المجموعة تبعها تطلع convergent
+
+337
+00:40:17,240 --> 00:40:23,180
+ومجموعة بساوي limit SN وهذا بساوي 1 على 1 minus R
+
+338
+00:40:24,110 --> 00:40:28,950
+إذن هذا بثبت الجزء الأول الجزء التاني ممكن اثباته
+
+339
+00:40:28,950 --> 00:40:34,190
+لو R بساوي واحد فبطل عندي بجمع واحد على واحد عدد
+
+340
+00:40:34,190 --> 00:40:37,730
+لا نهائي من المرات وبالتالي ال sequence of partial
+
+341
+00:40:37,730 --> 00:40:40,170
+sums ممكن اثبات أنها unbounded وبالتالي not
+
+342
+00:40:40,170 --> 00:40:44,330
+convergent إذن ال series not convergent نفس الحاجة
+
+343
+00:40:44,330 --> 00:40:47,210
+لو كان absolute R أكبر من واحد فممكن اثبات أن ال
+
+344
+00:40:47,210 --> 00:40:50,550
+sequence of partial sums is divergent وبالتالي ال
+
+345
+00:40:50,550 --> 00:40:57,170
+series is divergentتمام؟ في أي سؤال؟ إذا بنوقف هنا
+
+346
+00:40:57,170 --> 00:41:02,830
+و بنكمل ان شاء الله الموضوع اللي جاي في المحاضرة
+
+347
+00:41:02,830 --> 00:41:04,630
+القادمة يوم السبت
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..69cee197c637ec8f43ea1d5e3351f4cb5fbbb355
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/XjWoXKhuE-o_postprocess.srt
@@ -0,0 +1,1768 @@
+1
+00:00:21,090 --> 00:00:26,570
+إذن في المحاضرة هذه ان شاء الله هنحل بعض التمرين
+
+2
+00:00:26,570 --> 00:00:35,250
+لل homework اللي تابع ل section تلاتة واحد و تلاتة
+
+3
+00:00:35,250 --> 00:00:44,390
+اتنين فأزملتكم تسأل عن ال .. نحن نحل السؤال 13
+
+4
+00:00:44,390 --> 00:00:46,030
+section تلاتة واحد
+
+5
+00:00:49,430 --> 00:01:15,310
+نكتب السؤال على اللوحة سيكشن
+
+6
+00:01:15,310 --> 00:01:25,990
+السؤال 13section تلاتة واحد انا
+
+7
+00:01:25,990 --> 00:01:31,150
+عندي بي is real number اكبر من سفر اصغر من واحد
+
+8
+00:01:31,150 --> 00:01:36,010
+وبينا
+
+9
+00:01:36,010 --> 00:01:40,990
+نثبت show انه ال limit
+
+10
+00:01:44,880 --> 00:01:53,020
+للـ sequence اللي الحد العام تبعها n في d to n لما
+
+11
+00:01:53,020 --> 00:02:03,560
+n تقول infinity بساوي سفر والكتاب جايلكم
+
+12
+00:02:03,560 --> 00:02:08,840
+use ال binomial theorem كما في مثال 3 1 11 الجزء
+
+13
+00:02:08,840 --> 00:02:16,010
+ديفلو حاولتوا تستخدموا نفس أسلوب البرهان تبع
+
+14
+00:02:16,010 --> 00:02:20,350
+المثال اللي استخدمنا فيه ال binomial theorem
+
+15
+00:02:20,350 --> 00:02:28,550
+فهتصلوا للنتيجة فهي البرهان نشوف
+
+16
+00:02:28,550 --> 00:02:32,170
+كيف نستخدم ال binomial theorem في الوصول إلى
+
+17
+00:02:32,170 --> 00:02:40,300
+المطلوبأنا عندي من الفرض سفر أصغر من بي أصغر من
+
+18
+00:02:40,300 --> 00:02:48,300
+واحد هذا بيؤدي ان واحد على بي أكبر من واحد
+
+19
+00:02:48,300 --> 00:02:55,500
+وبالتالي هذا بيؤدي ان واحد على بي سالب واحد أكبر
+
+20
+00:02:55,500 --> 00:03:08,230
+من سفر اذا ناخد letlet a خلّيني اعرف عدد a على انه
+
+21
+00:03:08,230 --> 00:03:13,850
+العدد الموجب واحد على بي سالب واحد طبعا هذا عدد
+
+22
+00:03:13,850 --> 00:03:20,110
+موجب حسب ما شوفنا وهذا
+
+23
+00:03:20,110 --> 00:03:28,210
+بيقدّي انه العدد لو حليت المعادلة هذه في بي فهيطلع
+
+24
+00:03:28,210 --> 00:03:38,250
+بي بساوي واحدعلى واحد زائد ال a وبالتالي
+
+25
+00:03:38,250 --> 00:03:49,150
+so by ال binomial باستخدام
+
+26
+00:03:49,150 --> 00:03:59,120
+ال binomial theorem انا عنديواحد زائد a الكل قص n
+
+27
+00:03:59,120 --> 00:04:09,300
+بيساوي واحد زائد n في a زائد نص n في n سالب واحد
+
+28
+00:04:09,300 --> 00:04:17,800
+في a تردية زائد و هكذا تمام
+
+29
+00:04:17,800 --> 00:04:24,890
+إلى آخر حد طبعا هيكون a to nهذا بالظبط زي ما عملنا
+
+30
+00:04:24,890 --> 00:04:31,950
+في مثال سادة وبالتالي
+
+31
+00:04:31,950 --> 00:04:42,090
+هذا بيقدي من هنا هذا
+
+32
+00:04:42,090 --> 00:04:51,620
+المجموعة بيطلع أكبر من أو ساوي نصفي n في n زي سالب
+
+33
+00:04:51,620 --> 00:05:00,000
+واحد في ايه ترميها يعني أنا أخدت بس الحد التالت من
+
+34
+00:05:00,000 --> 00:05:05,120
+المجموعة ده المجموعة طبعا مجموعة أعداد موجة بكلها
+
+35
+00:05:05,120 --> 00:05:10,180
+فالمجموعة ده بالتأكيد أكبر من أو ساوي الحد التالت
+
+36
+00:05:10,180 --> 00:05:14,740
+في ايه هذا صحيح مافي مشكلة تمام
+
+37
+00:05:17,950 --> 00:05:33,450
+وبالتالي اذا n في b أس n ايش بيساوي بيساوي n على
+
+38
+00:05:33,450 --> 00:05:43,480
+واحد زائد a لكل أس n صح؟هذه B ف B أس N بساوي واحد
+
+39
+00:05:43,480 --> 00:05:50,900
+على واحد زاد A to N و أضرب في N فبصير هيك طيب
+
+40
+00:05:50,900 --> 00:05:58,560
+من هنا مقلوب واحد زاد A لكل أس N هيطلع أصغر من أوي
+
+41
+00:05:58,560 --> 00:06:09,580
+ساوي مقلوب العدد هذا اذا هذا أصغر من أوي ساوي N
+
+42
+00:06:14,820 --> 00:06:20,480
+على N في
+
+43
+00:06:20,480 --> 00:06:28,380
+N سالب واحد في .. في N سالب واحد في ايه تربية على
+
+44
+00:06:28,380 --> 00:06:38,980
+اتنين وفي عندنا كمان N العكس
+
+45
+00:06:38,980 --> 00:06:39,600
+العكس
+
+46
+00:06:46,180 --> 00:06:55,020
+هي عندى n ومقلوب هدا بطلع اتنين n في n سالب واحد
+
+47
+00:06:55,020 --> 00:07:01,760
+في a تربية تمام؟ إذا هدا إيجا من هنا الان بختصر ال
+
+48
+00:07:01,760 --> 00:07:12,620
+n مع ال n فهدا بطلع اتنين على n سالب واحد في a
+
+49
+00:07:12,620 --> 00:07:14,040
+تربية تمام؟
+
+50
+00:07:16,500 --> 00:07:23,140
+الان هذا الكلام صحيح لكل n أكبر من واحد طبعا ممنوع
+
+51
+00:07:23,140 --> 00:07:27,100
+أخد n بساوي واحد لأنه في الحالة هذه بيصير في قسم
+
+52
+00:07:27,100 --> 00:07:31,780
+على سفر لأن لكل الأعداد الطبيعية n أكبر من واحد n
+
+53
+00:07:31,780 --> 00:07:36,980
+في bios n بيطلع أصغر منه يساوي اثنين على n سالب
+
+54
+00:07:36,980 --> 00:07:44,420
+واحد في a تربية الان تعالوا نثبت ان ال limit لل
+
+55
+00:07:44,420 --> 00:07:46,240
+sequence هذه بساوي سفر
+
+56
+00:07:50,790 --> 00:07:57,390
+هنستخدم تعريف epsilon capital N لـ limit لذن let
+
+57
+00:07:57,390 --> 00:08:02,090
+epsilon let
+
+58
+00:08:02,090 --> 00:08:12,830
+epsilon أكبر من سبل be given by
+
+59
+00:08:12,830 --> 00:08:19,210
+Archimedean property by Archimedeanproperty حسب
+
+60
+00:08:19,210 --> 00:08:25,750
+خاصية Archimedes يوجد نقدر نلاقي عدد طبيعي capital
+
+61
+00:08:25,750 --> 00:08:34,530
+N ينتمي إلى N يعتمد طبعا على إبسلون بحيث أنه مقلوب
+
+62
+00:08:34,530 --> 00:08:41,590
+capital N أصغر من A تربية في إبسلون على 2
+
+63
+00:08:49,130 --> 00:08:54,230
+الـ A تربي عدد موجب إبسلون على 2 عدد موجب إذا هذا
+
+64
+00:08:54,230 --> 00:09:00,830
+عدد موجب الـ Archimedean property بتقول لأي عدد
+
+65
+00:09:00,830 --> 00:09:05,450
+موجب زي هذا بقدر ألاقي عدد طبيعي capital N مقلوب
+
+66
+00:09:05,450 --> 00:09:09,250
+وأصغر من العدد الموجب وبالتالي capital N هذا زي ما
+
+67
+00:09:09,250 --> 00:09:13,510
+أنتوا شايفين مرتبط بإبسلون بالمتباينة هذه وبالتالي
+
+68
+00:09:13,510 --> 00:09:18,970
+capital N هذا depends أو يعتمد على إبسلونOkay إذا
+
+69
+00:09:18,970 --> 00:09:22,510
+هذا من الـ Archimedean Property طب ليش أنا أختارت
+
+70
+00:09:22,510 --> 00:09:29,930
+هذا العدد عشان نخل المسافة بين Xm و 0 أصغر من Y
+
+71
+00:09:29,930 --> 00:09:38,290
+فرطبناها أو ركبناها عساس نصل لإيه الهدف هذا تعالى
+
+72
+00:09:38,290 --> 00:09:46,950
+نشوف إذا hence وبالتالي hence بناء على ذلكلو أخدت
+
+73
+00:09:46,950 --> 00:09:57,950
+n أكبر من capital N هذا بيقدي ان n سالب واحد أكبر
+
+74
+00:09:57,950 --> 00:10:08,270
+من أو ساوي capital N وهذا بيقدي ان absolute n في b
+
+75
+00:10:08,270 --> 00:10:16,690
+to n سالب صفرإيش هذا بيساوي بيساوي n في بيقص n لأن
+
+76
+00:10:16,690 --> 00:10:26,710
+هذا عدد موجب ومن هنا من هنا n في بيقص n أصغر من أو
+
+77
+00:10:26,710 --> 00:10:33,690
+يساوي اتنين على n
+
+78
+00:10:33,690 --> 00:10:40,750
+سالب واحد في a تربية وهذا
+
+79
+00:10:40,750 --> 00:10:42,390
+أصغر من أو يساوي
+
+80
+00:10:50,580 --> 00:11:00,000
+هذا أصغر من أوي ساوي واحد على capital M في اتنين
+
+81
+00:11:00,000 --> 00:11:11,660
+على A تربية يعني
+
+82
+00:11:11,660 --> 00:11:19,140
+أنا من هنا منواحد على N سالب واحد مقلوب N سالب
+
+83
+00:11:19,140 --> 00:11:27,000
+واحد هيطلع أعظم أسئلة مقلوب capital N وهذا
+
+84
+00:11:27,000 --> 00:11:33,400
+عبارة عن واحد على N سالب واحد اتنين على A تربية
+
+85
+00:11:36,970 --> 00:11:42,010
+فمقلوب n سالب واحد اصغر من او ساوي مقلوب capital N
+
+86
+00:11:42,010 --> 00:11:51,630
+في اتنين على ا تربية تمام؟ شفتوا من اين اتيت؟
+
+87
+00:11:51,630 --> 00:11:58,690
+طيب انا من هنا من هنا واحد مقلوب capital N اصغر من
+
+88
+00:11:58,690 --> 00:12:09,470
+ا تربية في ابسلون على اتنين ضربتنين على اي تربية
+
+89
+00:12:09,470 --> 00:12:13,630
+اذا شوفت ليه اخدت ان هنا اي تربية في ابسلون على
+
+90
+00:12:13,630 --> 00:12:19,210
+اتنين عشان اختصر اي تربية مع اي تربية واتنين مع
+
+91
+00:12:19,210 --> 00:12:26,870
+اتنين ويبقى ابسلون اذا
+
+92
+00:12:26,870 --> 00:12:35,170
+ماذا اثبتنا اثبتت انه لأي given ابسلون عدد موجب
+
+93
+00:12:35,980 --> 00:12:42,520
+يوجد capital N تعتمد على epsilon بحيث لكل N أكبر
+
+94
+00:12:42,520 --> 00:12:48,260
+من capital N طلع عندي المسافة بين الحد العام لل
+
+95
+00:12:48,260 --> 00:12:51,900
+sequence اللي هو N في BN وال limit المنشودة اللي
+
+96
+00:12:51,900 --> 00:12:57,860
+هي سفر المسافة بينهم طلعت أصغر من epsilonإذا حسب
+
+97
+00:12:57,860 --> 00:13:03,100
+تعريف epsilon capital N لل limit هذا معناه أنه ال
+
+98
+00:13:03,100 --> 00:13:06,720
+limit بما أن هذا صحيح لأي epsilon، epsilon was
+
+99
+00:13:06,720 --> 00:13:11,540
+arbitrary إذاً هيك ممكن أثبتنا إن limit N في B to
+
+100
+00:13:11,540 --> 00:13:16,760
+N as N tends to infinity بساوي سفر وهو المطلوب
+
+101
+00:13:16,760 --> 00:13:22,640
+okay تمام؟ إذاً
+
+102
+00:13:22,640 --> 00:13:27,950
+هنا استخدمنا ال binomial theorem ساعدتنيفي الوصول
+
+103
+00:13:27,950 --> 00:13:33,590
+للمتباينة هذه و الوصول للمتباينة هذه اللي احنا
+
+104
+00:13:33,590 --> 00:13:42,730
+استخدمناها في البرهان سهلة البرهان تمام بفهم
+
+105
+00:13:42,730 --> 00:13:45,970
+الخطوة هذه اقول ان ال limit يعني اخد ال limit
+
+106
+00:13:45,970 --> 00:13:49,750
+للترفين اقول ان واحد على n نقص الواحد ماهي cost
+
+107
+00:13:49,750 --> 00:13:55,840
+zero اذا ال limit المقدراتمن أنهي المتباينة؟ هذه؟
+
+108
+00:13:55,840 --> 00:14:02,160
+بنفع اه بنفع يعني انت عندك هنا ممكن واحد يستخدم ال
+
+109
+00:14:02,160 --> 00:14:08,980
+sandwich او ال squeeze theorem فبدل ما نستخدم
+
+110
+00:14:08,980 --> 00:14:15,680
+تعريف epsilon capital N نيجي نقول انه الان انا
+
+111
+00:14:15,680 --> 00:14:25,150
+عندي هاي Nفي b to n طلعت أصغر من أو ساوي اتنين على
+
+112
+00:14:25,150 --> 00:14:31,450
+n سالب واحد في a تردية وطبعا بالتأكيد هذا أكبر من
+
+113
+00:14:31,450 --> 00:14:35,390
+أو ساوي سفر لأن ال n عدد موجب و ال b to n عدد موجب
+
+114
+00:14:35,390 --> 00:14:43,530
+وهذا صحيح لكل n أكبر من واحدالان هذا عبارة عن
+
+115
+00:14:43,530 --> 00:14:47,410
+sequence هي الحد العام تبعها لما N تقول ل infinity
+
+116
+00:14:47,410 --> 00:14:52,230
+مقلوق N سالب واحد تقول ل infinity وبالتالي مقلوبها
+
+117
+00:14:52,230 --> 00:14:55,990
+يقول ل infinity فى ثابت موجة باتنين على A تربية
+
+118
+00:14:55,990 --> 00:15:01,570
+عفوا لما N تقول ل infinity المقام بيروح ل infinity
+
+119
+00:15:01,570 --> 00:15:07,110
+وبالتالي مقلوب وبروح ل سفر تمام؟
+
+120
+00:15:16,990 --> 00:15:22,190
+إذن هذه الـ sequence تقول لـ 0 نهايتها 0 وهذه ال
+
+121
+00:15:22,190 --> 00:15:26,970
+constant sequence 0 نهايتها 0 إذن by squeeze
+
+122
+00:15:26,970 --> 00:15:30,410
+theorem limit ال sequence هذه بيساوي 0 وبلاش
+
+123
+00:15:30,410 --> 00:15:35,350
+نستخدم تعريف epsilon capital N لكن هذا السؤال في
+
+124
+00:15:35,350 --> 00:15:39,750
+section 3-1 ماكناش ماخدين ال squeeze theorem فلازم
+
+125
+00:15:39,750 --> 00:15:43,770
+اتحاليها على طريقة باستخدام ال definition لكن لو
+
+126
+00:15:43,770 --> 00:15:48,910
+في الامتحانو ممكن ماتفرجش انت متعلم ال definition
+
+127
+00:15:48,910 --> 00:15:52,630
+و متعلم ال exquisite theorem و استخدم أي طريقة
+
+128
+00:15:52,630 --> 00:15:58,330
+تعجبك okay تمام في
+
+129
+00:15:58,330 --> 00:16:01,010
+أسئلة تانية في حد عنده أي سؤال تاني في section
+
+130
+00:16:01,010 --> 00:16:07,890
+تلاتة واحد و تلاتة اتنين تفضلي في أي section تلاتة
+
+131
+00:16:07,890 --> 00:16:10,510
+واحد طيب ماشي الحالة
+
+132
+00:16:50,410 --> 00:17:09,310
+السؤال عشرة section تلاتة واحد السؤال هذا بيقول if
+
+133
+00:17:09,310 --> 00:17:20,060
+limit sequence x in بساوي xوالـ X هذا أكبر من
+
+134
+00:17:20,060 --> 00:17:24,880
+السفر then
+
+135
+00:17:24,880 --> 00:17:29,340
+then
+
+136
+00:17:29,340 --> 00:17:36,780
+there exist يوجد capital N عدد طبيعي او capital M
+
+137
+00:17:36,780 --> 00:17:48,170
+natural number عدد طبيعي such thatxn أكبر من السفر
+
+138
+00:17:48,170 --> 00:18:08,950
+لكل n أكبر من أو ساوي م لت
+
+139
+00:18:08,950 --> 00:18:13,250
+y أكبر من السفر be given
+
+140
+00:18:17,620 --> 00:18:23,600
+خد أي إبسلون أكبر من الصفر إذن
+
+141
+00:18:23,600 --> 00:18:30,900
+إبسلون على اتنين برضه بيطلع عدد موجة طيب
+
+142
+00:18:30,900 --> 00:18:38,880
+احنا فرضين ان limit xn بيساوي x إذن since xn
+
+143
+00:18:38,880 --> 00:18:44,960
+converges to xوهي إبسلون على اتنين عدد أكبر من
+
+144
+00:18:44,960 --> 00:18:54,480
+السفر إذا يوجد capital M عدد طبيعي يعتمد على
+
+145
+00:18:54,480 --> 00:18:58,840
+إبسلون عدد
+
+146
+00:18:58,840 --> 00:19:05,140
+طبيعي بحيث أنه لكل N أكبر من أو ساوي capital M
+
+147
+00:19:05,140 --> 00:19:33,800
+تطلع المسافة من XNهو ال X أصغر من Y أتنين طيب
+
+148
+00:19:33,800 --> 00:19:34,980
+أنا ال epsilon هذا
+
+149
+00:19:37,520 --> 00:19:44,200
+ممكن اخده انا عندي من الفرض x اكبر من 0 فممكن اخد
+
+150
+00:19:44,200 --> 00:19:49,640
+ال epsilon هذا بساوي x بساوي
+
+151
+00:19:49,640 --> 00:19:56,480
+x انا
+
+152
+00:19:56,480 --> 00:20:04,400
+ممكن اخد ال epsilon بساوي x او حتى x ع 2 او x ع 2
+
+153
+00:20:04,400 --> 00:20:10,380
+هذا بالتأكيدالإبسلون هذا هي عدد موجب اعتبره هو
+
+154
+00:20:10,380 --> 00:20:15,660
+given وبالتالي
+
+155
+00:20:15,660 --> 00:20:20,580
+انا اخدت الأن إبسلون X عدد موجب إذا X عتنين عدد
+
+156
+00:20:20,580 --> 00:20:26,270
+موجب واخدت إبسلون عبارة عن X عتنينفاعتبر هذا given
+
+157
+00:20:26,270 --> 00:20:31,070
+إبسلون إبسلون مُعطى مُسبَخًا فحسب التعريف بما أن X
+
+158
+00:20:31,070 --> 00:20:34,390
+in converge ل X إذا يوجد عدد طبيعي يعتمد على
+
+159
+00:20:34,390 --> 00:20:38,710
+إبسلون بحيث لكل in أكبر من أو ساوي capital N
+
+160
+00:20:38,710 --> 00:20:45,730
+المسافة هذه أصغر من إبسلون الآن عوض عن إبسلون
+
+161
+00:20:45,730 --> 00:20:54,490
+بساوي X ع 2 فهذا بيؤديالان فك ال absolute value
+
+162
+00:20:54,490 --> 00:21:03,070
+فبطلع عندي xn سالب x أصغر من x ع 2 أكبر من سالب x
+
+163
+00:21:03,070 --> 00:21:08,570
+ع 2، مظبوط؟
+
+164
+00:21:08,570 --> 00:21:15,370
+طب
+
+165
+00:21:15,370 --> 00:21:17,790
+لو أخدت هذا الجزء من المتباينة
+
+166
+00:21:20,790 --> 00:21:28,770
+فبصير عندي xn أكبر من ودي x على الناحية التالية
+
+167
+00:21:28,770 --> 00:21:38,050
+أكبر من x سالب x على 2 وبالتالي
+
+168
+00:21:38,050 --> 00:21:46,710
+إذا أنا عندي هاي xn أكبر من x على 2 وهذا أكبر من
+
+169
+00:21:46,710 --> 00:21:57,210
+السفرتمام؟ وهذا صحيح إذا طلع عندي xn أكبر من السفر
+
+170
+00:21:57,210 --> 00:22:07,170
+وهذا صحيح لكل n أكبر من أو ساوي capital M وهو
+
+171
+00:22:07,170 --> 00:22:12,630
+المطلوبتمام إذا هنا استخدمنا تعريف epsilon capital
+
+172
+00:22:12,630 --> 00:22:19,690
+M وهنا استنتجنا إن لازم xn يطلع أكبر من السفر لكل
+
+173
+00:22:19,690 --> 00:22:32,210
+M أكبر من أو يساوي capital M تمام واضح البرهان طيب
+
+174
+00:22:32,210 --> 00:22:34,110
+في أي أسئلة تانية؟
+
+175
+00:22:37,830 --> 00:22:48,330
+section تلاتة اتنين مين
+
+176
+00:22:48,330 --> 00:22:54,390
+عنده سؤال اي سؤال في اي section تلاتة اتنين تلاتة
+
+177
+00:22:54,390 --> 00:23:03,070
+اتنين سبعتاش
+
+178
+00:23:03,070 --> 00:23:05,150
+section تلاتة اتنين
+
+179
+00:23:40,180 --> 00:23:44,200
+أنا في عندي هنا sequence of positive real numbers
+
+180
+00:23:44,200 --> 00:23:55,680
+إذا xn حدود عموجة بقى لكل n such
+
+181
+00:23:55,680 --> 00:24:00,560
+that limit ل
+
+182
+00:24:00,560 --> 00:24:11,550
+xn زاد واحد على xn لما n تقول infinityبساوي عدد ال
+
+183
+00:24:11,550 --> 00:24:20,550
+أكبر من واحد و المقلوب show اثبت في الحالة هذه ان
+
+184
+00:24:20,550 --> 00:24:25,750
+ال sequence
+
+185
+00:24:25,750 --> 00:24:30,170
+xm is
+
+186
+00:24:30,170 --> 00:24:34,350
+unbounded is not bounded
+
+187
+00:24:38,480 --> 00:24:46,100
+and hence not
+
+188
+00:24:46,100 --> 00:24:53,460
+convergent لأن لو كانت convergent تطلع bounded
+
+189
+00:25:13,370 --> 00:25:17,190
+يعني من الشرط هذا ممكن تباطم الـ sequence
+
+190
+00:25:17,190 --> 00:25:21,290
+increasing متزايدة
+
+191
+00:26:05,950 --> 00:26:08,750
+أه ..
+
+192
+00:26:31,500 --> 00:26:38,240
+ممكن نعمل برهان بال .. بالتناقض افرم
+
+193
+00:26:38,240 --> 00:26:48,640
+انها bounded وممكن نصل لتناقض من تعريف ال .. هنا
+
+194
+00:26:48,640 --> 00:26:56,680
+ال sequence هذه of quotient convergent لعدد L أكبر
+
+195
+00:26:56,680 --> 00:27:00,220
+من واحد ممكن باستخدامه
+
+196
+00:27:02,850 --> 00:27:14,250
+باستخدام تعريف الـ convergence زاد او
+
+197
+00:27:14,250 --> 00:27:18,390
+ممكن من الفرض هذا لثبت انه ال sequence unbounded
+
+198
+00:27:18,390 --> 00:27:22,870
+او ممكن بالتناقض اما باستخدام تعريف epsilon
+
+199
+00:27:22,870 --> 00:27:29,600
+capital N من ال convergence هذانعمل برهان بالتناقض
+
+200
+00:27:29,600 --> 00:27:35,560
+لنصل إلى هاجة يعني تتناقض مع الفرض اللي هنا على أي
+
+201
+00:27:35,560 --> 00:27:40,540
+حال انا هسيب في حد حل السؤال هذا طيب انا هسيبكم
+
+202
+00:27:40,540 --> 00:27:45,320
+تفكروا فيه و تقرؤوا برهان شوفوا برهان انا في
+
+203
+00:27:45,320 --> 00:27:49,380
+البرهان النظرية هذه اللي كانت قلتلكم اقرؤوا
+
+204
+00:27:49,380 --> 00:27:54,650
+فحاولوا انك تتسفيدوا من البرهان تبع النظريةاللي
+
+205
+00:27:54,650 --> 00:27:57,930
+كانت بتقول إن لو كانت ال limit هذه بساوي L أصغر من
+
+206
+00:27:57,930 --> 00:28:03,370
+واحد فبتطلع ال sequence convergent للصفر فإقرأوا
+
+207
+00:28:03,370 --> 00:28:08,710
+البرهان تبع النظرية هذه وشوفوا كيف يعني النظرية
+
+208
+00:28:08,710 --> 00:28:12,750
+هذه أثبتت وشوفوا لو كان ال L أكبر من واحد كيف
+
+209
+00:28:12,750 --> 00:28:17,450
+بيطلع البرهان إيش اللي بيخل البرهان هذا يبطل صحيح
+
+210
+00:28:18,870 --> 00:28:23,230
+أه فعيدوا قراءته و حالكم تحلوه و إذا ما حلتوهوش
+
+211
+00:28:23,230 --> 00:28:27,290
+يعني المرة الجاية ممكن تحلوا مع بعض أه ماشي الحال
+
+212
+00:28:27,290 --> 00:28:30,470
+فإقرأوا
+
+213
+00:28:30,470 --> 00:28:35,150
+برهان النظرية اللي سيبنا قولنالكم البرهانها موجود
+
+214
+00:28:35,150 --> 00:28:38,030
+في الكتاب و بدي أكم تقرأوا تفهموا هل قرأتوا
+
+215
+00:28:38,030 --> 00:28:45,010
+البرهان؟حاولوا تقرأ ايه حاولوا تتعملوا ايه تشوفوا
+
+216
+00:28:45,010 --> 00:28:50,070
+وين في البرهان ال ال اكبر من واحد بتخلي البرهان
+
+217
+00:28:50,070 --> 00:28:55,050
+يبطل صح وين المشكلة وشوفوا
+
+218
+00:28:55,050 --> 00:28:58,210
+اذا كانوا تقدروا تحلو ولا لأ اذا انا هاسيبكم
+
+219
+00:28:58,210 --> 00:29:02,610
+تفكروا فيه مرة تانية و تحاولوا تحلوه اذا ماعرفتهوش
+
+220
+00:29:02,610 --> 00:29:09,110
+ممكن نحله مرة تانية او في المرة القادمة نعم مين
+
+221
+00:29:09,110 --> 00:29:13,190
+اللي بتحكي هذهماحدش لو سمحته تحكي إلا غير ترفع
+
+222
+00:29:13,190 --> 00:29:18,790
+إيدها الأول و بعدين أقزمها طيب إذا هذا السؤال
+
+223
+00:29:18,790 --> 00:29:22,510
+هنسيبكم يتفكروا فيه مرة تانية في أي أسئلة تانية
+
+224
+00:29:22,510 --> 00:29:26,710
+section تلاتة اتنين أو تلاتة واحد
+
+225
+00:29:45,050 --> 00:29:50,450
+في حد عندها سؤال في نفس
+
+226
+00:29:50,450 --> 00:29:55,770
+ال section نعم فالقادة ماعطينا sequence انه احنا
+
+227
+00:29:55,770 --> 00:29:59,390
+نشوف اذا هي تتجوز و لا تتجوز استخدمت ال ratio test
+
+228
+00:29:59,390 --> 00:30:04,310
+نعم طلعت ال limit بتساوي واحد و احنا الشرط ان تكون
+
+229
+00:30:04,310 --> 00:30:09,790
+ال limit اقل من واحد صح فالقادة هذه بتطلع تطلع ال
+
+230
+00:30:09,790 --> 00:30:12,430
+limit ل sequence لو معطنيها تساوي zero
+
+231
+00:30:15,730 --> 00:30:21,110
+لأ لازم يكون أصغر من واحد مابتساويش الواحد معناته
+
+232
+00:30:21,110 --> 00:30:26,150
+ال test بيفشل لأ هي سوى واحد إذا بالساوية واحد
+
+233
+00:30:26,150 --> 00:30:33,430
+ارجعي لهي تمرين 16 بقول إذا كانت ال limit بالساوية
+
+234
+00:30:33,430 --> 00:30:38,710
+واحد فممكن
+
+235
+00:30:38,710 --> 00:30:41,650
+تكون ال sequence convergent أو divergent يعني هذا
+
+236
+00:30:41,650 --> 00:30:46,920
+ال test ال ratio test بيفشلهي في سؤال 16 هتجيب
+
+237
+00:30:46,920 --> 00:30:52,480
+بمثالين اول شي اذا كانت ال limit هذه بالساوي واحد
+
+238
+00:30:52,480 --> 00:30:59,740
+فهتجيب بمثالين ال limit تبع ال quotient تبع كل
+
+239
+00:30:59,740 --> 00:31:03,220
+واحدة بالساوي واحد لكن واحدة convergent واحدة
+
+240
+00:31:03,220 --> 00:31:08,140
+divergent وبالتالي ال test هذا بيفشل اذا كانت ال L
+
+241
+00:31:08,140 --> 00:31:12,420
+بالساوي واحد اما لو كانت ال L اصغر من واحدفال
+
+242
+00:31:12,420 --> 00:31:16,400
+sequence xn تطلع convergent للصفر إذا كان ال L
+
+243
+00:31:16,400 --> 00:31:21,740
+أكبر من 1 فال sequence تطلع divergent okay تمام
+
+244
+00:31:21,740 --> 00:31:30,340
+هذا هو ال ratio test فهل جبت أمثلة؟ كويس ممتاز طيب
+
+245
+00:31:30,340 --> 00:31:36,220
+إيش دخل دي؟ دي معناته بدك تستخدم طريقة تانية غير
+
+246
+00:31:36,220 --> 00:31:43,260
+ال ratio test صحيح لأن حسب سؤال 16الـ test بيفشل
+
+247
+00:31:43,260 --> 00:31:48,320
+إذا كانت limit ال ratio ال ratio test بيفشل إذا
+
+248
+00:31:48,320 --> 00:31:53,020
+كانت limit لل ratio بساوي واحد وبالتالي بدك تبحث
+
+249
+00:31:53,020 --> 00:31:54,300
+عن طريقة تانية
+
+250
+00:32:12,840 --> 00:32:31,940
+طيب في أسئلة تانية في
+
+251
+00:32:31,940 --> 00:32:35,300
+section تلاتة واحد و تلاتة اتنين في عندكم أي سؤال
+
+252
+00:32:35,300 --> 00:32:42,490
+مافيش أسئلة لسه مش دارسين مش محاضرينكان واحدة بس
+
+253
+00:32:42,490 --> 00:32:51,430
+لدرسة و هم اللي بيسألوا الأسئلة والباقي مستمع طيب
+
+254
+00:32:51,430 --> 00:32:54,930
+بتحبوا نرجع لأسئلة chapter اتنين في أسئلة في
+
+255
+00:32:54,930 --> 00:33:01,130
+chapter اتنين اذا
+
+256
+00:33:01,130 --> 00:33:10,470
+في عندكم أسئلة في section اتنين
+
+257
+00:33:10,470 --> 00:33:11,010
+اربعة
+
+258
+00:33:26,250 --> 00:33:35,710
+السؤال هذا يعني في الكتاب أعطيكم hint كيف
+
+259
+00:33:35,710 --> 00:33:41,790
+يعني تحلوه موجود في نهاية الكتاب فحاولوا تقرا أيه
+
+260
+00:33:41,790 --> 00:33:46,530
+تقرا ال hint هذا و تستفيدي منه و تشوفي يعني هذا
+
+261
+00:33:46,530 --> 00:33:54,780
+أكيد هساعدك في حل السؤال شفتيه قبل هيك؟طيب طلعي
+
+262
+00:33:54,780 --> 00:33:59,360
+خلف الكتاب فيه hint او ارشادات لبعض التمرين
+
+263
+00:33:59,360 --> 00:34:06,680
+بيعطيكي يعني طريقة مقتضبة لحل او بحط رجلك على طريق
+
+264
+00:34:06,680 --> 00:34:12,840
+الحل فحاولي تقرا ايه و تستفيدي منه و اذا فهمتي
+
+265
+00:34:12,840 --> 00:34:19,640
+الارشاد هذا ممكن تحل السؤال فانتي و زمايلكتطلعوا
+
+266
+00:34:19,640 --> 00:34:23,580
+على الإرشادات هذه تبعت التمرين أو بعض الحلول
+
+267
+00:34:23,580 --> 00:34:28,240
+المختصرة و حاولوا تستفيدوا منها و تفصلوها و تكتبوا
+
+268
+00:34:28,240 --> 00:34:35,340
+الحل بطريقة واضحة و كاملة فهسيبكم
+
+269
+00:34:35,340 --> 00:34:42,440
+تقرؤوا الإرشاد و تحاولوا تستفيدوا منه أي أسئلة
+
+270
+00:34:42,440 --> 00:34:49,980
+تانية في section 2 4 2 3 2 2إن واحد الجزء اللي
+
+271
+00:34:49,980 --> 00:34:56,460
+داخل الامتحان، في عندكم أي سؤال فيه؟ منين في عندها
+
+272
+00:34:56,460 --> 00:35:00,260
+سؤال؟
+
+273
+00:35:00,260 --> 00:35:07,020
+في أسئلة كتير حلوة ومهمة ويا بدوا أنكم مش مدرسين
+
+274
+00:35:07,020 --> 00:35:08,680
+ولا حتى مستعدين للامتحان
+
+275
+00:35:16,700 --> 00:35:20,800
+في اي اسلة في chapter 2 او chapter 3 الجزء الداخل
+
+276
+00:35:20,800 --> 00:35:21,960
+في الامتحان
+
+277
+00:36:04,610 --> 00:36:11,090
+فيش أسئلة؟ طيب
+
+278
+00:36:11,090 --> 00:36:15,390
+أنا هحللكم يعني كمان سؤالين واحد من section تلاتة
+
+279
+00:36:15,390 --> 00:36:21,070
+واحد وواحد من تلاتة اتنين
+
+280
+00:36:21,070 --> 00:36:28,670
+خليني
+
+281
+00:36:28,670 --> 00:36:29,830
+أحل السؤال
+
+282
+00:36:46,350 --> 00:36:58,770
+يعني مثلا يعني
+
+283
+00:36:58,770 --> 00:37:04,090
+مثلا السؤال الخامسة
+
+284
+00:37:04,090 --> 00:37:10,530
+السؤال
+
+285
+00:37:10,530 --> 00:37:16,320
+الخامسة الفرح دي section تلاتة واحدuse definition
+
+286
+00:37:16,320 --> 00:37:25,660
+use definition of limit to
+
+287
+00:37:25,660 --> 00:37:33,880
+establish انه
+
+288
+00:37:33,880 --> 00:37:37,800
+ال limit لإن
+
+289
+00:37:37,800 --> 00:37:44,970
+تربية سالب واحد علىتنين انتر بيه زائد تلاتة ال
+
+290
+00:37:44,970 --> 00:37:52,850
+sequence اللي حد العم تبعها الكاسر هذا بيساوي نص و
+
+291
+00:37:52,850 --> 00:37:56,410
+بيثبت ان ال sequence هذي convergence و نهايتها نص
+
+292
+00:37:56,410 --> 00:38:00,390
+بيستخدم ال definition ماهو ال definition المقصود
+
+293
+00:38:00,390 --> 00:38:06,700
+في هنااللي هو تعريف epsilon capital N لل limit أو
+
+294
+00:38:06,700 --> 00:38:21,360
+للنهاية تعريف epsilon capital N طيب انا
+
+295
+00:38:21,360 --> 00:38:27,300
+في النهاية في نهاية المطاف تعريف epsilon capital N
+
+296
+00:38:31,470 --> 00:38:36,710
+عايزني أثبت أن المسافة بين xn اللي هو enter بيها
+
+297
+00:38:36,710 --> 00:38:42,510
+سالب واحد على اتنين enter بيها زائد تلاتة سالب نص
+
+298
+00:38:42,510 --> 00:38:47,270
+بدنا هذا يكون أصغر من أي given epsilon عدد موجه
+
+299
+00:38:47,270 --> 00:38:53,950
+لكل n أكبر من أو ساوي capital N حيث capital N عدد
+
+300
+00:38:53,950 --> 00:39:00,410
+طبيعي هنجيبه ويعتمد على ال epsilonفنشوف مع بعض هذا
+
+301
+00:39:00,410 --> 00:39:07,410
+إيه من الآخر طيب إذا هنا solution إذا بقول أنا
+
+302
+00:39:07,410 --> 00:39:12,490
+عايز في النهاية absolute interview سالب واحد على
+
+303
+00:39:12,490 --> 00:39:17,970
+اتنين interview زائد تلاتة سالب مصر بسأل نفسي متى
+
+304
+00:39:17,970 --> 00:39:24,570
+هذا بيكون أصغر من أي epsilon موجب هذا بكافئ
+
+305
+00:39:27,220 --> 00:39:34,160
+الـ absolute value بين واحد المقامات هي اتنين في
+
+306
+00:39:34,160 --> 00:39:40,260
+اتنين انت ربيع الزائد تلاتة و بيصير عندنا اتنين
+
+307
+00:39:40,260 --> 00:39:46,720
+انت ربيع سالب اتنين تضرب هذا في اتنين سالب اتنين
+
+308
+00:39:46,720 --> 00:39:53,680
+انت ربيع موجة بتلاتة لان هذا المقدار اللي فوق
+
+309
+00:39:53,680 --> 00:39:55,580
+بيبقى اصغر من epsilon
+
+310
+00:39:58,630 --> 00:40:02,730
+طيب أنا عندي اتنين in تربية و هاي سالب اتنين in
+
+311
+00:40:02,730 --> 00:40:07,450
+تربية بروحوا مع بعض و عندي سالب اتنين و السالب
+
+312
+00:40:07,450 --> 00:40:10,630
+تلاتة بطلع خمسة يعني دلوقتي بصير absolute سالب
+
+313
+00:40:10,630 --> 00:40:16,330
+خمسة على اتنين في
+
+314
+00:40:16,330 --> 00:40:19,230
+اتنين in تربية زائد تلاتة
+
+315
+00:40:24,890 --> 00:40:31,830
+بدي هذا يكون أصغر من ي طيب
+
+316
+00:40:31,830 --> 00:40:38,990
+هاد عبارة عن خمسة هاد
+
+317
+00:40:38,990 --> 00:40:48,570
+عبارة عن خمسة على اتنين اتنين enter بيها زي
+
+318
+00:40:48,570 --> 00:40:49,370
+التلاتة
+
+319
+00:40:52,780 --> 00:41:02,080
+متى بيكون هذا أصغر من epsilon هذا
+
+320
+00:41:02,080 --> 00:41:09,220
+بكافئ هذا
+
+321
+00:41:09,220 --> 00:41:15,900
+بكافئ ان اقول واحد متى بيكون واحد على اتنين انتر
+
+322
+00:41:15,900 --> 00:41:30,390
+بيه زائد تلاتة أصغر منإتنين على خمسة إبسلون طيب
+
+323
+00:41:30,390 --> 00:41:35,550
+إذا
+
+324
+00:41:35,550 --> 00:41:42,470
+أنا ممكن أستخدم ال Archimedean property إذا هنا
+
+325
+00:41:42,470 --> 00:41:49,690
+let إبسلون أكبر من السفر بجبل
+
+326
+00:41:51,720 --> 00:41:57,880
+نبدأ بأبسلون أكبر من السفر تعريف epsilon capital N
+
+327
+00:41:57,880 --> 00:42:02,160
+بيقول ابدا بأبسلون أكبر من السفر و جيب capital N
+
+328
+00:42:03,440 --> 00:42:07,880
+بحيث أن المسافة بين XN و X أصغر من إمسون لكل N
+
+329
+00:42:07,880 --> 00:42:15,440
+أكبر من ما يستوى capital N بحيث أن المسافة بين XN
+
+330
+00:42:15,440 --> 00:42:17,600
+بحيث أن المسافة بين XN و X أصغر من إمسون لكل N
+
+331
+00:42:17,600 --> 00:42:17,940
+أكبر من ما يستوى capital N بحيث أن المسافة بين XN
+
+332
+00:42:17,940 --> 00:42:20,660
+و X أصغر من إمسون لكل N أكبر من ما يستوى capital N
+
+333
+00:42:20,660 --> 00:42:21,360
+بحيث أن المسافة بين XN و X أصغر من إمسون لكل N
+
+334
+00:42:21,360 --> 00:42:23,640
+أكبر من ما يستوى capital N بحيث أن المسافة بين XN
+
+335
+00:42:23,640 --> 00:42:28,200
+و X أصغر
+
+336
+00:42:28,200 --> 00:42:35,040
+من إمسون لكل N أكبر من ما يستوى capital N بحit
+
+337
+00:42:35,040 --> 00:42:43,620
+choose it choose طبعا
+
+338
+00:42:43,620 --> 00:42:51,500
+by Archimedean property capital
+
+339
+00:42:51,500 --> 00:43:01,200
+N عدد طبيعي بحيث انه واحد علىإتنين في capital N
+
+340
+00:43:01,200 --> 00:43:07,820
+تربية زائد تلاتة أصغر من اتنين على خمسة epsilon
+
+341
+00:43:07,820 --> 00:43:20,180
+ممكن
+
+342
+00:43:20,180 --> 00:43:26,070
+ألاقي capital N عدد طبيعيمقنوب 2 في مربع زائد
+
+343
+00:43:26,070 --> 00:43:32,170
+تلاتة طبعا تلاتة مش epsilon واحد على اتنين enter
+
+344
+00:43:32,170 --> 00:43:41,290
+key زائد تلاتة اصغر من اتنين على خمسة epsilon الان
+
+345
+00:43:41,290 --> 00:43:46,770
+اذا لو اخدت small n اكبر من أوسع ال capital N هذا
+
+346
+00:43:46,770 --> 00:44:00,110
+بيقدي انه واحد علىتنين انت ربيع زائد تلاتة او بلاش
+
+347
+00:44:00,110 --> 00:44:09,230
+absolute اه بيقدي ان absolute طيب
+
+348
+00:44:09,230 --> 00:44:16,750
+هذا بيقدي ان الكلام هذا اصغر من او يساوي واحد على
+
+349
+00:44:16,750 --> 00:44:25,510
+اتنين capital enter بيه زائد تلاتةوبالتالي هذا
+
+350
+00:44:25,510 --> 00:44:31,390
+بيقدي ان ال absolute value لان تربية سالب واحد على
+
+351
+00:44:31,390 --> 00:44:42,670
+اتنين ان تربية سالب تلاتة سالب نص طلع هذا
+
+352
+00:45:09,580 --> 00:45:16,580
+خمسة على اتنين
+
+353
+00:45:16,580 --> 00:45:20,140
+في اتنين enter بي عزائى التلاتة
+
+354
+00:45:28,400 --> 00:45:34,680
+وهذا هيطلع أصغر منه ويسوي خمسة على اتنين في اتنين
+
+355
+00:45:34,680 --> 00:45:42,000
+capital Interbias زاد تلاتة ومن هنا هذا أصغر من
+
+356
+00:45:42,000 --> 00:45:47,320
+خمسة
+
+357
+00:45:47,320 --> 00:45:54,280
+على اتنين ضرب اتنين على خمسة في epsilon اللي هو
+
+358
+00:45:54,280 --> 00:45:55,160
+بيطلع epsilon
+
+359
+00:45:59,840 --> 00:46:03,560
+أذن هذه لأي epsilon أكبر من صفر لجيت فيه capital N
+
+360
+00:46:03,560 --> 00:46:08,200
+مرتبطة لcapital N هي في epsilon depends on epsilon
+
+361
+00:46:08,200 --> 00:46:12,280
+بتعتمد على epsilon بحيث لكل n أكبر من او سوى
+
+362
+00:46:12,280 --> 00:46:17,920
+capital N طلع absolute xn minus x أصغر من epsilon
+
+363
+00:46:19,350 --> 00:46:24,350
+طبعا إذا هذا حسب تعريف by definition of epsilon
+
+364
+00:46:24,350 --> 00:46:29,770
+capital N of limit بطلع عندي limit N تربيع سالب
+
+365
+00:46:29,770 --> 00:46:34,750
+واحد على اتنين N تربيع زائد تلاتة لما N تقول
+
+366
+00:46:34,750 --> 00:46:37,830
+infinity بساوي نص
+
+367
+00:46:44,620 --> 00:46:48,560
+بالمثل ممكن نحل باقى التمرين اللى هى الفروع A وB
+
+368
+00:46:48,560 --> 00:46:54,940
+وC باستخدام التعريف فحاولوا تتدربوا على التمرين
+
+369
+00:46:54,940 --> 00:47:02,700
+هادى و تحلوا أسئلة زيها فى حد عنده أي سؤال تانى فى
+
+370
+00:47:02,700 --> 00:47:07,260
+هذا ال section طيب
+
+371
+00:47:07,260 --> 00:47:12,220
+نحل كمان سؤال فى section تلاتة اتنين
+
+372
+00:47:27,570 --> 00:47:34,750
+في انكم أي سؤال بسكتشن تلاتة اتنين اخر
+
+373
+00:47:34,750 --> 00:47:35,250
+سؤال
+
+374
+00:47:57,080 --> 00:48:03,480
+هي سؤال واحد وعشرين section تلاتة
+
+375
+00:48:03,480 --> 00:48:13,760
+اتنين suppose
+
+376
+00:48:13,760 --> 00:48:24,980
+افترضي ان ال sequence x in converge ل x and ال
+
+377
+00:48:24,980 --> 00:48:33,200
+sequence y inand yn is such that is a sequence
+
+378
+00:48:33,200 --> 00:48:40,900
+such that for any epsilon for
+
+379
+00:48:40,900 --> 00:48:46,240
+any epsilon أكبر من السفر يوجد
+
+380
+00:48:46,240 --> 00:48:53,780
+m بحيث يوجد عدد m such that
+
+381
+00:48:56,580 --> 00:49:06,460
+absolute xn minus yn أصغر من إبسلون لكل N أكبر من
+
+382
+00:49:06,460 --> 00:49:14,260
+أو ساو كابتل N فالسؤال
+
+383
+00:49:14,260 --> 00:49:19,060
+does it
+
+384
+00:49:19,060 --> 00:49:22,820
+follow هل
+
+385
+00:49:22,820 --> 00:49:34,030
+ينتج من ذلكهل ال sequence yn تطلع
+
+386
+00:49:34,030 --> 00:49:44,210
+convergent فنشوف
+
+387
+00:49:44,210 --> 00:49:44,930
+مع بعض
+
+388
+00:49:53,440 --> 00:49:59,260
+كمان مرة اندي two sequences واحدة x in واحدة y in
+
+389
+00:49:59,260 --> 00:50:04,280
+ال sequence x in مُعطَى انها convergent to some x
+
+390
+00:50:04,280 --> 00:50:08,880
+إلى عدد ما x ال limit تبقى تاكس وال sequence y in
+
+391
+00:50:08,880 --> 00:50:14,600
+بتحقق الشرط هذا وهو
+
+392
+00:50:14,600 --> 00:50:19,600
+انه لأي epsilon أكبر من سفر في عدد طبيعي حتى هذا
+
+393
+00:50:19,600 --> 00:50:27,790
+عدد طبيعي المفروض يكونبنشر ال number بحيث انه لكل
+
+394
+00:50:27,790 --> 00:50:31,810
+n أكبر من ما يستوى capital N المسافة بين xn وyn
+
+395
+00:50:31,810 --> 00:50:35,510
+أصغر من نفسها هل هذا بيقدم ال sequence yn
+
+396
+00:50:35,510 --> 00:50:40,870
+convergent؟ هنشوف الآن أن فعلا تطلع ال sequence yn
+
+397
+00:50:40,870 --> 00:50:46,130
+convergent ونهايتها هي نفس نهاية ال sequence xn
+
+398
+00:50:46,130 --> 00:50:51,270
+لأن هنا الإجابة yes
+
+399
+00:50:53,550 --> 00:51:01,270
+and y in converge to x لكن
+
+400
+00:51:01,270 --> 00:51:07,570
+هذا بيده برهان اذا
+
+401
+00:51:07,570 --> 00:51:11,370
+to see this
+
+402
+00:51:11,370 --> 00:51:16,610
+نبدأ
+
+403
+00:51:16,610 --> 00:51:18,610
+بإبسلون أكبر من السفر
+
+404
+00:51:36,810 --> 00:51:44,450
+let by hypothesis من الفرض من
+
+405
+00:51:44,450 --> 00:51:50,820
+الفرض من ال hypothesisأنا عندي absolute xn minus
+
+406
+00:51:50,820 --> 00:51:54,860
+yn أصغر
+
+407
+00:51:54,860 --> 00:52:03,700
+من إبسلون أكبر من أو ساوي سفر وهذا صحيح لكل n أكبر
+
+408
+00:52:03,700 --> 00:52:10,440
+من أو ساوي capital M وهذا
+
+409
+00:52:10,440 --> 00:52:15,380
+الكلام صحيح لكل إبسلون أكبر من السفر
+
+410
+00:52:24,820 --> 00:52:36,980
+فمن هنا فمن
+
+411
+00:52:36,980 --> 00:52:45,680
+هنا بهدف بيقدي ان ال limit ل xn minus yn لما n
+
+412
+00:52:45,680 --> 00:52:49,420
+تقول infinity بساوي سفر
+
+413
+00:52:54,150 --> 00:52:58,570
+مش شرط هذا انا
+
+414
+00:52:58,570 --> 00:53:03,950
+عندي ال ..
+
+415
+00:53:03,950 --> 00:53:08,010
+ما معناه ان limit ال sequence هذه بساوة سفر؟ معناه
+
+416
+00:53:08,010 --> 00:53:16,620
+لأي epsilon أكبر من السفر يوجد capital Mعدد طبيعي
+
+417
+00:53:16,620 --> 00:53:21,840
+يعتمد على إبسلن بحيث أنه لكل n أكبر من أو ساوي
+
+418
+00:53:21,840 --> 00:53:28,860
+capital N هذا بيقدي أن absolute xn minus yn minus
+
+419
+00:53:28,860 --> 00:53:34,700
+الصفر أصغر من إبسلنهي معنى ان limit ال sequence
+
+420
+00:53:34,700 --> 00:53:40,740
+للفرق بساوي سفر ايش معنى هذا لأي epsilon أكبر من
+
+421
+00:53:40,740 --> 00:53:46,660
+سفر يوجد capital M يعتمد على N عدد طبيعي يعتمد على
+
+422
+00:53:46,660 --> 00:53:51,020
+ال epsilon بحيث لكل N أكبر من أو ساوي capital N
+
+423
+00:53:51,020 --> 00:53:55,540
+المسافة بين الحد العام لل sequence و limit اللي هي
+
+424
+00:53:55,540 --> 00:54:00,140
+سفر أصغر من epsilon هذا الكلام هى متحقق هنا هى
+
+425
+00:54:00,140 --> 00:54:04,850
+متحققتامام؟ إذا هذا بنحصل عليه وبالتالي limit xn
+
+426
+00:54:04,850 --> 00:54:14,070
+minus yn بساوي سفر ومنها الآن أنا عندي ال yn ممكن
+
+427
+00:54:14,070 --> 00:54:20,870
+كتبتها على صورة yn
+
+428
+00:54:20,870 --> 00:54:32,610
+سالب xn موجب xnوهذا بيساوي سالب Xn سالب Yn زاد Xn
+
+429
+00:54:32,610 --> 00:54:40,630
+تمام؟ إذا ال limit ل Yn as n tends to infinity
+
+430
+00:54:40,630 --> 00:54:49,110
+بيساوي limit الطرف اليمين ف limit Xn سالب Yn
+
+431
+00:54:49,110 --> 00:54:56,410
+مضروبة في سالب واحد بيطلع برا ال limitزائد limit
+
+432
+00:54:56,410 --> 00:55:03,770
+xn لما n تقول لإنفينيتي وهنا
+
+433
+00:55:03,770 --> 00:55:08,770
+لسه احنا مثبتين هذا عبارة عن سالب limit sequence
+
+434
+00:55:08,770 --> 00:55:16,570
+xn minus yn بالساوية سفر، سالد واحد في سفر
+
+435
+00:55:19,990 --> 00:55:26,850
+زاد limit xn اللي هي x تمام اذا limit ال sequence
+
+436
+00:55:26,850 --> 00:55:32,370
+yn تطلع بالساوي x اذا
+
+437
+00:55:32,370 --> 00:55:37,210
+هنا اثبتنا ان ال sequence yn تطلع convergent وال
+
+438
+00:55:37,210 --> 00:55:44,210
+limit تبعتها بالساوي x تمامالبرهان هنا اعتمد على
+
+439
+00:55:44,210 --> 00:55:49,890
+انه من الفرض انا عندي المثال لأي epsilon هذا الفرض
+
+440
+00:55:49,890 --> 00:55:57,390
+معناه ان limit ال sequence x in minus y in بالساوي
+
+441
+00:55:57,390 --> 00:56:04,290
+سفر وهذا اللي ساعدنا في الحل وهذا ناتج هي من تعريف
+
+442
+00:56:04,290 --> 00:56:09,190
+epsilon capital N لل limit هذا هو البرهان
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/YiGM8L9BEY0.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/YiGM8L9BEY0.srt
new file mode 100644
index 0000000000000000000000000000000000000000..72dec1bc6fe5591fb82752bc4785fcd3b92a5b35
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/YiGM8L9BEY0.srt
@@ -0,0 +1,1331 @@
+1
+00:00:20,920 --> 00:00:26,360
+بسم الله الرحمن الرحيم اليوم هناخد آخر لقاء في ال
+
+2
+00:00:26,360 --> 00:00:31,460
+course وهو تكملة section خمسة أربعة في الكتاب
+
+3
+00:00:31,460 --> 00:00:38,850
+المقرر اللي بتكلم عن ال uniform continuity في
+
+4
+00:00:38,850 --> 00:00:45,330
+المحاضرة السابقة عرفنا الاتصال المنتظم وشوفنا 
+
+5
+00:00:45,330 --> 00:00:49,930
+أثبتنا نظريات
+
+6
+00:00:49,930 --> 00:00:54,170
+مهمة عن الاتصال المنتظم أو عدم الاتصال المنتظم
+
+7
+00:00:54,170 --> 00:00:58,850
+فأخدنا ال non uniform continuity criterion اللي 
+
+8
+00:00:58,850 --> 00:01:04,770
+حسبها أو ممكن نستخدمها في إثبات أن دالة محددة ليست
+
+9
+00:01:04,770 --> 00:01:09,750
+uniformly continuous على مجموعة جزئية محددة من 
+
+10
+00:01:09,750 --> 00:01:13,330
+الأعداد الحقيقية فكان في عندي non uniform
+
+11
+00:01:13,330 --> 00:01:18,270
+continuity criterion وآخر نظرية أثبتنا نظرية مهمة
+
+12
+00:01:18,270 --> 00:01:22,490
+وهي ال uniform continuity criterion اللي بتقول
+
+13
+00:01:22,490 --> 00:01:27,650
+أنه لو كانت ال function تبعتي متصلة
+
+14
+00:01:28,780 --> 00:01:31,940
+على المجال تبعها والمجال تبعها closed bounded
+
+15
+00:01:31,940 --> 00:01:38,760
+interval فالاتصال يتحول إلى اتصال منتظم طبعا احنا
+
+16
+00:01:38,760 --> 00:01:43,120
+شفنا في المحاضرة السابقة أنه دائما الاتصال المنتظم
+
+17
+00:01:43,120 --> 00:01:47,200
+أقوى من الاتصال العادي لو كانت الدالة uniformly 
+
+18
+00:01:47,200 --> 00:01:50,960
+continuous فبتكون continuous لكن العكس ليس صحيح
+
+19
+00:01:52,640 --> 00:02:00,080
+فخدنا مثال على دالة function دالة واحد على X شفنا
+
+20
+00:02:00,080 --> 00:02:04,920
+أنها متصلة continuous على الفترة المفتوحة من صفر
+
+21
+00:02:04,920 --> 00:02:09,620
+إلى ما لا نهاية but it was not uniformly continuous
+
+22
+00:02:09,620 --> 00:02:15,780
+على نفس الفترة وبالتالي الاتصال العادي لا يؤدي 
+
+23
+00:02:15,780 --> 00:02:22,820
+للاتصال المنتظم اليوم هنتعرف على نوع جديد من ال
+
+24
+00:02:22,820 --> 00:02:27,040
+functions وهو Lipschitz functions و ال functions هدول
+
+25
+00:02:27,040 --> 00:02:32,200
+هتكونوا دائما كلهم uniformly continuous فنعرف Lipschitz
+
+26
+00:02:32,200 --> 00:02:38,640
+function definition a
+
+27
+00:02:38,640 --> 00:02:42,680
+function f 
+
+28
+00:02:42,680 --> 00:02:44,740
+from a to r
+
+29
+00:02:47,770 --> 00:02:58,050
+إذ Lipschitz .. بنسميها
+
+30
+00:02:58,050 --> 00:03:03,010
+Lipschitz on
+
+31
+00:03:03,010 --> 00:03:10,310
+a إذا وجد if there exists k positive number such
+
+32
+00:03:10,310 --> 00:03:13,510
+that absolute f of x
+
+33
+00:03:36,970 --> 00:03:40,090
+وطبعا ممكن إثبات بكل سهولة
+
+34
+00:03:49,120 --> 00:03:55,860
+الآن هنثبت وهنشوف أنه كل Lipschitz function أو كل
+
+35
+00:03:55,860 --> 00:04:00,320
+function بتحقق Lipschitz condition اللي هو الشرط هذا
+
+36
+00:04:07,690 --> 00:04:12,250
+كل function بتحقق Lipschitz condition أو .. أو نسميها 
+
+37
+00:04:12,250 --> 00:04:17,910
+Lipschitz function بتكون uniformly continuous فنشوف
+
+38
+00:04:17,910 --> 00:04:23,390
+المرحلة دالك إذا هنا every أو
+
+39
+00:04:23,390 --> 00:04:33,270
+if .. if f from a to r is Lipschitz is 
+
+40
+00:04:33,270 --> 00:04:47,300
+Lipschitz on a then it is uniformly continuous
+
+41
+00:04:47,300 --> 00:04:56,080
+on a proof
+
+42
+00:04:56,080 --> 00:05:00,840
+assume
+
+43
+00:05:03,530 --> 00:05:10,310
+إذا كان Lipschitz على
+
+44
+00:05:10,310 --> 00:05:20,250
+a ثم حسب التعريف هناك كمية موجبة كمية كمية كامة
+
+45
+00:05:20,250 --> 00:05:20,470
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+46
+00:05:20,470 --> 00:05:20,690
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+47
+00:05:20,690 --> 00:05:21,490
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+48
+00:05:21,490 --> 00:05:21,530
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+49
+00:05:21,530 --> 00:05:21,690
+كمية كمية كمية كمية كمية كمية كمية كمية كمية كمية
+
+50
+00:05:21,690 --> 00:05:28,390
+كمية كمية كمية
+
+51
+00:05:28,390 --> 00:05:38,840
+k times absolute x minus u for all x where u and
+
+52
+00:05:38,840 --> 00:05:52,180
+a طيب
+
+53
+00:05:52,180 --> 00:05:55,760
+لتسمي ال condition هذا star
+
+54
+00:05:58,650 --> 00:06:02,570
+let epsilon أكبر من الصفر بيجبن let epsilon أكبر
+
+55
+00:06:02,570 --> 00:06:05,490
+من الصفر بيجبن let epsilon أكبر من الصفر بيجبن let
+
+56
+00:06:05,490 --> 00:06:06,290
+epsilon أكبر من الصفر بيجبن let epsilon أكبر من
+
+57
+00:06:06,290 --> 00:06:06,550
+الصفر بيجبن let epsilon أكبر من الصفر بيجبن let
+
+58
+00:06:06,550 --> 00:06:06,570
+epsilon أكبر من الصفر بيجبن let epsilon أكبر من
+
+59
+00:06:06,570 --> 00:06:07,810
+الصفر بيجبن let epsilon أكبر من الصفر بيجبن let
+
+60
+00:06:07,810 --> 00:06:09,990
+epsilon أكبر من الصفر بيجبن
+
+61
+00:06:23,960 --> 00:06:29,040
+عدد موجب إبسلون على K بيطلع عدد موجب وبالتالي إذن 
+
+62
+00:06:29,040 --> 00:06:35,920
+هنا أثبتت إن user Delta تعتمد على إبسلون فقط فلهذه
+
+63
+00:06:35,920 --> 00:06:42,280
+الإبسلون then لو كانت X و U موجودين في A و
+
+64
+00:06:42,280 --> 00:06:47,840
+Absolute X minus U أصغر من Delta فهذا هيقدّي إن
+
+65
+00:06:47,840 --> 00:07:01,000
+Absolute F of X-f of u باي star حسب المتباينة star
+
+66
+00:07:01,000 --> 00:07:06,600
+هذا بيطلع أصغر منه أو يساوي absolute x minus u
+
+67
+00:07:13,020 --> 00:07:18,320
+وأنا عندي absolute x minus u أصغر من دلتا إذا هذا
+
+68
+00:07:18,320 --> 00:07:25,240
+أصغر عفوا by star في عندي هنا k ضرب absolute x
+
+69
+00:07:25,240 --> 00:07:31,780
+minus u الآن أنا عندي absolute x minus u أصغر من
+
+70
+00:07:31,780 --> 00:07:36,840
+دلتا لأن هذا أصغر من k في دلتا وأنا عندي ماخد دلتا
+
+71
+00:07:36,840 --> 00:07:45,940
+بالمساوي إبسلون على k أصبح هذا أصغر من إبسلون لأي
+
+72
+00:07:45,940 --> 00:07:52,500
+إبسلون أكبر من 0 يوجد Delta تعتمد على إبسلون فقط
+
+73
+00:07:52,500 --> 00:07:59,340
+بحيث أنه لكل x و u في a المسافة بينهم أصغر من 
+
+74
+00:07:59,340 --> 00:08:02,820
+Delta طلع المسافة بين ال images أصغر من إبسلون
+
+75
+00:08:06,120 --> 00:08:11,800
+epsilon أكبر من الصفر was arbitrary، إذن هذا صحيح 
+
+76
+00:08:11,800 --> 00:08:15,580
+لكل epsilon وبالتالي by definition، إذن ال
+
+77
+00:08:15,580 --> 00:08:20,780
+function f is uniformly continuous
+
+78
+00:08:20,780 --> 00:08:30,340
+on E، وهو المطلوب إذن هنا أثبتنا إن كل Lipschitz
+
+79
+00:08:30,340 --> 00:08:34,120
+function is uniformly continuous
+
+80
+00:08:36,360 --> 00:08:45,020
+لكن العكس ليس صحيحا .. العكس ليس صحيحا remark ..
+
+81
+00:08:45,020 --> 00:08:55,400
+remark the
+
+82
+00:08:55,400 --> 00:09:00,740
+converse .. the converse of above theorem
+
+83
+00:09:05,350 --> 00:09:11,730
+is false for
+
+84
+00:09:11,730 --> 00:09:16,970
+example على سبيل المثال يعني معنى آخر لو كانت ال
+
+85
+00:09:16,970 --> 00:09:24,750
+function uniformly continuous مش شرط تكون Lipschitz على
+
+86
+00:09:24,750 --> 00:09:31,370
+نفس ال function على نفس ال .. for example consider
+
+87
+00:09:35,170 --> 00:09:48,950
+Consider الـ function f of x بساوي جذر الـ x هو 
+
+88
+00:09:48,950 --> 00:09:54,790
+x ينتمي ل I بساوي closed interval من صفر لاثنين
+
+89
+00:10:07,930 --> 00:10:14,030
+by exercise في exercise أخدناه اللي هو جبنالكم 
+
+90
+00:10:14,030 --> 00:10:23,090
+إياه سؤال في الامتحان ال exercise هذا كان .. خلينا
+
+91
+00:10:23,090 --> 00:10:23,730
+نشوف
+
+92
+00:10:39,170 --> 00:10:45,750
+أو ممكن إثبات أن الدالة هذه is continuous طيب
+
+93
+00:10:45,750 --> 00:10:52,030
+آه
+
+94
+00:10:52,030 --> 00:10:55,910
+by exercise
+
+95
+00:10:55,910 --> 00:11:04,250
+في chapter أربعة أربعة واحد question تمام آه أربعة
+
+96
+00:11:04,250 --> 00:11:11,540
+واحد مظبوط صحيح by exercise تماما section أربعة
+
+97
+00:11:11,540 --> 00:11:16,540
+واحد ال
+
+98
+00:11:16,540 --> 00:11:24,940
+function if is continuous على الفترة لأن في هديك
+
+99
+00:11:24,940 --> 00:11:32,480
+ال exercise هتثبتوا إنه limit جذر ال X لما X تؤول ل
+
+100
+00:11:32,480 --> 00:11:42,910
+C بساوي جذر ال C لكل C أكبر من أو يساوي الصفر طبعا
+
+101
+00:11:42,910 --> 00:11:48,030
+في ال exercise ماخد C أكبر من الصفر لكن لما C 
+
+102
+00:11:48,030 --> 00:11:52,950
+بساوي الصفر فهذا trivial وبالتالي هذا معناه أن
+
+103
+00:11:52,950 --> 00:11:59,890
+دالة F هذه 
+
+104
+00:11:59,890 --> 00:12:05,330
+معناه شرط الاتصال عند C متحقق فهذا معناه أن F is 
+
+105
+00:12:05,330 --> 00:12:12,720
+continuous at C وده صحيح لكل C أكبر من أو يساوي الصفر
+
+106
+00:12:12,720 --> 00:12:20,520
+وبالتالي إذا F is continuous على الفترة من صفر إلى 
+
+107
+00:12:20,520 --> 00:12:25,140
+ما لا نهاية وبالتالي متصلة على الفترة من صفر إلى 
+
+108
+00:12:25,140 --> 00:12:33,460
+اثنين اللي هي جزئية منها okay تمام طيب إذا 
+
+109
+00:12:40,220 --> 00:12:48,440
+إذا by طيب since I بساوي الفترة من الصفر للاثنين
+
+110
+00:12:48,440 --> 00:12:58,140
+الفترة هذه is closed and bounded و
+
+111
+00:12:58,140 --> 00:13:05,080
+if continuous عليها then by
+
+112
+00:13:05,080 --> 00:13:09,520
+uniform continuity theorem
+
+113
+00:13:11,440 --> 00:13:14,900
+نظرية الاتصال المنتظم بتقول إذا كان في عندي 
+
+114
+00:13:14,900 --> 00:13:20,060
+function f متصلة على فترة مغلقة أو محدودة فالاتصال
+
+115
+00:13:20,060 --> 00:13:25,320
+هذا بيكون اتصال منتظم uniform continuity ففي عندي 
+
+116
+00:13:25,320 --> 00:13:32,800
+by uniform continuity theorem تطلع f is uniformly
+
+117
+00:13:32,800 --> 00:13:39,840
+continuous
+
+118
+00:13:41,540 --> 00:13:48,120
+على الفترة I إذاً هي مثال على function uniformly
+
+119
+00:13:48,120 --> 00:13:52,880
+continuous على المجال تبعها هنشوف الآن إن هذه ال
+
+120
+00:13:52,880 --> 00:14:07,060
+function ما هيش Lipschitz على نفس الفترة إذا
+
+121
+00:14:07,060 --> 00:14:07,640
+ال claim
+
+122
+00:14:13,370 --> 00:14:23,650
+f is not .. f is not Lipschitz على 
+
+123
+00:14:23,650 --> 00:14:28,590
+الفترة I فلبرهان
+
+124
+00:14:28,590 --> 00:14:33,990
+ذلك assume
+
+125
+00:14:33,990 --> 00:14:37,950
+on
+
+126
+00:14:37,950 --> 00:14:38,670
+contrary
+
+127
+00:14:43,190 --> 00:14:48,550
+assume on contrary that
+
+128
+00:14:48,550 --> 00:15:01,650
+f is Lipschitz on
+
+129
+00:15:01,650 --> 00:15:04,090
+I then
+
+130
+00:15:06,570 --> 00:15:14,610
+there exists k أكبر من الصفر بحيث أنه absolute f
+
+131
+00:15:14,610 --> 00:15:28,610
+of x minus f of u أصغر من أو يساوي k في absolute x
+
+132
+00:15:28,610 --> 00:15:36,950
+minus u لكل x و you تنتمي للفترة I اللي هي الفترة 
+
+133
+00:15:36,950 --> 00:15:39,970
+المغلقة من صفر للاثنين
+
+134
+00:15:59,060 --> 00:16:04,840
+إذا هنا فرضنا ال contrary ويطلع إن بيطلع عندي
+
+135
+00:16:04,840 --> 00:16:09,160
+فيه huge العدد موجب بحيث كان أنا ادم اتحقق وهذا
+
+136
+00:16:09,160 --> 00:16:15,340
+بيقدر إن absolute f of x لو اخذنا u بساوي صفر minus
+
+137
+00:16:15,340 --> 00:16:24,910
+f of 0 أصغر من أو يساوي k فabsolute x وهذا صحيح لكل x
+
+138
+00:16:24,910 --> 00:16:32,150
+تنتمي للفترة I إذا أنا هنا أخدت U بساوي صفر و
+
+139
+00:16:32,150 --> 00:16:39,070
+الصفر ينتمي للفترة I طيب أنا عندي F صفر بساوي صفر
+
+140
+00:16:39,070 --> 00:16:44,030
+إذا 
+
+141
+00:16:44,030 --> 00:16:47,130
+بطلع عندي absolute
+
+142
+00:16:48,770 --> 00:16:58,510
+f of x أصغر من أو يساوي k في absolute  الـ X وهذا
+
+143
+00:16:58,510 --> 00:17:04,470
+صحيح لكل X  الذي ينتمي لفترة I هي الفترة المغلقة من
+
+144
+00:17:04,470 --> 00:17:09,690
+الصفر لفترة بس
+
+145
+00:17:09,690 --> 00:17:14,810
+هذا هيدي للتناقض طيب
+
+146
+00:17:15,530 --> 00:17:28,330
+تخيل لو أخدت x بساوي واحد على n تربيع فهذا
+
+147
+00:17:28,330 --> 00:17:33,990
+عبارة عن .. هذا ينتمي للفترة .. للفترة المغلقة من
+
+148
+00:17:33,990 --> 00:17:40,410
+الصفر إلى اثنين اللي هي I لأن هذا عدد موجب لكل n ينتمي
+
+149
+00:17:40,410 --> 00:17:46,820
+لـ N لكل عدد طبيعي هذا بطلع ينتمي للفترة هذه
+
+150
+00:17:46,820 --> 00:17:54,300
+وبالتالي إذا المفروض يطلع absolute  f لواحد على N
+
+151
+00:17:54,300 --> 00:18:02,060
+تربيع أصغر من أو يساوي K في absolute واحد على N
+
+152
+00:18:02,060 --> 00:18:10,840
+تربيع هذا صحيح لكل N في N طيب if واحد على n تربيع
+
+153
+00:18:10,840 --> 00:18:16,760
+بيطلع بساوي الجذر التربيعي لواحد على n تربيع
+
+154
+00:18:16,760 --> 00:18:20,200
+اللي 
+
+155
+00:18:20,200 --> 00:18:26,600
+هو عبارة عن واحد على n فـ absolute واحد على n
+
+156
+00:18:26,600 --> 00:18:34,100
+بيطلع واحد على n أصغر من أو يساوي K في واحد على n
+
+157
+00:18:34,100 --> 00:18:43,520
+تربيع هذا صحيح لكل N في N اضرب
+
+158
+00:18:43,520 --> 00:18:50,880
+الطرفين هذه في n تربيع فبطلع عندي n أصغر من أو
+
+159
+00:18:50,880 --> 00:18:59,320
+يساوي K for all N في N وهذا يتناقض مع ال
+
+160
+00:18:59,320 --> 00:19:06,040
+Archimedean property which contradicts
+
+161
+00:19:07,330 --> 00:19:11,130
+التي تتناقض
+
+162
+00:19:11,130 --> 00:19:19,250
+مع مين؟ التي تتناقض مع الـ Archimedean property
+
+163
+00:19:23,770 --> 00:19:28,110
+خاصية Archimedes لأن خاصية Archimedes بتقول لي لأي
+
+164
+00:19:28,110 --> 00:19:35,690
+عدد K عدد موجب أو أي عدد حقيقي K يوجد N0 عدد طبيعي
+
+165
+00:19:35,690 --> 00:19:45,660
+لحيث أن N0 أكبر من K صح؟ ومن هنا كل الأعداد
+
+166
+00:19:45,660 --> 00:19:53,360
+الطبيعية من ضمنها N0  أشملها أصغر من أو يساوي ال K
+
+167
+00:19:53,360 --> 00:20:00,740
+فبطلع N0 أكبر من N0 contradiction إذا السبب ال
+
+168
+00:20:00,740 --> 00:20:04,640
+contradiction هذا أنه إيه ال assumption الفرض
+
+169
+00:20:04,640 --> 00:20:12,440
+تبعنا ال assumption تبعنا أن F is ليس bounded on I okay
+
+170
+00:20:14,100 --> 00:20:19,120
+إذاً هذا بتثبت هذا ال contradiction بتثبت أن الـ f is
+
+171
+00:20:19,120 --> 00:20:29,820
+عفواً f is not bounded on
+
+172
+00:20:29,820 --> 00:20:37,120
+a أو i وهو المطلوب إذاً هذا مثال على function
+
+173
+00:20:37,120 --> 00:20:44,810
+uniformly continuous على set معينة لكنها ليست bounded 
+
+174
+00:20:44,810 --> 00:20:50,450
+لكن أثبتنا قبل إيه إن كل bounded function is always
+
+175
+00:20:50,450 --> 00:20:57,750
+uniformly continuous ناخذ
+
+176
+00:20:57,750 --> 00:20:58,770
+بعض الأمثلة
+
+177
+00:21:26,120 --> 00:21:35,220
+example let f of x بساوي x تربيع و x ينتمي
+
+178
+00:21:35,220 --> 00:21:41,800
+للمجموعة a اللي هي الفترة المغلقة من صفر إلى b
+
+179
+00:21:41,800 --> 00:21:50,410
+حيث b أي عدد موجب b أي عدد موجب بنثبت أن ال
+
+180
+00:21:50,410 --> 00:21:59,950
+function هذه تطلع uniformly continuous show
+
+181
+00:21:59,950 --> 00:22:05,950
+that show
+
+182
+00:22:05,950 --> 00:22:11,670
+أن f is uniformly continuous
+
+183
+00:22:11,670 --> 00:22:23,720
+on a ففيه برهانين حالين proof one حال الأول since
+
+184
+00:22:23,720 --> 00:22:28,400
+f
+
+185
+00:22:28,400 --> 00:22:37,400
+is continuous on a being
+
+186
+00:22:37,400 --> 00:22:39,440
+a polynomial
+
+187
+00:22:44,780 --> 00:22:47,300
+لأنها polynomial و احنا قلنا كل polynomial
+
+188
+00:22:47,300 --> 00:22:51,800
+function متصلة على R وبالتالي على أي مجموعة جزئية
+
+189
+00:22:51,800 --> 00:22:59,760
+من R زي المجموعة A اللي هي الفترة المغلقة من صفر
+
+190
+00:22:59,760 --> 00:23:05,620
+إلى الـ B ف
+
+191
+00:23:05,620 --> 00:23:10,920
+f is continuous على A كونها polynomial and بما
+
+192
+00:23:10,920 --> 00:23:19,480
+أنّه and since الـ set A هذه اللي هي عبارة عن الفترة
+
+193
+00:23:19,480 --> 00:23:29,320
+المغلقة من صفر لـ B is closed and bounded and
+
+194
+00:23:29,320 --> 00:23:34,500
+bounded interval
+
+195
+00:23:34,500 --> 00:23:44,980
+then by uniform continuity theorem
+
+196
+00:23:47,180 --> 00:23:53,380
+حسب نظرية الاتصال المنتظم اللي بتقول لو كان في 
+
+197
+00:23:53,380 --> 00:23:57,140
+function مجالها closed bounded interval و ال
+
+198
+00:23:57,140 --> 00:24:03,120
+function متصلة عليها فالاتصال بتحول الى اتصال منتظم
+
+199
+00:24:03,120 --> 00:24:08,040
+إذا ال function f is uniformly
+
+200
+00:24:10,270 --> 00:24:16,870
+continuous on a وهذا برهان لأنه ممكن نستخدم ال
+
+201
+00:24:16,870 --> 00:24:20,550
+uniform continuity theorem لإثبات أنه function
+
+202
+00:24:20,550 --> 00:24:24,970
+اللي زي هذه الدالة التربيعية uniform continuous على
+
+203
+00:24:24,970 --> 00:24:32,550
+أي فترة مغلقة زي الفترة هذه الحل
+
+204
+00:24:32,550 --> 00:24:37,830
+التاني ممكن نثبت أن الدالة هذه bounded برضه و أستخدم
+
+205
+00:24:37,830 --> 00:24:38,710
+نظرية هذه
+
+206
+00:24:41,510 --> 00:24:48,950
+نشوف مع بعض، هنا البرهان الثاني أو برهان رقم اثنين،
+
+207
+00:24:48,950 --> 00:24:58,810
+proof اثنين claim
+
+208
+00:24:58,810 --> 00:25:03,950
+أنّ f is bounded
+
+209
+00:25:08,200 --> 00:25:18,740
+on a التي هي الفترة المغلقة من صفر إلى b
+
+210
+00:25:18,740 --> 00:25:26,260
+فالإثبات هذا الكلام تعال نشوف هي absolute f of x
+
+211
+00:25:26,260 --> 00:25:35,000
+minus f of u إيش بيساوي absolute x
+
+212
+00:25:35,820 --> 00:25:43,960
+تربيع minus u تربيع بيساوي absolute x زائد u في
+
+213
+00:25:43,960 --> 00:25:51,860
+absolute x minus u وهذا بيساوي absolute x زائد u في
+
+214
+00:25:51,860 --> 00:25:58,640
+absolute x ناقص u و by triangle inequality
+
+215
+00:25:58,640 --> 00:26:04,330
+absolute x زائد u أصغر من أو يساوي absolute x زائد
+
+216
+00:26:04,330 --> 00:26:13,910
+absolute u كل هذا مضروب في absolute x minus u الآن
+
+217
+00:26:13,910 --> 00:26:19,970
+ال u و ال x ينتموا للمجال تبع الدالة وبالتالي
+
+218
+00:26:19,970 --> 00:26:29,990
+كلاهما عدد غير سالب و كلاهما أصغر من أو يساوي ال
+
+219
+00:26:29,990 --> 00:26:37,320
+b صح؟ إن هذا أصغر من أو يساوي b زائد b في
+
+220
+00:26:37,320 --> 00:26:45,420
+absolute x minus u for all x و u ينتموا للمجموعة
+
+221
+00:26:45,420 --> 00:26:52,520
+اللي هي الفترة المغلقة من صفر إلى b طبعاً هذا
+
+222
+00:26:52,520 --> 00:27:01,660
+بساوٍ 2 b في absolute x minus u for all x و u
+
+223
+00:27:01,660 --> 00:27:08,820
+تنتمي إلى a إذا هذا شرط الـ bounded تحقق with k بيساوي
+
+224
+00:27:08,820 --> 00:27:17,340
+2 b عدد موجب إذا هنا take k
+
+225
+00:27:17,340 --> 00:27:21,800
+بيساوٍ 2 b عدد موجب
+
+226
+00:27:36,690 --> 00:27:38,510
+Okay طبعا
+
+227
+00:27:54,180 --> 00:28:09,940
+واضح البرهان في أي سؤال أو استفسار في
+
+228
+00:28:09,940 --> 00:28:17,160
+عندي نظرية تتعلق بالـ uniform لها علاقة بال
+
+229
+00:28:17,160 --> 00:28:22,180
+uniform continuity وهي النظرية التالية
+
+230
+00:28:38,260 --> 00:28:49,440
+Theorem if f from a to r is uniformly is uniformly
+
+231
+00:28:49,440 --> 00:28:52,540
+continuous
+
+232
+00:28:52,540 --> 00:29:00,780
+on a then
+
+233
+00:29:03,000 --> 00:29:09,640
+For any Cauchy Sequence
+
+234
+00:29:09,640 --> 00:29:18,780
+xn contained in A The
+
+235
+00:29:18,780 --> 00:29:22,700
+sequence f
+
+236
+00:29:22,700 --> 00:29:31,580
+of xn اللي هي ال image لسيكنس xn is Cauchy
+
+237
+00:29:33,110 --> 00:29:40,090
+in R that
+
+238
+00:29:40,090 --> 00:29:45,950
+is that
+
+239
+00:29:45,950 --> 00:29:56,030
+is هذا يعني هذا يعني هذا يعني أنّه uniformly
+
+240
+00:29:56,030 --> 00:30:01,090
+uniformly continuous
+
+241
+00:30:04,960 --> 00:30:18,040
+functions preserve Cauchy
+
+242
+00:30:18,040 --> 00:30:22,460
+sequences
+
+243
+00:30:27,960 --> 00:30:34,420
+يعني الدوال اللي بتكون متصلة اتصال منتظم بتحافظ على
+
+244
+00:30:34,420 --> 00:30:39,960
+Cauchy sequences بمعنى أنّه لو كانت xn Cauchy
+
+245
+00:30:39,960 --> 00:30:46,260
+sequence في المجال تبع الدالة A فصورتها هتطلع
+
+246
+00:30:46,260 --> 00:30:52,080
+Cauchy sequence في المجال المقابل R والبرهان سهل
+
+247
+00:30:53,210 --> 00:30:57,390
+طبعاً هذا بس صحيح للـ uniform لـ continuous functions
+
+248
+00:30:57,390 --> 00:31:02,350
+أما لو كانت ال function بس continuous فمش شرط
+
+249
+00:31:02,350 --> 00:31:07,510
+تحافظ على Cauchy sequences والبرهان
+
+250
+00:31:07,510 --> 00:31:18,670
+سهل بسيط prove let
+
+251
+00:31:18,670 --> 00:31:29,880
+f from A to R be uniformly continuous on
+
+252
+00:31:29,880 --> 00:31:44,240
+a and let x in contained in a,b كوشي كوشي sequence
+
+253
+00:31:44,240 --> 00:31:50,420
+و بدنا نثبت أن ال image لل sequence x in بتطلع
+
+254
+00:31:50,420 --> 00:31:57,060
+كوشي طيب to show ان
+
+255
+00:31:57,060 --> 00:32:09,340
+ال image لسيكوينس XN is Cauchy للبرهان
+
+256
+00:32:09,340 --> 00:32:15,180
+أن ال sequence هذه ال image لسيكوينس XN is Cauchy
+
+257
+00:32:15,180 --> 00:32:24,130
+نحاول نطبق تعريف Cauchy sequence أو نحاول نحقق شرط
+
+258
+00:32:24,130 --> 00:32:31,290
+كوشي فكيف نحقق قولت epsilon أكبر من الصفر بيجي بنا
+
+259
+00:32:31,290 --> 00:32:39,210
+وبدنا نرد عليها بـ Capital N تحقق لي شرط كوشي طيب
+
+260
+00:32:39,210 --> 00:32:44,110
+since f
+
+261
+00:32:44,110 --> 00:32:45,390
+is uniformly
+
+262
+00:32:47,670 --> 00:32:55,510
+continuous on a إذا لأي إبسلون موجبة زي هذه يوجد
+
+263
+00:32:55,510 --> 00:33:02,650
+إذا
+
+264
+00:33:02,650 --> 00:33:09,410
+لأي إبسلون زي هذِ مع أنّه f uniform continuous إذا
+
+265
+00:33:09,410 --> 00:33:13,670
+لأي epsilon حسب تعريف ال uniform continuity يوجد
+
+266
+00:33:13,670 --> 00:33:21,510
+delta تعتمد على epsilon عدد موجب بحيث أنّه لو كان x
+
+267
+00:33:24,390 --> 00:33:30,090
+و u موجودين في A و Absolute x minus u أصغر من
+
+268
+00:33:30,090 --> 00:33:37,570
+delta فهذا يعني أن Absolute f of x minus f of u
+
+269
+00:33:37,570 --> 00:33:47,250
+أصغر من epsilon نسمي ال implication هذه star الآن
+
+270
+00:33:47,250 --> 00:33:53,610
+since ال sequence xn is Cauchy
+
+271
+00:33:58,410 --> 00:34:02,810
+then و delta and
+
+272
+00:34:02,810 --> 00:34:11,090
+delta أكبر من الصفر طبعاً هذه given is given ال
+
+273
+00:34:11,090 --> 00:34:13,850
+delta هذه قلنا يوجد delta عدد موجب بما أن هذه
+
+274
+00:34:13,850 --> 00:34:20,650
+تعتبر given delta فلل delta هذه اللي هنا عدد موجب
+
+275
+00:34:20,650 --> 00:34:27,960
+بما أن xn is Cauchy إذا there exist يوجد Capital N
+
+276
+00:34:27,960 --> 00:34:37,220
+يعتمد على delta عدد طبيعي بحيث أنّه شرط كوشي يتحقق
+
+277
+00:34:37,220 --> 00:34:43,140
+وهو لكل n و m أكبر من أو يساوي Capital N بطلع عندي
+
+278
+00:34:43,140 --> 00:34:49,040
+absolute xn minus xm أصغر من delta
+
+279
+00:34:52,280 --> 00:35:01,060
+بنسمي هذه double star now star and double star
+
+280
+00:35:01,060 --> 00:35:14,240
+بيقدّوا أنّه يوجد Capital N يعتمد على epsilon لأن 
+
+281
+00:35:14,240 --> 00:35:18,800
+الـ delta بتعتمد على الـ epsilon
+
+282
+00:35:18,800 --> 00:35:26,260
+إبسلون، ملاحظة الحال فـ N هذه نفسها N of delta
+
+283
+00:35:26,260 --> 00:35:33,360
+بيساوي N of إبسلون بتتمي لـ N بحيث أنه لو كان N و M
+
+284
+00:35:33,360 --> 00:35:40,480
+أكبر من أو يساوي capital N فهذا بيقدّي أنه by
+
+285
+00:35:40,480 --> 00:35:48,270
+double star هذا بيقدّم |xn - xm| أصغر من
+
+286
+00:35:48,270 --> 00:35:55,450
+دلتا وحسب الـ star by star الـ star بتقول لو كان
+
+287
+00:35:55,450 --> 00:36:01,030
+عندي x و u المسافة بينهم أصغر من دلتا فالمسافة بين
+
+288
+00:36:01,030 --> 00:36:08,790
+صورهم اللي هي xn هنا وصورة الـ xm تطلع أصغر من إبسلون
+
+289
+00:36:10,550 --> 00:36:16,550
+تمام؟ إذا هنا أثبتت لأي إبسلون أكبر من الصفر يوجد
+
+290
+00:36:16,550 --> 00:36:20,830
+capital N يعتمد على إبسلون عدد طبيعي بحيث لكل M و M
+
+291
+00:36:20,830 --> 00:36:25,510
+أكبر من أو يساوي capital N طلع المسافة بين F of X M
+
+292
+00:36:25,510 --> 00:36:31,650
+و F of X N أصغر من إبسلون إذا بما أنه since إبسلون
+
+293
+00:36:31,650 --> 00:36:39,100
+أكبر من الصفر was arbitrary إذا الـ sequence f of x
+
+294
+00:36:39,100 --> 00:36:45,120
+is Cauchy تطلع الـ sequence هذه Cauchy وهو
+
+295
+00:36:45,120 --> 00:36:54,080
+المطلوب okay تمام ممكن نستخدم النظرية هذه ممكن
+
+296
+00:36:54,080 --> 00:37:01,400
+نستخدم النظرية هذه في الـ ..
+
+297
+00:37:01,400 --> 00:37:06,320
+أن نثبت أن function معينة ليست uniform and
+
+298
+00:37:06,320 --> 00:37:16,200
+continuous هاي example use
+
+299
+00:37:16,200 --> 00:37:20,100
+above theorem
+
+300
+00:37:20,100 --> 00:37:31,860
+to show الـ function f of x بسعر واحد على x is
+
+301
+00:37:31,860 --> 00:37:32,280
+not
+
+302
+00:37:35,480 --> 00:37:43,220
+uniformly continuous on a بساوي الفترة المفتوحة من
+
+303
+00:37:43,220 --> 00:37:44,680
+صفر إلى ما لا نهاية
+
+304
+00:37:57,800 --> 00:38:01,060
+لحظة أن النظرية دي إيش بتقول لو كانت الـ function
+
+305
+00:38:01,060 --> 00:38:05,140
+uniformly continuous فلازم تحافظ على Cauchy sequence
+
+306
+00:38:05,140 --> 00:38:09,440
+طب لو محافظتش على Cauchy sequence مش ممكن تكون
+
+307
+00:38:09,440 --> 00:38:17,260
+uniformly continuous صح؟ مظبوط؟ إذا هنا proof
+
+308
+00:38:17,260 --> 00:38:25,680
+by above theorem حسب النظرية على it suffices
+
+309
+00:38:28,360 --> 00:38:36,220
+to show يكفي إثبات أن f is .. if does not .. if
+
+310
+00:38:36,220 --> 00:38:46,960
+does .. does not preserve .. preserve Cauchy
+
+311
+00:38:46,960 --> 00:38:52,860
+sequences ف
+
+312
+00:38:52,860 --> 00:38:53,500
+consider
+
+313
+00:38:56,930 --> 00:39:03,270
+consider الـ sequence xn اللي هي بساوي واحد على n
+
+314
+00:39:03,270 --> 00:39:11,470
+الـ sequence هذه converge لصفر وبالتالي
+
+315
+00:39:11,470 --> 00:39:25,130
+إذا xn is Cauchy تمام but صورة الـ xn
+
+316
+00:39:28,660 --> 00:39:37,460
+إيش بتطلع؟ صورة الواحد على n تطلع الـ sequence n
+
+317
+00:39:37,460 --> 00:39:43,620
+صح؟ و الـ sequence هذه properly divergent to
+
+318
+00:39:43,620 --> 00:39:49,760
+infinity، إذا it's divergent، إذا it's not Cauchy
+
+319
+00:39:49,760 --> 00:39:54,220
+تمام؟
+
+320
+00:39:57,130 --> 00:40:01,450
+Okay؟ وبالتالي إذا هاي في عندي .. هاي في عندي ..
+
+321
+00:40:01,450 --> 00:40:08,790
+إذا if لا تحافظ على الـ Cauchy sequences إذا if does
+
+322
+00:40:08,790 --> 00:40:13,210
+not preserve
+
+323
+00:40:13,210 --> 00:40:19,050
+.. preserve Cauchy
+
+324
+00:40:26,330 --> 00:40:31,610
+sequences وبالتالي حسب النظرية الأخيرة ما بتكونش
+
+325
+00:40:31,610 --> 00:40:34,630
+uniformly continuous لأن لو كانت uniformly
+
+326
+00:40:34,630 --> 00:40:38,370
+continuous فالمفروض تاخد Cauchy sequence زي هذه
+
+327
+00:40:38,370 --> 00:40:42,730
+تعطينا صورتها Cauchy sequence وهذا مستحيل okay تمام
+
+328
+00:40:42,730 --> 00:40:47,970
+واضح في أي سؤال أي استفسار إذا هيك نكتفي بهذا
+
+329
+00:40:47,970 --> 00:40:52,540
+القدر من section خمسة أربعة وزي ما حكينا سابقًا هذا
+
+330
+00:40:52,540 --> 00:40:57,260
+كان آخر section هناخده في المقرر وبالتالي هيكون
+
+331
+00:40:57,260 --> 00:41:03,400
+يعني .. يعني إن شاء الله أنهينا الـ course كما هو
+
+332
+00:41:03,400 --> 00:41:10,600
+موضح على الـ syllabus فشكرًا لكم و شكرًا لحسن إصغائكم
+
+333
+00:41:10,600 --> 00:41:13,580
+و يعطيكم ألف عافية
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..21306c3129c830dfbf990959e53d226be05855e9
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M_postprocess.srt
@@ -0,0 +1,1932 @@
+1
+00:00:20,670 --> 00:00:26,650
+السلام عليكم اليوم ان شاء الله هنكمل section أربعة
+
+2
+00:00:26,650 --> 00:00:35,990
+واحد اللي عرفنا فيه ال limits و ال functions أخدنا
+
+3
+00:00:35,990 --> 00:00:40,650
+المرة الأولى تلتة تعريف epsilon delta ل limit of
+
+4
+00:00:40,650 --> 00:00:45,990
+function وشوفنا أن هذا بكافة تعريف في neighborhood
+
+5
+00:00:45,990 --> 00:00:51,040
+definition لlimit of function على النقطةو بدنا
+
+6
+00:00:51,040 --> 00:00:57,500
+ناخد أمثلة كيف نستخدم تعريف epsilon delta في إثبات
+
+7
+00:00:57,500 --> 00:01:03,360
+أن ال limit لدالة معينة عن مقطة معددة بتساوي عدد
+
+8
+00:01:03,360 --> 00:01:08,310
+محدد فاخدنا بعض الأمثلة اليوم هنستمرهنعطي مزيد من
+
+9
+00:01:08,310 --> 00:01:12,930
+الأمثلة و بندرس خواص ال limits ل ال functions
+
+10
+00:01:12,930 --> 00:01:19,490
+فالمثال اللى وصلناه له رقم تلاتة عايزين نثبت ان ال
+
+11
+00:01:19,490 --> 00:01:25,730
+limit لدلة تربية X تربية لما X تقول ل C بساوي C
+
+12
+00:01:25,730 --> 00:01:29,330
+تربية ف solution
+
+13
+00:01:33,240 --> 00:01:40,260
+ناخد f of x بالساوي x تربية هيفر اكس ينتمي الى r
+
+14
+00:01:40,260 --> 00:01:43,880
+واحنا
+
+15
+00:01:43,880 --> 00:01:50,600
+عايزين من الآخر نثبت ان ال absolute value ل f of x
+
+16
+00:01:50,600 --> 00:01:58,400
+minus c تربية أصغر من أي given epsilon عدد موجة
+
+17
+00:01:59,300 --> 00:02:04,780
+عندما الـ X تكون قريبة من المقطة C أو تقع فيه جوار
+
+18
+00:02:04,780 --> 00:02:13,600
+Delta معينة للعدد C طيب هذا عبارة عن Absolute X
+
+19
+00:02:13,600 --> 00:02:22,620
+تربية سالم C تربية بتحلل إلى Absolute X minus C في
+
+20
+00:02:22,620 --> 00:02:24,620
+X موجة بالـ C
+
+21
+00:02:27,570 --> 00:02:33,430
+إذاً هذا عبارة عن absolute X زائد C في absolute X
+
+22
+00:02:33,430 --> 00:02:37,830
+minus C الأن
+
+23
+00:02:37,830 --> 00:02:42,430
+بدي أحاول أخلي هذا أصغر من أو ساوي عدد موجة بم
+
+24
+00:02:42,430 --> 00:02:47,290
+فبحاول
+
+25
+00:02:47,290 --> 00:02:52,830
+أخد فيه ملة Delta لت دلتا بالساوي واحد
+
+26
+00:03:00,710 --> 00:03:12,430
+then أنا عندي absolute x زائد c هذا أصغر
+
+27
+00:03:12,430 --> 00:03:19,910
+من او ساوي absolute x زايد absolute cفي absolute x
+
+28
+00:03:19,910 --> 00:03:25,450
+minus c استخدام ال triangle inequality absolute x
+
+29
+00:03:25,450 --> 00:03:30,430
+زاد c أصلا لو ساوي absolute x زاد absolute c الآن
+
+30
+00:03:30,430 --> 00:03:39,530
+absolute x بساوي absolute x سالب c زاد زاد
+
+31
+00:03:39,530 --> 00:03:44,270
+c ممكن أطرح من ال xc و أرجعهاوباستخدام الـ
+
+32
+00:03:44,270 --> 00:03:49,030
+triangular equality هذا أصغر لو يساوي absolute x
+
+33
+00:03:49,030 --> 00:03:58,150
+ثالث c زائد absolute c فلو كان absolute x minus c
+
+34
+00:03:58,150 --> 00:04:04,070
+أصغر من دلتا اللي هي بساوية واحد إذا كان خلّينا
+
+35
+00:04:04,070 --> 00:04:07,190
+ناخد دلتا بساوية واحد إذا كان absolute x minus c
+
+36
+00:04:07,190 --> 00:04:13,370
+أصغر من دلتا اللي أنا ماخدها واحدفهذا بيطلع أصغر
+
+37
+00:04:13,370 --> 00:04:21,490
+من واحد زائد أبسليوت C وبالتالي
+
+38
+00:04:21,490 --> 00:04:32,370
+أبسليوت X تربية سالب C تربية بيطلع أصغر من أبسليوت
+
+39
+00:04:32,370 --> 00:04:35,150
+X اللي هي واحد زائد
+
+40
+00:04:37,400 --> 00:04:43,720
+أتنين في absolute C في absolute X minus C
+
+41
+00:04:48,510 --> 00:04:51,770
+كمان مرة احنا توصلنا إلى ان ال absolute value
+
+42
+00:04:51,770 --> 00:04:57,150
+للفرق هذا أصغر من أو ساوي absolute x زاد absolute
+
+43
+00:04:57,150 --> 00:05:01,290
+c في absolute x ثالث c أخدنا delta بالساوي واحد
+
+44
+00:05:01,290 --> 00:05:05,190
+وقلنا لو كان absolute x minus t أصغر من delta اللي
+
+45
+00:05:05,190 --> 00:05:09,190
+هي واحد بتطلع absolute x أصغر من واحد زاد absolute
+
+46
+00:05:09,190 --> 00:05:13,830
+c وبالتالي absolute الفرق هذا هي أصغر من أو ساوي
+
+47
+00:05:14,180 --> 00:05:18,500
+absolute x هي أصغر من واحد زاد absolute c وانتي
+
+48
+00:05:18,500 --> 00:05:23,820
+absolute c فأصغر من واحد زاد اتنين فabsolute c ضرب
+
+49
+00:05:23,820 --> 00:05:29,460
+absolute x minus c الان بدي أخلي هذا أصغر من
+
+50
+00:05:29,460 --> 00:05:39,160
+epsilon هذا بدي أخليه أصغر من epsilon لما
+
+51
+00:05:39,160 --> 00:05:40,920
+يكون هذا أصغر من delta
+
+52
+00:05:48,400 --> 00:05:52,700
+فباخد إذا لما يكون هذا أصغر من دلتا فهذا بصير أصغر
+
+53
+00:05:52,700 --> 00:05:58,880
+من واحد زي اتنين absolute c في دلتا لما يكون ال
+
+54
+00:05:58,880 --> 00:06:03,000
+absolute value ل X معناه C أصغر من دلتا فهذا بطلع
+
+55
+00:06:03,000 --> 00:06:07,740
+أصغر من واحد زي اتنين في absolute c في دلتا الآن
+
+56
+00:06:07,740 --> 00:06:16,040
+متى بيكون هذا أصغر من إبسرن لما دلتاإذا كانت delta
+
+57
+00:06:16,040 --> 00:06:25,240
+هذه أصغر من أوي ساوي epsilon على واحد زايد اتنين
+
+58
+00:06:25,240 --> 00:06:29,960
+في absolute of c إذا هاي قيمة تانية ل delta هاي
+
+59
+00:06:29,960 --> 00:06:35,660
+اندي delta بساوي واحد و delta أصغر من أوي ساوي
+
+60
+00:06:35,660 --> 00:06:39,760
+epsilon على واحد زايد اتنين في absolute of c إذا
+
+61
+00:06:39,760 --> 00:06:49,590
+باجي بقولlet epsilon أكبر من السفر be given a
+
+62
+00:06:49,590 --> 00:06:57,690
+choose delta بتساوي ال minimum الأصغر بين القمتين
+
+63
+00:06:57,690 --> 00:07:07,050
+واحد وepsilon على واحد زاد اتنين في absolute c ال
+
+64
+00:07:07,050 --> 00:07:13,660
+delta هذه الآن عدد موجب ويعتمد على epsilonإذا لهذه
+
+65
+00:07:13,660 --> 00:07:24,140
+الـ Delta لو كان X ينتمي ل R اللي هو مجال الدالة و
+
+66
+00:07:24,140 --> 00:07:31,780
+Absolute X minus C أكبر من صفر أصغر من Delta فهذا
+
+67
+00:07:31,780 --> 00:07:36,960
+بيؤدي طبعا ال Delta هذه هي الأصغر من العددين هذوله
+
+68
+00:07:36,960 --> 00:07:40,920
+وبالتالي أصغر من أو يساوي واحد وأصغر من أو يساوي
+
+69
+00:07:40,920 --> 00:07:46,820
+كسر هذافالـ delta أكيد أصغر من أو يساوي الواحد،
+
+70
+00:07:46,820 --> 00:07:52,360
+لما الـ delta أصغر من أو يساوي الواحد، هذا بيقدر
+
+71
+00:07:52,360 --> 00:07:56,940
+أن absolute X
+
+72
+00:07:56,940 --> 00:08:04,800
+أصغر من واحد زائد absolute Cوكمان هذا بيقدي أنه
+
+73
+00:08:04,800 --> 00:08:11,000
+absolute x تربية سالب c تربية أصغر من أوي ساوي
+
+74
+00:08:11,000 --> 00:08:23,400
+absolute x زاد absolute c absolute
+
+75
+00:08:23,400 --> 00:08:28,640
+x سالب c وبالتالي هذا أصغر من أوي ساوي واحد زاد
+
+76
+00:08:28,640 --> 00:08:39,840
+اتنين absolute cو هذا اصغر من delta و
+
+77
+00:08:39,840 --> 00:08:49,520
+الان ال delta هذه طبعا
+
+78
+00:08:49,520 --> 00:08:55,420
+هذا اصغر من delta و ال delta قلنا اصغر منها و
+
+79
+00:08:55,420 --> 00:08:58,600
+يساوي epsilon هاي واحد زي اتنين
+
+80
+00:09:19,770 --> 00:09:26,670
+بشكل صحيح بما أن ابسلون أكبر من السفر was
+
+81
+00:09:26,670 --> 00:09:27,550
+arbitrary
+
+82
+00:09:31,810 --> 00:09:35,410
+إذاً هيك بنكون أثباتنا لكل epsilon أكبر من السفر
+
+83
+00:09:35,410 --> 00:09:41,150
+يوجد delta تعتمد على epsilon عدد موجة ب half لكل x
+
+84
+00:09:41,150 --> 00:09:46,710
+المسافة مختلفة عن ال c والمسافة بينها وبين ال c
+
+85
+00:09:46,710 --> 00:09:52,590
+أصغر من delta بتطلع المسافة بين f of x وc تربيه
+
+86
+00:09:52,590 --> 00:10:01,190
+أصغر من epsilon إذاً we haveBy definition إن ال
+
+87
+00:10:01,190 --> 00:10:11,390
+limit ل X تربيه لما X تقول إلى C بساوي C تربيه وهو
+
+88
+00:10:11,390 --> 00:10:17,410
+المطلوب، okay؟ إذن هذا هو برهان إن ال limit للدالة
+
+89
+00:10:17,410 --> 00:10:23,300
+التربيهية عن C بساوي C تربيهاستخدمنا تعريف epsilon
+
+90
+00:10:23,300 --> 00:10:28,260
+دلتا وشوفنا ان دلتا هنا لازم تكون الأصغر من
+
+91
+00:10:28,260 --> 00:10:34,520
+الكمتين اللي هو الواحد والكاسر اللي هناك هي اي
+
+92
+00:10:34,520 --> 00:10:41,020
+سخصار اي سؤال خلينا ناخد كمان مثال مشابه لهذا وفيه
+
+93
+00:10:41,020 --> 00:10:44,560
+ال delta برضه بتساوي ال minimum لكمتين
+
+94
+00:10:53,440 --> 00:11:02,620
+المثال الرقم أربعة show أنه ال limit لواحد على x
+
+95
+00:11:02,620 --> 00:11:14,020
+لما x تقول إلى zero لأ
+
+96
+00:11:14,020 --> 00:11:20,340
+ال limit لواحد على x لما x تقول إلى أي عدد cبساوي
+
+97
+00:11:20,340 --> 00:11:27,420
+واحد على C حيث C أكبر من 7 فهنا
+
+98
+00:11:27,420 --> 00:11:30,520
+بناخد ال function تبعتي الدالة اللي بتجميلها ال
+
+99
+00:11:30,520 --> 00:11:36,800
+limit هي عبارة عن F of X بساوي واحد على X حيث X
+
+100
+00:11:36,800 --> 00:11:41,720
+موجبة إذا المجال تبع الدالة هذه الفترة المفتوحة من
+
+101
+00:11:41,720 --> 00:11:49,410
+سفر إلى ما لا نهاية واند C عدد موجبةطيب انا عايز
+
+102
+00:11:49,410 --> 00:11:57,190
+اثبت ان absolute f of x minus واحد على c بدي هذا
+
+103
+00:11:57,190 --> 00:12:02,470
+يكون اصغر من اي given epsilon عندما x تكون قريبة
+
+104
+00:12:02,470 --> 00:12:12,230
+من ال c او في جوار delta لل cفهذا طبعا اش بساوي هي
+
+105
+00:12:12,230 --> 00:12:17,790
+absolute واحد على X minus واحد على C وهذا بساوي
+
+106
+00:12:17,790 --> 00:12:27,950
+absolute C minus X على X في C وهذا بساوي واحد على
+
+107
+00:12:27,950 --> 00:12:33,270
+X في C ضرب absolute X minus C
+
+108
+00:12:38,060 --> 00:12:45,780
+الان بدي أحاول أجيب upper bound عدد
+
+109
+00:12:45,780 --> 00:12:53,720
+موجة ب M بحيث ال 1 على X في C يكون أصغر من أو ساوي
+
+110
+00:12:53,720 --> 00:12:57,360
+ال M تعالوا نشوف كيف نجيب ال upper bound هذا أو ال
+
+111
+00:12:57,360 --> 00:13:06,330
+boundأنا عندى ال take الاول take انا عندى ال c عدد
+
+112
+00:13:06,330 --> 00:13:12,190
+موجب take delta بساوي c على اتنين هذا عدد موجب
+
+113
+00:13:12,190 --> 00:13:16,050
+then
+
+114
+00:13:16,050 --> 00:13:23,510
+absolute x minus c اصغر من delta اللى هو بساويC ع
+
+115
+00:13:23,510 --> 00:13:32,030
+2 بيقدي ان X أصغر من ثلاثة C ع 2 أكبر من C ع 2
+
+116
+00:13:32,030 --> 00:13:43,430
+وهذا بيقدي ان واحد على X في C أصغر من اتنين على C
+
+117
+00:13:43,430 --> 00:13:44,270
+ترمية
+
+118
+00:13:50,000 --> 00:13:54,920
+ال X أكبر من C على 2 إذا مقلوب ال X أصغر من 2 على
+
+119
+00:13:54,920 --> 00:14:01,220
+C مقلوب ال X و أضربها في 1 على C بيطلع أصغر من 2
+
+120
+00:14:01,220 --> 00:14:07,100
+على C تربيه وبالتالي
+
+121
+00:14:07,100 --> 00:14:13,540
+هذا العدد هذا هو ال M عدد
+
+122
+00:14:13,540 --> 00:14:16,320
+موجة إذا
+
+123
+00:14:18,670 --> 00:14:28,290
+في الحالة هذه في الحالة
+
+124
+00:14:28,290 --> 00:14:34,910
+هذه بصير عندي هذا أصغر من اتنين على C تربية وطبعا
+
+125
+00:14:34,910 --> 00:14:39,430
+هذا أصغر من Delta Absolute X minus C طبعا بيكون
+
+126
+00:14:39,430 --> 00:14:44,690
+أصغر من Delta الآن عشان يكون هذا أصغر من أو ساوي
+
+127
+00:14:44,690 --> 00:14:52,820
+Epsilonفنختار choose الـ delta أصغر من أو ساوي حل
+
+128
+00:14:52,820 --> 00:14:57,140
+المتباينة هذه في الـ delta فالـ delta ستصبح أصغر
+
+129
+00:14:57,140 --> 00:15:04,540
+من أو ساوي C تربيع على 2 تلصق فهي قيمة تانية لـ
+
+130
+00:15:04,540 --> 00:15:09,440
+delta فبأخد الـ delta ال minimum للقيمة الأولى
+
+131
+00:15:10,560 --> 00:15:16,040
+والقيمة التانية هذا هيخلي انه لكل x المسافة بين او
+
+132
+00:15:16,040 --> 00:15:20,320
+بين c اصغر من delta هتخلي المسافة بين f of x واحد
+
+133
+00:15:20,320 --> 00:15:26,280
+على c اصغر من ال given epsilon نكتب الكلام هذا let
+
+134
+00:15:26,280 --> 00:15:29,240
+epsilon be given choose delta بالساوي ال minimum
+
+135
+00:15:29,240 --> 00:15:36,260
+نختار
+
+136
+00:15:36,260 --> 00:15:42,030
+delta ال minimumللعدد الموجة بـ c ع 2، والعدد
+
+137
+00:15:42,030 --> 00:15:49,000
+التاني ده هو c تربيه ع 2 في epsilonطبعا هذا عدد
+
+138
+00:15:49,000 --> 00:15:52,740
+أكيد عدد موجب لأن هذا موجب وهذا موجب والأصغر بينهم
+
+139
+00:15:52,740 --> 00:15:56,820
+هيطلع موجب واتنين بيعتمدوا على epsilon إذن delta
+
+140
+00:15:56,820 --> 00:16:00,620
+عدد موجب بيعتمد على epsilon إذا لأي epsilon أكبر
+
+141
+00:16:00,620 --> 00:16:04,360
+من سفر هين أثبتت يوجد delta تعتمد على epsilon عدد
+
+142
+00:16:04,360 --> 00:16:11,680
+موجب بحيث أنه لكل x ينتمي لإيه المجال هنا اللي هو
+
+143
+00:16:11,680 --> 00:16:19,300
+الفترة المفتوحة من سفر إلى دالة نهايةو absolute x
+
+144
+00:16:19,300 --> 00:16:25,400
+minus c أكبر من سفر أصغر من ال delta هذا بيقدي أن
+
+145
+00:16:25,400 --> 00:16:33,260
+ال delta هذه أصغر من أو يساوي c ع 2 فلما ال delta
+
+146
+00:16:33,260 --> 00:16:39,280
+تكون أصغر من أو يساوي c ع 2 هذا بيقدي أنه واحد على
+
+147
+00:16:39,280 --> 00:16:42,460
+واحد
+
+148
+00:16:42,460 --> 00:16:52,060
+على xفى c أصغر من اتنين على c تربية وهذا بدوره
+
+149
+00:16:52,060 --> 00:17:00,940
+بيقدم absolute واحد على x minus واحد على c بساوي
+
+150
+00:17:00,940 --> 00:17:06,480
+واحد على x في c في absolute x minus c أصغر من
+
+151
+00:17:06,480 --> 00:17:14,850
+اتنين على c تربية فى deltaوالـ delta هذه الأن أصغر
+
+152
+00:17:14,850 --> 00:17:19,310
+من أو يساوي الـ delta هذه هي الـ delta اللي فوق
+
+153
+00:17:19,310 --> 00:17:25,010
+أصغر من أو يساوي العدد هذا أيه والعدد التاني لأنها
+
+154
+00:17:25,010 --> 00:17:31,130
+الأصغر بين اتنين لأن هي اتنين على c تربيع ضرب c
+
+155
+00:17:31,130 --> 00:17:36,390
+تربيع اتنين في epsilon هذا بروح مع هذا مخلوق بعض
+
+156
+00:17:36,390 --> 00:17:41,030
+بيضل عندي epsilon since
+
+157
+00:17:43,190 --> 00:17:50,970
+Y أكبر من السفر was arbitrary إذا أنا لكل Y أكبر
+
+158
+00:17:50,970 --> 00:17:56,850
+من السفر جبت Delta تعتمد على Y بحيث لكل X مختلفة
+
+159
+00:17:56,850 --> 00:18:00,570
+عن الـC المسافة بينها وبين الـC أصغر من Delta كل
+
+160
+00:18:00,570 --> 00:18:05,450
+المسافة بين F of X و1 على C أصغر من Y إذا by
+
+161
+00:18:05,450 --> 00:18:06,010
+definition
+
+162
+00:18:09,260 --> 00:18:14,820
+by definition of limit بيطلع عند ال limit لل
+
+163
+00:18:14,820 --> 00:18:20,740
+function واحد على X لما X تقول إلى C بيساوي واحد
+
+164
+00:18:20,740 --> 00:18:24,240
+على C وهو المطلوب
+
+165
+00:18:26,860 --> 00:18:31,720
+واضح في أي سؤال؟ في كمان مثال آخر زي هدف الكتاب،
+
+166
+00:18:31,720 --> 00:18:37,860
+هسيبكم تقرؤوه لأن الفكرة شبيهة بالفكرة في المثال
+
+167
+00:18:37,860 --> 00:18:47,720
+الأخير وبالتالي مافيش إشي جديد ننتقل إلى دراسة
+
+168
+00:18:52,750 --> 00:18:56,370
+ال sequential criterion هي حاجة اسمها sequential
+
+169
+00:18:56,370 --> 00:19:09,570
+criterion حاجة .. حاجة بتكافئ التعريف sequential
+
+170
+00:19:09,570 --> 00:19:12,750
+criterion
+
+171
+00:19:45,930 --> 00:19:53,970
+العبارات التالية متكافعةLimit f of x as x tends to
+
+172
+00:19:53,970 --> 00:20:03,290
+c بساوي عدد L بغند عدد حقيقي اتنين for every
+
+173
+00:20:06,410 --> 00:20:14,330
+for every sequence xn contained in A وحدودها
+
+174
+00:20:14,330 --> 00:20:25,090
+مختلفة عن الـC such that limit xn بالساوي C we
+
+175
+00:20:25,090 --> 00:20:31,470
+have limit الـimage لسيكوينس xn as n tends to
+
+176
+00:20:31,470 --> 00:20:34,410
+infinity بالساوي العدد القليل
+
+177
+00:20:39,210 --> 00:20:42,690
+إن الـ sequential criterion هذه بتقول إن عشان أثبت
+
+178
+00:20:42,690 --> 00:20:46,470
+إن ال limit لل function f and x بساوي c بساوي
+
+179
+00:20:46,470 --> 00:20:52,090
+العدد L هدى بكافة إن أنا أثبت إنه لو أخدت أي
+
+180
+00:20:52,090 --> 00:20:59,010
+sequence نهايتها C فلازم يكون نهاية صورتها بساوي
+
+181
+00:20:59,010 --> 00:21:04,140
+العدد Lلو اقدرت اعمل هذا في الكلام فبقى هذا بيكافئ
+
+182
+00:21:04,140 --> 00:21:08,880
+ان احنا نقول ان ال limit ل f of x يعني ال x بيساوي
+
+183
+00:21:08,880 --> 00:21:15,060
+c بيساوي العدد ال .. نثبت النظرية هذه تروف one
+
+184
+00:21:15,060 --> 00:21:23,020
+implies two assume one
+
+185
+00:21:23,020 --> 00:21:28,480
+IE
+
+186
+00:21:30,950 --> 00:21:37,470
+الـ limit لأخب X لما X تقول لـ C بساوي العدد M
+
+187
+00:21:37,470 --> 00:21:44,070
+عايزين
+
+188
+00:21:44,070 --> 00:21:48,450
+نثبت عشان
+
+189
+00:21:48,450 --> 00:21:55,250
+نثبت اتنين عشان نثبت اتنين صحيح to
+
+190
+00:21:55,250 --> 00:21:58,190
+prove two holes
+
+191
+00:22:00,720 --> 00:22:05,500
+to prove two holds let
+
+192
+00:22:05,500 --> 00:22:17,380
+Xn be a sequence in A هدودها مختلفة عن الـC such
+
+193
+00:22:17,380 --> 00:22:26,960
+that limit Xn بالساوي C we claim
+
+194
+00:22:30,360 --> 00:22:45,320
+بت ال limit ل f of x ل f of x n لما
+
+195
+00:22:45,320 --> 00:22:52,340
+n تقول ل infinity دي ساوي L لبرهان ذلك let epsilon
+
+196
+00:22:52,340 --> 00:22:55,400
+أكبر
+
+197
+00:22:55,400 --> 00:22:57,020
+من السفر be given
+
+198
+00:23:02,180 --> 00:23:08,440
+سنس اكس اكس اكس اكس
+
+199
+00:23:10,770 --> 00:23:16,490
+بما أننا فرضين limit f of x لما x تقوله c بالساوي
+
+200
+00:23:16,490 --> 00:23:21,450
+L من تعريف epsilon دلتا لل limit إذا يوجد دلتا
+
+201
+00:23:21,450 --> 00:23:27,770
+تعتمد على epsilon عدد موجب بحيث أنه لكل x ينتمي
+
+202
+00:23:27,770 --> 00:23:33,730
+إلى a وabsolute x minus c أكبر من صفر أصغر من دلتا
+
+203
+00:23:42,740 --> 00:23:52,080
+أبسلون دلتا للنهايات نسمي
+
+204
+00:23:52,080 --> 00:23:53,760
+ال implication هذه star
+
+205
+00:24:01,580 --> 00:24:07,300
+And the limit xn بالساوي سي احنا فرضين ان في انديو
+
+206
+00:24:07,300 --> 00:24:14,840
+سيكوينس xn ونهايتها c then
+
+207
+00:24:14,840 --> 00:24:26,910
+for the aboveدلتا الموجبة يوجد دلتا موجبة خدت دلتا
+
+208
+00:24:26,910 --> 00:24:31,910
+هذه الموجبة وطبق تعريف epsilon capital N لlimit of
+
+209
+00:24:31,910 --> 00:24:36,590
+sequence فبما ان ال sequence هذه نهايتها C إذا لأي
+
+210
+00:24:36,590 --> 00:24:42,070
+دلتا أو epsilon عدد موجبThere exists capital N
+
+211
+00:24:42,070 --> 00:24:46,710
+يعتمد على الـ Delta طبعا الـ Delta تعتمد على
+
+212
+00:24:46,710 --> 00:24:51,370
+إبسلون، إذا الـ N هذه يعتمد على إبسلون عدد طبيعي،
+
+213
+00:24:51,370 --> 00:24:56,650
+بحيث أنه لكل N أكبر من أو ساوي capital N، تطلع
+
+214
+00:24:56,650 --> 00:25:02,130
+عندي absolute X N minus C أصغر من Delta، نسمي ال
+
+215
+00:25:02,130 --> 00:25:03,990
+implication هذه double star
+
+216
+00:25:07,580 --> 00:25:16,300
+now star and double star بيؤدوا إلى ما يلي لو كان
+
+217
+00:25:16,300 --> 00:25:23,340
+M أكبر من أو ساوي capital M هذا بيؤدي انه absolute
+
+218
+00:25:23,340 --> 00:25:27,740
+XM
+
+219
+00:25:27,740 --> 00:25:36,440
+minus C أصغر من دلتا هذا باستخدام double star صح؟
+
+220
+00:25:39,750 --> 00:25:44,230
+لو كانت n أكبر من أو ساوي capital N فبطلع absolute
+
+221
+00:25:44,230 --> 00:25:52,050
+xn minus c أصغر من delta و من ال star لو كان عندى
+
+222
+00:25:52,050 --> 00:25:59,130
+xn طبعا xn هذا موجود في a ال xn موجود في a مختلف
+
+223
+00:25:59,130 --> 00:25:59,810
+عن ال c
+
+224
+00:26:02,830 --> 00:26:07,750
+فلو كان absolute of xn minus c badly except xn
+
+225
+00:26:07,750 --> 00:26:13,850
+أصغر من delta فحسب الstar هذا بقدر absolute of f
+
+226
+00:26:13,850 --> 00:26:22,590
+of xn minus L أصغر من إبسلون الان بما أن هذا صحيح
+
+227
+00:26:22,590 --> 00:26:28,270
+بما أن since إبسلون أكبر من الصفر was arbitrary
+
+228
+00:26:30,740 --> 00:26:42,380
+إن إحنا أثبتنا هيك لكل إبسلون يوجد
+
+229
+00:26:42,380 --> 00:26:50,250
+capital N يعتمد على إبسلون عدد طبيعيبكل n أكبر من
+
+230
+00:26:50,250 --> 00:26:55,310
+أوي سوى capital N absolute f of xn minus L أصغر من
+
+231
+00:26:55,310 --> 00:27:00,150
+إبسلون إذا by إبسلون capital N definition لل limit
+
+232
+00:27:00,150 --> 00:27:06,050
+of sequence بطلع عندي limit لsequence
+
+233
+00:27:06,050 --> 00:27:12,910
+f of xn as n tends to infinity بساوية L وبالتالي
+
+234
+00:27:12,910 --> 00:27:21,850
+هيك بيكون إذا two holesهكذا أثبتنا أن واحد يؤدي
+
+235
+00:27:21,850 --> 00:27:26,610
+إلى اتنين اتنين
+
+236
+00:27:26,610 --> 00:27:30,270
+بيقول for every sequence فهي اللي أخدت arbitrary
+
+237
+00:27:30,270 --> 00:27:36,810
+sequence في a minus c وبشرط بحيث ان ال sequence هي
+
+238
+00:27:36,810 --> 00:27:37,710
+اللي نهيتها c
+
+239
+00:27:42,350 --> 00:27:46,210
+و اثبتنا ان ال limit لل image لل sequence بساوي L
+
+240
+00:27:46,210 --> 00:27:51,710
+هذا بالظبط اللي هو الابارة اتنين لان هيك يكون
+
+241
+00:27:51,710 --> 00:27:58,270
+اثبتنا واحد بيقدي لاتنين واضح مفهوم اللي هو نثبت
+
+242
+00:27:58,270 --> 00:28:02,510
+العكس نثبت ان اتنين بيقدي لواحد
+
+243
+00:28:16,210 --> 00:28:22,870
+بالنسبة العبارة اثنين بتقدي للعبارة واحد فالاثنان
+
+244
+00:28:22,870 --> 00:28:27,270
+ذالف بالمناسبة الأخوات اللي قاعدات ورا دولة إيش
+
+245
+00:28:27,270 --> 00:28:31,510
+بتعملوا انتوا؟ ماعليش أوقف تصوير إيش مجاعتكم انتوا
+
+246
+00:28:31,510 --> 00:28:34,830
+أنا أول حاجة و تاني حاجة؟ إيش بتتكلمون؟ دكتور معاك
+
+247
+00:28:34,830 --> 00:28:38,070
+لأ لأ لما هم بتتكلم عامليننا أزعاج لأ باحكوا إذا
+
+248
+00:28:38,070 --> 00:28:40,550
+انتوا بتتكلموا لأ بحكي على اندر ده ليش مصورة أن
+
+249
+00:28:40,550 --> 00:28:44,510
+الوضع هو وضع نفسه لأ بنتكلميش لأ باحكي عن البرادة
+
+250
+00:28:44,510 --> 00:28:49,530
+اللي ورا دولةفي بنات بتتكلموا، أنتوا اللي ورا
+
+251
+00:28:49,530 --> 00:28:55,390
+بتتكلموا ولا في ناس غيرك؟ في حد بتتكلم و أنا بشرح
+
+252
+00:28:55,390 --> 00:29:00,090
+تتكلم و هذا عمللي أزعاج كتير، فلو سمحتوا إذا أنتوا
+
+253
+00:29:00,090 --> 00:29:04,890
+قاعدين تتكلموا ورا اطلعوا في حديقة اتكلموا فيها،
+
+254
+00:29:04,890 --> 00:29:10,670
+حتى لو باسم المحاضرة ممنوح تتكلموا، شوية أزعاجهو
+
+255
+00:29:10,670 --> 00:29:13,850
+مين اللي بتتكلم؟ إذاً أنت اللي بتتكلم من قعدته
+
+256
+00:29:13,850 --> 00:29:20,350
+وراك بتتكلم ما تتكلمش لإن غير ترفع يدك، ارفع يدك و
+
+257
+00:29:20,350 --> 00:29:24,670
+تقعد لسانك، ما تتكلمي مع الجنك بدون اسم، لإن هذا
+
+258
+00:29:24,670 --> 00:29:28,150
+عندنا قاعدة في المحاضرة، ممنوع حد يتكلم مع الجنك و
+
+259
+00:29:28,150 --> 00:29:34,030
+تتحدث مع حد شخص آخر إلا إذا عندك سؤال، ترفع يدك،
+
+260
+00:29:34,030 --> 00:29:37,990
+تستنى لما أقول من عنده سؤال من عنده حاسب صار، ترفع
+
+261
+00:29:37,990 --> 00:29:41,790
+يدك و بجاوبكانا مابتقدر انت تعمليني قصة مع اللغة،
+
+262
+00:29:41,790 --> 00:29:48,330
+قوم انت .. انت .. قوم يقعد في مطعم، يبقين عالم،
+
+263
+00:29:48,330 --> 00:29:51,190
+فلو سرحت انك تتكلم مش مع بعض، هانديك السفسة
+
+264
+00:30:01,910 --> 00:30:04,950
+ممنوع حد يتكلم مع الجنب في المحاضرة، أنا بعمل
+
+265
+00:30:04,950 --> 00:30:08,850
+إزعاج، بدك أنت في السفسار، عندك أي إيش أنا بواجب،
+
+266
+00:30:08,850 --> 00:30:13,990
+بقول من عنده سؤال، من عنده حاجة، اتفضل يسأل
+
+267
+00:30:13,990 --> 00:30:21,850
+ساعتها، بس لا تتكلم وأنا ضايق طرابك،
+
+268
+00:30:21,850 --> 00:30:24,230
+يقولنا الكلام قدر مئة مرة في المحاضرة، ممنوع
+
+269
+00:30:24,230 --> 00:30:25,690
+الكلام الجامل
+
+270
+00:30:35,880 --> 00:30:40,860
+تفضل يا أبو حمزي إذا
+
+271
+00:30:40,860 --> 00:30:45,380
+الأن بدنا نكمل البرنامج بإثبات الأثنين بأد لواحد
+
+272
+00:30:45,380 --> 00:30:51,740
+الإثبات الأثنين بأد لواحد بدنا نثبت we prove ال
+
+273
+00:30:51,740 --> 00:30:59,120
+contrapositive we prove not واحد implies not two
+
+274
+00:31:01,070 --> 00:31:04,730
+هذا هو ال contrapositive للعبارة لل implication
+
+275
+00:31:04,730 --> 00:31:16,630
+هذه فإذا assume .. assume not one ف not one معناته
+
+276
+00:31:16,630 --> 00:31:27,190
+ال limit ل F of X لما X تقول ل C لا تساوي L
+
+277
+00:31:30,200 --> 00:31:32,020
+this means هذا يعني
+
+278
+00:31:35,090 --> 00:31:40,190
+الان نرجع لتعريف ال limit أو ال function شوفنا
+
+279
+00:31:40,190 --> 00:31:42,530
+المرة السادسة في تعريفين في epsilon delta
+
+280
+00:31:42,530 --> 00:31:46,270
+definition و في neighborhood definition ال
+
+281
+00:31:46,270 --> 00:31:49,610
+neighborhood definition بيقول اذا كان عشان تكون
+
+282
+00:31:49,610 --> 00:31:53,770
+limit ل f of x من x او ل c بالساوي عدد L هذا
+
+283
+00:31:53,770 --> 00:31:57,210
+بيكافئ انه لكل epsilon neighborhood ل L يوجد delta
+
+284
+00:31:57,210 --> 00:32:01,130
+neighborhood لل C بحيث لكل x في ال delta
+
+285
+00:32:01,130 --> 00:32:04,630
+neighborhoodصورته لازم تطلع في الـ epsilon
+
+286
+00:32:04,630 --> 00:32:08,290
+neighborhood الان ان في الكلام هذا ما معنى ان ال
+
+287
+00:32:08,290 --> 00:32:13,570
+limit and c بيستويش لعدد L معناته بدل لكل epsilon
+
+288
+00:32:13,570 --> 00:32:17,930
+neighborhood ل L there exist there exist epsilon
+
+289
+00:32:17,930 --> 00:32:25,330
+zero neighborhood of L بسميه
+
+290
+00:32:25,330 --> 00:32:32,110
+V epsilon zero neighborhood ل L بحيث انه لكل
+
+291
+00:32:33,900 --> 00:32:43,060
+Delta neighborhood V Delta أو C يوجد X يعتمد على
+
+292
+00:32:43,060 --> 00:32:50,540
+Delta ينتمي إلى A ومختلف عن الـ C وموجود في الـ
+
+293
+00:32:50,540 --> 00:32:55,560
+Delta neighborhood بحيث
+
+294
+00:32:55,560 --> 00:33:01,100
+أن صورة الـ X Delta
+
+295
+00:33:05,360 --> 00:33:16,380
+لا تنتمي للإبسلون zero neighborhood ل LL طيب
+
+296
+00:33:16,380 --> 00:33:26,140
+لو أخدنا take لكل N في N take delta بساوي واحد على
+
+297
+00:33:26,140 --> 00:33:31,600
+N then
+
+298
+00:33:31,600 --> 00:33:32,540
+they exist
+
+299
+00:33:37,520 --> 00:33:47,100
+دلتا تعتمد على n دلتا تعتمد على n دلتا تعتمد على n
+
+300
+00:33:47,100 --> 00:33:50,360
+دلتا
+
+301
+00:33:50,360 --> 00:33:56,520
+تعتمد
+
+302
+00:33:56,520 --> 00:33:57,620
+على
+
+303
+00:34:00,880 --> 00:34:10,260
+و بحيث ان F ل Xm لا ينتمي لإبسلون Zero
+
+304
+00:34:10,260 --> 00:34:18,300
+neighborhood ل L طب ما هذا الأخير معناه أو بيقدّي
+
+305
+00:34:26,330 --> 00:34:34,390
+this implies هذا بيقدّي نكون أثبتنا ان لكل n يوجد
+
+306
+00:34:34,390 --> 00:34:42,490
+xn في a إذا يوجد sequence xn موجودة في ال set A
+
+307
+00:34:42,490 --> 00:34:47,170
+حدودها مختلفة عن ال C كل ال xn مختلفة عن ال C
+
+308
+00:34:47,170 --> 00:35:01,980
+وموجودة فيv1 على n of c بحيث ان f ل xn لا تنتمي ل
+
+309
+00:35:01,980 --> 00:35:11,900
+v epsilon zero ل n لكل n هذا معناه ان يوجد
+
+310
+00:35:11,900 --> 00:35:20,580
+sequence xn contained in a minus c بحيث انلاحظوا
+
+311
+00:35:20,580 --> 00:35:26,240
+الـ sequence Xn تنتمي ل V 1 على N of C اللي هو
+
+312
+00:35:26,240 --> 00:35:30,960
+عبارة عن الفترة C سالف واحد على N C موجب واحد على
+
+313
+00:35:30,960 --> 00:35:37,360
+N لكل N هذا معناه ان absolute Xn minus C أصغر من
+
+314
+00:35:37,360 --> 00:35:42,420
+واحد على N أصغر
+
+315
+00:35:42,420 --> 00:35:47,020
+من واحد على N لكل N في N and
+
+316
+00:35:50,460 --> 00:35:55,500
+F of Xn لا تنتمي للـY0 neighborhood الـY0
+
+317
+00:35:55,500 --> 00:35:59,720
+neighborhood هذا عبارة عن الفترة المفتوحة L minus
+
+318
+00:35:59,720 --> 00:36:08,000
+Y0 L زائد Y0 فF of Xn لا تنتمي للفترة المفتوحة هذه
+
+319
+00:36:08,000 --> 00:36:15,720
+معناه absolute المسافة بين F of Xn وL أكبر من أو
+
+320
+00:36:15,720 --> 00:36:18,460
+ساوي Y0 لكل N
+
+321
+00:36:21,430 --> 00:36:26,750
+هذا الكلام معناه أن
+
+322
+00:36:26,750 --> 00:36:32,190
+يوجد sequence x in موجودة في a حدودها مختلفة عن ال
+
+323
+00:36:32,190 --> 00:36:41,030
+c وهذا الكلام معناه such that limit x in بساوي c
+
+324
+00:36:43,330 --> 00:36:51,410
+حسب نظرية اتنين اربعة اتنين
+
+325
+00:36:51,410 --> 00:36:54,970
+اربعة اتنين اربعة اتنين اربعة اتنين اربع اتنين
+
+326
+00:36:54,970 --> 00:36:55,590
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+327
+00:36:55,590 --> 00:36:55,730
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+328
+00:36:55,730 --> 00:36:56,990
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+329
+00:36:56,990 --> 00:36:59,130
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+330
+00:36:59,130 --> 00:37:08,450
+اتنين اربع اتنين اربع اتنين اربع ا
+
+331
+00:37:12,640 --> 00:37:16,920
+الـ limit لـ
+
+332
+00:37:16,920 --> 00:37:20,760
+sequence f of xn لما n تقول الـ infinity مش ممكن
+
+333
+00:37:20,760 --> 00:37:26,540
+تساوي العدد L لأن لو ال limit ل f of xn بيساوي
+
+334
+00:37:26,540 --> 00:37:30,220
+العدد L، المفروض ال absolute value للفرق ده تكون
+
+335
+00:37:30,220 --> 00:37:36,660
+أصغر من أي epsilon zero لكل N من capital N و انت
+
+336
+00:37:36,660 --> 00:37:40,930
+طالع، لكن هذا الكلام مش صحيحOkay إن هذا بالظبط
+
+337
+00:37:40,930 --> 00:37:48,210
+العبارة الأخيرة which which
+
+338
+00:37:48,210 --> 00:37:55,690
+is نفي العبارة اتنين هذه
+
+339
+00:37:55,690 --> 00:37:58,450
+العبارة الأخيرة هي نفي العبارة اتنين هذه العبارة
+
+340
+00:37:58,450 --> 00:38:06,320
+اتنين ال statement اتنينبقول لكل sequence بحيث ان
+
+341
+00:38:06,320 --> 00:38:09,100
+ال limit بتاعتها C، ال limit لل image بتاعتها
+
+342
+00:38:09,100 --> 00:38:13,660
+بالساولة L هنا اتوصلنا ان there exist بدل for all
+
+343
+00:38:13,660 --> 00:38:18,660
+there exist sequence نهايتها C لكن نهاية صورتها
+
+344
+00:38:18,660 --> 00:38:25,020
+لاتساول L إذا هيك بنكون أثبتنا أنه لا إذا we
+
+345
+00:38:25,020 --> 00:38:29,800
+proved not
+
+346
+00:38:31,130 --> 00:38:39,390
+not one implies not two therefore two implies one
+
+347
+00:38:39,390 --> 00:38:46,610
+وهذا يكمل البرهان واضح؟ في أي سؤال؟ في أي استفسار؟
+
+348
+00:38:46,610 --> 00:38:53,590
+يبدو أننا كملنا برهان النظرية في أي استفسار؟
+
+349
+00:38:55,700 --> 00:39:03,780
+الان من النظرية هذه ينتج مباشرة نظرية مهمة لتقل
+
+350
+00:39:03,780 --> 00:39:13,660
+عنها أهمية ويلها اسم divergence
+
+351
+00:39:13,660 --> 00:39:16,900
+criteria
+
+352
+00:39:25,650 --> 00:39:36,650
+لت if the function from A to R and see the cluster
+
+353
+00:39:36,650 --> 00:39:39,750
+point
+
+354
+00:39:39,750 --> 00:39:45,850
+of A then واحد
+
+355
+00:39:47,360 --> 00:39:54,460
+الـ limit ل f of x لما x تقول ل c لا تساوي ال f
+
+356
+00:39:54,460 --> 00:40:01,440
+and only f there exist a sequence xm contained in
+
+357
+00:40:01,440 --> 00:40:10,180
+a حدودها مختلفة عن ال c such that limit xm بتساوي
+
+358
+00:40:10,180 --> 00:40:20,490
+c butLimit f of x in لاتساوي n الكرتيريا
+
+359
+00:40:20,490 --> 00:40:25,750
+التانية اللي هي عشان
+
+360
+00:40:25,750 --> 00:40:31,930
+نقول limit f of x لما x تقولها c does not exist in
+
+361
+00:40:31,930 --> 00:40:43,690
+Rهذا بكافئ أن هناك سيكوانس Xn محتوى A حدودها
+
+362
+00:40:43,690 --> 00:40:50,870
+مختلفة عن C بحيث أن نهايتها بساوي
+
+363
+00:40:50,870 --> 00:41:00,310
+C بط نهاية صورتها لا
+
+364
+00:41:00,310 --> 00:41:02,670
+توجد في R
+
+365
+00:41:16,230 --> 00:41:21,250
+كمان النظرية هذه مرهانها ينتج مباشرة من النظرية
+
+366
+00:41:21,250 --> 00:41:27,990
+اللي فوق مثلا هي عندي لإثبات ال band الأول عشان
+
+367
+00:41:27,990 --> 00:41:31,130
+أثبت limit f of x مستويش L and C
+
+368
+00:41:34,380 --> 00:41:38,880
+يعني كإني بقول نفي العبارة واحد هذا هو نفي العبارة
+
+369
+00:41:38,880 --> 00:41:42,560
+واحد طب احنا لسه بثبتين ان واحد بكافي اتنين
+
+370
+00:41:42,560 --> 00:41:46,560
+وبالتالي نفي العبارة واحد بكافي نفي الاتنين فنفي
+
+371
+00:41:46,560 --> 00:41:51,100
+الاتنين هذا هو يوجد a sequence تتقارب ل C لكن صورة
+
+372
+00:41:51,100 --> 00:41:56,720
+تلاتة تتقارب لL إذا برهان الجزء الأول نتيجة مباشرة
+
+373
+00:41:56,720 --> 00:42:02,130
+على مضارية ال form والجزء التاني زيه بدل هناعشان
+
+374
+00:42:02,130 --> 00:42:06,070
+اقول ان ال limit هذه does not exist يعني لو اخدت
+
+375
+00:42:06,070 --> 00:42:12,650
+اي عدد L فال limit هنا لا تساوي L معناته انه في
+
+376
+00:42:12,650 --> 00:42:18,050
+sequence و الكلام هذا ال limit هذه ماسويش اي L اي
+
+377
+00:42:18,050 --> 00:42:23,890
+عدد حقيقي اذا النظرية هذه نتيجة مباشرة على النظرية
+
+378
+00:42:23,890 --> 00:42:27,880
+sequential criterion النظرية التي سبقتهاالان هذه
+
+379
+00:42:27,880 --> 00:42:31,560
+النظرية هنستخدمها في إثبات إن ال limit لدالة
+
+380
+00:42:31,560 --> 00:42:36,000
+معينة، عن نقطة معينة غير موجودة، فهي بعض الأمثلة
+
+381
+00:42:36,000 --> 00:42:39,140
+كيف
+
+382
+00:42:39,140 --> 00:42:42,500
+نستخدم ال divergence كتير، كيف نثبت ال divergence
+
+383
+00:42:42,500 --> 00:42:48,020
+أو عدم وجود limit لدالة معينة عن نقطة معينة، فمثلا
+
+384
+00:42:48,020 --> 00:43:02,210
+ناخد أول مثالshow that limit ل 1 على x لما x تقول
+
+385
+00:43:02,210 --> 00:43:09,470
+إلى السفر does not exist in R فلبرهان
+
+386
+00:43:09,470 --> 00:43:16,870
+ذلك let
+
+387
+00:43:16,870 --> 00:43:24,050
+f of x بساوي 1 على x و ده أخد الـ x موجبةيعني
+
+388
+00:43:24,050 --> 00:43:27,130
+نعتبر أن ال domain للدالة هذه اللي هو الفترة A
+
+389
+00:43:27,130 --> 00:43:31,270
+بساوي الفترة مفتوحة من الصفر لما لا نهاية و نثبت
+
+390
+00:43:31,270 --> 00:43:34,750
+أن الدالة هذه ماليهاش limit عند الصفر أو عند الصفر
+
+391
+00:43:34,750 --> 00:43:40,650
+من اليمين فلإثبات أن ال limit للدالة هذه عند الصفر
+
+392
+00:43:40,650 --> 00:43:44,470
+ماهياش موجودة حسب ال divergence criteria يعني بدي
+
+393
+00:43:44,470 --> 00:43:48,210
+أثبت أن يوجه .. بدي أجيب sequence نهايتها صفر لكن
+
+394
+00:43:48,210 --> 00:43:52,490
+نهاية صورتها مش موجودة فال sequence إذا هنا
+
+395
+00:43:52,490 --> 00:43:59,560
+considerالـ sequence التي تفي بهذا الغرض اللي هي
+
+396
+00:43:59,560 --> 00:44:06,400
+xn بالساوي واحد على n لكل n في n فواضح أنه limit
+
+397
+00:44:06,400 --> 00:44:16,720
+xn بالساوي limit واحد على n بتساوي سفر وواضح أنه
+
+398
+00:44:16,720 --> 00:44:22,800
+xn contained in a اللي هي الفترة هذه معدى السفر
+
+399
+00:44:22,800 --> 00:44:31,310
+صح؟وعندي ال limit لل image لل sequence xn بساوي ال
+
+400
+00:44:31,310 --> 00:44:38,250
+limit ل 1 على xn لما n تقوى ل infinity بساوي ال
+
+401
+00:44:38,250 --> 00:44:43,410
+limit ل n لما n تقوى ل infinity بساوي infinity
+
+402
+00:44:43,410 --> 00:44:49,730
+وهذه طبعا ال infinity does not exist in R ليست عدد
+
+403
+00:44:49,730 --> 00:44:55,980
+حقيقيالنهاية نجحت في إيجاد sequence موجودة في A
+
+404
+00:44:55,980 --> 00:45:00,840
+وحدودها مختلفة عن السفر ونهايتها سفر لكن نهاية
+
+405
+00:45:00,840 --> 00:45:06,960
+صورتها مش موجودة في R وبالتالي therefore by
+
+406
+00:45:06,960 --> 00:45:14,020
+divergence criterion limit
+
+407
+00:45:14,020 --> 00:45:23,240
+ل F of X أو واحد على Xلما x سقول إلى 0 does not
+
+408
+00:45:23,240 --> 00:45:28,860
+exist in R وفي حقيقة الأمر اثبتنا ان limit 1 على x
+
+409
+00:45:28,860 --> 00:45:34,180
+لما x سقول إلى 0 من اليمين غير موجودة لان اخذنا
+
+410
+00:45:34,180 --> 00:45:42,620
+المجال كل الاعداد الموجودة بالمثل ممكن اثبات ان
+
+411
+00:45:42,620 --> 00:45:50,390
+limit ل1 على xلمّا X تقول إلى سفر من اليسار does
+
+412
+00:45:50,390 --> 00:45:55,540
+not existان انا اخد المرة هذه ال X هنا في الدالة
+
+413
+00:45:55,540 --> 00:46:00,560
+هذه ال domain تبعها الفترة من سالب ماله نهاية الى
+
+414
+00:46:00,560 --> 00:46:05,720
+سفر و اقول ان ال X هنا أصغر من سفر و نفس البرهان
+
+415
+00:46:05,720 --> 00:46:09,820
+هيطلع عندى ال limit لما X تقوله سفر من اليسار does
+
+416
+00:46:09,820 --> 00:46:13,500
+not exist وبالتالي ال limit عند ال X من الجهتين
+
+417
+00:46:13,500 --> 00:46:19,820
+does not exist تمام okay هذا مثال مثال تاني واضح
+
+418
+00:46:19,820 --> 00:46:21,440
+فيه اي سفصار فيه اي سؤال
+
+419
+00:46:25,390 --> 00:46:35,810
+ناخد مثال تاني show
+
+420
+00:46:35,810 --> 00:46:42,590
+that limit للـ signum function signum x لما x تقول
+
+421
+00:46:42,590 --> 00:46:48,930
+إلى سفر does not exist where حيث و ال signum
+
+422
+00:46:48,930 --> 00:46:52,450
+function where
+
+423
+00:46:57,580 --> 00:47:02,460
+where signum x هي عبارة عن function في x بنعرفها
+
+424
+00:47:02,460 --> 00:47:07,000
+على أنها واحد إذا كان x أكبر من سفر سفر إذا كان x
+
+425
+00:47:07,000 --> 00:47:12,300
+بساول سفر سالب واحد إذا كان x أصغر من سفر وهي
+
+426
+00:47:12,300 --> 00:47:13,360
+الرسمة تبعتها
+
+427
+00:47:24,680 --> 00:47:28,920
+فالدالة لما x أكبر من صفر بيستوي ثابت واحد عند
+
+428
+00:47:28,920 --> 00:47:34,400
+الصفر بيستوي صفر و لما x أصغر من واحد بيستوي سالب
+
+429
+00:47:34,400 --> 00:47:38,040
+واحد طيب
+
+430
+00:47:38,040 --> 00:47:48,360
+note that لاحظوا أن الدالة هذه sigma of x بتساوي
+
+431
+00:47:48,360 --> 00:47:52,120
+x على absolute x fx
+
+432
+00:47:53,900 --> 00:47:59,820
+لا تساوي سفر إذا كان x بساوي سفر فدالة sigma بها
+
+433
+00:47:59,820 --> 00:48:07,900
+نفس x على absolute xنفس .. نفس الحاجة طيب الان
+
+434
+00:48:07,900 --> 00:48:13,400
+اثبات ان ال limit لدالها جاند سفر مش موجودة طبعا
+
+435
+00:48:13,400 --> 00:48:17,440
+في تفاضل ألف في برهان في تفاضل ألف بيقول ان هى
+
+436
+00:48:17,440 --> 00:48:21,380
+الدالة لما X اولا سفر من اليمين ال limit لها واحد
+
+437
+00:48:21,380 --> 00:48:25,500
+لما X اولا سفر من اليمين نهيتها سالب واحد ال limit
+
+438
+00:48:25,500 --> 00:48:28,040
+من اليمين مستويش ال limit من اليسار اذا ال limit
+
+439
+00:48:28,040 --> 00:48:33,690
+لدالها جاند سفر does not exist برهانaccurate صحيح
+
+440
+00:48:33,690 --> 00:48:37,030
+مية المية مافي مشكلة لكن لو بدنا نعطي برهان
+
+441
+00:48:37,030 --> 00:48:41,810
+باستخدام ال divergence criterion فالبرهان هيكون
+
+442
+00:48:41,810 --> 00:48:46,270
+كالتالي consider
+
+443
+00:48:46,270 --> 00:48:51,410
+بدنا نجيب sequence xn
+
+444
+00:48:54,550 --> 00:48:58,490
+Rدودها مختلفة عن السفر نهايتها سفر لكن نهايت
+
+445
+00:48:58,490 --> 00:49:03,950
+صورتها بساوي سفر ف consider ال sequence اللي هي Xn
+
+446
+00:49:03,950 --> 00:49:09,110
+الحد اللي عام تبعها Xn بساوي سالف واحد أس ان على N
+
+447
+00:49:09,110 --> 00:49:19,190
+لكل N في N ال sequence هذه تنتمي إلى A اللي هو R
+
+448
+00:49:19,190 --> 00:49:21,890
+بعد السفر
+
+449
+00:49:26,050 --> 00:49:29,530
+موجودة في المجال تبع الدالة المجال تبع الدالة دي
+
+450
+00:49:29,530 --> 00:49:37,570
+كل الأعداد اللي حصلت فيها معدد C صح؟ وعندي و ال
+
+451
+00:49:37,570 --> 00:49:44,610
+limit و ال limit ل XM as M tends to infinity بسوى
+
+452
+00:49:44,610 --> 00:49:50,150
+و ال limitلسالب واحد قص ان على ان لما ان تقول
+
+453
+00:49:50,150 --> 00:49:55,110
+infinity ال limit لل sequence دي ايش بيساوي بيساوي
+
+454
+00:49:55,110 --> 00:50:03,470
+سفر by squeeze theorem او
+
+455
+00:50:03,470 --> 00:50:08,050
+by sandwich theorem but
+
+456
+00:50:08,050 --> 00:50:15,650
+لكن تعالوا نشوف ال limitلـ f of xn as n tends to
+
+457
+00:50:15,650 --> 00:50:19,810
+infinity شو بيساوي؟ بيساوي الـ limit as n tends to
+
+458
+00:50:19,810 --> 00:50:25,750
+infinity احنا عندي الـ xn هنا بيستويش صفر وبالتالي
+
+459
+00:50:25,750 --> 00:50:30,250
+الـ f of x تبعتي اللي هي الـ signum function فهذا
+
+460
+00:50:30,250 --> 00:50:34,050
+بيساوي limit signum xn
+
+461
+00:50:36,620 --> 00:50:41,540
+مظبوط و ال x in قلنا هنا بسويش 0 وبالتالي هذا
+
+462
+00:50:41,540 --> 00:50:47,000
+عبارة عن limit as n tends to infinity ال signal ل
+
+463
+00:50:47,000 --> 00:50:55,420
+x in بساوي x in على absolute x in فهذا
+
+464
+00:50:55,420 --> 00:51:02,210
+بساوي ال limitas n tends to infinity لـ xn عبارة
+
+465
+00:51:02,210 --> 00:51:09,190
+عن سالب واحد قص n على n على absolute xn absolute
+
+466
+00:51:09,190 --> 00:51:16,530
+xn بساوي واحد على n أصبت؟ الآن نجسم ونبسط ال limit
+
+467
+00:51:16,530 --> 00:51:23,750
+as n tends to infinity بطلع سالب واحد قص n وال
+
+468
+00:51:23,750 --> 00:51:27,210
+sequence هذه ال limit تبعتها أثبتنا قبل هيك
+
+469
+00:51:28,730 --> 00:51:33,410
+بطريقتين على الأقل ان ال limit هذه does not exist
+
+470
+00:51:33,410 --> 00:51:44,830
+does not exist وبالتالي اذا either by the
+
+471
+00:51:44,830 --> 00:51:47,630
+divergence criterion
+
+472
+00:51:50,230 --> 00:51:54,070
+هي اثبتت ان الـ use and sequence موجودة في المجال
+
+473
+00:51:54,070 --> 00:51:58,970
+تبع الدالة معدى السفر نهايتها سفر لكن نهاية صورتها
+
+474
+00:51:58,970 --> 00:52:03,270
+does not exist اذا by ال band التاني من ال
+
+475
+00:52:03,270 --> 00:52:11,590
+divergence criterion ال limit لل
+
+476
+00:52:11,590 --> 00:52:17,490
+signum function لما X تقول السفر does not exist
+
+477
+00:52:17,490 --> 00:52:18,570
+غير موجودة
+
+478
+00:52:20,890 --> 00:52:26,890
+Okay تمام واضح واضح البرهان في اي استفسار في اي
+
+479
+00:52:26,890 --> 00:52:34,470
+سؤال Okay
+
+480
+00:52:34,470 --> 00:52:39,470
+نوقف هنا وان شاء الله بنكمل المرة الجاية في بعض
+
+481
+00:52:39,470 --> 00:52:45,290
+مثالين الموجودة في الكتاب تحاولوا تقرؤهم او مثال
+
+482
+00:52:46,220 --> 00:52:50,740
+الشباب بالمثال هذا تحاولوا تقرؤوا و المرة الجاية
+
+483
+00:52:50,740 --> 00:52:52,580
+هنبدأ section جديد
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ac5dc78f782a7e5d778f4c312536054ed70e9352
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/ZfnDnf4RR5M_raw.srt
@@ -0,0 +1,1936 @@
+1
+00:00:20,670 --> 00:00:26,650
+السلام عليكم اليوم ان شاء الله هنكمل section أربعة
+
+2
+00:00:26,650 --> 00:00:35,990
+واحد اللي عرفنا فيه ال limits و ال functions أخدنا
+
+3
+00:00:35,990 --> 00:00:40,650
+المرة الأولى تلتة تعريف epsilon delta ل limit of
+
+4
+00:00:40,650 --> 00:00:45,990
+function وشوفنا أن هذا بكافة تعريف في neighborhood
+
+5
+00:00:45,990 --> 00:00:51,040
+definition لlimit of function على النقطةو بدنا
+
+6
+00:00:51,040 --> 00:00:57,500
+ناخد أمثلة كيف نستخدم تعريف epsilon delta في إثبات
+
+7
+00:00:57,500 --> 00:01:03,360
+أن ال limit لدالة معينة عن مقطة معددة بتساوي عدد
+
+8
+00:01:03,360 --> 00:01:08,310
+محدد فاخدنا بعض الأمثلة اليوم هنستمرهنعطي مزيد من
+
+9
+00:01:08,310 --> 00:01:12,930
+الأمثلة و بندرس خواص ال limits ل ال functions
+
+10
+00:01:12,930 --> 00:01:19,490
+فالمثال اللى وصلناه له رقم تلاتة عايزين نثبت ان ال
+
+11
+00:01:19,490 --> 00:01:25,730
+limit لدلة تربية X تربية لما X تقول ل C بساوي C
+
+12
+00:01:25,730 --> 00:01:29,330
+تربية ف solution
+
+13
+00:01:33,240 --> 00:01:40,260
+ناخد f of x بالساوي x تربية هيفر اكس ينتمي الى r
+
+14
+00:01:40,260 --> 00:01:43,880
+واحنا
+
+15
+00:01:43,880 --> 00:01:50,600
+عايزين من الآخر نثبت ان ال absolute value ل f of x
+
+16
+00:01:50,600 --> 00:01:58,400
+minus c تربية أصغر من أي given epsilon عدد موجة
+
+17
+00:01:59,300 --> 00:02:04,780
+عندما الـ X تكون قريبة من المقطة C أو تقع فيه جوار
+
+18
+00:02:04,780 --> 00:02:13,600
+Delta معينة للعدد C طيب هذا عبارة عن Absolute X
+
+19
+00:02:13,600 --> 00:02:22,620
+تربية سالم C تربية بتحلل إلى Absolute X minus C في
+
+20
+00:02:22,620 --> 00:02:24,620
+X موجة بالـ C
+
+21
+00:02:27,570 --> 00:02:33,430
+إذاً هذا عبارة عن absolute X زائد C في absolute X
+
+22
+00:02:33,430 --> 00:02:37,830
+minus C الأن
+
+23
+00:02:37,830 --> 00:02:42,430
+بدي أحاول أخلي هذا أصغر من أو ساوي عدد موجة بم
+
+24
+00:02:42,430 --> 00:02:47,290
+فبحاول
+
+25
+00:02:47,290 --> 00:02:52,830
+أخد فيه ملة Delta لت دلتا بالساوي واحد
+
+26
+00:03:00,710 --> 00:03:12,430
+then أنا عندي absolute x زائد c هذا أصغر
+
+27
+00:03:12,430 --> 00:03:19,910
+من او ساوي absolute x زايد absolute cفي absolute x
+
+28
+00:03:19,910 --> 00:03:25,450
+minus c استخدام ال triangle inequality absolute x
+
+29
+00:03:25,450 --> 00:03:30,430
+زاد c أصلا لو ساوي absolute x زاد absolute c الآن
+
+30
+00:03:30,430 --> 00:03:39,530
+absolute x بساوي absolute x سالب c زاد زاد
+
+31
+00:03:39,530 --> 00:03:44,270
+c ممكن أطرح من ال xc و أرجعهاوباستخدام الـ
+
+32
+00:03:44,270 --> 00:03:49,030
+triangular equality هذا أصغر لو يساوي absolute x
+
+33
+00:03:49,030 --> 00:03:58,150
+ثالث c زائد absolute c فلو كان absolute x minus c
+
+34
+00:03:58,150 --> 00:04:04,070
+أصغر من دلتا اللي هي بساوية واحد إذا كان خلّينا
+
+35
+00:04:04,070 --> 00:04:07,190
+ناخد دلتا بساوية واحد إذا كان absolute x minus c
+
+36
+00:04:07,190 --> 00:04:13,370
+أصغر من دلتا اللي أنا ماخدها واحدفهذا بيطلع أصغر
+
+37
+00:04:13,370 --> 00:04:21,490
+من واحد زائد أبسليوت C وبالتالي
+
+38
+00:04:21,490 --> 00:04:32,370
+أبسليوت X تربية سالب C تربية بيطلع أصغر من أبسليوت
+
+39
+00:04:32,370 --> 00:04:35,150
+X اللي هي واحد زائد
+
+40
+00:04:37,400 --> 00:04:43,720
+أتنين في absolute C في absolute X minus C
+
+41
+00:04:48,510 --> 00:04:51,770
+كمان مرة احنا توصلنا إلى ان ال absolute value
+
+42
+00:04:51,770 --> 00:04:57,150
+للفرق هذا أصغر من أو ساوي absolute x زاد absolute
+
+43
+00:04:57,150 --> 00:05:01,290
+c في absolute x ثالث c أخدنا delta بالساوي واحد
+
+44
+00:05:01,290 --> 00:05:05,190
+وقلنا لو كان absolute x minus t أصغر من delta اللي
+
+45
+00:05:05,190 --> 00:05:09,190
+هي واحد بتطلع absolute x أصغر من واحد زاد absolute
+
+46
+00:05:09,190 --> 00:05:13,830
+c وبالتالي absolute الفرق هذا هي أصغر من أو ساوي
+
+47
+00:05:14,180 --> 00:05:18,500
+absolute x هي أصغر من واحد زاد absolute c وانتي
+
+48
+00:05:18,500 --> 00:05:23,820
+absolute c فأصغر من واحد زاد اتنين فabsolute c ضرب
+
+49
+00:05:23,820 --> 00:05:29,460
+absolute x minus c الان بدي أخلي هذا أصغر من
+
+50
+00:05:29,460 --> 00:05:39,160
+epsilon هذا بدي أخليه أصغر من epsilon لما
+
+51
+00:05:39,160 --> 00:05:40,920
+يكون هذا أصغر من delta
+
+52
+00:05:48,400 --> 00:05:52,700
+فباخد إذا لما يكون هذا أصغر من دلتا فهذا بصير أصغر
+
+53
+00:05:52,700 --> 00:05:58,880
+من واحد زي اتنين absolute c في دلتا لما يكون ال
+
+54
+00:05:58,880 --> 00:06:03,000
+absolute value ل X معناه C أصغر من دلتا فهذا بطلع
+
+55
+00:06:03,000 --> 00:06:07,740
+أصغر من واحد زي اتنين في absolute c في دلتا الآن
+
+56
+00:06:07,740 --> 00:06:16,040
+متى بيكون هذا أصغر من إبسرن لما دلتاإذا كانت delta
+
+57
+00:06:16,040 --> 00:06:25,240
+هذه أصغر من أوي ساوي epsilon على واحد زايد اتنين
+
+58
+00:06:25,240 --> 00:06:29,960
+في absolute of c إذا هاي قيمة تانية ل delta هاي
+
+59
+00:06:29,960 --> 00:06:35,660
+اندي delta بساوي واحد و delta أصغر من أوي ساوي
+
+60
+00:06:35,660 --> 00:06:39,760
+epsilon على واحد زايد اتنين في absolute of c إذا
+
+61
+00:06:39,760 --> 00:06:49,590
+باجي بقولlet epsilon أكبر من السفر be given a
+
+62
+00:06:49,590 --> 00:06:57,690
+choose delta بتساوي ال minimum الأصغر بين القمتين
+
+63
+00:06:57,690 --> 00:07:07,050
+واحد وepsilon على واحد زاد اتنين في absolute c ال
+
+64
+00:07:07,050 --> 00:07:13,660
+delta هذه الآن عدد موجب ويعتمد على epsilonإذا لهذه
+
+65
+00:07:13,660 --> 00:07:24,140
+الـ Delta لو كان X ينتمي ل R اللي هو مجال الدالة و
+
+66
+00:07:24,140 --> 00:07:31,780
+Absolute X minus C أكبر من صفر أصغر من Delta فهذا
+
+67
+00:07:31,780 --> 00:07:36,960
+بيؤدي طبعا ال Delta هذه هي الأصغر من العددين هذوله
+
+68
+00:07:36,960 --> 00:07:40,920
+وبالتالي أصغر من أو يساوي واحد وأصغر من أو يساوي
+
+69
+00:07:40,920 --> 00:07:46,820
+كسر هذافالـ delta أكيد أصغر من أو يساوي الواحد،
+
+70
+00:07:46,820 --> 00:07:52,360
+لما الـ delta أصغر من أو يساوي الواحد، هذا بيقدر
+
+71
+00:07:52,360 --> 00:07:56,940
+أن absolute X
+
+72
+00:07:56,940 --> 00:08:04,800
+أصغر من واحد زائد absolute Cوكمان هذا بيقدي أنه
+
+73
+00:08:04,800 --> 00:08:11,000
+absolute x تربية سالب c تربية أصغر من أوي ساوي
+
+74
+00:08:11,000 --> 00:08:23,400
+absolute x زاد absolute c absolute
+
+75
+00:08:23,400 --> 00:08:28,640
+x سالب c وبالتالي هذا أصغر من أوي ساوي واحد زاد
+
+76
+00:08:28,640 --> 00:08:39,840
+اتنين absolute cو هذا اصغر من delta و
+
+77
+00:08:39,840 --> 00:08:49,520
+الان ال delta هذه طبعا
+
+78
+00:08:49,520 --> 00:08:55,420
+هذا اصغر من delta و ال delta قلنا اصغر منها و
+
+79
+00:08:55,420 --> 00:08:58,600
+يساوي epsilon هاي واحد زي اتنين
+
+80
+00:09:19,770 --> 00:09:26,670
+بشكل صحيح بما أن ابسلون أكبر من السفر was
+
+81
+00:09:26,670 --> 00:09:27,550
+arbitrary
+
+82
+00:09:31,810 --> 00:09:35,410
+إذاً هيك بنكون أثباتنا لكل epsilon أكبر من السفر
+
+83
+00:09:35,410 --> 00:09:41,150
+يوجد delta تعتمد على epsilon عدد موجة ب half لكل x
+
+84
+00:09:41,150 --> 00:09:46,710
+المسافة مختلفة عن ال c والمسافة بينها وبين ال c
+
+85
+00:09:46,710 --> 00:09:52,590
+أصغر من delta بتطلع المسافة بين f of x وc تربيه
+
+86
+00:09:52,590 --> 00:10:01,190
+أصغر من epsilon إذاً we haveBy definition إن ال
+
+87
+00:10:01,190 --> 00:10:11,390
+limit ل X تربيه لما X تقول إلى C بساوي C تربيه وهو
+
+88
+00:10:11,390 --> 00:10:17,410
+المطلوب، okay؟ إذن هذا هو برهان إن ال limit للدالة
+
+89
+00:10:17,410 --> 00:10:23,300
+التربيهية عن C بساوي C تربيهاستخدمنا تعريف epsilon
+
+90
+00:10:23,300 --> 00:10:28,260
+دلتا وشوفنا ان دلتا هنا لازم تكون الأصغر من
+
+91
+00:10:28,260 --> 00:10:34,520
+الكمتين اللي هو الواحد والكاسر اللي هناك هي اي
+
+92
+00:10:34,520 --> 00:10:41,020
+سخصار اي سؤال خلينا ناخد كمان مثال مشابه لهذا وفيه
+
+93
+00:10:41,020 --> 00:10:44,560
+ال delta برضه بتساوي ال minimum لكمتين
+
+94
+00:10:53,440 --> 00:11:02,620
+المثال الرقم أربعة show أنه ال limit لواحد على x
+
+95
+00:11:02,620 --> 00:11:14,020
+لما x تقول إلى zero لأ
+
+96
+00:11:14,020 --> 00:11:20,340
+ال limit لواحد على x لما x تقول إلى أي عدد cبساوي
+
+97
+00:11:20,340 --> 00:11:27,420
+واحد على C حيث C أكبر من 7 فهنا
+
+98
+00:11:27,420 --> 00:11:30,520
+بناخد ال function تبعتي الدالة اللي بتجميلها ال
+
+99
+00:11:30,520 --> 00:11:36,800
+limit هي عبارة عن F of X بساوي واحد على X حيث X
+
+100
+00:11:36,800 --> 00:11:41,720
+موجبة إذا المجال تبع الدالة هذه الفترة المفتوحة من
+
+101
+00:11:41,720 --> 00:11:49,410
+سفر إلى ما لا نهاية واند C عدد موجبةطيب انا عايز
+
+102
+00:11:49,410 --> 00:11:57,190
+اثبت ان absolute f of x minus واحد على c بدي هذا
+
+103
+00:11:57,190 --> 00:12:02,470
+يكون اصغر من اي given epsilon عندما x تكون قريبة
+
+104
+00:12:02,470 --> 00:12:12,230
+من ال c او في جوار delta لل cفهذا طبعا اش بساوي هي
+
+105
+00:12:12,230 --> 00:12:17,790
+absolute واحد على X minus واحد على C وهذا بساوي
+
+106
+00:12:17,790 --> 00:12:27,950
+absolute C minus X على X في C وهذا بساوي واحد على
+
+107
+00:12:27,950 --> 00:12:33,270
+X في C ضرب absolute X minus C
+
+108
+00:12:38,060 --> 00:12:45,780
+الان بدي أحاول أجيب upper bound عدد
+
+109
+00:12:45,780 --> 00:12:53,720
+موجة ب M بحيث ال 1 على X في C يكون أصغر من أو ساوي
+
+110
+00:12:53,720 --> 00:12:57,360
+ال M تعالوا نشوف كيف نجيب ال upper bound هذا أو ال
+
+111
+00:12:57,360 --> 00:13:06,330
+boundأنا عندى ال take الاول take انا عندى ال c عدد
+
+112
+00:13:06,330 --> 00:13:12,190
+موجب take delta بساوي c على اتنين هذا عدد موجب
+
+113
+00:13:12,190 --> 00:13:16,050
+then
+
+114
+00:13:16,050 --> 00:13:23,510
+absolute x minus c اصغر من delta اللى هو بساويC ع
+
+115
+00:13:23,510 --> 00:13:32,030
+2 بيقدي ان X أصغر من ثلاثة C ع 2 أكبر من C ع 2
+
+116
+00:13:32,030 --> 00:13:43,430
+وهذا بيقدي ان واحد على X في C أصغر من اتنين على C
+
+117
+00:13:43,430 --> 00:13:44,270
+ترمية
+
+118
+00:13:50,000 --> 00:13:54,920
+ال X أكبر من C على 2 إذا مقلوب ال X أصغر من 2 على
+
+119
+00:13:54,920 --> 00:14:01,220
+C مقلوب ال X و أضربها في 1 على C بيطلع أصغر من 2
+
+120
+00:14:01,220 --> 00:14:07,100
+على C تربيه وبالتالي
+
+121
+00:14:07,100 --> 00:14:13,540
+هذا العدد هذا هو ال M عدد
+
+122
+00:14:13,540 --> 00:14:16,320
+موجة إذا
+
+123
+00:14:18,670 --> 00:14:28,290
+في الحالة هذه في الحالة
+
+124
+00:14:28,290 --> 00:14:34,910
+هذه بصير عندي هذا أصغر من اتنين على C تربية وطبعا
+
+125
+00:14:34,910 --> 00:14:39,430
+هذا أصغر من Delta Absolute X minus C طبعا بيكون
+
+126
+00:14:39,430 --> 00:14:44,690
+أصغر من Delta الآن عشان يكون هذا أصغر من أو ساوي
+
+127
+00:14:44,690 --> 00:14:52,820
+Epsilonفنختار choose الـ delta أصغر من أو ساوي حل
+
+128
+00:14:52,820 --> 00:14:57,140
+المتباينة هذه في الـ delta فالـ delta ستصبح أصغر
+
+129
+00:14:57,140 --> 00:15:04,540
+من أو ساوي C تربيع على 2 تلصق فهي قيمة تانية لـ
+
+130
+00:15:04,540 --> 00:15:09,440
+delta فبأخد الـ delta ال minimum للقيمة الأولى
+
+131
+00:15:10,560 --> 00:15:16,040
+والقيمة التانية هذا هيخلي انه لكل x المسافة بين او
+
+132
+00:15:16,040 --> 00:15:20,320
+بين c اصغر من delta هتخلي المسافة بين f of x واحد
+
+133
+00:15:20,320 --> 00:15:26,280
+على c اصغر من ال given epsilon نكتب الكلام هذا let
+
+134
+00:15:26,280 --> 00:15:29,240
+epsilon be given choose delta بالساوي ال minimum
+
+135
+00:15:29,240 --> 00:15:36,260
+نختار
+
+136
+00:15:36,260 --> 00:15:42,030
+delta ال minimumللعدد الموجة بـ c ع 2، والعدد
+
+137
+00:15:42,030 --> 00:15:49,000
+التاني ده هو c تربيه ع 2 في epsilonطبعا هذا عدد
+
+138
+00:15:49,000 --> 00:15:52,740
+أكيد عدد موجب لأن هذا موجب وهذا موجب والأصغر بينهم
+
+139
+00:15:52,740 --> 00:15:56,820
+هيطلع موجب واتنين بيعتمدوا على epsilon إذن delta
+
+140
+00:15:56,820 --> 00:16:00,620
+عدد موجب بيعتمد على epsilon إذا لأي epsilon أكبر
+
+141
+00:16:00,620 --> 00:16:04,360
+من سفر هين أثبتت يوجد delta تعتمد على epsilon عدد
+
+142
+00:16:04,360 --> 00:16:11,680
+موجب بحيث أنه لكل x ينتمي لإيه المجال هنا اللي هو
+
+143
+00:16:11,680 --> 00:16:19,300
+الفترة المفتوحة من سفر إلى دالة نهايةو absolute x
+
+144
+00:16:19,300 --> 00:16:25,400
+minus c أكبر من سفر أصغر من ال delta هذا بيقدي أن
+
+145
+00:16:25,400 --> 00:16:33,260
+ال delta هذه أصغر من أو يساوي c ع 2 فلما ال delta
+
+146
+00:16:33,260 --> 00:16:39,280
+تكون أصغر من أو يساوي c ع 2 هذا بيقدي أنه واحد على
+
+147
+00:16:39,280 --> 00:16:42,460
+واحد
+
+148
+00:16:42,460 --> 00:16:52,060
+على xفى c أصغر من اتنين على c تربية وهذا بدوره
+
+149
+00:16:52,060 --> 00:17:00,940
+بيقدم absolute واحد على x minus واحد على c بساوي
+
+150
+00:17:00,940 --> 00:17:06,480
+واحد على x في c في absolute x minus c أصغر من
+
+151
+00:17:06,480 --> 00:17:14,850
+اتنين على c تربية فى deltaوالـ delta هذه الأن أصغر
+
+152
+00:17:14,850 --> 00:17:19,310
+من أو يساوي الـ delta هذه هي الـ delta اللي فوق
+
+153
+00:17:19,310 --> 00:17:25,010
+أصغر من أو يساوي العدد هذا أيه والعدد التاني لأنها
+
+154
+00:17:25,010 --> 00:17:31,130
+الأصغر بين اتنين لأن هي اتنين على c تربيع ضرب c
+
+155
+00:17:31,130 --> 00:17:36,390
+تربيع اتنين في epsilon هذا بروح مع هذا مخلوق بعض
+
+156
+00:17:36,390 --> 00:17:41,030
+بيضل عندي epsilon since
+
+157
+00:17:43,190 --> 00:17:50,970
+Y أكبر من السفر was arbitrary إذا أنا لكل Y أكبر
+
+158
+00:17:50,970 --> 00:17:56,850
+من السفر جبت Delta تعتمد على Y بحيث لكل X مختلفة
+
+159
+00:17:56,850 --> 00:18:00,570
+عن الـC المسافة بينها وبين الـC أصغر من Delta كل
+
+160
+00:18:00,570 --> 00:18:05,450
+المسافة بين F of X و1 على C أصغر من Y إذا by
+
+161
+00:18:05,450 --> 00:18:06,010
+definition
+
+162
+00:18:09,260 --> 00:18:14,820
+by definition of limit بيطلع عند ال limit لل
+
+163
+00:18:14,820 --> 00:18:20,740
+function واحد على X لما X تقول إلى C بيساوي واحد
+
+164
+00:18:20,740 --> 00:18:24,240
+على C وهو المطلوب
+
+165
+00:18:26,860 --> 00:18:31,720
+واضح في أي سؤال؟ في كمان مثال آخر زي هدف الكتاب،
+
+166
+00:18:31,720 --> 00:18:37,860
+هسيبكم تقرؤوه لأن الفكرة شبيهة بالفكرة في المثال
+
+167
+00:18:37,860 --> 00:18:47,720
+الأخير وبالتالي مافيش إشي جديد ننتقل إلى دراسة
+
+168
+00:18:52,750 --> 00:18:56,370
+ال sequential criterion هي حاجة اسمها sequential
+
+169
+00:18:56,370 --> 00:19:09,570
+criterion حاجة .. حاجة بتكافئ التعريف sequential
+
+170
+00:19:09,570 --> 00:19:12,750
+criterion
+
+171
+00:19:45,930 --> 00:19:53,970
+العبارات التالية متكافعةLimit f of x as x tends to
+
+172
+00:19:53,970 --> 00:20:03,290
+c بساوي عدد L بغند عدد حقيقي اتنين for every
+
+173
+00:20:06,410 --> 00:20:14,330
+for every sequence xn contained in A وحدودها
+
+174
+00:20:14,330 --> 00:20:25,090
+مختلفة عن الـC such that limit xn بالساوي C we
+
+175
+00:20:25,090 --> 00:20:31,470
+have limit الـimage لسيكوينس xn as n tends to
+
+176
+00:20:31,470 --> 00:20:34,410
+infinity بالساوي العدد القليل
+
+177
+00:20:39,210 --> 00:20:42,690
+إن الـ sequential criterion هذه بتقول إن عشان أثبت
+
+178
+00:20:42,690 --> 00:20:46,470
+إن ال limit لل function f and x بساوي c بساوي
+
+179
+00:20:46,470 --> 00:20:52,090
+العدد L هدى بكافة إن أنا أثبت إنه لو أخدت أي
+
+180
+00:20:52,090 --> 00:20:59,010
+sequence نهايتها C فلازم يكون نهاية صورتها بساوي
+
+181
+00:20:59,010 --> 00:21:04,140
+العدد Lلو اقدرت اعمل هذا في الكلام فبقى هذا بيكافئ
+
+182
+00:21:04,140 --> 00:21:08,880
+ان احنا نقول ان ال limit ل f of x يعني ال x بيساوي
+
+183
+00:21:08,880 --> 00:21:15,060
+c بيساوي العدد ال .. نثبت النظرية هذه تروف one
+
+184
+00:21:15,060 --> 00:21:23,020
+implies two assume one
+
+185
+00:21:23,020 --> 00:21:28,480
+IE
+
+186
+00:21:30,950 --> 00:21:37,470
+الـ limit لأخب X لما X تقول لـ C بساوي العدد M
+
+187
+00:21:37,470 --> 00:21:44,070
+عايزين
+
+188
+00:21:44,070 --> 00:21:48,450
+نثبت عشان
+
+189
+00:21:48,450 --> 00:21:55,250
+نثبت اتنين عشان نثبت اتنين صحيح to
+
+190
+00:21:55,250 --> 00:21:58,190
+prove two holes
+
+191
+00:22:00,720 --> 00:22:05,500
+to prove two holds let
+
+192
+00:22:05,500 --> 00:22:17,380
+Xn be a sequence in A هدودها مختلفة عن الـC such
+
+193
+00:22:17,380 --> 00:22:26,960
+that limit Xn بالساوي C we claim
+
+194
+00:22:30,360 --> 00:22:45,320
+بت ال limit ل f of x ل f of x n لما
+
+195
+00:22:45,320 --> 00:22:52,340
+n تقول ل infinity دي ساوي L لبرهان ذلك let epsilon
+
+196
+00:22:52,340 --> 00:22:55,400
+أكبر
+
+197
+00:22:55,400 --> 00:22:57,020
+من السفر be given
+
+198
+00:23:02,180 --> 00:23:08,440
+سنس اكس اكس اكس اكس
+
+199
+00:23:10,770 --> 00:23:16,490
+بما أننا فرضين limit f of x لما x تقوله c بالساوي
+
+200
+00:23:16,490 --> 00:23:21,450
+L من تعريف epsilon دلتا لل limit إذا يوجد دلتا
+
+201
+00:23:21,450 --> 00:23:27,770
+تعتمد على epsilon عدد موجب بحيث أنه لكل x ينتمي
+
+202
+00:23:27,770 --> 00:23:33,730
+إلى a وabsolute x minus c أكبر من صفر أصغر من دلتا
+
+203
+00:23:42,740 --> 00:23:52,080
+أبسلون دلتا للنهايات نسمي
+
+204
+00:23:52,080 --> 00:23:53,760
+ال implication هذه star
+
+205
+00:24:01,580 --> 00:24:07,300
+And the limit xn بالساوي سي احنا فرضين ان في انديو
+
+206
+00:24:07,300 --> 00:24:14,840
+سيكوينس xn ونهايتها c then
+
+207
+00:24:14,840 --> 00:24:26,910
+for the aboveدلتا الموجبة يوجد دلتا موجبة خدت دلتا
+
+208
+00:24:26,910 --> 00:24:31,910
+هذه الموجبة وطبق تعريف epsilon capital N لlimit of
+
+209
+00:24:31,910 --> 00:24:36,590
+sequence فبما ان ال sequence هذه نهايتها C إذا لأي
+
+210
+00:24:36,590 --> 00:24:42,070
+دلتا أو epsilon عدد موجبThere exists capital N
+
+211
+00:24:42,070 --> 00:24:46,710
+يعتمد على الـ Delta طبعا الـ Delta تعتمد على
+
+212
+00:24:46,710 --> 00:24:51,370
+إبسلون، إذا الـ N هذه يعتمد على إبسلون عدد طبيعي،
+
+213
+00:24:51,370 --> 00:24:56,650
+بحيث أنه لكل N أكبر من أو ساوي capital N، تطلع
+
+214
+00:24:56,650 --> 00:25:02,130
+عندي absolute X N minus C أصغر من Delta، نسمي ال
+
+215
+00:25:02,130 --> 00:25:03,990
+implication هذه double star
+
+216
+00:25:07,580 --> 00:25:16,300
+now star and double star بيؤدوا إلى ما يلي لو كان
+
+217
+00:25:16,300 --> 00:25:23,340
+M أكبر من أو ساوي capital M هذا بيؤدي انه absolute
+
+218
+00:25:23,340 --> 00:25:27,740
+XM
+
+219
+00:25:27,740 --> 00:25:36,440
+minus C أصغر من دلتا هذا باستخدام double star صح؟
+
+220
+00:25:39,750 --> 00:25:44,230
+لو كانت n أكبر من أو ساوي capital N فبطلع absolute
+
+221
+00:25:44,230 --> 00:25:52,050
+xn minus c أصغر من delta و من ال star لو كان عندى
+
+222
+00:25:52,050 --> 00:25:59,130
+xn طبعا xn هذا موجود في a ال xn موجود في a مختلف
+
+223
+00:25:59,130 --> 00:25:59,810
+عن ال c
+
+224
+00:26:02,830 --> 00:26:07,750
+فلو كان absolute of xn minus c badly except xn
+
+225
+00:26:07,750 --> 00:26:13,850
+أصغر من delta فحسب الstar هذا بقدر absolute of f
+
+226
+00:26:13,850 --> 00:26:22,590
+of xn minus L أصغر من إبسلون الان بما أن هذا صحيح
+
+227
+00:26:22,590 --> 00:26:28,270
+بما أن since إبسلون أكبر من الصفر was arbitrary
+
+228
+00:26:30,740 --> 00:26:42,380
+إن إحنا أثبتنا هيك لكل إبسلون يوجد
+
+229
+00:26:42,380 --> 00:26:50,250
+capital N يعتمد على إبسلون عدد طبيعيبكل n أكبر من
+
+230
+00:26:50,250 --> 00:26:55,310
+أوي سوى capital N absolute f of xn minus L أصغر من
+
+231
+00:26:55,310 --> 00:27:00,150
+إبسلون إذا by إبسلون capital N definition لل limit
+
+232
+00:27:00,150 --> 00:27:06,050
+of sequence بطلع عندي limit لsequence
+
+233
+00:27:06,050 --> 00:27:12,910
+f of xn as n tends to infinity بساوية L وبالتالي
+
+234
+00:27:12,910 --> 00:27:21,850
+هيك بيكون إذا two holesهكذا أثبتنا أن واحد يؤدي
+
+235
+00:27:21,850 --> 00:27:26,610
+إلى اتنين اتنين
+
+236
+00:27:26,610 --> 00:27:30,270
+بيقول for every sequence فهي اللي أخدت arbitrary
+
+237
+00:27:30,270 --> 00:27:36,810
+sequence في a minus c وبشرط بحيث ان ال sequence هي
+
+238
+00:27:36,810 --> 00:27:37,710
+اللي نهيتها c
+
+239
+00:27:42,350 --> 00:27:46,210
+و اثبتنا ان ال limit لل image لل sequence بساوي L
+
+240
+00:27:46,210 --> 00:27:51,710
+هذا بالظبط اللي هو الابارة اتنين لان هيك يكون
+
+241
+00:27:51,710 --> 00:27:58,270
+اثبتنا واحد بيقدي لاتنين واضح مفهوم اللي هو نثبت
+
+242
+00:27:58,270 --> 00:28:02,510
+العكس نثبت ان اتنين بيقدي لواحد
+
+243
+00:28:16,210 --> 00:28:22,870
+بالنسبة العبارة اثنين بتقدي للعبارة واحد فالاثنان
+
+244
+00:28:22,870 --> 00:28:27,270
+ذالف بالمناسبة الأخوات اللي قاعدات ورا دولة إيش
+
+245
+00:28:27,270 --> 00:28:31,510
+بتعملوا انتوا؟ ماعليش أوقف تصوير إيش مجاعتكم انتوا
+
+246
+00:28:31,510 --> 00:28:34,830
+أنا أول حاجة و تاني حاجة؟ إيش بتتكلمون؟ دكتور معاك
+
+247
+00:28:34,830 --> 00:28:38,070
+لأ لأ لما هم بتتكلم عامليننا أزعاج لأ باحكوا إذا
+
+248
+00:28:38,070 --> 00:28:40,550
+انتوا بتتكلموا لأ بحكي على اندر ده ليش مصورة أن
+
+249
+00:28:40,550 --> 00:28:44,510
+الوضع هو وضع نفسه لأ بنتكلميش لأ باحكي عن البرادة
+
+250
+00:28:44,510 --> 00:28:49,530
+اللي ورا دولةفي بنات بتتكلموا، أنتوا اللي ورا
+
+251
+00:28:49,530 --> 00:28:55,390
+بتتكلموا ولا في ناس غيرك؟ في حد بتتكلم و أنا بشرح
+
+252
+00:28:55,390 --> 00:29:00,090
+تتكلم و هذا عمللي أزعاج كتير، فلو سمحتوا إذا أنتوا
+
+253
+00:29:00,090 --> 00:29:04,890
+قاعدين تتكلموا ورا اطلعوا في حديقة اتكلموا فيها،
+
+254
+00:29:04,890 --> 00:29:10,670
+حتى لو باسم المحاضرة ممنوح تتكلموا، شوية أزعاجهو
+
+255
+00:29:10,670 --> 00:29:13,850
+مين اللي بتتكلم؟ إذاً أنت اللي بتتكلم من قعدته
+
+256
+00:29:13,850 --> 00:29:20,350
+وراك بتتكلم ما تتكلمش لإن غير ترفع يدك، ارفع يدك و
+
+257
+00:29:20,350 --> 00:29:24,670
+تقعد لسانك، ما تتكلمي مع الجنك بدون اسم، لإن هذا
+
+258
+00:29:24,670 --> 00:29:28,150
+عندنا قاعدة في المحاضرة، ممنوع حد يتكلم مع الجنك و
+
+259
+00:29:28,150 --> 00:29:34,030
+تتحدث مع حد شخص آخر إلا إذا عندك سؤال، ترفع يدك،
+
+260
+00:29:34,030 --> 00:29:37,990
+تستنى لما أقول من عنده سؤال من عنده حاسب صار، ترفع
+
+261
+00:29:37,990 --> 00:29:41,790
+يدك و بجاوبكانا مابتقدر انت تعمليني قصة مع اللغة،
+
+262
+00:29:41,790 --> 00:29:48,330
+قوم انت .. انت .. قوم يقعد في مطعم، يبقين عالم،
+
+263
+00:29:48,330 --> 00:29:51,190
+فلو سرحت انك تتكلم مش مع بعض، هانديك السفسة
+
+264
+00:30:01,910 --> 00:30:04,950
+ممنوع حد يتكلم مع الجنب في المحاضرة، أنا بعمل
+
+265
+00:30:04,950 --> 00:30:08,850
+إزعاج، بدك أنت في السفسار، عندك أي إيش أنا بواجب،
+
+266
+00:30:08,850 --> 00:30:13,990
+بقول من عنده سؤال، من عنده حاجة، اتفضل يسأل
+
+267
+00:30:13,990 --> 00:30:21,850
+ساعتها، بس لا تتكلم وأنا ضايق طرابك،
+
+268
+00:30:21,850 --> 00:30:24,230
+يقولنا الكلام قدر مئة مرة في المحاضرة، ممنوع
+
+269
+00:30:24,230 --> 00:30:25,690
+الكلام الجامل
+
+270
+00:30:35,880 --> 00:30:40,860
+تفضل يا أبو حمزي إذا
+
+271
+00:30:40,860 --> 00:30:45,380
+الأن بدنا نكمل البرنامج بإثبات الأثنين بأد لواحد
+
+272
+00:30:45,380 --> 00:30:51,740
+الإثبات الأثنين بأد لواحد بدنا نثبت we prove ال
+
+273
+00:30:51,740 --> 00:30:59,120
+contrapositive we prove not واحد implies not two
+
+274
+00:31:01,070 --> 00:31:04,730
+هذا هو ال contrapositive للعبارة لل implication
+
+275
+00:31:04,730 --> 00:31:16,630
+هذه فإذا assume .. assume not one ف not one معناته
+
+276
+00:31:16,630 --> 00:31:27,190
+ال limit ل F of X لما X تقول ل C لا تساوي L
+
+277
+00:31:30,200 --> 00:31:32,020
+this means هذا يعني
+
+278
+00:31:35,090 --> 00:31:40,190
+الان نرجع لتعريف ال limit أو ال function شوفنا
+
+279
+00:31:40,190 --> 00:31:42,530
+المرة السادسة في تعريفين في epsilon delta
+
+280
+00:31:42,530 --> 00:31:46,270
+definition و في neighborhood definition ال
+
+281
+00:31:46,270 --> 00:31:49,610
+neighborhood definition بيقول اذا كان عشان تكون
+
+282
+00:31:49,610 --> 00:31:53,770
+limit ل f of x من x او ل c بالساوي عدد L هذا
+
+283
+00:31:53,770 --> 00:31:57,210
+بيكافئ انه لكل epsilon neighborhood ل L يوجد delta
+
+284
+00:31:57,210 --> 00:32:01,130
+neighborhood لل C بحيث لكل x في ال delta
+
+285
+00:32:01,130 --> 00:32:04,630
+neighborhoodصورته لازم تطلع في الـ epsilon
+
+286
+00:32:04,630 --> 00:32:08,290
+neighborhood الان ان في الكلام هذا ما معنى ان ال
+
+287
+00:32:08,290 --> 00:32:13,570
+limit and c بيستويش لعدد L معناته بدل لكل epsilon
+
+288
+00:32:13,570 --> 00:32:17,930
+neighborhood ل L there exist there exist epsilon
+
+289
+00:32:17,930 --> 00:32:25,330
+zero neighborhood of L بسميه
+
+290
+00:32:25,330 --> 00:32:32,110
+V epsilon zero neighborhood ل L بحيث انه لكل
+
+291
+00:32:33,900 --> 00:32:43,060
+Delta neighborhood V Delta أو C يوجد X يعتمد على
+
+292
+00:32:43,060 --> 00:32:50,540
+Delta ينتمي إلى A ومختلف عن الـ C وموجود في الـ
+
+293
+00:32:50,540 --> 00:32:55,560
+Delta neighborhood بحيث
+
+294
+00:32:55,560 --> 00:33:01,100
+أن صورة الـ X Delta
+
+295
+00:33:05,360 --> 00:33:16,380
+لا تنتمي للإبسلون zero neighborhood ل LL طيب
+
+296
+00:33:16,380 --> 00:33:26,140
+لو أخدنا take لكل N في N take delta بساوي واحد على
+
+297
+00:33:26,140 --> 00:33:31,600
+N then
+
+298
+00:33:31,600 --> 00:33:32,540
+they exist
+
+299
+00:33:37,520 --> 00:33:47,100
+دلتا تعتمد على n دلتا تعتمد على n دلتا تعتمد على n
+
+300
+00:33:47,100 --> 00:33:50,360
+دلتا
+
+301
+00:33:50,360 --> 00:33:56,520
+تعتمد
+
+302
+00:33:56,520 --> 00:33:57,620
+على
+
+303
+00:34:00,880 --> 00:34:10,260
+و بحيث ان F ل Xm لا ينتمي لإبسلون Zero
+
+304
+00:34:10,260 --> 00:34:18,300
+neighborhood ل L طب ما هذا الأخير معناه أو بيقدّي
+
+305
+00:34:26,330 --> 00:34:34,390
+this implies هذا بيقدّي نكون أثبتنا ان لكل n يوجد
+
+306
+00:34:34,390 --> 00:34:42,490
+xn في a إذا يوجد sequence xn موجودة في ال set A
+
+307
+00:34:42,490 --> 00:34:47,170
+حدودها مختلفة عن ال C كل ال xn مختلفة عن ال C
+
+308
+00:34:47,170 --> 00:35:01,980
+وموجودة فيv1 على n of c بحيث ان f ل xn لا تنتمي ل
+
+309
+00:35:01,980 --> 00:35:11,900
+v epsilon zero ل n لكل n هذا معناه ان يوجد
+
+310
+00:35:11,900 --> 00:35:20,580
+sequence xn contained in a minus c بحيث انلاحظوا
+
+311
+00:35:20,580 --> 00:35:26,240
+الـ sequence Xn تنتمي ل V 1 على N of C اللي هو
+
+312
+00:35:26,240 --> 00:35:30,960
+عبارة عن الفترة C سالف واحد على N C موجب واحد على
+
+313
+00:35:30,960 --> 00:35:37,360
+N لكل N هذا معناه ان absolute Xn minus C أصغر من
+
+314
+00:35:37,360 --> 00:35:42,420
+واحد على N أصغر
+
+315
+00:35:42,420 --> 00:35:47,020
+من واحد على N لكل N في N and
+
+316
+00:35:50,460 --> 00:35:55,500
+F of Xn لا تنتمي للـY0 neighborhood الـY0
+
+317
+00:35:55,500 --> 00:35:59,720
+neighborhood هذا عبارة عن الفترة المفتوحة L minus
+
+318
+00:35:59,720 --> 00:36:08,000
+Y0 L زائد Y0 فF of Xn لا تنتمي للفترة المفتوحة هذه
+
+319
+00:36:08,000 --> 00:36:15,720
+معناه absolute المسافة بين F of Xn وL أكبر من أو
+
+320
+00:36:15,720 --> 00:36:18,460
+ساوي Y0 لكل N
+
+321
+00:36:21,430 --> 00:36:26,750
+هذا الكلام معناه أن
+
+322
+00:36:26,750 --> 00:36:32,190
+يوجد sequence x in موجودة في a حدودها مختلفة عن ال
+
+323
+00:36:32,190 --> 00:36:41,030
+c وهذا الكلام معناه such that limit x in بساوي c
+
+324
+00:36:43,330 --> 00:36:51,410
+حسب نظرية اتنين اربعة اتنين
+
+325
+00:36:51,410 --> 00:36:54,970
+اربعة اتنين اربعة اتنين اربعة اتنين اربع اتنين
+
+326
+00:36:54,970 --> 00:36:55,590
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+327
+00:36:55,590 --> 00:36:55,590
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+328
+00:36:55,590 --> 00:36:55,730
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+329
+00:36:55,730 --> 00:36:56,990
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+330
+00:36:56,990 --> 00:36:59,130
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+331
+00:36:59,130 --> 00:37:08,450
+اتنين اربع اتنين اربع اتنين اربع ا
+
+332
+00:37:12,640 --> 00:37:16,920
+الـ limit لـ
+
+333
+00:37:16,920 --> 00:37:20,760
+sequence f of xn لما n تقول الـ infinity مش ممكن
+
+334
+00:37:20,760 --> 00:37:26,540
+تساوي العدد L لأن لو ال limit ل f of xn بيساوي
+
+335
+00:37:26,540 --> 00:37:30,220
+العدد L، المفروض ال absolute value للفرق ده تكون
+
+336
+00:37:30,220 --> 00:37:36,660
+أصغر من أي epsilon zero لكل N من capital N و انت
+
+337
+00:37:36,660 --> 00:37:40,930
+طالع، لكن هذا الكلام مش صحيحOkay إن هذا بالظبط
+
+338
+00:37:40,930 --> 00:37:48,210
+العبارة الأخيرة which which
+
+339
+00:37:48,210 --> 00:37:55,690
+is نفي العبارة اتنين هذه
+
+340
+00:37:55,690 --> 00:37:58,450
+العبارة الأخيرة هي نفي العبارة اتنين هذه العبارة
+
+341
+00:37:58,450 --> 00:38:06,320
+اتنين ال statement اتنينبقول لكل sequence بحيث ان
+
+342
+00:38:06,320 --> 00:38:09,100
+ال limit بتاعتها C، ال limit لل image بتاعتها
+
+343
+00:38:09,100 --> 00:38:13,660
+بالساولة L هنا اتوصلنا ان there exist بدل for all
+
+344
+00:38:13,660 --> 00:38:18,660
+there exist sequence نهايتها C لكن نهاية صورتها
+
+345
+00:38:18,660 --> 00:38:25,020
+لاتساول L إذا هيك بنكون أثبتنا أنه لا إذا we
+
+346
+00:38:25,020 --> 00:38:29,800
+proved not
+
+347
+00:38:31,130 --> 00:38:39,390
+not one implies not two therefore two implies one
+
+348
+00:38:39,390 --> 00:38:46,610
+وهذا يكمل البرهان واضح؟ في أي سؤال؟ في أي استفسار؟
+
+349
+00:38:46,610 --> 00:38:53,590
+يبدو أننا كملنا برهان النظرية في أي استفسار؟
+
+350
+00:38:55,700 --> 00:39:03,780
+الان من النظرية هذه ينتج مباشرة نظرية مهمة لتقل
+
+351
+00:39:03,780 --> 00:39:13,660
+عنها أهمية ويلها اسم divergence
+
+352
+00:39:13,660 --> 00:39:16,900
+criteria
+
+353
+00:39:25,650 --> 00:39:36,650
+لت if the function from A to R and see the cluster
+
+354
+00:39:36,650 --> 00:39:39,750
+point
+
+355
+00:39:39,750 --> 00:39:45,850
+of A then واحد
+
+356
+00:39:47,360 --> 00:39:54,460
+الـ limit ل f of x لما x تقول ل c لا تساوي ال f
+
+357
+00:39:54,460 --> 00:40:01,440
+and only f there exist a sequence xm contained in
+
+358
+00:40:01,440 --> 00:40:10,180
+a حدودها مختلفة عن ال c such that limit xm بتساوي
+
+359
+00:40:10,180 --> 00:40:20,490
+c butLimit f of x in لاتساوي n الكرتيريا
+
+360
+00:40:20,490 --> 00:40:25,750
+التانية اللي هي عشان
+
+361
+00:40:25,750 --> 00:40:31,930
+نقول limit f of x لما x تقولها c does not exist in
+
+362
+00:40:31,930 --> 00:40:43,690
+Rهذا بكافئ أن هناك سيكوانس Xn محتوى A حدودها
+
+363
+00:40:43,690 --> 00:40:50,870
+مختلفة عن C بحيث أن نهايتها بساوي
+
+364
+00:40:50,870 --> 00:41:00,310
+C بط نهاية صورتها لا
+
+365
+00:41:00,310 --> 00:41:02,670
+توجد في R
+
+366
+00:41:16,230 --> 00:41:21,250
+كمان النظرية هذه مرهانها ينتج مباشرة من النظرية
+
+367
+00:41:21,250 --> 00:41:27,990
+اللي فوق مثلا هي عندي لإثبات ال band الأول عشان
+
+368
+00:41:27,990 --> 00:41:31,130
+أثبت limit f of x مستويش L and C
+
+369
+00:41:34,380 --> 00:41:38,880
+يعني كإني بقول نفي العبارة واحد هذا هو نفي العبارة
+
+370
+00:41:38,880 --> 00:41:42,560
+واحد طب احنا لسه بثبتين ان واحد بكافي اتنين
+
+371
+00:41:42,560 --> 00:41:46,560
+وبالتالي نفي العبارة واحد بكافي نفي الاتنين فنفي
+
+372
+00:41:46,560 --> 00:41:51,100
+الاتنين هذا هو يوجد a sequence تتقارب ل C لكن صورة
+
+373
+00:41:51,100 --> 00:41:56,720
+تلاتة تتقارب لL إذا برهان الجزء الأول نتيجة مباشرة
+
+374
+00:41:56,720 --> 00:42:02,130
+على مضارية ال form والجزء التاني زيه بدل هناعشان
+
+375
+00:42:02,130 --> 00:42:06,070
+اقول ان ال limit هذه does not exist يعني لو اخدت
+
+376
+00:42:06,070 --> 00:42:12,650
+اي عدد L فال limit هنا لا تساوي L معناته انه في
+
+377
+00:42:12,650 --> 00:42:18,050
+sequence و الكلام هذا ال limit هذه ماسويش اي L اي
+
+378
+00:42:18,050 --> 00:42:23,890
+عدد حقيقي اذا النظرية هذه نتيجة مباشرة على النظرية
+
+379
+00:42:23,890 --> 00:42:27,880
+sequential criterion النظرية التي سبقتهاالان هذه
+
+380
+00:42:27,880 --> 00:42:31,560
+النظرية هنستخدمها في إثبات إن ال limit لدالة
+
+381
+00:42:31,560 --> 00:42:36,000
+معينة، عن نقطة معينة غير موجودة، فهي بعض الأمثلة
+
+382
+00:42:36,000 --> 00:42:39,140
+كيف
+
+383
+00:42:39,140 --> 00:42:42,500
+نستخدم ال divergence كتير، كيف نثبت ال divergence
+
+384
+00:42:42,500 --> 00:42:48,020
+أو عدم وجود limit لدالة معينة عن نقطة معينة، فمثلا
+
+385
+00:42:48,020 --> 00:43:02,210
+ناخد أول مثالshow that limit ل 1 على x لما x تقول
+
+386
+00:43:02,210 --> 00:43:09,470
+إلى السفر does not exist in R فلبرهان
+
+387
+00:43:09,470 --> 00:43:16,870
+ذلك let
+
+388
+00:43:16,870 --> 00:43:24,050
+f of x بساوي 1 على x و ده أخد الـ x موجبةيعني
+
+389
+00:43:24,050 --> 00:43:27,130
+نعتبر أن ال domain للدالة هذه اللي هو الفترة A
+
+390
+00:43:27,130 --> 00:43:31,270
+بساوي الفترة مفتوحة من الصفر لما لا نهاية و نثبت
+
+391
+00:43:31,270 --> 00:43:34,750
+أن الدالة هذه ماليهاش limit عند الصفر أو عند الصفر
+
+392
+00:43:34,750 --> 00:43:40,650
+من اليمين فلإثبات أن ال limit للدالة هذه عند الصفر
+
+393
+00:43:40,650 --> 00:43:44,470
+ماهياش موجودة حسب ال divergence criteria يعني بدي
+
+394
+00:43:44,470 --> 00:43:48,210
+أثبت أن يوجه .. بدي أجيب sequence نهايتها صفر لكن
+
+395
+00:43:48,210 --> 00:43:52,490
+نهاية صورتها مش موجودة فال sequence إذا هنا
+
+396
+00:43:52,490 --> 00:43:59,560
+considerالـ sequence التي تفي بهذا الغرض اللي هي
+
+397
+00:43:59,560 --> 00:44:06,400
+xn بالساوي واحد على n لكل n في n فواضح أنه limit
+
+398
+00:44:06,400 --> 00:44:16,720
+xn بالساوي limit واحد على n بتساوي سفر وواضح أنه
+
+399
+00:44:16,720 --> 00:44:22,800
+xn contained in a اللي هي الفترة هذه معدى السفر
+
+400
+00:44:22,800 --> 00:44:31,310
+صح؟وعندي ال limit لل image لل sequence xn بساوي ال
+
+401
+00:44:31,310 --> 00:44:38,250
+limit ل 1 على xn لما n تقوى ل infinity بساوي ال
+
+402
+00:44:38,250 --> 00:44:43,410
+limit ل n لما n تقوى ل infinity بساوي infinity
+
+403
+00:44:43,410 --> 00:44:49,730
+وهذه طبعا ال infinity does not exist in R ليست عدد
+
+404
+00:44:49,730 --> 00:44:55,980
+حقيقيالنهاية نجحت في إيجاد sequence موجودة في A
+
+405
+00:44:55,980 --> 00:45:00,840
+وحدودها مختلفة عن السفر ونهايتها سفر لكن نهاية
+
+406
+00:45:00,840 --> 00:45:06,960
+صورتها مش موجودة في R وبالتالي therefore by
+
+407
+00:45:06,960 --> 00:45:14,020
+divergence criterion limit
+
+408
+00:45:14,020 --> 00:45:23,240
+ل F of X أو واحد على Xلما x سقول إلى 0 does not
+
+409
+00:45:23,240 --> 00:45:28,860
+exist in R وفي حقيقة الأمر اثبتنا ان limit 1 على x
+
+410
+00:45:28,860 --> 00:45:34,180
+لما x سقول إلى 0 من اليمين غير موجودة لان اخذنا
+
+411
+00:45:34,180 --> 00:45:42,620
+المجال كل الاعداد الموجودة بالمثل ممكن اثبات ان
+
+412
+00:45:42,620 --> 00:45:50,390
+limit ل1 على xلمّا X تقول إلى سفر من اليسار does
+
+413
+00:45:50,390 --> 00:45:55,540
+not existان انا اخد المرة هذه ال X هنا في الدالة
+
+414
+00:45:55,540 --> 00:46:00,560
+هذه ال domain تبعها الفترة من سالب ماله نهاية الى
+
+415
+00:46:00,560 --> 00:46:05,720
+سفر و اقول ان ال X هنا أصغر من سفر و نفس البرهان
+
+416
+00:46:05,720 --> 00:46:09,820
+هيطلع عندى ال limit لما X تقوله سفر من اليسار does
+
+417
+00:46:09,820 --> 00:46:13,500
+not exist وبالتالي ال limit عند ال X من الجهتين
+
+418
+00:46:13,500 --> 00:46:19,820
+does not exist تمام okay هذا مثال مثال تاني واضح
+
+419
+00:46:19,820 --> 00:46:21,440
+فيه اي سفصار فيه اي سؤال
+
+420
+00:46:25,390 --> 00:46:35,810
+ناخد مثال تاني show
+
+421
+00:46:35,810 --> 00:46:42,590
+that limit للـ signum function signum x لما x تقول
+
+422
+00:46:42,590 --> 00:46:48,930
+إلى سفر does not exist where حيث و ال signum
+
+423
+00:46:48,930 --> 00:46:52,450
+function where
+
+424
+00:46:57,580 --> 00:47:02,460
+where signum x هي عبارة عن function في x بنعرفها
+
+425
+00:47:02,460 --> 00:47:07,000
+على أنها واحد إذا كان x أكبر من سفر سفر إذا كان x
+
+426
+00:47:07,000 --> 00:47:12,300
+بساول سفر سالب واحد إذا كان x أصغر من سفر وهي
+
+427
+00:47:12,300 --> 00:47:13,360
+الرسمة تبعتها
+
+428
+00:47:24,680 --> 00:47:28,920
+فالدالة لما x أكبر من صفر بيستوي ثابت واحد عند
+
+429
+00:47:28,920 --> 00:47:34,400
+الصفر بيستوي صفر و لما x أصغر من واحد بيستوي سالب
+
+430
+00:47:34,400 --> 00:47:38,040
+واحد طيب
+
+431
+00:47:38,040 --> 00:47:48,360
+note that لاحظوا أن الدالة هذه sigma of x بتساوي
+
+432
+00:47:48,360 --> 00:47:52,120
+x على absolute x fx
+
+433
+00:47:53,900 --> 00:47:59,820
+لا تساوي سفر إذا كان x بساوي سفر فدالة sigma بها
+
+434
+00:47:59,820 --> 00:48:07,900
+نفس x على absolute xنفس .. نفس الحاجة طيب الان
+
+435
+00:48:07,900 --> 00:48:13,400
+اثبات ان ال limit لدالها جاند سفر مش موجودة طبعا
+
+436
+00:48:13,400 --> 00:48:17,440
+في تفاضل ألف في برهان في تفاضل ألف بيقول ان هى
+
+437
+00:48:17,440 --> 00:48:21,380
+الدالة لما X اولا سفر من اليمين ال limit لها واحد
+
+438
+00:48:21,380 --> 00:48:25,500
+لما X اولا سفر من اليمين نهيتها سالب واحد ال limit
+
+439
+00:48:25,500 --> 00:48:28,040
+من اليمين مستويش ال limit من اليسار اذا ال limit
+
+440
+00:48:28,040 --> 00:48:33,690
+لدالها جاند سفر does not exist برهانaccurate صحيح
+
+441
+00:48:33,690 --> 00:48:37,030
+مية المية مافي مشكلة لكن لو بدنا نعطي برهان
+
+442
+00:48:37,030 --> 00:48:41,810
+باستخدام ال divergence criterion فالبرهان هيكون
+
+443
+00:48:41,810 --> 00:48:46,270
+كالتالي consider
+
+444
+00:48:46,270 --> 00:48:51,410
+بدنا نجيب sequence xn
+
+445
+00:48:54,550 --> 00:48:58,490
+Rدودها مختلفة عن السفر نهايتها سفر لكن نهايت
+
+446
+00:48:58,490 --> 00:49:03,950
+صورتها بساوي سفر ف consider ال sequence اللي هي Xn
+
+447
+00:49:03,950 --> 00:49:09,110
+الحد اللي عام تبعها Xn بساوي سالف واحد أس ان على N
+
+448
+00:49:09,110 --> 00:49:19,190
+لكل N في N ال sequence هذه تنتمي إلى A اللي هو R
+
+449
+00:49:19,190 --> 00:49:21,890
+بعد السفر
+
+450
+00:49:26,050 --> 00:49:29,530
+موجودة في المجال تبع الدالة المجال تبع الدالة دي
+
+451
+00:49:29,530 --> 00:49:37,570
+كل الأعداد اللي حصلت فيها معدد C صح؟ وعندي و ال
+
+452
+00:49:37,570 --> 00:49:44,610
+limit و ال limit ل XM as M tends to infinity بسوى
+
+453
+00:49:44,610 --> 00:49:50,150
+و ال limitلسالب واحد قص ان على ان لما ان تقول
+
+454
+00:49:50,150 --> 00:49:55,110
+infinity ال limit لل sequence دي ايش بيساوي بيساوي
+
+455
+00:49:55,110 --> 00:50:03,470
+سفر by squeeze theorem او
+
+456
+00:50:03,470 --> 00:50:08,050
+by sandwich theorem but
+
+457
+00:50:08,050 --> 00:50:15,650
+لكن تعالوا نشوف ال limitلـ f of xn as n tends to
+
+458
+00:50:15,650 --> 00:50:19,810
+infinity شو بيساوي؟ بيساوي الـ limit as n tends to
+
+459
+00:50:19,810 --> 00:50:25,750
+infinity احنا عندي الـ xn هنا بيستويش صفر وبالتالي
+
+460
+00:50:25,750 --> 00:50:30,250
+الـ f of x تبعتي اللي هي الـ signum function فهذا
+
+461
+00:50:30,250 --> 00:50:34,050
+بيساوي limit signum xn
+
+462
+00:50:36,620 --> 00:50:41,540
+مظبوط و ال x in قلنا هنا بسويش 0 وبالتالي هذا
+
+463
+00:50:41,540 --> 00:50:47,000
+عبارة عن limit as n tends to infinity ال signal ل
+
+464
+00:50:47,000 --> 00:50:55,420
+x in بساوي x in على absolute x in فهذا
+
+465
+00:50:55,420 --> 00:51:02,210
+بساوي ال limitas n tends to infinity لـ xn عبارة
+
+466
+00:51:02,210 --> 00:51:09,190
+عن سالب واحد قص n على n على absolute xn absolute
+
+467
+00:51:09,190 --> 00:51:16,530
+xn بساوي واحد على n أصبت؟ الآن نجسم ونبسط ال limit
+
+468
+00:51:16,530 --> 00:51:23,750
+as n tends to infinity بطلع سالب واحد قص n وال
+
+469
+00:51:23,750 --> 00:51:27,210
+sequence هذه ال limit تبعتها أثبتنا قبل هيك
+
+470
+00:51:28,730 --> 00:51:33,410
+بطريقتين على الأقل ان ال limit هذه does not exist
+
+471
+00:51:33,410 --> 00:51:44,830
+does not exist وبالتالي اذا either by the
+
+472
+00:51:44,830 --> 00:51:47,630
+divergence criterion
+
+473
+00:51:50,230 --> 00:51:54,070
+هي اثبتت ان الـ use and sequence موجودة في المجال
+
+474
+00:51:54,070 --> 00:51:58,970
+تبع الدالة معدى السفر نهايتها سفر لكن نهاية صورتها
+
+475
+00:51:58,970 --> 00:52:03,270
+does not exist اذا by ال band التاني من ال
+
+476
+00:52:03,270 --> 00:52:11,590
+divergence criterion ال limit لل
+
+477
+00:52:11,590 --> 00:52:17,490
+signum function لما X تقول السفر does not exist
+
+478
+00:52:17,490 --> 00:52:18,570
+غير موجودة
+
+479
+00:52:20,890 --> 00:52:26,890
+Okay تمام واضح واضح البرهان في اي استفسار في اي
+
+480
+00:52:26,890 --> 00:52:34,470
+سؤال Okay
+
+481
+00:52:34,470 --> 00:52:39,470
+نوقف هنا وان شاء الله بنكمل المرة الجاية في بعض
+
+482
+00:52:39,470 --> 00:52:45,290
+مثالين الموجودة في الكتاب تحاولوا تقرؤهم او مثال
+
+483
+00:52:46,220 --> 00:52:50,740
+الشباب بالمثال هذا تحاولوا تقرؤوا و المرة الجاية
+
+484
+00:52:50,740 --> 00:52:52,580
+هنبدأ section جديد
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b6932196cf6b6e070e528084a93d79d9877abcd6
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_postprocess.srt
@@ -0,0 +1,1232 @@
+1
+00:00:20,670 --> 00:00:27,570
+بسم الله الرحمن الرحيم و السلام عليكم هنكمل
+
+2
+00:00:27,570 --> 00:00:33,570
+ان شاء الله اليوم ال .. المثال رقم اتنين اللي
+
+3
+00:00:33,570 --> 00:00:39,530
+بدناه في المحاضرة السابقة و ماكملناهوش فنرجعوا مع
+
+4
+00:00:39,530 --> 00:00:48,020
+بعض بسرعة و نحاول نكمل البرهانلهذا المثال وهو ان
+
+5
+00:00:48,020 --> 00:00:53,900
+ال sequence المعرفة بطريقة استقرائية هنا بنثبت
+
+6
+00:00:53,900 --> 00:00:58,560
+انها convergence و ال limit تبعتها بساوي العدد
+
+7
+00:00:58,560 --> 00:01:04,700
+خمسة على تلاتة تمام فبدينا البرهان المرة اللي فاتت
+
+8
+00:01:04,700 --> 00:01:11,900
+و أثبتنا claim .. claim واحد و كان في ال claim هذا
+
+9
+00:01:11,900 --> 00:01:13,680
+أثبتنا ان ال
+
+10
+00:01:17,150 --> 00:01:22,690
+إن حدود الـ sequence bounded below by one and
+
+11
+00:01:22,690 --> 00:01:29,450
+bounded above by two هذا صحيح لكل إن وشوفنا هذا
+
+12
+00:01:29,450 --> 00:01:36,570
+البراني هذا ممكن يعني ممكن إعطاءه by induction
+
+13
+00:01:36,570 --> 00:01:41,450
+therefore by claim one
+
+14
+00:01:45,770 --> 00:01:52,510
+السيكوانس xn is bounded واضح من ال claim ان
+
+15
+00:01:52,510 --> 00:01:59,230
+السيكوانس is bounded المرة التي اثبتنا claim رقم 2
+
+16
+00:01:59,230 --> 00:02:03,470
+اثبتنا
+
+17
+00:02:03,470 --> 00:02:07,470
+ان السيكوانس
+
+18
+00:02:07,470 --> 00:02:15,960
+xn بتحققالمعادلة absolute xn minus xn زايد واحد
+
+19
+00:02:15,960 --> 00:02:22,960
+هذا بيساوي واحد على اتنين أس n negative one for
+
+20
+00:02:22,960 --> 00:02:30,360
+every natural number in it وشوفنا برهنة المعادلة
+
+21
+00:02:30,360 --> 00:02:36,020
+هذه او العبارة هذه لكل عدد طبيعي by induction okay
+
+22
+00:02:38,610 --> 00:02:45,630
+اليوم باستخدام ال claim 2 and
+
+23
+00:02:45,630 --> 00:02:51,330
+triangle
+
+24
+00:02:51,330 --> 00:03:01,490
+inequality متباينة مثلث نرى
+
+25
+00:03:01,490 --> 00:03:02,070
+ان
+
+26
+00:03:05,510 --> 00:03:16,950
+وإذا M أكتر من N، M نموات طبيعية، و M أكتر من N،
+
+27
+00:03:16,950 --> 00:03:23,190
+فلدينا أكتر من X M أكتر من X
+
+28
+00:03:29,470 --> 00:03:35,150
+طبعا هذا ممكن نكتبه على صورة هي absolute xn هترح
+
+29
+00:03:35,150 --> 00:03:43,230
+xn زاد واحد و هرجعها و
+
+30
+00:03:43,230 --> 00:03:55,190
+هترح xn زاد اتنين و هرجعها و
+
+31
+00:03:55,190 --> 00:03:58,250
+هكذا الى ان اصل الى
+
+32
+00:04:03,240 --> 00:04:13,220
+x m negative one سالب x m في الآخر خالص هاطرح x m
+
+33
+00:04:13,220 --> 00:04:19,960
+سالب واحد ورجعها فكأني أنا يعني ماعملتش ماغيرتش
+
+34
+00:04:19,960 --> 00:04:24,600
+حاجة فالمخضر اللي على اليمين هو نفسه اللي على
+
+35
+00:04:24,600 --> 00:04:30,360
+الشمال لأن طرحة had وضفته طرحة had وضفتها كذلك
+
+36
+00:04:30,360 --> 00:04:36,550
+فكأني ضفت سفرالان ناخد الحدين هذول مع بعض و هذول
+
+37
+00:04:36,550 --> 00:04:47,810
+مع بعض و هكذا و هذول مع بعض و هذول اخر حدين مع بعض
+
+38
+00:04:47,810 --> 00:04:54,110
+و بنستخدم ال triangle inequality ف by triangle
+
+39
+00:04:54,110 --> 00:05:00,330
+inequality ال absolute value لمجموعةLess than or
+
+40
+00:05:00,330 --> 00:05:06,890
+equal مجموع الـ absolute values فهذا absolute xn
+
+41
+00:05:06,890 --> 00:05:12,850
+minus xn زاد واحد زاد absolute xn زاد واحد minus
+
+42
+00:05:12,850 --> 00:05:21,510
+xn زاد اتو وهكذا
+
+43
+00:05:21,510 --> 00:05:27,170
+إلى absolute xm negative one minus xm
+
+44
+00:05:30,780 --> 00:05:37,120
+الان باستخدام claim اتنين الحد الاول هذا عبارة عن
+
+45
+00:05:37,120 --> 00:05:44,100
+واحد على two أس in negative one الحد اللي بعده one
+
+46
+00:05:44,100 --> 00:05:53,000
+over two أس in و الحد اللي بعده هكذا و
+
+47
+00:05:53,000 --> 00:06:03,820
+الحد الأخير الحد الأخير هذا هيكون واحد علىتو اص ام
+
+48
+00:06:03,820 --> 00:06:07,580
+ماينوس
+
+49
+00:06:07,580 --> 00:06:14,460
+اتنين اذا
+
+50
+00:06:14,460 --> 00:06:21,040
+هذا من المعادلة اللي هنا ناخد
+
+51
+00:06:21,040 --> 00:06:33,760
+عامل مشترك one over two اص n negative oneفبيبقى
+
+52
+00:06:33,760 --> 00:06:38,000
+إذا من الحد الأول دي بقى اللي عندي واحد من الحد
+
+53
+00:06:38,000 --> 00:06:46,740
+التاني بيبقى عندي نص و هكذا إلى الحد الأخير اللي
+
+54
+00:06:46,740 --> 00:06:56,120
+بيبقى عندي two أُس M negative M negative one الآن
+
+55
+00:06:56,120 --> 00:07:01,080
+المجموعة هذا اللي بين جثين هذا المجموعةأصغر من
+
+56
+00:07:01,080 --> 00:07:06,220
+اتنين لأن هذا المجموع لاحظوا
+
+57
+00:07:06,220 --> 00:07:16,080
+انتوا واحد زائد نص زائد إلى one over two to m
+
+58
+00:07:16,080 --> 00:07:23,020
+negative n negative one this is less than one زائد
+
+59
+00:07:23,020 --> 00:07:25,600
+نص زائد
+
+60
+00:07:29,510 --> 00:07:37,230
+زائد واحد على اتنين أس ان زائد إلى مالة نهائية
+
+61
+00:07:37,230 --> 00:07:44,610
+اللي هو مجموعة series sigma from k equals zero to
+
+62
+00:07:44,610 --> 00:07:54,430
+infinity ال one over two to k هذا
+
+63
+00:07:54,430 --> 00:07:58,010
+جزء من ال infinite series
+
+64
+00:08:00,560 --> 00:08:07,300
+هذه الـ inference series هذه أول يعني M سالب N
+
+65
+00:08:07,300 --> 00:08:12,700
+مايرس واحد من حدودها هذه
+
+66
+00:08:12,700 --> 00:08:17,380
+ال series معروفة هي geometric series geometric
+
+67
+00:08:17,380 --> 00:08:25,320
+series with a الحد الأول واحد وال ratio and ال
+
+68
+00:08:25,320 --> 00:08:31,860
+ratio بساوي نصففي تفاضل بقى اتعلمتوا انه اي
+
+69
+00:08:31,860 --> 00:08:35,200
+geometric series اذا ال ratio ال absolute value ل
+
+70
+00:08:35,200 --> 00:08:40,940
+R أصغر من واحد فال series تطلع convergent ومجموعة
+
+71
+00:08:40,940 --> 00:08:48,520
+.. مجموعة بساوي a على one minus الأساس وهذا بطلع a
+
+72
+00:08:48,520 --> 00:08:53,720
+اللي هو واحد على واحد minus الأساس نص بطلع بساوي
+
+73
+00:08:53,720 --> 00:08:57,060
+اتنين okay اذا ال ..
+
+74
+00:09:00,710 --> 00:09:07,850
+إذا المجموع هذا بيطلع أصغر من اتنين إذا هي حاول
+
+75
+00:09:07,850 --> 00:09:14,690
+أصغر من واحد على اتنين أسن negative واحد ضرب اتنين
+
+76
+00:09:14,690 --> 00:09:25,210
+وهذا بيساوي واحد على اتنين أسن سالب اتنين تمام؟
+
+77
+00:09:36,740 --> 00:09:42,940
+الان بنا نثبت احنا ان ال sequence
+
+78
+00:09:42,940 --> 00:09:48,860
+احنا كان بنا نثبت ان ال sequence x in convergent
+
+79
+00:09:48,860 --> 00:09:52,820
+وقلنا في بداية البرهان المرة اللى فات عشان نثبت
+
+80
+00:09:52,820 --> 00:09:57,380
+انها convergent يكفي ان احنا نثبت انها Cauchy صح؟
+
+81
+00:09:57,380 --> 00:10:01,400
+لأن اذا كانت Cauchy بتكون convergent by Cauchy a
+
+82
+00:10:01,400 --> 00:10:03,920
+criterion اذا هنا to show
+
+83
+00:10:08,040 --> 00:10:13,900
+إن X in convergence it
+
+84
+00:10:13,900 --> 00:10:18,100
+suffices يعني
+
+85
+00:10:18,100 --> 00:10:29,100
+يكفي إثبات to show it is Cauchy إذا يكفي إثبات
+
+86
+00:10:29,100 --> 00:10:36,510
+إنها Cauchyطيب هاي عندي .. الآن هستفيد من المتباين
+
+87
+00:10:36,510 --> 00:10:45,890
+هذه الأخيرة لإثبات إنها كوشي طيب
+
+88
+00:10:45,890 --> 00:10:54,790
+أنا عندي .. أنا عندي four .. قولنا M أكبر من N
+
+89
+00:10:57,610 --> 00:11:04,270
+أثبتنا أن أبسليوت xn نيجاتيف xm less than one over
+
+90
+00:11:04,270 --> 00:11:16,970
+two to n minus two نسمي الانيقواليتي هذه star الان
+
+91
+00:11:16,970 --> 00:11:21,010
+let
+
+92
+00:11:21,010 --> 00:11:28,960
+epsilon أكبر من السفر be givenأنا بدأ أثبت إن الـ
+
+93
+00:11:28,960 --> 00:11:32,980
+sequence تبعتي كوشي فعشان أثبت إنها كوشي ببدأ
+
+94
+00:11:32,980 --> 00:11:38,840
+بإمسون أكبر من سفر برد عليها بcapital N بحيث إنه
+
+95
+00:11:38,840 --> 00:11:44,280
+لكل M و N أكبر من أو يساوي capital N لازم المسافة
+
+96
+00:11:44,280 --> 00:11:50,000
+بين XN وXM أصغر من إبسون ف let إبسون أكبر من سفر
+
+97
+00:11:50,000 --> 00:11:59,040
+be given by Archimedean propertyمن خاصية
+
+98
+00:11:59,040 --> 00:12:08,700
+Archimedes choose ممكن نختار capital N عدد طبيعي
+
+99
+00:12:08,700 --> 00:12:19,020
+بحيث أنه واحد على N أصغر من epsilon على أربعةوهذا
+
+100
+00:12:19,020 --> 00:12:23,440
+صحيح by the Archimedean property إبسلون عدد موجب
+
+101
+00:12:23,440 --> 00:12:26,920
+بتعني إبسلون على أربع عدد موجب لهذا العدد الموجب
+
+102
+00:12:26,920 --> 00:12:31,740
+بقدر ألاقي عدد طبيعه عدد طبيعي مقلوب وأصغر من عدد
+
+103
+00:12:31,740 --> 00:12:44,060
+الموجب تمام؟ وبالتالي هذا بيقدر أنه ال ..واحد على
+
+104
+00:12:44,060 --> 00:12:52,020
+two to n أصغر من epsilon على أربعة لأن
+
+105
+00:12:52,020 --> 00:13:02,640
+since لأن two to n أكبر من n صح؟ وبالتالي مقلوب
+
+106
+00:13:02,640 --> 00:13:07,560
+هذا أصغر من مقلوب ال n اللي هو أصغر من epsilon على
+
+107
+00:13:07,560 --> 00:13:10,720
+أربعة okay تمام؟ طيب
+
+108
+00:13:14,350 --> 00:13:25,310
+إذن this .. this and star بيؤدوا إنه لو كان M أكبر
+
+109
+00:13:25,310 --> 00:13:30,450
+من أو يساوي N أكبر من أو يساوي capital N فهذا
+
+110
+00:13:30,450 --> 00:13:40,050
+بيؤدي إنه absolute xn negative xm أصغر
+
+111
+00:13:40,050 --> 00:13:51,740
+من1 على 2 to N minus 2 وهذا أصغر من أو ساوي 1 على
+
+112
+00:13:51,740 --> 00:13:58,200
+2 to capital N minus 2 لأن small n أكبر من أو ساوي
+
+113
+00:13:58,200 --> 00:14:04,700
+capital N ف2
+
+114
+00:14:04,700 --> 00:14:11,200
+N سالب 2 أصغر يعني مقلوب هذهيعني أنا عندي هنا
+
+115
+00:14:11,200 --> 00:14:16,460
+اتنين أس ان سالب اتنين بطلع أكبر من أو ساوي two
+
+116
+00:14:16,460 --> 00:14:21,340
+two capital N سالب اتنين لأن ان أكبر من أو ساوي
+
+117
+00:14:21,340 --> 00:14:25,200
+capital N وبالتالي مقلوب الكبير أصغر من أو ساوي
+
+118
+00:14:25,200 --> 00:14:35,420
+مقلوب الكبير أو مقلوب الصغير فهذا صح ومن هنا هذا
+
+119
+00:14:35,420 --> 00:14:39,280
+أصغر هذا من هنا أصغر من epsilon
+
+120
+00:14:42,800 --> 00:14:48,200
+لأن هذا عبارة عن .. هذا بساوي .. أيوه بساوي أربعة
+
+121
+00:14:48,200 --> 00:14:53,500
+على اتنين أسن و أربعة على اتنين أسن أصغر من
+
+122
+00:14:53,500 --> 00:15:01,740
+إبسلون، صح؟ إذن هذه أثبتت for any given .. for any
+
+123
+00:15:01,740 --> 00:15:06,700
+given إبسلون يوجد capital N يعتمد على إبسلون، هذا
+
+124
+00:15:06,700 --> 00:15:15,590
+هويوجد capital N غير مرتبط بـY بحيث أنه لكل M و N
+
+125
+00:15:15,590 --> 00:15:20,350
+أكبر من أو ساوي capital N فالمسافة بين XN و XM
+
+126
+00:15:20,350 --> 00:15:26,650
+أصغر من Y وبالتالي هذا بثبت أن ال sequence is
+
+127
+00:15:26,650 --> 00:15:36,270
+Cauchy بس ال sequence XN is Cauchy
+
+128
+00:15:39,020 --> 00:15:43,660
+and therefore x
+
+129
+00:15:43,660 --> 00:15:50,900
+in converges say
+
+130
+00:15:50,900 --> 00:16:02,740
+ال limit ل x in بساوي some x ينتمي ال R هنا
+
+131
+00:16:02,740 --> 00:16:08,130
+أثبتنا أن ال sequence x in convergentby Cauchy
+
+132
+00:16:08,130 --> 00:16:13,790
+criterion وفرضنا ان ال limit تبعتها بساوي X عشان
+
+133
+00:16:13,790 --> 00:16:15,870
+كام اذا هين اثبتنا ان ال sequence تبعتنا
+
+134
+00:16:15,870 --> 00:16:20,070
+convergent ال limit تبعتها عدد X بقى هينثبت ان ال
+
+135
+00:16:20,070 --> 00:16:25,790
+X اللي هو limit ل X in بساوي خمسة على تلاتة بساوي
+
+136
+00:16:25,790 --> 00:16:29,350
+خمسة على تلاتة okay اذا هينثبت
+
+137
+00:16:34,190 --> 00:16:42,010
+الجزء الأخير هذا وهو نسميه
+
+138
+00:16:42,010 --> 00:16:50,790
+claim تلاتة claim three ال X بساوي five over three
+
+139
+00:16:50,790 --> 00:17:00,130
+لبرهان ذلك first
+
+140
+00:17:03,450 --> 00:17:08,530
+use induction on
+
+141
+00:17:08,530 --> 00:17:20,170
+n to show الإثبات إن x to n plus one بساوي واحد
+
+142
+00:17:20,170 --> 00:17:28,610
+زائد نص زائد واحد على اتنين تكعيب زائدو هكذا one
+
+143
+00:17:28,610 --> 00:17:35,410
+over two to two n سالب واحد وهذا صحيح لكل natural
+
+144
+00:17:35,410 --> 00:17:42,910
+number n المعادلة هذه ممكن اثباتها by induction on
+
+145
+00:17:42,910 --> 00:17:54,130
+n سهل جدا طبعا ممكن تستخدم .. تحتاج ال inductive
+
+146
+00:17:54,130 --> 00:17:59,300
+definition في البرهنزي ما شوفنا في برهان claim 2
+
+147
+00:17:59,300 --> 00:18:08,440
+طيب الآن افرض ان احنا هذا أثبتناها hence وبالتالي
+
+148
+00:18:08,440 --> 00:18:15,500
+من هنا بطلع عندي x2n plus one بساوي واحد زائد هاخد
+
+149
+00:18:15,500 --> 00:18:21,820
+من المجموع هذا هاخد
+
+150
+00:18:21,820 --> 00:18:28,200
+نص عام المشتركفبيبقى عندي واحد زائد واحد على اتنين
+
+151
+00:18:28,200 --> 00:18:36,880
+تربية زائد واحد على اتنين تربية لكل تربية زائد و
+
+152
+00:18:36,880 --> 00:18:44,260
+هكذا زائد واحد على اتنين تربية to end negative one
+
+153
+00:18:44,260 --> 00:18:52,220
+اذا انا خدت من هاي الواحد نزلته واخدتنص عام
+
+154
+00:18:52,220 --> 00:18:57,320
+المشترك من باقي الحدود هذه فطل عند المجموع هذا هذا
+
+155
+00:18:57,320 --> 00:19:04,120
+مجموع متوالية هندسية geometric progression لأن هذا
+
+156
+00:19:04,120 --> 00:19:08,080
+بشكل geometric
+
+157
+00:19:08,080 --> 00:19:19,740
+.. geometric progression متوالية
+
+158
+00:19:19,740 --> 00:19:20,480
+هندسية
+
+159
+00:19:23,660 --> 00:19:30,740
+with الحد الأول a بساوي واحد وال ratio بساوي واحد
+
+160
+00:19:30,740 --> 00:19:37,080
+على اتنين تربيات اللي هو ربعها ف ال geometric
+
+161
+00:19:37,080 --> 00:19:40,820
+progression فيه قانون لإيجاد مجموعة المتوالية
+
+162
+00:19:40,820 --> 00:19:46,080
+الهندسية فيه قانون لإيجاد مجموعة فالقانون هذا
+
+163
+00:19:46,080 --> 00:19:54,830
+عبارة عن الحد الأول واحد سالبالحد الأخير مضروب في
+
+164
+00:19:54,830 --> 00:20:02,650
+الأساس اللي هو واحد على اتنين تربية الكل قسم على
+
+165
+00:20:02,650 --> 00:20:08,150
+واحد minus الأساس على واحد minus الأساس اللي هو
+
+166
+00:20:08,150 --> 00:20:14,270
+واحد على اتنين تربية وهذا
+
+167
+00:20:14,270 --> 00:20:22,130
+بساوي اي واحد زائد المقام هذا عبارة عن تلت تربعة
+
+168
+00:20:23,720 --> 00:20:32,140
+هذا عبارة عن تلات اربعة فنص على تلات اربعة بطلع
+
+169
+00:20:32,140 --> 00:20:37,180
+اتنين على تلاتة و ال bust هذا هو في ال bust اللي
+
+170
+00:20:37,180 --> 00:20:45,520
+هو واحد سالب واحد على اتنين اص اتنين in او اربعة
+
+171
+00:20:45,520 --> 00:20:47,440
+اص in
+
+172
+00:20:52,700 --> 00:21:08,740
+الان ناخد ال limit للطرفين اذا
+
+173
+00:21:08,740 --> 00:21:14,380
+ناخد .. لو أخدنا ال limit للطرفين فبطلع limit x to
+
+174
+00:21:14,380 --> 00:21:20,800
+n plus one as n tends to infinityبساوي واحد زاد
+
+175
+00:21:20,800 --> 00:21:28,040
+اتنين على التلاتة في ال limit الجوس واحد سالب
+
+176
+00:21:28,040 --> 00:21:34,720
+limit واحد على اربعة أس in as in tends to infinity
+
+177
+00:21:34,720 --> 00:21:41,640
+وهذا بساوي واحد زاد اتنين على تلاتة في واحد سالب
+
+178
+00:21:41,640 --> 00:21:48,440
+limit واحد على اربعة in بساوي سفر فبطلع بساوي واحد
+
+179
+00:21:50,170 --> 00:21:57,150
+زاد اتنين على تلاتة بساوي خمسة على تلاتة طيب
+
+180
+00:21:57,150 --> 00:22:01,070
+هذه عبارة عن sub sequence من ال sequence xn هذه
+
+181
+00:22:01,070 --> 00:22:05,930
+الحدود الفردية ل sequence xn طيب و انا عندي ال
+
+182
+00:22:05,930 --> 00:22:09,750
+sequence تبعتي convergence هاي أثبتنا ان xn
+
+183
+00:22:09,750 --> 00:22:13,930
+convergent و ال limit تبعتها بساوي العدد x
+
+184
+00:22:20,160 --> 00:22:26,000
+إذا أنا في عندي هنا since x2n
+
+185
+00:22:26,000 --> 00:22:38,600
+plus one is a subsequence of a sequence xn and xn
+
+186
+00:22:38,600 --> 00:22:45,540
+converges to xthen by previous theorem حسب نظرية
+
+187
+00:22:45,540 --> 00:22:50,120
+السابقة إذا كانت ال sequence convergent ل x فأي
+
+188
+00:22:50,120 --> 00:22:54,960
+subsequence منها بتكون convergent لنفس ال x إذا
+
+189
+00:22:54,960 --> 00:23:01,460
+limit x اتنين n plus one as n tends to infinity
+
+190
+00:23:01,460 --> 00:23:07,640
+بساوي x وبالتالي إذا x بساوي limit
+
+191
+00:23:12,130 --> 00:23:19,830
+x2n زائد واحد وهذه أثبتنا في السطر الأخير هنا هذه
+
+192
+00:23:19,830 --> 00:23:23,750
+بساوي خمسة على تلاتة وهذا اللي بدنا يعني إذا هيك
+
+193
+00:23:23,750 --> 00:23:30,770
+بتكون أثبتنا ان ال sequence x in converges to x و
+
+194
+00:23:30,770 --> 00:23:36,370
+limit x تبعتها هي طلعت ساوي خمسة على تلاتة كما هو
+
+195
+00:23:36,370 --> 00:23:41,830
+مطلوبOkay إذا هيك بنكون إحنا برهننا إن ال sequence
+
+196
+00:23:41,830 --> 00:23:46,490
+في المثال هذا اللي معرفة بطريقة استقرائية is
+
+197
+00:23:46,490 --> 00:23:52,870
+convergent ونهايتها خمسة تلاتة Okay تمام؟ المفهوم
+
+198
+00:23:52,870 --> 00:23:59,680
+واضح؟ في أي استفسار؟ في أي سؤال؟طيب ناخد كمان مثال
+
+199
+00:23:59,680 --> 00:24:02,520
+يمكن يبقى المثال طويل شوية لكن احنا زى ما شوفته
+
+200
+00:24:02,520 --> 00:24:08,660
+احنا جزقناه الى تلاتة claims او three claims وكل
+
+201
+00:24:08,660 --> 00:24:12,340
+claim كان برهانه by induction مش صعب شفنا برهان
+
+202
+00:24:12,340 --> 00:24:18,020
+واحد منهم المرة اللى فاتت التانين برضه اسأل كمان
+
+203
+00:24:18,020 --> 00:24:23,060
+كل claim بيخطو خطوة الى الامام بيجربنى اكتر من
+
+204
+00:24:23,060 --> 00:24:23,540
+البرهان
+
+205
+00:24:27,870 --> 00:24:45,850
+ناخد مثال آخر، تالت إذا
+
+206
+00:24:45,850 --> 00:24:54,090
+example three consider
+
+207
+00:24:57,250 --> 00:25:02,550
+الحد العام بحيث ال sequence xn
+
+208
+00:25:02,550 --> 00:25:09,510
+where حيث ال term of the sequence الحد العام by
+
+209
+00:25:09,510 --> 00:25:14,070
+definition بساوي one over one plus one over two
+
+210
+00:25:15,270 --> 00:25:22,310
+plus one over three و هكذا and so on until we get
+
+211
+00:25:22,310 --> 00:25:31,730
+one over N حيث N ينتمي إلى N لكل N في N بنعرف XN
+
+212
+00:25:31,730 --> 00:25:36,530
+على أنه المجموع أو مجموعة أول N
+
+213
+00:25:49,690 --> 00:25:57,950
+مجموع أول n من حدود ال harmonic series فهذا بنسميه
+
+214
+00:25:57,950 --> 00:26:02,210
+ال nth partial sum هذا عبارة في تفاضل باسم منها ال
+
+215
+00:26:02,210 --> 00:26:14,410
+nth partial ال nth partial sum of ال harmonic ال
+
+216
+00:26:14,410 --> 00:26:16,110
+harmonic series
+
+217
+00:26:20,920 --> 00:26:26,800
+اللي هي summation from k equals one to infinity ل
+
+218
+00:26:26,800 --> 00:26:27,880
+one over k
+
+219
+00:26:31,950 --> 00:26:39,910
+المطلوب show أن سيكوينس XN divergence ليست
+
+220
+00:26:39,910 --> 00:26:46,070
+convergent، is not convergent بنثبت أن سيكوينس of
+
+221
+00:26:46,070 --> 00:26:51,090
+partial sums متتالية المجاميع الجزئية لل harmonic
+
+222
+00:26:51,090 --> 00:26:54,070
+series بتشكل divergence sequence
+
+223
+00:27:00,340 --> 00:27:19,020
+by cushy criterion اذا حسب cushy criterion by
+
+224
+00:27:19,020 --> 00:27:25,740
+cushy criterion it suffices to
+
+225
+00:27:25,740 --> 00:27:32,760
+show يكفي اثباتيكفي اثبات ان ال sequence عشان نثبت
+
+226
+00:27:32,760 --> 00:27:36,440
+ان ال sequence is divergent it suffices to show ان
+
+227
+00:27:36,440 --> 00:27:47,920
+ال sequence xn is not كوشي لأن
+
+228
+00:27:47,920 --> 00:27:52,980
+كوشي criterion بتقول ان ال sequence is convergent
+
+229
+00:27:52,980 --> 00:27:58,700
+if and only if it is كوشيوبالتالي هذا بكافي ان
+
+230
+00:27:58,700 --> 00:28:02,400
+احنا نقول ان ال sequence is not convergent if and
+
+231
+00:28:02,400 --> 00:28:06,440
+only if it is not Cauchy عشان نثبت ان ال sequence
+
+232
+00:28:06,440 --> 00:28:12,780
+is divergent ممكن نثبت انها is not Cauchy طيب ال
+
+233
+00:28:12,780 --> 00:28:22,900
+.. الاثبات انها not Cauchy هنستخدم indeed
+
+234
+00:28:29,180 --> 00:28:41,120
+في حقيقة الأمر لو أخدنا لو كان M أكبر من N فهذا
+
+235
+00:28:41,120 --> 00:28:48,320
+بيقدي ان XM minus
+
+236
+00:28:48,320 --> 00:28:53,660
+XN ايش بيساوي؟
+
+237
+00:28:53,660 --> 00:28:55,500
+أنا هي عندي ال ..
+
+238
+00:29:06,360 --> 00:29:18,240
+هي عندي xn و لو بدك تكتب xm ف xm هيكون بساوي واحد
+
+239
+00:29:18,240 --> 00:29:25,640
+اول حد زائد نص زائد
+
+240
+00:29:25,640 --> 00:29:33,140
+تلت و هكذا زائد
+
+241
+00:29:33,140 --> 00:29:40,460
+واحد على nو لسه كمان هكمل .. هنكمل لإن ال M أكبر
+
+242
+00:29:40,460 --> 00:29:49,380
+من N هنا ال M .. ال M أكبر من N لما تكون M أكبر من
+
+243
+00:29:49,380 --> 00:29:55,640
+N فهيكون الحد اللي بعدها ده واحد على M زايد واحد
+
+244
+00:30:04,780 --> 00:30:10,120
+واحد عال ان انا
+
+245
+00:30:10,120 --> 00:30:19,580
+مش نافع one
+
+246
+00:30:19,580 --> 00:30:29,190
+over n plus one و هكذا إلى one over nالآن لما أطرح
+
+247
+00:30:29,190 --> 00:30:37,890
+xn من xm فالحدود المتشابهة هتروح مع بعضها لحد واحد
+
+248
+00:30:37,890 --> 00:30:44,690
+على n بروح مع واحد على n بيبقى الفرق بين الإتنين
+
+249
+00:30:44,690 --> 00:30:54,930
+الفرق بين الإتنين هيكون عبارة عن واحد
+
+250
+00:30:56,990 --> 00:31:04,630
+على n زائد واحد زائد واحد على n plus two and so on
+
+251
+00:31:04,630 --> 00:31:12,230
+until we get one over m تمام الحدود هدول عددهم كم
+
+252
+00:31:12,230 --> 00:31:23,270
+حد m negative n terms عدد الحدود في المجموع هذا m
+
+253
+00:31:23,270 --> 00:31:37,060
+negative nهذول حدود عددهم M و هذول عددهم N فالفرق
+
+254
+00:31:37,060 --> 00:31:43,460
+بينهم هيطلع M negative N الآن
+
+255
+00:31:43,460 --> 00:31:46,700
+هذا المجموع أول حد
+
+256
+00:31:49,570 --> 00:31:57,510
+لاحظوا ان ان ال M أكبر من N هذا بيقدي ان M أكبر من
+
+257
+00:31:57,510 --> 00:32:04,230
+أو ساوي N زي واحد وهذا بيقدي ان مخلوق واحد على N
+
+258
+00:32:04,230 --> 00:32:10,210
+زي واحد بطلع أكبر من أو ساوي واحد على N
+
+259
+00:32:13,150 --> 00:32:17,850
+وبالتالي إذا واحد على N زاد واحد أكبر من أو ساوي
+
+260
+00:32:17,850 --> 00:32:22,850
+واحد على M بالمثل واحد على N زاد اتنين لحظة هيكون
+
+261
+00:32:22,850 --> 00:32:29,570
+ال M ال M أكبر من أو ساوي N زاد اتنين فمقلوب N زاد
+
+262
+00:32:29,570 --> 00:32:37,230
+اتنين هيطلع أكبر من أو ساوي واحد على Mو هكذا إذا
+
+263
+00:32:37,230 --> 00:32:41,670
+كل الحدود هذه كل واحد فيهم أكبر من أو ساوي واحد
+
+264
+00:32:41,670 --> 00:32:46,070
+على M إلى أن نصل لآخر حد واحد على M طبعا أكبر من
+
+265
+00:32:46,070 --> 00:32:51,070
+أو ساوي نفسه عدد الحدود هذه لازال M negative in
+
+266
+00:32:51,070 --> 00:32:51,810
+terms
+
+267
+00:32:55,640 --> 00:33:00,980
+طيب أنا لما بجمع عدد على نفسه M minus N من المرات،
+
+268
+00:33:00,980 --> 00:33:06,200
+إيش بيعطيني المجموعة؟ بيطلع بساوي M سالب N في
+
+269
+00:33:06,200 --> 00:33:08,540
+العدد الثابت، صح؟
+
+270
+00:33:14,860 --> 00:33:21,620
+إذن المجموعة الأخيرة هدا هيطلع بساوي M negative N
+
+271
+00:33:21,620 --> 00:33:30,660
+في واحد على M وهذا بساوي على
+
+272
+00:33:30,660 --> 00:33:40,520
+M هذي اه M وهذا بساوي واحد negative N على M طيب
+
+273
+00:33:40,520 --> 00:33:48,050
+انا في التحليل هذا ماخدال M أكبر من N يعني هذا
+
+274
+00:33:48,050 --> 00:33:54,670
+الكلام صحيح إذا كان M أكبر من N طيب الآن take M
+
+275
+00:33:54,670 --> 00:34:00,230
+بساوة 2N بالتأكيد 2N أي عدد طبيعي N لأي عدد طبيعي
+
+276
+00:34:00,230 --> 00:34:08,890
+N 2N أكبر من Nإذا لو عوضت عن M بتنين N في المتباين
+
+277
+00:34:08,890 --> 00:34:18,610
+الأخير هذه هيطلع عندي XM أو X to N negative XN
+
+278
+00:34:18,610 --> 00:34:28,670
+بتطلع أكبر من أو يساوي واحد negative two N على
+
+279
+00:34:28,670 --> 00:34:35,450
+اتنين N صح؟اللي هو واحد negative one-half بطلع one
+
+280
+00:34:35,450 --> 00:34:45,930
+-half نص الكلام هذا صحيح لكل n ينتمي
+
+281
+00:34:45,930 --> 00:34:50,470
+إلى n تمام؟
+
+282
+00:34:52,310 --> 00:34:58,910
+إذا أنا أصبح في عندي المتباينة x to n negative xn
+
+283
+00:34:58,910 --> 00:35:06,750
+أكبر من أو ساوي one half for all n belong to a now
+
+284
+00:35:06,750 --> 00:35:14,710
+you can easily show
+
+285
+00:35:19,500 --> 00:35:28,860
+ممكن بسهولة اثبات انه this implies المتباينة هذه
+
+286
+00:35:28,860 --> 00:35:37,540
+الأخيرة بتقدي this implies that
+
+287
+00:35:37,540 --> 00:35:43,620
+هذا بيقدي ان ال sequence xn is not Cauchy
+
+288
+00:35:46,720 --> 00:36:01,500
+is not Cauchy as desired كما هو مطلوب تمام؟
+
+289
+00:36:01,500 --> 00:36:07,220
+من المتباين هذا ممكن نثبت أن ال sequence تبعتنا
+
+290
+00:36:07,220 --> 00:36:12,820
+ليست Cauchyيمكن هذا مش واضح كيف ان هذا بيعدى ان ال
+
+291
+00:36:12,820 --> 00:36:17,520
+sequence not Cauchy لكن ممكن نعمل برهان بالتناقض
+
+292
+00:36:17,520 --> 00:36:22,680
+افرض ان ال sequence Cauchy واستخدم الشرط هذا او
+
+293
+00:36:22,680 --> 00:36:26,700
+المتبين هذى فيه الوصول الى تناقض هسيكم تكتبوا
+
+294
+00:36:26,700 --> 00:36:30,580
+البرهان هذا وكل واحدة بتكتب البرهان في ورقة
+
+295
+00:36:30,580 --> 00:36:35,840
+وبيسلمنيها في الأيام القادمة هتاخد علامتين يضافوا
+
+296
+00:36:35,840 --> 00:36:43,000
+للامتحان علامة امتحان النص فيه الأولOkay تمام فال
+
+297
+00:36:43,000 --> 00:36:43,360
+..
+
+298
+00:36:55,700 --> 00:37:00,340
+إذن باقي إثبات إن عشان البرهان يكون كامل بإنه يثبت
+
+299
+00:37:00,340 --> 00:37:04,460
+إن المتباينة الأخيرة هذه بتأدى إن ال sequence
+
+300
+00:37:04,460 --> 00:37:12,840
+بتاعتنا لا تكون شيء فأنا بقول إذا حابين ممكن إنكم
+
+301
+00:37:12,840 --> 00:37:19,620
+تكتبوا البرهان على ورقة خارجية وتعطوني إذا برهانكم
+
+302
+00:37:19,620 --> 00:37:23,400
+بيكون صح بعطيه لكم علامتين يضافوا إلى امتحان النصف
+
+303
+00:37:23,400 --> 00:37:31,150
+الأولOkay تمام اتفقنا ولا اعطيكم البرهان و بلاش
+
+304
+00:37:31,150 --> 00:37:42,990
+خلاص؟Okay إذا بنوقف هنا نكتفي بالأمثلة هذه وإن شاء
+
+305
+00:37:42,990 --> 00:37:46,870
+الله المرة الجاية هناخد .. ندخل في الموضوع ال
+
+306
+00:37:46,870 --> 00:37:51,570
+contractive sequences و بعدين نبدأ section جديد إن
+
+307
+00:37:51,570 --> 00:37:58,010
+شاء الله فشكرا لكم و ال .. نكمل إن شاء الله في
+
+308
+00:37:58,010 --> 00:37:58,830
+اللقاء القادم
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b6932196cf6b6e070e528084a93d79d9877abcd6
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/_mc9oZHzNxs_raw.srt
@@ -0,0 +1,1232 @@
+1
+00:00:20,670 --> 00:00:27,570
+بسم الله الرحمن الرحيم و السلام عليكم هنكمل
+
+2
+00:00:27,570 --> 00:00:33,570
+ان شاء الله اليوم ال .. المثال رقم اتنين اللي
+
+3
+00:00:33,570 --> 00:00:39,530
+بدناه في المحاضرة السابقة و ماكملناهوش فنرجعوا مع
+
+4
+00:00:39,530 --> 00:00:48,020
+بعض بسرعة و نحاول نكمل البرهانلهذا المثال وهو ان
+
+5
+00:00:48,020 --> 00:00:53,900
+ال sequence المعرفة بطريقة استقرائية هنا بنثبت
+
+6
+00:00:53,900 --> 00:00:58,560
+انها convergence و ال limit تبعتها بساوي العدد
+
+7
+00:00:58,560 --> 00:01:04,700
+خمسة على تلاتة تمام فبدينا البرهان المرة اللي فاتت
+
+8
+00:01:04,700 --> 00:01:11,900
+و أثبتنا claim .. claim واحد و كان في ال claim هذا
+
+9
+00:01:11,900 --> 00:01:13,680
+أثبتنا ان ال
+
+10
+00:01:17,150 --> 00:01:22,690
+إن حدود الـ sequence bounded below by one and
+
+11
+00:01:22,690 --> 00:01:29,450
+bounded above by two هذا صحيح لكل إن وشوفنا هذا
+
+12
+00:01:29,450 --> 00:01:36,570
+البراني هذا ممكن يعني ممكن إعطاءه by induction
+
+13
+00:01:36,570 --> 00:01:41,450
+therefore by claim one
+
+14
+00:01:45,770 --> 00:01:52,510
+السيكوانس xn is bounded واضح من ال claim ان
+
+15
+00:01:52,510 --> 00:01:59,230
+السيكوانس is bounded المرة التي اثبتنا claim رقم 2
+
+16
+00:01:59,230 --> 00:02:03,470
+اثبتنا
+
+17
+00:02:03,470 --> 00:02:07,470
+ان السيكوانس
+
+18
+00:02:07,470 --> 00:02:15,960
+xn بتحققالمعادلة absolute xn minus xn زايد واحد
+
+19
+00:02:15,960 --> 00:02:22,960
+هذا بيساوي واحد على اتنين أس n negative one for
+
+20
+00:02:22,960 --> 00:02:30,360
+every natural number in it وشوفنا برهنة المعادلة
+
+21
+00:02:30,360 --> 00:02:36,020
+هذه او العبارة هذه لكل عدد طبيعي by induction okay
+
+22
+00:02:38,610 --> 00:02:45,630
+اليوم باستخدام ال claim 2 and
+
+23
+00:02:45,630 --> 00:02:51,330
+triangle
+
+24
+00:02:51,330 --> 00:03:01,490
+inequality متباينة مثلث نرى
+
+25
+00:03:01,490 --> 00:03:02,070
+ان
+
+26
+00:03:05,510 --> 00:03:16,950
+وإذا M أكتر من N، M نموات طبيعية، و M أكتر من N،
+
+27
+00:03:16,950 --> 00:03:23,190
+فلدينا أكتر من X M أكتر من X
+
+28
+00:03:29,470 --> 00:03:35,150
+طبعا هذا ممكن نكتبه على صورة هي absolute xn هترح
+
+29
+00:03:35,150 --> 00:03:43,230
+xn زاد واحد و هرجعها و
+
+30
+00:03:43,230 --> 00:03:55,190
+هترح xn زاد اتنين و هرجعها و
+
+31
+00:03:55,190 --> 00:03:58,250
+هكذا الى ان اصل الى
+
+32
+00:04:03,240 --> 00:04:13,220
+x m negative one سالب x m في الآخر خالص هاطرح x m
+
+33
+00:04:13,220 --> 00:04:19,960
+سالب واحد ورجعها فكأني أنا يعني ماعملتش ماغيرتش
+
+34
+00:04:19,960 --> 00:04:24,600
+حاجة فالمخضر اللي على اليمين هو نفسه اللي على
+
+35
+00:04:24,600 --> 00:04:30,360
+الشمال لأن طرحة had وضفته طرحة had وضفتها كذلك
+
+36
+00:04:30,360 --> 00:04:36,550
+فكأني ضفت سفرالان ناخد الحدين هذول مع بعض و هذول
+
+37
+00:04:36,550 --> 00:04:47,810
+مع بعض و هكذا و هذول مع بعض و هذول اخر حدين مع بعض
+
+38
+00:04:47,810 --> 00:04:54,110
+و بنستخدم ال triangle inequality ف by triangle
+
+39
+00:04:54,110 --> 00:05:00,330
+inequality ال absolute value لمجموعةLess than or
+
+40
+00:05:00,330 --> 00:05:06,890
+equal مجموع الـ absolute values فهذا absolute xn
+
+41
+00:05:06,890 --> 00:05:12,850
+minus xn زاد واحد زاد absolute xn زاد واحد minus
+
+42
+00:05:12,850 --> 00:05:21,510
+xn زاد اتو وهكذا
+
+43
+00:05:21,510 --> 00:05:27,170
+إلى absolute xm negative one minus xm
+
+44
+00:05:30,780 --> 00:05:37,120
+الان باستخدام claim اتنين الحد الاول هذا عبارة عن
+
+45
+00:05:37,120 --> 00:05:44,100
+واحد على two أس in negative one الحد اللي بعده one
+
+46
+00:05:44,100 --> 00:05:53,000
+over two أس in و الحد اللي بعده هكذا و
+
+47
+00:05:53,000 --> 00:06:03,820
+الحد الأخير الحد الأخير هذا هيكون واحد علىتو اص ام
+
+48
+00:06:03,820 --> 00:06:07,580
+ماينوس
+
+49
+00:06:07,580 --> 00:06:14,460
+اتنين اذا
+
+50
+00:06:14,460 --> 00:06:21,040
+هذا من المعادلة اللي هنا ناخد
+
+51
+00:06:21,040 --> 00:06:33,760
+عامل مشترك one over two اص n negative oneفبيبقى
+
+52
+00:06:33,760 --> 00:06:38,000
+إذا من الحد الأول دي بقى اللي عندي واحد من الحد
+
+53
+00:06:38,000 --> 00:06:46,740
+التاني بيبقى عندي نص و هكذا إلى الحد الأخير اللي
+
+54
+00:06:46,740 --> 00:06:56,120
+بيبقى عندي two أُس M negative M negative one الآن
+
+55
+00:06:56,120 --> 00:07:01,080
+المجموعة هذا اللي بين جثين هذا المجموعةأصغر من
+
+56
+00:07:01,080 --> 00:07:06,220
+اتنين لأن هذا المجموع لاحظوا
+
+57
+00:07:06,220 --> 00:07:16,080
+انتوا واحد زائد نص زائد إلى one over two to m
+
+58
+00:07:16,080 --> 00:07:23,020
+negative n negative one this is less than one زائد
+
+59
+00:07:23,020 --> 00:07:25,600
+نص زائد
+
+60
+00:07:29,510 --> 00:07:37,230
+زائد واحد على اتنين أس ان زائد إلى مالة نهائية
+
+61
+00:07:37,230 --> 00:07:44,610
+اللي هو مجموعة series sigma from k equals zero to
+
+62
+00:07:44,610 --> 00:07:54,430
+infinity ال one over two to k هذا
+
+63
+00:07:54,430 --> 00:07:58,010
+جزء من ال infinite series
+
+64
+00:08:00,560 --> 00:08:07,300
+هذه الـ inference series هذه أول يعني M سالب N
+
+65
+00:08:07,300 --> 00:08:12,700
+مايرس واحد من حدودها هذه
+
+66
+00:08:12,700 --> 00:08:17,380
+ال series معروفة هي geometric series geometric
+
+67
+00:08:17,380 --> 00:08:25,320
+series with a الحد الأول واحد وال ratio and ال
+
+68
+00:08:25,320 --> 00:08:31,860
+ratio بساوي نصففي تفاضل بقى اتعلمتوا انه اي
+
+69
+00:08:31,860 --> 00:08:35,200
+geometric series اذا ال ratio ال absolute value ل
+
+70
+00:08:35,200 --> 00:08:40,940
+R أصغر من واحد فال series تطلع convergent ومجموعة
+
+71
+00:08:40,940 --> 00:08:48,520
+.. مجموعة بساوي a على one minus الأساس وهذا بطلع a
+
+72
+00:08:48,520 --> 00:08:53,720
+اللي هو واحد على واحد minus الأساس نص بطلع بساوي
+
+73
+00:08:53,720 --> 00:08:57,060
+اتنين okay اذا ال ..
+
+74
+00:09:00,710 --> 00:09:07,850
+إذا المجموع هذا بيطلع أصغر من اتنين إذا هي حاول
+
+75
+00:09:07,850 --> 00:09:14,690
+أصغر من واحد على اتنين أسن negative واحد ضرب اتنين
+
+76
+00:09:14,690 --> 00:09:25,210
+وهذا بيساوي واحد على اتنين أسن سالب اتنين تمام؟
+
+77
+00:09:36,740 --> 00:09:42,940
+الان بنا نثبت احنا ان ال sequence
+
+78
+00:09:42,940 --> 00:09:48,860
+احنا كان بنا نثبت ان ال sequence x in convergent
+
+79
+00:09:48,860 --> 00:09:52,820
+وقلنا في بداية البرهان المرة اللى فات عشان نثبت
+
+80
+00:09:52,820 --> 00:09:57,380
+انها convergent يكفي ان احنا نثبت انها Cauchy صح؟
+
+81
+00:09:57,380 --> 00:10:01,400
+لأن اذا كانت Cauchy بتكون convergent by Cauchy a
+
+82
+00:10:01,400 --> 00:10:03,920
+criterion اذا هنا to show
+
+83
+00:10:08,040 --> 00:10:13,900
+إن X in convergence it
+
+84
+00:10:13,900 --> 00:10:18,100
+suffices يعني
+
+85
+00:10:18,100 --> 00:10:29,100
+يكفي إثبات to show it is Cauchy إذا يكفي إثبات
+
+86
+00:10:29,100 --> 00:10:36,510
+إنها Cauchyطيب هاي عندي .. الآن هستفيد من المتباين
+
+87
+00:10:36,510 --> 00:10:45,890
+هذه الأخيرة لإثبات إنها كوشي طيب
+
+88
+00:10:45,890 --> 00:10:54,790
+أنا عندي .. أنا عندي four .. قولنا M أكبر من N
+
+89
+00:10:57,610 --> 00:11:04,270
+أثبتنا أن أبسليوت xn نيجاتيف xm less than one over
+
+90
+00:11:04,270 --> 00:11:16,970
+two to n minus two نسمي الانيقواليتي هذه star الان
+
+91
+00:11:16,970 --> 00:11:21,010
+let
+
+92
+00:11:21,010 --> 00:11:28,960
+epsilon أكبر من السفر be givenأنا بدأ أثبت إن الـ
+
+93
+00:11:28,960 --> 00:11:32,980
+sequence تبعتي كوشي فعشان أثبت إنها كوشي ببدأ
+
+94
+00:11:32,980 --> 00:11:38,840
+بإمسون أكبر من سفر برد عليها بcapital N بحيث إنه
+
+95
+00:11:38,840 --> 00:11:44,280
+لكل M و N أكبر من أو يساوي capital N لازم المسافة
+
+96
+00:11:44,280 --> 00:11:50,000
+بين XN وXM أصغر من إبسون ف let إبسون أكبر من سفر
+
+97
+00:11:50,000 --> 00:11:59,040
+be given by Archimedean propertyمن خاصية
+
+98
+00:11:59,040 --> 00:12:08,700
+Archimedes choose ممكن نختار capital N عدد طبيعي
+
+99
+00:12:08,700 --> 00:12:19,020
+بحيث أنه واحد على N أصغر من epsilon على أربعةوهذا
+
+100
+00:12:19,020 --> 00:12:23,440
+صحيح by the Archimedean property إبسلون عدد موجب
+
+101
+00:12:23,440 --> 00:12:26,920
+بتعني إبسلون على أربع عدد موجب لهذا العدد الموجب
+
+102
+00:12:26,920 --> 00:12:31,740
+بقدر ألاقي عدد طبيعه عدد طبيعي مقلوب وأصغر من عدد
+
+103
+00:12:31,740 --> 00:12:44,060
+الموجب تمام؟ وبالتالي هذا بيقدر أنه ال ..واحد على
+
+104
+00:12:44,060 --> 00:12:52,020
+two to n أصغر من epsilon على أربعة لأن
+
+105
+00:12:52,020 --> 00:13:02,640
+since لأن two to n أكبر من n صح؟ وبالتالي مقلوب
+
+106
+00:13:02,640 --> 00:13:07,560
+هذا أصغر من مقلوب ال n اللي هو أصغر من epsilon على
+
+107
+00:13:07,560 --> 00:13:10,720
+أربعة okay تمام؟ طيب
+
+108
+00:13:14,350 --> 00:13:25,310
+إذن this .. this and star بيؤدوا إنه لو كان M أكبر
+
+109
+00:13:25,310 --> 00:13:30,450
+من أو يساوي N أكبر من أو يساوي capital N فهذا
+
+110
+00:13:30,450 --> 00:13:40,050
+بيؤدي إنه absolute xn negative xm أصغر
+
+111
+00:13:40,050 --> 00:13:51,740
+من1 على 2 to N minus 2 وهذا أصغر من أو ساوي 1 على
+
+112
+00:13:51,740 --> 00:13:58,200
+2 to capital N minus 2 لأن small n أكبر من أو ساوي
+
+113
+00:13:58,200 --> 00:14:04,700
+capital N ف2
+
+114
+00:14:04,700 --> 00:14:11,200
+N سالب 2 أصغر يعني مقلوب هذهيعني أنا عندي هنا
+
+115
+00:14:11,200 --> 00:14:16,460
+اتنين أس ان سالب اتنين بطلع أكبر من أو ساوي two
+
+116
+00:14:16,460 --> 00:14:21,340
+two capital N سالب اتنين لأن ان أكبر من أو ساوي
+
+117
+00:14:21,340 --> 00:14:25,200
+capital N وبالتالي مقلوب الكبير أصغر من أو ساوي
+
+118
+00:14:25,200 --> 00:14:35,420
+مقلوب الكبير أو مقلوب الصغير فهذا صح ومن هنا هذا
+
+119
+00:14:35,420 --> 00:14:39,280
+أصغر هذا من هنا أصغر من epsilon
+
+120
+00:14:42,800 --> 00:14:48,200
+لأن هذا عبارة عن .. هذا بساوي .. أيوه بساوي أربعة
+
+121
+00:14:48,200 --> 00:14:53,500
+على اتنين أسن و أربعة على اتنين أسن أصغر من
+
+122
+00:14:53,500 --> 00:15:01,740
+إبسلون، صح؟ إذن هذه أثبتت for any given .. for any
+
+123
+00:15:01,740 --> 00:15:06,700
+given إبسلون يوجد capital N يعتمد على إبسلون، هذا
+
+124
+00:15:06,700 --> 00:15:15,590
+هويوجد capital N غير مرتبط بـY بحيث أنه لكل M و N
+
+125
+00:15:15,590 --> 00:15:20,350
+أكبر من أو ساوي capital N فالمسافة بين XN و XM
+
+126
+00:15:20,350 --> 00:15:26,650
+أصغر من Y وبالتالي هذا بثبت أن ال sequence is
+
+127
+00:15:26,650 --> 00:15:36,270
+Cauchy بس ال sequence XN is Cauchy
+
+128
+00:15:39,020 --> 00:15:43,660
+and therefore x
+
+129
+00:15:43,660 --> 00:15:50,900
+in converges say
+
+130
+00:15:50,900 --> 00:16:02,740
+ال limit ل x in بساوي some x ينتمي ال R هنا
+
+131
+00:16:02,740 --> 00:16:08,130
+أثبتنا أن ال sequence x in convergentby Cauchy
+
+132
+00:16:08,130 --> 00:16:13,790
+criterion وفرضنا ان ال limit تبعتها بساوي X عشان
+
+133
+00:16:13,790 --> 00:16:15,870
+كام اذا هين اثبتنا ان ال sequence تبعتنا
+
+134
+00:16:15,870 --> 00:16:20,070
+convergent ال limit تبعتها عدد X بقى هينثبت ان ال
+
+135
+00:16:20,070 --> 00:16:25,790
+X اللي هو limit ل X in بساوي خمسة على تلاتة بساوي
+
+136
+00:16:25,790 --> 00:16:29,350
+خمسة على تلاتة okay اذا هينثبت
+
+137
+00:16:34,190 --> 00:16:42,010
+الجزء الأخير هذا وهو نسميه
+
+138
+00:16:42,010 --> 00:16:50,790
+claim تلاتة claim three ال X بساوي five over three
+
+139
+00:16:50,790 --> 00:17:00,130
+لبرهان ذلك first
+
+140
+00:17:03,450 --> 00:17:08,530
+use induction on
+
+141
+00:17:08,530 --> 00:17:20,170
+n to show الإثبات إن x to n plus one بساوي واحد
+
+142
+00:17:20,170 --> 00:17:28,610
+زائد نص زائد واحد على اتنين تكعيب زائدو هكذا one
+
+143
+00:17:28,610 --> 00:17:35,410
+over two to two n سالب واحد وهذا صحيح لكل natural
+
+144
+00:17:35,410 --> 00:17:42,910
+number n المعادلة هذه ممكن اثباتها by induction on
+
+145
+00:17:42,910 --> 00:17:54,130
+n سهل جدا طبعا ممكن تستخدم .. تحتاج ال inductive
+
+146
+00:17:54,130 --> 00:17:59,300
+definition في البرهنزي ما شوفنا في برهان claim 2
+
+147
+00:17:59,300 --> 00:18:08,440
+طيب الآن افرض ان احنا هذا أثبتناها hence وبالتالي
+
+148
+00:18:08,440 --> 00:18:15,500
+من هنا بطلع عندي x2n plus one بساوي واحد زائد هاخد
+
+149
+00:18:15,500 --> 00:18:21,820
+من المجموع هذا هاخد
+
+150
+00:18:21,820 --> 00:18:28,200
+نص عام المشتركفبيبقى عندي واحد زائد واحد على اتنين
+
+151
+00:18:28,200 --> 00:18:36,880
+تربية زائد واحد على اتنين تربية لكل تربية زائد و
+
+152
+00:18:36,880 --> 00:18:44,260
+هكذا زائد واحد على اتنين تربية to end negative one
+
+153
+00:18:44,260 --> 00:18:52,220
+اذا انا خدت من هاي الواحد نزلته واخدتنص عام
+
+154
+00:18:52,220 --> 00:18:57,320
+المشترك من باقي الحدود هذه فطل عند المجموع هذا هذا
+
+155
+00:18:57,320 --> 00:19:04,120
+مجموع متوالية هندسية geometric progression لأن هذا
+
+156
+00:19:04,120 --> 00:19:08,080
+بشكل geometric
+
+157
+00:19:08,080 --> 00:19:19,740
+.. geometric progression متوالية
+
+158
+00:19:19,740 --> 00:19:20,480
+هندسية
+
+159
+00:19:23,660 --> 00:19:30,740
+with الحد الأول a بساوي واحد وال ratio بساوي واحد
+
+160
+00:19:30,740 --> 00:19:37,080
+على اتنين تربيات اللي هو ربعها ف ال geometric
+
+161
+00:19:37,080 --> 00:19:40,820
+progression فيه قانون لإيجاد مجموعة المتوالية
+
+162
+00:19:40,820 --> 00:19:46,080
+الهندسية فيه قانون لإيجاد مجموعة فالقانون هذا
+
+163
+00:19:46,080 --> 00:19:54,830
+عبارة عن الحد الأول واحد سالبالحد الأخير مضروب في
+
+164
+00:19:54,830 --> 00:20:02,650
+الأساس اللي هو واحد على اتنين تربية الكل قسم على
+
+165
+00:20:02,650 --> 00:20:08,150
+واحد minus الأساس على واحد minus الأساس اللي هو
+
+166
+00:20:08,150 --> 00:20:14,270
+واحد على اتنين تربية وهذا
+
+167
+00:20:14,270 --> 00:20:22,130
+بساوي اي واحد زائد المقام هذا عبارة عن تلت تربعة
+
+168
+00:20:23,720 --> 00:20:32,140
+هذا عبارة عن تلات اربعة فنص على تلات اربعة بطلع
+
+169
+00:20:32,140 --> 00:20:37,180
+اتنين على تلاتة و ال bust هذا هو في ال bust اللي
+
+170
+00:20:37,180 --> 00:20:45,520
+هو واحد سالب واحد على اتنين اص اتنين in او اربعة
+
+171
+00:20:45,520 --> 00:20:47,440
+اص in
+
+172
+00:20:52,700 --> 00:21:08,740
+الان ناخد ال limit للطرفين اذا
+
+173
+00:21:08,740 --> 00:21:14,380
+ناخد .. لو أخدنا ال limit للطرفين فبطلع limit x to
+
+174
+00:21:14,380 --> 00:21:20,800
+n plus one as n tends to infinityبساوي واحد زاد
+
+175
+00:21:20,800 --> 00:21:28,040
+اتنين على التلاتة في ال limit الجوس واحد سالب
+
+176
+00:21:28,040 --> 00:21:34,720
+limit واحد على اربعة أس in as in tends to infinity
+
+177
+00:21:34,720 --> 00:21:41,640
+وهذا بساوي واحد زاد اتنين على تلاتة في واحد سالب
+
+178
+00:21:41,640 --> 00:21:48,440
+limit واحد على اربعة in بساوي سفر فبطلع بساوي واحد
+
+179
+00:21:50,170 --> 00:21:57,150
+زاد اتنين على تلاتة بساوي خمسة على تلاتة طيب
+
+180
+00:21:57,150 --> 00:22:01,070
+هذه عبارة عن sub sequence من ال sequence xn هذه
+
+181
+00:22:01,070 --> 00:22:05,930
+الحدود الفردية ل sequence xn طيب و انا عندي ال
+
+182
+00:22:05,930 --> 00:22:09,750
+sequence تبعتي convergence هاي أثبتنا ان xn
+
+183
+00:22:09,750 --> 00:22:13,930
+convergent و ال limit تبعتها بساوي العدد x
+
+184
+00:22:20,160 --> 00:22:26,000
+إذا أنا في عندي هنا since x2n
+
+185
+00:22:26,000 --> 00:22:38,600
+plus one is a subsequence of a sequence xn and xn
+
+186
+00:22:38,600 --> 00:22:45,540
+converges to xthen by previous theorem حسب نظرية
+
+187
+00:22:45,540 --> 00:22:50,120
+السابقة إذا كانت ال sequence convergent ل x فأي
+
+188
+00:22:50,120 --> 00:22:54,960
+subsequence منها بتكون convergent لنفس ال x إذا
+
+189
+00:22:54,960 --> 00:23:01,460
+limit x اتنين n plus one as n tends to infinity
+
+190
+00:23:01,460 --> 00:23:07,640
+بساوي x وبالتالي إذا x بساوي limit
+
+191
+00:23:12,130 --> 00:23:19,830
+x2n زائد واحد وهذه أثبتنا في السطر الأخير هنا هذه
+
+192
+00:23:19,830 --> 00:23:23,750
+بساوي خمسة على تلاتة وهذا اللي بدنا يعني إذا هيك
+
+193
+00:23:23,750 --> 00:23:30,770
+بتكون أثبتنا ان ال sequence x in converges to x و
+
+194
+00:23:30,770 --> 00:23:36,370
+limit x تبعتها هي طلعت ساوي خمسة على تلاتة كما هو
+
+195
+00:23:36,370 --> 00:23:41,830
+مطلوبOkay إذا هيك بنكون إحنا برهننا إن ال sequence
+
+196
+00:23:41,830 --> 00:23:46,490
+في المثال هذا اللي معرفة بطريقة استقرائية is
+
+197
+00:23:46,490 --> 00:23:52,870
+convergent ونهايتها خمسة تلاتة Okay تمام؟ المفهوم
+
+198
+00:23:52,870 --> 00:23:59,680
+واضح؟ في أي استفسار؟ في أي سؤال؟طيب ناخد كمان مثال
+
+199
+00:23:59,680 --> 00:24:02,520
+يمكن يبقى المثال طويل شوية لكن احنا زى ما شوفته
+
+200
+00:24:02,520 --> 00:24:08,660
+احنا جزقناه الى تلاتة claims او three claims وكل
+
+201
+00:24:08,660 --> 00:24:12,340
+claim كان برهانه by induction مش صعب شفنا برهان
+
+202
+00:24:12,340 --> 00:24:18,020
+واحد منهم المرة اللى فاتت التانين برضه اسأل كمان
+
+203
+00:24:18,020 --> 00:24:23,060
+كل claim بيخطو خطوة الى الامام بيجربنى اكتر من
+
+204
+00:24:23,060 --> 00:24:23,540
+البرهان
+
+205
+00:24:27,870 --> 00:24:45,850
+ناخد مثال آخر، تالت إذا
+
+206
+00:24:45,850 --> 00:24:54,090
+example three consider
+
+207
+00:24:57,250 --> 00:25:02,550
+الحد العام بحيث ال sequence xn
+
+208
+00:25:02,550 --> 00:25:09,510
+where حيث ال term of the sequence الحد العام by
+
+209
+00:25:09,510 --> 00:25:14,070
+definition بساوي one over one plus one over two
+
+210
+00:25:15,270 --> 00:25:22,310
+plus one over three و هكذا and so on until we get
+
+211
+00:25:22,310 --> 00:25:31,730
+one over N حيث N ينتمي إلى N لكل N في N بنعرف XN
+
+212
+00:25:31,730 --> 00:25:36,530
+على أنه المجموع أو مجموعة أول N
+
+213
+00:25:49,690 --> 00:25:57,950
+مجموع أول n من حدود ال harmonic series فهذا بنسميه
+
+214
+00:25:57,950 --> 00:26:02,210
+ال nth partial sum هذا عبارة في تفاضل باسم منها ال
+
+215
+00:26:02,210 --> 00:26:14,410
+nth partial ال nth partial sum of ال harmonic ال
+
+216
+00:26:14,410 --> 00:26:16,110
+harmonic series
+
+217
+00:26:20,920 --> 00:26:26,800
+اللي هي summation from k equals one to infinity ل
+
+218
+00:26:26,800 --> 00:26:27,880
+one over k
+
+219
+00:26:31,950 --> 00:26:39,910
+المطلوب show أن سيكوينس XN divergence ليست
+
+220
+00:26:39,910 --> 00:26:46,070
+convergent، is not convergent بنثبت أن سيكوينس of
+
+221
+00:26:46,070 --> 00:26:51,090
+partial sums متتالية المجاميع الجزئية لل harmonic
+
+222
+00:26:51,090 --> 00:26:54,070
+series بتشكل divergence sequence
+
+223
+00:27:00,340 --> 00:27:19,020
+by cushy criterion اذا حسب cushy criterion by
+
+224
+00:27:19,020 --> 00:27:25,740
+cushy criterion it suffices to
+
+225
+00:27:25,740 --> 00:27:32,760
+show يكفي اثباتيكفي اثبات ان ال sequence عشان نثبت
+
+226
+00:27:32,760 --> 00:27:36,440
+ان ال sequence is divergent it suffices to show ان
+
+227
+00:27:36,440 --> 00:27:47,920
+ال sequence xn is not كوشي لأن
+
+228
+00:27:47,920 --> 00:27:52,980
+كوشي criterion بتقول ان ال sequence is convergent
+
+229
+00:27:52,980 --> 00:27:58,700
+if and only if it is كوشيوبالتالي هذا بكافي ان
+
+230
+00:27:58,700 --> 00:28:02,400
+احنا نقول ان ال sequence is not convergent if and
+
+231
+00:28:02,400 --> 00:28:06,440
+only if it is not Cauchy عشان نثبت ان ال sequence
+
+232
+00:28:06,440 --> 00:28:12,780
+is divergent ممكن نثبت انها is not Cauchy طيب ال
+
+233
+00:28:12,780 --> 00:28:22,900
+.. الاثبات انها not Cauchy هنستخدم indeed
+
+234
+00:28:29,180 --> 00:28:41,120
+في حقيقة الأمر لو أخدنا لو كان M أكبر من N فهذا
+
+235
+00:28:41,120 --> 00:28:48,320
+بيقدي ان XM minus
+
+236
+00:28:48,320 --> 00:28:53,660
+XN ايش بيساوي؟
+
+237
+00:28:53,660 --> 00:28:55,500
+أنا هي عندي ال ..
+
+238
+00:29:06,360 --> 00:29:18,240
+هي عندي xn و لو بدك تكتب xm ف xm هيكون بساوي واحد
+
+239
+00:29:18,240 --> 00:29:25,640
+اول حد زائد نص زائد
+
+240
+00:29:25,640 --> 00:29:33,140
+تلت و هكذا زائد
+
+241
+00:29:33,140 --> 00:29:40,460
+واحد على nو لسه كمان هكمل .. هنكمل لإن ال M أكبر
+
+242
+00:29:40,460 --> 00:29:49,380
+من N هنا ال M .. ال M أكبر من N لما تكون M أكبر من
+
+243
+00:29:49,380 --> 00:29:55,640
+N فهيكون الحد اللي بعدها ده واحد على M زايد واحد
+
+244
+00:30:04,780 --> 00:30:10,120
+واحد عال ان انا
+
+245
+00:30:10,120 --> 00:30:19,580
+مش نافع one
+
+246
+00:30:19,580 --> 00:30:29,190
+over n plus one و هكذا إلى one over nالآن لما أطرح
+
+247
+00:30:29,190 --> 00:30:37,890
+xn من xm فالحدود المتشابهة هتروح مع بعضها لحد واحد
+
+248
+00:30:37,890 --> 00:30:44,690
+على n بروح مع واحد على n بيبقى الفرق بين الإتنين
+
+249
+00:30:44,690 --> 00:30:54,930
+الفرق بين الإتنين هيكون عبارة عن واحد
+
+250
+00:30:56,990 --> 00:31:04,630
+على n زائد واحد زائد واحد على n plus two and so on
+
+251
+00:31:04,630 --> 00:31:12,230
+until we get one over m تمام الحدود هدول عددهم كم
+
+252
+00:31:12,230 --> 00:31:23,270
+حد m negative n terms عدد الحدود في المجموع هذا m
+
+253
+00:31:23,270 --> 00:31:37,060
+negative nهذول حدود عددهم M و هذول عددهم N فالفرق
+
+254
+00:31:37,060 --> 00:31:43,460
+بينهم هيطلع M negative N الآن
+
+255
+00:31:43,460 --> 00:31:46,700
+هذا المجموع أول حد
+
+256
+00:31:49,570 --> 00:31:57,510
+لاحظوا ان ان ال M أكبر من N هذا بيقدي ان M أكبر من
+
+257
+00:31:57,510 --> 00:32:04,230
+أو ساوي N زي واحد وهذا بيقدي ان مخلوق واحد على N
+
+258
+00:32:04,230 --> 00:32:10,210
+زي واحد بطلع أكبر من أو ساوي واحد على N
+
+259
+00:32:13,150 --> 00:32:17,850
+وبالتالي إذا واحد على N زاد واحد أكبر من أو ساوي
+
+260
+00:32:17,850 --> 00:32:22,850
+واحد على M بالمثل واحد على N زاد اتنين لحظة هيكون
+
+261
+00:32:22,850 --> 00:32:29,570
+ال M ال M أكبر من أو ساوي N زاد اتنين فمقلوب N زاد
+
+262
+00:32:29,570 --> 00:32:37,230
+اتنين هيطلع أكبر من أو ساوي واحد على Mو هكذا إذا
+
+263
+00:32:37,230 --> 00:32:41,670
+كل الحدود هذه كل واحد فيهم أكبر من أو ساوي واحد
+
+264
+00:32:41,670 --> 00:32:46,070
+على M إلى أن نصل لآخر حد واحد على M طبعا أكبر من
+
+265
+00:32:46,070 --> 00:32:51,070
+أو ساوي نفسه عدد الحدود هذه لازال M negative in
+
+266
+00:32:51,070 --> 00:32:51,810
+terms
+
+267
+00:32:55,640 --> 00:33:00,980
+طيب أنا لما بجمع عدد على نفسه M minus N من المرات،
+
+268
+00:33:00,980 --> 00:33:06,200
+إيش بيعطيني المجموعة؟ بيطلع بساوي M سالب N في
+
+269
+00:33:06,200 --> 00:33:08,540
+العدد الثابت، صح؟
+
+270
+00:33:14,860 --> 00:33:21,620
+إذن المجموعة الأخيرة هدا هيطلع بساوي M negative N
+
+271
+00:33:21,620 --> 00:33:30,660
+في واحد على M وهذا بساوي على
+
+272
+00:33:30,660 --> 00:33:40,520
+M هذي اه M وهذا بساوي واحد negative N على M طيب
+
+273
+00:33:40,520 --> 00:33:48,050
+انا في التحليل هذا ماخدال M أكبر من N يعني هذا
+
+274
+00:33:48,050 --> 00:33:54,670
+الكلام صحيح إذا كان M أكبر من N طيب الآن take M
+
+275
+00:33:54,670 --> 00:34:00,230
+بساوة 2N بالتأكيد 2N أي عدد طبيعي N لأي عدد طبيعي
+
+276
+00:34:00,230 --> 00:34:08,890
+N 2N أكبر من Nإذا لو عوضت عن M بتنين N في المتباين
+
+277
+00:34:08,890 --> 00:34:18,610
+الأخير هذه هيطلع عندي XM أو X to N negative XN
+
+278
+00:34:18,610 --> 00:34:28,670
+بتطلع أكبر من أو يساوي واحد negative two N على
+
+279
+00:34:28,670 --> 00:34:35,450
+اتنين N صح؟اللي هو واحد negative one-half بطلع one
+
+280
+00:34:35,450 --> 00:34:45,930
+-half نص الكلام هذا صحيح لكل n ينتمي
+
+281
+00:34:45,930 --> 00:34:50,470
+إلى n تمام؟
+
+282
+00:34:52,310 --> 00:34:58,910
+إذا أنا أصبح في عندي المتباينة x to n negative xn
+
+283
+00:34:58,910 --> 00:35:06,750
+أكبر من أو ساوي one half for all n belong to a now
+
+284
+00:35:06,750 --> 00:35:14,710
+you can easily show
+
+285
+00:35:19,500 --> 00:35:28,860
+ممكن بسهولة اثبات انه this implies المتباينة هذه
+
+286
+00:35:28,860 --> 00:35:37,540
+الأخيرة بتقدي this implies that
+
+287
+00:35:37,540 --> 00:35:43,620
+هذا بيقدي ان ال sequence xn is not Cauchy
+
+288
+00:35:46,720 --> 00:36:01,500
+is not Cauchy as desired كما هو مطلوب تمام؟
+
+289
+00:36:01,500 --> 00:36:07,220
+من المتباين هذا ممكن نثبت أن ال sequence تبعتنا
+
+290
+00:36:07,220 --> 00:36:12,820
+ليست Cauchyيمكن هذا مش واضح كيف ان هذا بيعدى ان ال
+
+291
+00:36:12,820 --> 00:36:17,520
+sequence not Cauchy لكن ممكن نعمل برهان بالتناقض
+
+292
+00:36:17,520 --> 00:36:22,680
+افرض ان ال sequence Cauchy واستخدم الشرط هذا او
+
+293
+00:36:22,680 --> 00:36:26,700
+المتبين هذى فيه الوصول الى تناقض هسيكم تكتبوا
+
+294
+00:36:26,700 --> 00:36:30,580
+البرهان هذا وكل واحدة بتكتب البرهان في ورقة
+
+295
+00:36:30,580 --> 00:36:35,840
+وبيسلمنيها في الأيام القادمة هتاخد علامتين يضافوا
+
+296
+00:36:35,840 --> 00:36:43,000
+للامتحان علامة امتحان النص فيه الأولOkay تمام فال
+
+297
+00:36:43,000 --> 00:36:43,360
+..
+
+298
+00:36:55,700 --> 00:37:00,340
+إذن باقي إثبات إن عشان البرهان يكون كامل بإنه يثبت
+
+299
+00:37:00,340 --> 00:37:04,460
+إن المتباينة الأخيرة هذه بتأدى إن ال sequence
+
+300
+00:37:04,460 --> 00:37:12,840
+بتاعتنا لا تكون شيء فأنا بقول إذا حابين ممكن إنكم
+
+301
+00:37:12,840 --> 00:37:19,620
+تكتبوا البرهان على ورقة خارجية وتعطوني إذا برهانكم
+
+302
+00:37:19,620 --> 00:37:23,400
+بيكون صح بعطيه لكم علامتين يضافوا إلى امتحان النصف
+
+303
+00:37:23,400 --> 00:37:31,150
+الأولOkay تمام اتفقنا ولا اعطيكم البرهان و بلاش
+
+304
+00:37:31,150 --> 00:37:42,990
+خلاص؟Okay إذا بنوقف هنا نكتفي بالأمثلة هذه وإن شاء
+
+305
+00:37:42,990 --> 00:37:46,870
+الله المرة الجاية هناخد .. ندخل في الموضوع ال
+
+306
+00:37:46,870 --> 00:37:51,570
+contractive sequences و بعدين نبدأ section جديد إن
+
+307
+00:37:51,570 --> 00:37:58,010
+شاء الله فشكرا لكم و ال .. نكمل إن شاء الله في
+
+308
+00:37:58,010 --> 00:37:58,830
+اللقاء القادم
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/a-utq7LmSIM_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/a-utq7LmSIM_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..69dd4e727acaa6777c22edb3a592a169c6cb8cdd
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/a-utq7LmSIM_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 5042, "start": 21.06, "end": 50.42, "text": "اليوم ان شاء الله هنحاول نحل امتحان نصفي سابق كمراجعة للامتحان النصفي الأول اللي هناخده ان شاء الله غدا اول سؤال في الامتحان هذا عبارة عن سؤال true or false اذا العبارة صح فبنعلم عليها صح او true واذا خطأ بنعلم عليها خطأ فأول عبارة", "tokens": [6027, 1829, 20498, 16472, 13412, 16606, 21984, 8032, 1863, 5016, 995, 12610, 8717, 5016, 1211, 1975, 2304, 2655, 5016, 7649, 8717, 9381, 41185, 8608, 16758, 4587, 9122, 2304, 2288, 26108, 27884, 24976, 10943, 2655, 5016, 7649, 28239, 9381, 41185, 16247, 12610, 13672, 1829, 8032, 1863, 47283, 3215, 3224, 16472, 13412, 16606, 21984, 32771, 28259, 1975, 12610, 8608, 33604, 6027, 8978, 2423, 10943, 2655, 5016, 7649, 23758, 6225, 3555, 9640, 3660, 18871, 8608, 33604, 6027, 2074, 420, 7908, 1975, 15730, 18863, 3555, 9640, 3660, 20328, 5016, 6156, 3555, 1863, 3615, 19528, 25894, 11296, 20328, 5016, 1975, 2407, 2074, 4032, 45636, 995, 16490, 9566, 10721, 44945, 3615, 19528, 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واحد اثنان وكمان ال sequence التانية تكون سالب واحد اثنان او ممكن تكون سالب واحد اثنان زائد واحد نفس الشغل بتطلع convergent ال product", "tokens": [6027, 3615, 3555, 9640, 3660, 8032, 18513, 5172, 7908, 23032, 3555, 5296, 15042, 12174, 1211, 8315, 5296, 4117, 14407, 10874, 1829, 3555, 14407, 5682, 1365, 3714, 43020, 6055, 7435, 1829, 3555, 8608, 6027, 3555, 36764, 24401, 1975, 12984, 1863, 7649, 4032, 24793, 7649, 2423, 8310, 16712, 7649, 10632, 6055, 30544, 8608, 6027, 3555, 36764, 24401, 1975, 12984, 1863, 7649, 1975, 2407, 3714, 43020, 6055, 30544, 8608, 6027, 3555, 36764, 24401, 1975, 12984, 1863, 7649, 30767, 16373, 3215, 36764, 24401, 8717, 36178, 25124, 17082, 1211, 39894, 9566, 1211, 3615, 9652, 6930, 2423, 1674], "avg_logprob": -0.27116935099324874, "compression_ratio": 1.7486910994764397, "no_speech_prob": 0.0, "words": [{"start": 428.09000000000003, "end": 428.81, "word": "العبارة", "probability": 0.88515625}, {"start": 428.81, "end": 429.17, "word": " هادف", "probability": 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0.979736328125}, {"start": 434.39, "end": 435.07, "word": " اثنان", "probability": 0.50323486328125}, {"start": 435.07, "end": 437.19, "word": " وكمان", "probability": 0.8601888020833334}, {"start": 437.19, "end": 437.31, "word": " ال", "probability": 0.8857421875}, {"start": 437.31, "end": 437.67, "word": " sequence", "probability": 0.890625}, {"start": 437.67, "end": 438.01, "word": " التانية", "probability": 0.86865234375}, {"start": 438.01, "end": 438.25, "word": " تكون", "probability": 0.957763671875}, {"start": 438.25, "end": 438.55, "word": " سالب", "probability": 0.91650390625}, {"start": 438.55, "end": 438.85, "word": " واحد", "probability": 0.9921875}, {"start": 438.85, "end": 439.45, "word": " اثنان", "probability": 0.909912109375}, {"start": 439.45, "end": 443.29, "word": " او", "probability": 0.917236328125}, {"start": 443.29, "end": 443.51, "word": " ممكن", "probability": 0.954833984375}, {"start": 443.51, "end": 443.75, "word": " تكون", "probability": 0.9931640625}, {"start": 443.75, "end": 444.05, "word": " سالب", "probability": 0.89208984375}, {"start": 444.05, "end": 444.31, "word": " واحد", "probability": 0.994873046875}, {"start": 444.31, "end": 444.71, "word": " اثنان", "probability": 0.9189453125}, {"start": 444.71, "end": 444.95, "word": " زائد", "probability": 0.745361328125}, {"start": 444.95, "end": 445.77, "word": " واحد", "probability": 0.86767578125}, {"start": 445.77, "end": 446.47, "word": " نفس", "probability": 0.939697265625}, {"start": 446.47, "end": 446.83, "word": " الشغل", "probability": 0.8821614583333334}, {"start": 446.83, "end": 447.21, "word": " بتطلع", "probability": 0.72735595703125}, {"start": 447.21, "end": 447.85, "word": " convergent", "probability": 0.533935546875}, {"start": 447.85, "end": 448.19, "word": " ال", "probability": 0.3564453125}, {"start": 448.19, "end": 448.53, "word": " product", "probability": 0.95849609375}], "temperature": 1.0}, {"id": 16, "seek": 47945, "start": 450.65, "end": 479.45, "text": "هذه الـ divergent وهذه الـ divergent لكن X in في Y in هساوي ال sequence سلب واحد واست اتنين in لما نضربهم في بعض فهذا بيعطيني ال sequence ثابت واحد وهذه converge لواحد اذا هذه في عندي example of two divergent sequences لكن حصل ضربهم بيطلع", "tokens": [3224, 24192, 2423, 39184, 18558, 6930, 37037, 24192, 2423, 39184, 18558, 6930, 44381, 1783, 294, 8978, 398, 294, 8032, 3794, 995, 45865, 2423, 8310, 8608, 46152, 36764, 24401, 4032, 32277, 2655, 1975, 2655, 1863, 9957, 294, 5296, 15042, 8717, 11242, 25513, 16095, 8978, 45030, 11242, 6156, 3224, 15730, 4724, 1829, 3615, 9566, 1829, 22653, 2423, 8310, 38637, 16758, 2655, 36764, 24401, 37037, 24192, 41881, 5296, 14407, 24401, 1975, 15730, 29538, 8978, 18871, 16254, 1365, 295, 732, 18558, 6930, 22978, 44381, 11331, 36520, 48812, 25513, 16095, 4724, 1829, 9566, 1211, 3615], "avg_logprob": -0.39217031919039214, "compression_ratio": 1.729064039408867, "no_speech_prob": 0.0, "words": [{"start": 450.65, "end": 451.19, "word": "هذه", 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0.625732421875}, {"start": 457.39, "end": 457.61, "word": " ال", "probability": 0.599609375}, {"start": 457.61, "end": 458.21, "word": " sequence", "probability": 0.76953125}, {"start": 458.21, "end": 459.99, "word": " سلب", "probability": 0.460693359375}, {"start": 459.99, "end": 460.35, "word": " واحد", "probability": 0.950439453125}, {"start": 460.35, "end": 460.63, "word": " واست", "probability": 0.3516438802083333}, {"start": 460.63, "end": 461.03, "word": " اتنين", "probability": 0.9036865234375}, {"start": 461.03, "end": 461.39, "word": " in", "probability": 0.81103515625}, {"start": 461.39, "end": 464.65, "word": " لما", "probability": 0.865478515625}, {"start": 464.65, "end": 465.13, "word": " نضربهم", "probability": 0.923828125}, {"start": 465.13, "end": 465.31, "word": " في", "probability": 0.8896484375}, {"start": 465.31, "end": 465.67, "word": " بعض", "probability": 0.992431640625}, {"start": 465.67, "end": 467.13, "word": " فهذا", "probability": 0.791015625}, {"start": 467.13, "end": 467.71, "word": " بيعطيني", "probability": 0.767333984375}, {"start": 467.71, "end": 467.91, "word": " ال", "probability": 0.9013671875}, {"start": 467.91, "end": 468.35, "word": " sequence", "probability": 0.96435546875}, {"start": 468.35, "end": 468.83, "word": " ثابت", "probability": 0.9873046875}, {"start": 468.83, "end": 469.37, "word": " واحد", "probability": 0.96337890625}, {"start": 469.37, "end": 470.29, "word": " وهذه", "probability": 0.80224609375}, {"start": 470.29, "end": 470.91, "word": " converge", "probability": 0.07000732421875}, {"start": 470.91, "end": 471.49, "word": " لواحد", "probability": 0.81591796875}, {"start": 471.49, "end": 473.53, "word": " اذا", "probability": 0.6136474609375}, {"start": 473.53, "end": 473.83, "word": " هذه", "probability": 0.361083984375}, {"start": 473.83, "end": 473.99, "word": " في", "probability": 0.5361328125}, {"start": 473.99, "end": 474.39, "word": " عندي", "probability": 0.8232421875}, {"start": 474.39, "end": 475.23, "word": " example", "probability": 0.73974609375}, {"start": 475.23, "end": 475.51, "word": " of", "probability": 0.9169921875}, {"start": 475.51, "end": 475.73, "word": " two", "probability": 0.931640625}, {"start": 475.73, "end": 476.35, "word": " divergent", "probability": 0.7841796875}, {"start": 476.35, "end": 477.11, "word": " sequences", "probability": 0.951171875}, {"start": 477.11, "end": 477.99, "word": " لكن", "probability": 0.90869140625}, {"start": 477.99, "end": 478.41, "word": " حصل", "probability": 0.818115234375}, {"start": 478.41, "end": 478.93, "word": " ضربهم", "probability": 0.97998046875}, {"start": 478.93, "end": 479.45, "word": " بيطلع", "probability": 0.786083984375}], "temperature": 1.0}, {"id": 17, "seek": 50851, "start": 480.21, "end": 508.51, "text": "convergent وليس divergent طيب لو كانت S bounded S bounded subset of R وS0 subset من S هل هذا بيقدّي ان انفمم", "tokens": [1671, 331, 6930, 4032, 20292, 3794, 18558, 6930, 23032, 1829, 3555, 45164, 25961, 2655, 318, 37498, 318, 37498, 25993, 295, 497, 4032, 50, 15, 25993, 9154, 318, 8032, 1211, 23758, 4724, 1829, 28543, 11703, 1829, 16472, 16472, 5172, 2304, 2304], "avg_logprob": -0.28315549943505264, "compression_ratio": 1.1932773109243697, "no_speech_prob": 0.0, "words": [{"start": 480.21, "end": 481.09, "word": "convergent", "probability": 0.8377278645833334}, {"start": 481.09, "end": 481.69, "word": " وليس", "probability": 0.83447265625}, {"start": 481.69, "end": 482.69, "word": " divergent", "probability": 0.6734619140625}, {"start": 482.69, "end": 486.23, "word": " طيب", "probability": 0.8849283854166666}, {"start": 486.23, "end": 486.97, "word": " لو", "probability": 0.8583984375}, {"start": 486.97, "end": 488.89, "word": " كانت", "probability": 0.9833984375}, {"start": 488.89, "end": 489.73, "word": " S", "probability": 0.70947265625}, {"start": 489.73, "end": 490.47, "word": " bounded", "probability": 0.9404296875}, {"start": 490.47, "end": 492.69, "word": " S", "probability": 0.436767578125}, {"start": 492.69, "end": 493.59, "word": " bounded", "probability": 0.9619140625}, {"start": 493.59, "end": 496.15, "word": " subset", "probability": 0.87646484375}, {"start": 496.15, "end": 496.97, "word": " of", "probability": 0.974609375}, {"start": 496.97, "end": 497.69, "word": " R", "probability": 0.9560546875}, {"start": 497.69, "end": 501.47, "word": " وS0", "probability": 0.531494140625}, {"start": 501.47, "end": 502.03, "word": " subset", "probability": 0.9072265625}, {"start": 502.03, "end": 502.45, "word": " من", "probability": 0.9921875}, {"start": 502.45, "end": 502.91, "word": " S", "probability": 0.9853515625}, {"start": 502.91, "end": 504.87, "word": " هل", "probability": 0.864501953125}, {"start": 504.87, "end": 505.17, "word": " هذا", "probability": 0.91845703125}, {"start": 505.17, "end": 505.79, "word": " بيقدّي", "probability": 0.6695068359375}, {"start": 505.79, "end": 506.07, "word": " ان", "probability": 0.73876953125}, {"start": 506.07, "end": 508.51, "word": " انفمم", "probability": 0.686279296875}], "temperature": 1.0}, {"id": 18, "seek": 53482, "start": 509.76, "end": 534.82, "text": "S0 subset أصغر من أو يساوي انفمه من ال S هل هذه العبارة صحيحة؟ أيه رأيكم؟ العبارة هذه false", "tokens": [50, 15, 25993, 5551, 9381, 17082, 2288, 9154, 34051, 7251, 3794, 995, 45865, 16472, 5172, 2304, 3224, 9154, 2423, 318, 8032, 1211, 29538, 18863, 3555, 9640, 3660, 20328, 5016, 1829, 5016, 3660, 22807, 36632, 3224, 12602, 10721, 1829, 24793, 22807, 18863, 3555, 9640, 3660, 29538, 7908], "avg_logprob": -0.20728058764275084, "compression_ratio": 1.25, "no_speech_prob": 0.0, "words": [{"start": 509.76, "end": 510.88, "word": "S0", "probability": 0.681884765625}, {"start": 510.88, "end": 511.76, "word": " subset", "probability": 0.7216796875}, {"start": 511.76, "end": 513.64, "word": " أصغر", "probability": 0.9100341796875}, {"start": 513.64, "end": 513.94, "word": " من", "probability": 0.9736328125}, {"start": 513.94, "end": 514.22, "word": " أو", "probability": 0.91162109375}, {"start": 514.22, "end": 514.9, "word": " يساوي", "probability": 0.83941650390625}, {"start": 514.9, "end": 517.34, "word": " انفمه", "probability": 0.56390380859375}, {"start": 517.34, "end": 517.48, "word": " من", "probability": 0.91259765625}, {"start": 517.48, "end": 517.66, "word": " ال", "probability": 0.791015625}, {"start": 517.66, "end": 518.08, "word": " S", "probability": 0.62548828125}, {"start": 518.08, "end": 522.48, "word": " هل", "probability": 0.919921875}, {"start": 522.48, "end": 522.7, "word": " هذه", "probability": 0.87451171875}, {"start": 522.7, "end": 523.2, "word": " العبارة", "probability": 0.9442138671875}, {"start": 523.2, "end": 526.1, "word": " صحيحة؟", "probability": 0.8980305989583334}, {"start": 526.1, "end": 531.02, "word": " أيه", "probability": 0.6510009765625}, {"start": 531.02, "end": 531.58, "word": " رأيكم؟", "probability": 0.91943359375}, {"start": 531.58, "end": 533.98, "word": " العبارة", "probability": 0.927734375}, {"start": 533.98, "end": 534.34, "word": " هذه", "probability": 0.830078125}, {"start": 534.34, "end": 534.82, "word": " false", "probability": 0.7197265625}], "temperature": 1.0}, {"id": 19, "seek": 55732, "start": 536.54, "end": 557.32, "text": "لما مجموعة تصغر ال inform تباعها بيكبر، لكن اللي بيكون صح أنه ال supremum لو أخدت ال supremum للمجموعة الجزئية S0 فهذا بيطلع أصغر من أو ساوي ال supremum للمجموع S، هذه العبارة true", "tokens": [1211, 15042, 3714, 7435, 2304, 2407, 27884, 6055, 9381, 17082, 2288, 2423, 1356, 6055, 3555, 45761, 11296, 4724, 1829, 4117, 26890, 12399, 44381, 13672, 1829, 4724, 1829, 30544, 20328, 5016, 14739, 3224, 2423, 23710, 449, 45164, 5551, 9778, 3215, 2655, 2423, 23710, 449, 5296, 19528, 7435, 2304, 45367, 3660, 25724, 11622, 19986, 10632, 318, 15, 6156, 3224, 15730, 4724, 1829, 9566, 1211, 3615, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 23710, 449, 5296, 19528, 7435, 2304, 45367, 318, 12399, 29538, 18863, 3555, 9640, 3660, 2074], "avg_logprob": -0.24859551901227972, "compression_ratio": 1.6571428571428573, "no_speech_prob": 0.0, "words": [{"start": 536.54, "end": 537.08, "word": "لما", "probability": 0.7100830078125}, {"start": 537.08, "end": 538.1, "word": " مجموعة", "probability": 0.82216796875}, {"start": 538.1, "end": 538.74, "word": " تصغر", "probability": 0.9644775390625}, {"start": 538.74, "end": 538.92, "word": " ال", "probability": 0.912109375}, {"start": 538.92, "end": 539.32, "word": " inform", "probability": 0.11175537109375}, {"start": 539.32, "end": 540.02, "word": " تباعها", "probability": 0.876220703125}, {"start": 540.02, "end": 541.4, "word": " بيكبر،", "probability": 0.7787109375}, {"start": 541.4, "end": 542.92, "word": " لكن", "probability": 0.9736328125}, {"start": 542.92, "end": 544.54, "word": " اللي", "probability": 0.841552734375}, {"start": 544.54, "end": 544.98, "word": " بيكون", "probability": 0.9650065104166666}, {"start": 544.98, "end": 545.44, "word": " صح", "probability": 0.993896484375}, {"start": 545.44, "end": 546.08, "word": " أنه", "probability": 0.35626220703125}, {"start": 546.08, "end": 546.2, "word": " ال", "probability": 0.5810546875}, {"start": 546.2, "end": 546.94, "word": " supremum", "probability": 0.32171630859375}, {"start": 546.94, "end": 547.94, "word": " لو", "probability": 0.8017578125}, {"start": 547.94, "end": 548.38, "word": " أخدت", "probability": 0.7921142578125}, {"start": 548.38, "end": 548.54, "word": " ال", "probability": 0.87646484375}, {"start": 548.54, "end": 549.0, "word": " supremum", "probability": 0.871826171875}, {"start": 549.0, "end": 549.7, "word": " للمجموعة", "probability": 0.863525390625}, {"start": 549.7, "end": 550.16, "word": " الجزئية", "probability": 0.8603515625}, {"start": 550.16, "end": 550.82, "word": " S0", "probability": 0.797119140625}, {"start": 550.82, "end": 551.84, "word": " فهذا", "probability": 0.9054361979166666}, {"start": 551.84, "end": 552.6, "word": " بيطلع", "probability": 0.87900390625}, {"start": 552.6, "end": 553.08, "word": " أصغر", "probability": 0.9700927734375}, {"start": 553.08, "end": 553.22, "word": " من", "probability": 0.9345703125}, {"start": 553.22, "end": 553.4, "word": " أو", "probability": 0.81884765625}, {"start": 553.4, "end": 553.9, "word": " ساوي", "probability": 0.9163411458333334}, {"start": 553.9, "end": 554.08, "word": " ال", "probability": 0.84375}, {"start": 554.08, "end": 555.22, "word": " supremum", "probability": 0.894775390625}, {"start": 555.22, "end": 555.94, "word": " للمجموع", "probability": 0.89501953125}, {"start": 555.94, "end": 556.26, "word": " S،", "probability": 0.7314453125}, {"start": 556.26, "end": 556.48, "word": " هذه", "probability": 0.626953125}, {"start": 556.48, "end": 557.04, "word": " العبارة", "probability": 0.781890869140625}, {"start": 557.04, "end": 557.32, "word": " true", "probability": 0.8291015625}], "temperature": 1.0}, {"id": 20, "seek": 57811, "start": 560.87, "end": 578.11, "text": "ال supremum لما المجموعه تكبر بيكبر لكن ال infimum لما المجموعه تكبر بيصغر عكس بعض طيب", "tokens": [6027, 23710, 449, 5296, 15042, 9673, 7435, 2304, 45367, 3224, 6055, 4117, 26890, 4724, 1829, 4117, 26890, 44381, 2423, 1536, 332, 449, 5296, 15042, 9673, 7435, 2304, 45367, 3224, 6055, 4117, 26890, 4724, 1829, 9381, 17082, 2288, 6225, 4117, 3794, 45030, 11242, 23032, 1829, 3555], "avg_logprob": -0.16635529696941376, "compression_ratio": 1.5106382978723405, "no_speech_prob": 0.0, "words": [{"start": 560.87, "end": 561.19, "word": "ال", "probability": 0.89697265625}, {"start": 561.19, "end": 561.93, "word": " supremum", "probability": 0.5238037109375}, {"start": 561.93, "end": 562.79, "word": " لما", "probability": 0.939453125}, {"start": 562.79, "end": 563.31, "word": " المجموعه", "probability": 0.831884765625}, {"start": 563.31, "end": 563.73, "word": " تكبر", "probability": 0.9637044270833334}, {"start": 563.73, "end": 564.21, "word": " بيكبر", "probability": 0.892333984375}, {"start": 564.21, "end": 564.57, "word": " لكن", "probability": 0.72412109375}, {"start": 564.57, "end": 565.17, "word": " ال", "probability": 0.90673828125}, {"start": 565.17, "end": 565.71, "word": " infimum", "probability": 0.5775553385416666}, {"start": 565.71, "end": 565.93, "word": " لما", "probability": 0.983642578125}, {"start": 565.93, "end": 566.35, "word": " المجموعه", "probability": 0.9859375}, {"start": 566.35, "end": 566.79, "word": " تكبر", "probability": 0.9754231770833334}, {"start": 566.79, "end": 567.51, "word": " بيصغر", "probability": 0.9685546875}, {"start": 567.51, "end": 568.81, "word": " عكس", "probability": 0.92138671875}, {"start": 568.81, "end": 569.21, "word": " بعض", "probability": 0.9951171875}, {"start": 569.21, "end": 578.11, "word": " طيب", "probability": 0.9544270833333334}], "temperature": 1.0}, {"id": 21, "seek": 61174, "start": 583.2, "end": 611.74, "text": "لو كانت x in sequence of positive real numbers و limit x in plus one over x in بساوي واحد فهذا بيقدي ان ال sequence x in diverges هل هذه العبارة صح ولا خطأ؟", "tokens": [1211, 2407, 25961, 2655, 2031, 294, 8310, 295, 3353, 957, 3547, 4032, 4948, 2031, 294, 1804, 472, 670, 2031, 294, 4724, 3794, 995, 45865, 36764, 24401, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 16472, 2423, 8310, 2031, 294, 18558, 2880, 8032, 1211, 29538, 18863, 3555, 9640, 3660, 20328, 5016, 49429, 16490, 9566, 10721, 22807], "avg_logprob": -0.17571022998202931, "compression_ratio": 1.2875, "no_speech_prob": 0.0, "words": [{"start": 583.2, "end": 583.58, "word": "لو", "probability": 0.855712890625}, {"start": 583.58, "end": 584.16, "word": " كانت", "probability": 0.974853515625}, {"start": 584.16, "end": 584.52, "word": " x", "probability": 0.5390625}, {"start": 584.52, "end": 585.0, "word": " in", "probability": 0.353515625}, {"start": 585.0, "end": 586.68, "word": " sequence", "probability": 0.76220703125}, {"start": 586.68, "end": 587.38, "word": " of", "probability": 0.96533203125}, {"start": 587.38, "end": 587.86, "word": " positive", "probability": 0.90380859375}, {"start": 587.86, "end": 588.24, "word": " real", "probability": 0.958984375}, {"start": 588.24, "end": 588.82, "word": " numbers", "probability": 0.87109375}, {"start": 588.82, "end": 590.1, "word": " و", "probability": 0.7646484375}, {"start": 590.1, "end": 590.74, "word": " limit", "probability": 0.89306640625}, {"start": 590.74, "end": 592.86, "word": " x", "probability": 0.865234375}, {"start": 592.86, "end": 593.38, "word": " in", "probability": 0.9052734375}, {"start": 593.38, "end": 593.92, "word": " plus", "probability": 0.85693359375}, {"start": 593.92, "end": 594.52, "word": " one", "probability": 0.90185546875}, {"start": 594.52, "end": 595.16, "word": " over", "probability": 0.9130859375}, {"start": 595.16, "end": 595.72, "word": " x", "probability": 0.990234375}, {"start": 595.72, "end": 596.5, "word": " in", "probability": 0.9111328125}, {"start": 596.5, "end": 597.8, "word": " بساوي", "probability": 0.8458251953125}, {"start": 597.8, "end": 598.42, "word": " واحد", "probability": 0.975830078125}, {"start": 598.42, "end": 602.06, "word": " فهذا", "probability": 0.9503580729166666}, {"start": 602.06, "end": 602.62, "word": " بيقدي", "probability": 0.63690185546875}, {"start": 602.62, "end": 602.98, "word": " ان", "probability": 0.66015625}, {"start": 602.98, "end": 603.36, "word": " ال", "probability": 0.623046875}, {"start": 603.36, "end": 603.96, "word": " sequence", "probability": 0.98291015625}, {"start": 603.96, "end": 604.42, "word": " x", "probability": 0.916015625}, {"start": 604.42, "end": 605.02, "word": " in", "probability": 0.9453125}, {"start": 605.02, "end": 608.18, "word": " diverges", "probability": 0.7225341796875}, {"start": 608.18, "end": 609.92, "word": " هل", "probability": 0.87353515625}, {"start": 609.92, "end": 610.14, "word": " هذه", "probability": 0.8154296875}, {"start": 610.14, "end": 610.62, "word": " العبارة", "probability": 0.9136962890625}, {"start": 610.62, "end": 610.92, "word": " صح", "probability": 0.997314453125}, {"start": 610.92, "end": 611.12, "word": " ولا", "probability": 0.943359375}, {"start": 611.12, "end": 611.74, "word": " خطأ؟", "probability": 0.9351806640625}], "temperature": 1.0}, {"id": 22, "seek": 63835, "start": 612.75, "end": 638.35, "text": "العبارة هذه خطأ لأنه احنا فينا في تمرين سبتاشر في section تلاتة اتنين بقول اذا كانت ال sequence .. اذا كان ال limit ratio هذا بساوي واحد فال sequence ممكن تكون convergent او divergent وفي السؤال هذا اعطينا مثلين واحد ل sequence convergent", "tokens": [6027, 3615, 3555, 9640, 3660, 29538, 16490, 9566, 10721, 5296, 33456, 3224, 1975, 5016, 8315, 8978, 8315, 8978, 6055, 29973, 9957, 8608, 3555, 2655, 33599, 2288, 8978, 3541, 6055, 1211, 9307, 3660, 1975, 2655, 1863, 9957, 4724, 39648, 1975, 15730, 25961, 2655, 2423, 8310, 4386, 1975, 15730, 25961, 2423, 4948, 8509, 23758, 4724, 3794, 995, 45865, 36764, 24401, 6156, 6027, 8310, 3714, 43020, 6055, 30544, 9652, 6930, 1975, 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{"start": 620.69, "end": 620.91, "word": " في", "probability": 0.7373046875}, {"start": 620.91, "end": 621.29, "word": " section", "probability": 0.60791015625}, {"start": 621.29, "end": 621.83, "word": " تلاتة", "probability": 0.81884765625}, {"start": 621.83, "end": 622.23, "word": " اتنين", "probability": 0.9705810546875}, {"start": 622.23, "end": 623.77, "word": " بقول", "probability": 0.812744140625}, {"start": 623.77, "end": 624.07, "word": " اذا", "probability": 0.775146484375}, {"start": 624.07, "end": 624.55, "word": " كانت", "probability": 0.9296875}, {"start": 624.55, "end": 624.69, "word": " ال", "probability": 0.91015625}, {"start": 624.69, "end": 625.29, "word": " sequence", "probability": 0.83251953125}, {"start": 625.29, "end": 625.57, "word": " ..", "probability": 0.223388671875}, {"start": 625.57, "end": 625.71, "word": " اذا", "probability": 0.899169921875}, {"start": 625.71, "end": 625.93, "word": " كان", "probability": 0.9931640625}, {"start": 625.93, "end": 626.03, "word": " ال", "probability": 0.41650390625}, {"start": 626.03, "end": 626.23, "word": " limit", "probability": 0.91845703125}, {"start": 626.23, "end": 626.67, "word": " ratio", "probability": 0.316162109375}, {"start": 626.67, "end": 626.87, "word": " هذا", "probability": 0.5595703125}, {"start": 626.87, "end": 627.27, "word": " بساوي", "probability": 0.68963623046875}, {"start": 627.27, "end": 627.75, "word": " واحد", "probability": 0.9873046875}, {"start": 627.75, "end": 628.61, "word": " فال", "probability": 0.90380859375}, {"start": 628.61, "end": 628.89, "word": " sequence", "probability": 0.98193359375}, {"start": 628.89, "end": 629.29, "word": " ممكن", "probability": 0.9833984375}, {"start": 629.29, "end": 629.63, "word": " تكون", "probability": 0.970947265625}, {"start": 629.63, "end": 630.35, "word": " convergent", "probability": 0.88232421875}, {"start": 630.35, "end": 630.55, "word": " او", "probability": 0.8671875}, {"start": 630.55, "end": 631.29, "word": " divergent", "probability": 0.837158203125}, {"start": 631.29, "end": 631.99, "word": " وفي", "probability": 0.779296875}, {"start": 631.99, "end": 632.41, "word": " السؤال", "probability": 0.9949544270833334}, {"start": 632.41, "end": 632.77, "word": " هذا", "probability": 0.95263671875}, {"start": 632.77, "end": 633.89, "word": " اعطينا", "probability": 0.8900390625}, {"start": 633.89, "end": 634.57, "word": " مثلين", "probability": 0.95068359375}, {"start": 634.57, "end": 636.29, "word": " واحد", "probability": 0.924072265625}, {"start": 636.29, "end": 636.81, "word": " ل", "probability": 0.9287109375}, {"start": 636.81, "end": 637.27, "word": " sequence", "probability": 0.9443359375}, {"start": 637.27, "end": 638.35, "word": " convergent", "probability": 0.95947265625}], "temperature": 1.0}, {"id": 23, "seek": 66834, "start": 639.76, "end": 668.34, "text": "ال ratio limit ال ratio تبعها بيساوي واحد لكنها convergent ومثال تاني ل sequence limit ال ratio تبعها أيضا بيساوي واحد لكنها divergent إذا لو كان limit ال ratio بيساوي واحد فال sequence إما converge أو divergent مالضايش نجزم إنها convergent أو نجزم إنها divergent okay إذا العبارة هذه خطأ أو false إذا العبارة هذه false", "tokens": [6027, 8509, 4948, 2423, 8509, 6055, 3555, 3615, 11296, 4724, 1829, 3794, 995, 45865, 36764, 24401, 44381, 11296, 9652, 6930, 4032, 2304, 12984, 6027, 6055, 7649, 1829, 5296, 8310, 4948, 2423, 8509, 6055, 3555, 3615, 11296, 36632, 11242, 995, 4724, 1829, 3794, 995, 45865, 36764, 24401, 44381, 11296, 18558, 6930, 11933, 15730, 45164, 25961, 4948, 2423, 8509, 4724, 1829, 3794, 995, 45865, 36764, 24401, 6156, 6027, 8310, 11933, 15042, 41881, 34051, 18558, 6930, 3714, 6027, 11242, 47302, 8592, 8717, 7435, 11622, 2304, 36145, 11296, 9652, 6930, 34051, 8717, 7435, 11622, 2304, 36145, 11296, 18558, 6930, 1392, 11933, 15730, 18863, 3555, 9640, 3660, 29538, 16490, 9566, 10721, 34051, 7908, 11933, 15730, 18863, 3555, 9640, 3660, 29538, 7908], "avg_logprob": -0.21941774217491475, "compression_ratio": 2.2807881773399017, "no_speech_prob": 0.0, "words": [{"start": 639.76, "end": 640.06, "word": "ال", "probability": 0.35498046875}, {"start": 640.06, "end": 640.52, "word": " ratio", "probability": 0.257080078125}, {"start": 640.52, "end": 640.96, "word": " limit", "probability": 0.50439453125}, {"start": 640.96, "end": 641.24, "word": " ال", "probability": 0.63232421875}, {"start": 641.24, "end": 641.5, "word": " ratio", "probability": 0.98876953125}, {"start": 641.5, "end": 641.88, "word": " تبعها", "probability": 0.8426513671875}, {"start": 641.88, "end": 642.28, "word": " بيساوي", "probability": 0.6823974609375}, {"start": 642.28, "end": 642.7, "word": " واحد", "probability": 0.955078125}, {"start": 642.7, "end": 643.24, "word": " لكنها", "probability": 0.874755859375}, {"start": 643.24, "end": 644.02, "word": " convergent", "probability": 0.855224609375}, {"start": 644.02, "end": 645.1, "word": " ومثال", "probability": 0.806640625}, {"start": 645.1, "end": 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"probability": 0.763916015625}, {"start": 652.6, "end": 652.84, "word": " لو", "probability": 0.9912109375}, {"start": 652.84, "end": 653.06, "word": " كان", "probability": 0.9873046875}, {"start": 653.06, "end": 653.28, "word": " limit", "probability": 0.9111328125}, {"start": 653.28, "end": 653.46, "word": " ال", "probability": 0.97900390625}, {"start": 653.46, "end": 653.7, "word": " ratio", "probability": 0.99609375}, {"start": 653.7, "end": 654.08, "word": " بيساوي", "probability": 0.98505859375}, {"start": 654.08, "end": 654.4, "word": " واحد", "probability": 0.993408203125}, {"start": 654.4, "end": 654.64, "word": " فال", "probability": 0.4534912109375}, {"start": 654.64, "end": 654.98, "word": " sequence", "probability": 0.97021484375}, {"start": 654.98, "end": 655.28, "word": " إما", "probability": 0.7822265625}, {"start": 655.28, "end": 655.9, "word": " converge", "probability": 0.90625}, {"start": 655.9, "end": 656.38, "word": " أو", "probability": 0.6884765625}, {"start": 656.38, "end": 657.1, "word": " divergent", "probability": 0.797607421875}, {"start": 657.1, "end": 658.06, "word": " مالضايش", "probability": 0.526171875}, {"start": 658.06, "end": 658.46, "word": " نجزم", "probability": 0.7840576171875}, {"start": 658.46, "end": 658.68, "word": " إنها", "probability": 0.6900634765625}, {"start": 658.68, "end": 659.46, "word": " convergent", "probability": 0.904541015625}, {"start": 659.46, "end": 659.9, "word": " أو", "probability": 0.93896484375}, {"start": 659.9, "end": 661.0, "word": " نجزم", "probability": 0.9857177734375}, {"start": 661.0, "end": 661.4, "word": " إنها", "probability": 0.948974609375}, {"start": 661.4, "end": 662.14, "word": " divergent", "probability": 0.89501953125}, {"start": 662.14, "end": 662.58, "word": " okay", "probability": 0.642578125}, {"start": 662.58, "end": 663.42, "word": " إذا", "probability": 0.7392578125}, {"start": 663.42, "end": 663.86, "word": " العبارة", "probability": 0.917724609375}, {"start": 663.86, "end": 664.12, "word": " هذه", "probability": 0.7744140625}, {"start": 664.12, "end": 664.66, "word": " خطأ", "probability": 0.9319661458333334}, {"start": 664.66, "end": 665.52, "word": " أو", "probability": 0.86572265625}, {"start": 665.52, "end": 666.1, "word": " false", "probability": 0.87109375}, {"start": 666.1, "end": 667.02, "word": " إذا", "probability": 0.766357421875}, {"start": 667.02, "end": 667.6, "word": " العبارة", "probability": 0.9921875}, {"start": 667.6, "end": 667.86, "word": " هذه", "probability": 0.927734375}, {"start": 667.86, "end": 668.34, "word": " false", "probability": 0.9052734375}], "temperature": 1.0}, {"id": 24, "seek": 68833, "start": 670.83, "end": 688.33, "text": "لو بدلت divergent بconvergent برضه false الصح انها may converge or may diverge طيب العبارة الأخيرة تسعة", "tokens": [1211, 2407, 47525, 1211, 2655, 18558, 6930, 4724, 1671, 331, 6930, 4724, 43042, 3224, 7908, 31767, 5016, 16472, 11296, 815, 41881, 420, 815, 18558, 432, 23032, 1829, 3555, 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"text": "إذن هذا برهان الجزء التاني من السؤال الأول في عند هنا السؤال التاني question 2 الفرع a show S&T", "tokens": [28814, 8848, 1863, 23758, 4724, 2288, 3224, 7649, 25724, 11622, 38207, 16712, 7649, 1829, 9154, 21136, 33604, 6027, 16247, 12610, 8978, 43242, 34105, 21136, 33604, 6027, 16712, 7649, 1829, 1168, 568, 27188, 2288, 3615, 257, 855, 318, 5, 51], "avg_logprob": -0.28437499031424524, "compression_ratio": 1.3478260869565217, "no_speech_prob": 0.0, "words": [{"start": 973.89, "end": 974.25, "word": "إذن", "probability": 0.4686279296875}, {"start": 974.25, "end": 974.49, "word": " هذا", "probability": 0.92724609375}, {"start": 974.49, "end": 975.07, "word": " برهان", "probability": 0.8612060546875}, {"start": 975.07, "end": 975.61, "word": " الجزء", "probability": 0.876953125}, {"start": 975.61, "end": 976.45, "word": " التاني", "probability": 0.9241536458333334}, {"start": 976.45, "end": 976.91, "word": " من", "probability": 0.98974609375}, {"start": 976.91, "end": 977.65, 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"probability": 0.6461181640625}, {"start": 1194.21, "end": 1194.73, "word": " تربية", "probability": 0.9599609375}, {"start": 1194.73, "end": 1195.47, "word": " بالتأكيد", "probability": 0.96474609375}, {"start": 1195.47, "end": 1195.79, "word": " عدد", "probability": 0.994140625}, {"start": 1195.79, "end": 1196.25, "word": " موجب", "probability": 0.994140625}], "temperature": 1.0}, {"id": 43, "seek": 122224, "start": 1196.96, "end": 1222.24, "text": "فانا بطبّق الـ Archimedean property مش على w وعلى w تربية العدد الموجب w square فبقدر ألاقي by Archimedean property natural number n0 مقلوب وأصغر من w تربية طبعا هذا بيقدّي أن واحد على جذر n0 أصغر من w ومن ال", "tokens": [5172, 7649, 995, 4724, 9566, 3555, 11703, 4587, 2423, 39184, 10984, 332, 4858, 282, 4707, 37893, 15844, 261, 4032, 3615, 23942, 261, 6055, 25513, 10632, 18863, 3215, 3215, 9673, 29245, 3555, 261, 3732, 6156, 3555, 28543, 2288, 5551, 15040, 38436, 538, 10984, 332, 4858, 282, 4707, 3303, 1230, 297, 15, 3714, 4587, 1211, 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" w", "probability": 0.450439453125}, {"start": 1199.94, "end": 1200.22, "word": " وعلى", "probability": 0.6293131510416666}, {"start": 1200.22, "end": 1200.54, "word": " w", "probability": 0.7626953125}, {"start": 1200.54, "end": 1201.02, "word": " تربية", "probability": 0.8199869791666666}, {"start": 1201.02, "end": 1201.36, "word": " العدد", "probability": 0.8310546875}, {"start": 1201.36, "end": 1201.8, "word": " الموجب", "probability": 0.9519856770833334}, {"start": 1201.8, "end": 1202.18, "word": " w", "probability": 0.51611328125}, {"start": 1202.18, "end": 1202.84, "word": " square", "probability": 0.351806640625}, {"start": 1202.84, "end": 1204.68, "word": " فبقدر", "probability": 0.8797607421875}, {"start": 1204.68, "end": 1205.12, "word": " ألاقي", "probability": 0.74072265625}, {"start": 1205.12, "end": 1205.36, "word": " by", "probability": 0.56396484375}, {"start": 1205.36, "end": 1206.06, "word": " Archimedean", "probability": 0.9556884765625}, {"start": 1206.06, "end": 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"end": 1214.02, "word": " أن", "probability": 0.440185546875}, {"start": 1214.02, "end": 1214.86, "word": " واحد", "probability": 0.89697265625}, {"start": 1214.86, "end": 1215.22, "word": " على", "probability": 0.88916015625}, {"start": 1215.22, "end": 1215.9, "word": " جذر", "probability": 0.75537109375}, {"start": 1215.9, "end": 1218.08, "word": " n0", "probability": 0.811767578125}, {"start": 1218.08, "end": 1218.68, "word": " أصغر", "probability": 0.9739990234375}, {"start": 1218.68, "end": 1218.92, "word": " من", "probability": 0.99462890625}, {"start": 1218.92, "end": 1219.36, "word": " w", "probability": 0.8798828125}, {"start": 1219.36, "end": 1221.94, "word": " ومن", "probability": 0.90966796875}, {"start": 1221.94, "end": 1222.24, "word": " ال", "probability": 0.90869140625}], "temperature": 1.0}, {"id": 44, "seek": 125032, "start": 1226.3, "end": 1250.32, "text": "من الفرض الـ W هذا lower bound للست S وبالتالي هذا أصغر من أو سوى واحد على الجدر التربيهي ل N0 لأن هذا ينتمي ل S، هذا عنصر في S، صح؟ واحنا فرضين أن الـ W أشمل lower bound", "tokens": [27842, 27188, 43042, 2423, 39184, 343, 23758, 3126, 5472, 24976, 14851, 318, 46599, 6027, 2655, 6027, 1829, 23758, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 2407, 7578, 36764, 24401, 15844, 25724, 3215, 2288, 16712, 2288, 21292, 3224, 1829, 5296, 426, 15, 5296, 33456, 23758, 7251, 29399, 2304, 1829, 5296, 318, 12399, 23758, 18871, 9381, 2288, 8978, 318, 12399, 20328, 5016, 22807, 4032, 39319, 8315, 6156, 43042, 9957, 14739, 2423, 39184, 343, 5551, 8592, 42213, 2295, 260, 5472], "avg_logprob": -0.27243590660584277, "compression_ratio": 1.5449438202247192, "no_speech_prob": 0.0, "words": [{"start": 1226.3, "end": 1226.66, "word": "من", "probability": 0.6181640625}, {"start": 1226.66, "end": 1227.26, "word": " الفرض", "probability": 0.95556640625}, {"start": 1227.26, "end": 1228.1, "word": " الـ", "probability": 0.627197265625}, {"start": 1228.1, "end": 1228.44, "word": " W", "probability": 0.33837890625}, 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في المجموع S إذا ال W كونه lower bound ل S أصغر من أو ساوي واحد على square root ل N0 فطبعا هذا بيدي ان واحد على square root ل N0 أصغر من واحد على square root ل N0 وهذا مدينة تناقض", "tokens": [1211, 19528, 7435, 2304, 45367, 318, 37037, 15730, 1867, 8978, 9673, 7435, 2304, 45367, 318, 11933, 15730, 2423, 343, 9122, 11536, 3224, 3126, 5472, 5296, 318, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 36764, 24401, 15844, 3732, 5593, 5296, 426, 15, 6156, 9566, 3555, 3615, 995, 23758, 4724, 25708, 1829, 16472, 36764, 24401, 15844, 3732, 5593, 5296, 426, 15, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 15844, 3732, 5593, 5296, 426, 15, 37037, 15730, 3714, 3215, 9957, 3660, 6055, 8315, 4587, 11242], "avg_logprob": -0.18382353502161364, "compression_ratio": 1.739884393063584, "no_speech_prob": 0.0, "words": [{"start": 1251.23, "end": 1252.01, "word": "للمجموع", "probability": 0.818701171875}, {"start": 1252.01, "end": 1252.41, "word": " S", "probability": 0.7802734375}, {"start": 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"word": " على", "probability": 0.89111328125}, {"start": 1268.69, "end": 1269.15, "word": " square", "probability": 0.89013671875}, {"start": 1269.15, "end": 1269.69, "word": " root", "probability": 0.92724609375}, {"start": 1269.69, "end": 1271.05, "word": " ل", "probability": 0.974609375}, {"start": 1271.05, "end": 1272.45, "word": " N0", "probability": 0.9765625}, {"start": 1272.45, "end": 1273.97, "word": " وهذا", "probability": 0.8994140625}, {"start": 1273.97, "end": 1274.45, "word": " مدينة", "probability": 0.71478271484375}, {"start": 1274.45, "end": 1275.01, "word": " تناقض", "probability": 0.82275390625}], "temperature": 1.0}, {"id": 46, "seek": 130575, "start": 1276.61, "end": 1305.75, "text": "كيف عدد بيطلع أصغر من نفسه هذا تناقض لأن هذا التناقض بيقول لإن ال assumption تبعنا ان ال W أكبر من السفر خطأ وبالتالي ال W لازم يكون أصغر بدل ما هو أكبر من السفر يطلع أصغر من أو يساوي سفر وبالتالي ال W السفر هو أكبر lower bound okay إذا هذا بيكمل البرهان تمام", "tokens": [4117, 33911, 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"end": 1442.29, "word": " ألاقي", "probability": 0.8323567708333334}, {"start": 1442.29, "end": 1442.69, "word": " عدد", "probability": 0.9930013020833334}, {"start": 1442.69, "end": 1443.13, "word": " طبيعي", "probability": 0.95458984375}, {"start": 1443.13, "end": 1443.55, "word": " capital", "probability": 0.677734375}, {"start": 1443.55, "end": 1443.75, "word": " N", "probability": 0.87548828125}, {"start": 1443.75, "end": 1444.19, "word": " مقلوب", "probability": 0.935791015625}, {"start": 1444.19, "end": 1444.53, "word": " وأصغر", "probability": 0.863037109375}, {"start": 1444.53, "end": 1444.67, "word": " من", "probability": 0.99169921875}, {"start": 1444.67, "end": 1444.93, "word": " epsilon", "probability": 0.85498046875}, {"start": 1444.93, "end": 1445.47, "word": " تربية", "probability": 0.9837239583333334}, {"start": 1445.47, "end": 1447.55, "word": " ناشي", "probability": 0.7849934895833334}], "temperature": 1.0}, {"id": 51, "seek": 147676, "start": 1448.92, "end": 1476.76, "text": "بدا هنا given epsilon أكبر من السفر use الarchimedean property to choose n عدد طبيعي بحيث أن واحد على n أصغر من epsilon", "tokens": [3555, 28259, 34105, 2212, 17889, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 764, 2423, 1178, 332, 4858, 282, 4707, 281, 2826, 297, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 14739, 36764, 24401, 15844, 297, 5551, 9381, 17082, 2288, 9154, 17889], "avg_logprob": -0.3484374973509047, "compression_ratio": 1.1785714285714286, "no_speech_prob": 0.0, "words": [{"start": 1448.92, "end": 1449.7, "word": "بدا", "probability": 0.21490478515625}, {"start": 1449.7, "end": 1450.02, "word": " هنا", "probability": 0.87939453125}, {"start": 1450.02, "end": 1451.72, "word": " given", "probability": 0.517578125}, {"start": 1451.72, "end": 1455.32, "word": " epsilon", "probability": 0.249755859375}, {"start": 1455.32, "end": 1455.8, "word": " أكبر", "probability": 0.9171549479166666}, {"start": 1455.8, "end": 1456.0, "word": " من", "probability": 0.982421875}, {"start": 1456.0, "end": 1456.56, "word": " السفر", "probability": 0.8181966145833334}, {"start": 1456.56, "end": 1458.1, "word": " use", "probability": 0.57080078125}, {"start": 1458.1, "end": 1460.84, "word": " الarchimedean", "probability": 0.661492919921875}, {"start": 1460.84, "end": 1463.48, "word": " property", "probability": 0.97314453125}, {"start": 1463.48, "end": 1466.22, "word": " to", "probability": 0.87939453125}, {"start": 1466.22, "end": 1467.32, "word": " choose", "probability": 0.95654296875}, {"start": 1467.32, "end": 1469.88, "word": " n", "probability": 0.67724609375}, {"start": 1469.88, "end": 1470.4, "word": " عدد", "probability": 0.9820963541666666}, {"start": 1470.4, "end": 1471.28, "word": " طبيعي", "probability": 0.96630859375}, {"start": 1471.28, "end": 1473.3, "word": " بحيث", "probability": 0.9539794921875}, {"start": 1473.3, "end": 1473.5, "word": " أن", "probability": 0.365966796875}, {"start": 1473.5, "end": 1474.06, "word": " واحد", "probability": 0.936279296875}, {"start": 1474.06, "end": 1474.32, "word": " على", "probability": 0.71240234375}, {"start": 1474.32, "end": 1474.8, "word": " n", "probability": 0.4326171875}, {"start": 1474.8, "end": 1476.04, "word": " أصغر", "probability": 0.9859619140625}, {"start": 1476.04, "end": 1476.26, "word": " من", "probability": 0.99267578125}, {"start": 1476.26, "end": 1476.76, "word": " epsilon", "probability": 0.84228515625}], "temperature": 1.0}, {"id": 52, "seek": 150019, "start": 1479.75, "end": 1500.19, "text": "الان لو خدت small n أكبر من أو ساوي capital N فهذا بيقدي أن واحد على small n أصغر لو ساوي واحد على capital N", "tokens": [6027, 7649, 45164, 16490, 3215, 2655, 1359, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 14739, 36764, 24401, 15844, 1359, 297, 5551, 9381, 17082, 2288, 45164, 8608, 995, 45865, 36764, 24401, 15844, 4238, 426], "avg_logprob": -0.239062496026357, "compression_ratio": 1.3666666666666667, "no_speech_prob": 0.0, "words": [{"start": 1479.7499999999998, "end": 1480.6299999999999, "word": "الان", "probability": 0.3876953125}, {"start": 1480.6299999999999, "end": 1481.51, "word": " لو", "probability": 0.79443359375}, {"start": 1481.51, "end": 1481.97, "word": " خدت", "probability": 0.8058268229166666}, {"start": 1481.97, "end": 1482.43, "word": " small", "probability": 0.587890625}, {"start": 1482.43, "end": 1482.85, "word": " n", "probability": 0.68212890625}, {"start": 1482.85, "end": 1483.95, "word": " أكبر", "probability": 0.875}, {"start": 1483.95, "end": 1484.15, "word": " من", "probability": 0.841796875}, {"start": 1484.15, "end": 1484.31, "word": " أو", "probability": 0.72607421875}, {"start": 1484.31, "end": 1484.77, "word": " ساوي", "probability": 0.83349609375}, {"start": 1484.77, "end": 1485.21, "word": " capital", "probability": 0.59375}, {"start": 1485.21, "end": 1485.65, "word": " N", "probability": 0.81396484375}, {"start": 1485.65, "end": 1494.59, "word": " فهذا", "probability": 0.9583333333333334}, {"start": 1494.59, "end": 1495.21, "word": " بيقدي", "probability": 0.68408203125}, {"start": 1495.21, "end": 1496.37, "word": " أن", "probability": 0.409912109375}, {"start": 1496.37, "end": 1497.03, "word": " واحد", "probability": 0.841064453125}, {"start": 1497.03, "end": 1497.23, "word": " على", "probability": 0.7900390625}, {"start": 1497.23, "end": 1497.65, "word": " small", "probability": 0.9248046875}, {"start": 1497.65, "end": 1497.97, "word": " n", "probability": 0.833984375}, {"start": 1497.97, "end": 1498.43, "word": " أصغر", "probability": 0.9796142578125}, {"start": 1498.43, "end": 1498.65, "word": " لو", "probability": 0.822265625}, {"start": 1498.65, "end": 1498.97, "word": " ساوي", "probability": 0.93994140625}, {"start": 1498.97, "end": 1499.25, "word": " واحد", "probability": 0.986328125}, {"start": 1499.25, "end": 1499.41, "word": " على", "probability": 0.82177734375}, {"start": 1499.41, "end": 1499.81, "word": " capital", "probability": 0.73095703125}, {"start": 1499.81, "end": 1500.19, "word": " N", "probability": 0.97607421875}], "temperature": 1.0}, {"id": 53, "seek": 153425, "start": 1505.57, "end": 1534.25, "text": "بيقدي واحد على square root ل N أصغر من لو يساوي واحد على square root ل capital N وهذا بيقدي ان ال absolute value لواحد على square root ل N minus صفر بيساوي واحد على square root ل N وهذا أصغر من لو يساوي واحد على square root ل capital N وهذا من هنا", "tokens": [3555, 1829, 4587, 16254, 36764, 24401, 15844, 3732, 5593, 5296, 426, 5551, 9381, 17082, 2288, 9154, 45164, 7251, 3794, 995, 45865, 36764, 24401, 15844, 3732, 5593, 5296, 4238, 426, 37037, 15730, 4724, 1829, 4587, 16254, 16472, 2423, 8236, 2158, 5296, 14407, 24401, 15844, 3732, 5593, 5296, 426, 3175, 20328, 5172, 2288, 4724, 1829, 3794, 995, 45865, 36764, 24401, 15844, 3732, 5593, 5296, 426, 37037, 15730, 5551, 9381, 17082, 2288, 9154, 45164, 7251, 3794, 995, 45865, 36764, 24401, 15844, 3732, 5593, 5296, 4238, 426, 37037, 15730, 9154, 34105], "avg_logprob": -0.1738281227986921, "compression_ratio": 2.36, "no_speech_prob": 0.0, "words": [{"start": 1505.57, "end": 1506.21, "word": "بيقدي", "probability": 0.4837646484375}, {"start": 1506.21, "end": 1506.79, "word": " واحد", "probability": 0.79833984375}, {"start": 1506.79, "end": 1507.01, "word": " على", "probability": 0.65869140625}, {"start": 1507.01, "end": 1507.41, "word": " square", "probability": 0.79931640625}, {"start": 1507.41, "end": 1507.97, "word": " root", "probability": 0.90673828125}, {"start": 1507.97, "end": 1508.23, "word": " ل", "probability": 0.90869140625}, {"start": 1508.23, "end": 1508.51, "word": " N", "probability": 0.37646484375}, {"start": 1508.51, "end": 1509.01, "word": " أصغر", "probability": 0.9100341796875}, {"start": 1509.01, "end": 1509.13, "word": " من", "probability": 0.87158203125}, {"start": 1509.13, "end": 1509.27, "word": " لو", "probability": 0.38671875}, {"start": 1509.27, "end": 1509.57, "word": " يساوي", "probability": 0.7838134765625}, {"start": 1509.57, "end": 1510.07, "word": " واحد", "probability": 0.970947265625}, {"start": 1510.07, "end": 1511.25, "word": " على", "probability": 0.81396484375}, {"start": 1511.25, "end": 1511.65, "word": " square", "probability": 0.89990234375}, {"start": 1511.65, "end": 1512.03, "word": " root", "probability": 0.943359375}, {"start": 1512.03, "end": 1512.21, "word": " ل", "probability": 0.962890625}, {"start": 1512.21, "end": 1512.55, "word": " capital", "probability": 0.37744140625}, {"start": 1512.55, "end": 1512.87, "word": " N", "probability": 0.9560546875}, {"start": 1512.87, "end": 1515.31, "word": " وهذا", "probability": 0.847900390625}, {"start": 1515.31, "end": 1515.95, "word": " بيقدي", "probability": 0.8468017578125}, {"start": 1515.95, "end": 1516.31, "word": " ان", "probability": 0.49658203125}, {"start": 1516.31, "end": 1516.45, "word": " ال", "probability": 0.8681640625}, {"start": 1516.45, "end": 1516.89, "word": " absolute", "probability": 0.736328125}, {"start": 1516.89, "end": 1517.77, "word": " value", "probability": 0.99267578125}, {"start": 1517.77, "end": 1520.05, "word": " لواحد", "probability": 0.82666015625}, {"start": 1520.05, "end": 1520.31, "word": " على", "probability": 0.85498046875}, {"start": 1520.31, "end": 1520.81, "word": " square", "probability": 0.90966796875}, {"start": 1520.81, "end": 1521.33, "word": " root", "probability": 0.93408203125}, {"start": 1521.33, "end": 1521.59, "word": " ل", "probability": 0.9697265625}, {"start": 1521.59, "end": 1521.83, "word": " N", "probability": 0.89111328125}, {"start": 1521.83, "end": 1522.37, "word": " minus", "probability": 0.8349609375}, {"start": 1522.37, "end": 1523.65, "word": " صفر", "probability": 0.888671875}, {"start": 1523.65, "end": 1524.59, "word": " بيساوي", "probability": 0.8841796875}, {"start": 1524.59, "end": 1525.15, "word": " واحد", "probability": 0.9873046875}, {"start": 1525.15, "end": 1525.31, "word": " على", "probability": 0.86474609375}, {"start": 1525.31, "end": 1525.73, "word": " square", "probability": 0.9248046875}, {"start": 1525.73, "end": 1526.15, "word": " root", "probability": 0.93017578125}, {"start": 1526.15, "end": 1526.39, "word": " ل", "probability": 0.97705078125}, {"start": 1526.39, "end": 1526.63, "word": " N", "probability": 0.93359375}, {"start": 1526.63, "end": 1527.05, "word": " وهذا", "probability": 0.91357421875}, {"start": 1527.05, "end": 1527.51, "word": " أصغر", "probability": 0.9525146484375}, {"start": 1527.51, "end": 1527.65, "word": " من", "probability": 0.96923828125}, {"start": 1527.65, "end": 1527.79, "word": " لو", "probability": 0.98681640625}, {"start": 1527.79, "end": 1528.41, "word": " يساوي", "probability": 0.982666015625}, {"start": 1528.41, "end": 1529.61, "word": " واحد", "probability": 0.986572265625}, {"start": 1529.61, "end": 1529.77, "word": " على", "probability": 0.87646484375}, {"start": 1529.77, "end": 1530.17, "word": " square", "probability": 0.927734375}, {"start": 1530.17, "end": 1530.59, "word": " root", "probability": 0.92333984375}, {"start": 1530.59, "end": 1530.77, "word": " ل", "probability": 0.98681640625}, {"start": 1530.77, "end": 1531.17, "word": " capital", "probability": 0.72216796875}, {"start": 1531.17, "end": 1531.67, "word": " N", "probability": 0.99560546875}, {"start": 1531.67, "end": 1533.71, "word": " وهذا", "probability": 0.95654296875}, {"start": 1533.71, "end": 1533.93, "word": " من", "probability": 0.98046875}, {"start": 1533.93, "end": 1534.25, "word": " هنا", "probability": 0.99755859375}], "temperature": 1.0}, {"id": 54, "seek": 156390, "start": 1536.26, "end": 1563.9, "text": "لو سمينا الـ inequality هذه الـ star إذا by star واحد على square root ل N أصغر من epsilon إذا هاي نحققنا تعريف epsilon capital N للنهايات for any given epsilon أثبتنا إنه يوجد capital N عدد طبيعي وهذا العدد الطبيعي يعتمد على epsilon هاي مرتبط بepsilon", "tokens": [1211, 2407, 8608, 2304, 1829, 8315, 2423, 39184, 16970, 29538, 2423, 39184, 3543, 11933, 15730, 538, 3543, 36764, 24401, 15844, 3732, 5593, 5296, 426, 5551, 9381, 17082, 2288, 9154, 17889, 11933, 15730, 8032, 47302, 8717, 5016, 4587, 4587, 8315, 37279, 16572, 5172, 17889, 4238, 426, 24976, 1863, 11296, 1829, 9307, 337, 604, 2212, 17889, 5551, 12984, 3555, 2655, 8315, 36145, 3224, 7251, 29245, 3215, 4238, 426, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 37037, 15730, 18863, 3215, 3215, 41950, 21292, 3615, 1829, 7251, 34268, 2304, 3215, 15844, 17889, 8032, 47302, 3714, 43500, 3555, 9566, 4724, 10653, 15754], "avg_logprob": -0.23007014925990785, "compression_ratio": 1.6133333333333333, "no_speech_prob": 0.0, "words": [{"start": 1536.26, "end": 1536.62, "word": "لو", "probability": 0.743408203125}, {"start": 1536.62, "end": 1537.06, "word": " سمينا", "probability": 0.79901123046875}, {"start": 1537.06, "end": 1537.18, "word": " الـ", "probability": 0.6502685546875}, {"start": 1537.18, "end": 1537.62, "word": " inequality", "probability": 0.7763671875}, {"start": 1537.62, "end": 1538.0, "word": " هذه", "probability": 0.242431640625}, {"start": 1538.0, "end": 1538.16, "word": " الـ", "probability": 0.745361328125}, {"start": 1538.16, "end": 1538.52, "word": " star", "probability": 0.595703125}, {"start": 1538.52, "end": 1539.66, "word": " إذا", "probability": 0.5701904296875}, {"start": 1539.66, "end": 1540.04, "word": " by", "probability": 0.495361328125}, {"start": 1540.04, "end": 1540.76, "word": " star", "probability": 0.71728515625}, {"start": 1540.76, "end": 1542.76, "word": " واحد", "probability": 0.76513671875}, {"start": 1542.76, "end": 1542.92, "word": " على", "probability": 0.71533203125}, {"start": 1542.92, "end": 1543.24, "word": " square", "probability": 0.7294921875}, {"start": 1543.24, "end": 1543.64, "word": " root", "probability": 0.92333984375}, {"start": 1543.64, "end": 1543.82, "word": " ل", "probability": 0.93017578125}, {"start": 1543.82, "end": 1544.02, "word": " N", "probability": 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0.8828125}, {"start": 1553.44, "end": 1553.7, "word": " any", "probability": 0.93310546875}, {"start": 1553.7, "end": 1554.04, "word": " given", "probability": 0.90576171875}, {"start": 1554.04, "end": 1554.54, "word": " epsilon", "probability": 0.95849609375}, {"start": 1554.54, "end": 1555.14, "word": " أثبتنا", "probability": 0.9875}, {"start": 1555.14, "end": 1555.42, "word": " إنه", "probability": 0.4730224609375}, {"start": 1555.42, "end": 1555.84, "word": " يوجد", "probability": 0.9847005208333334}, {"start": 1555.84, "end": 1556.28, "word": " capital", "probability": 0.74462890625}, {"start": 1556.28, "end": 1556.74, "word": " N", "probability": 0.96435546875}, {"start": 1556.74, "end": 1558.6, "word": " عدد", "probability": 0.7327473958333334}, {"start": 1558.6, "end": 1559.28, "word": " طبيعي", "probability": 0.982421875}, {"start": 1559.28, "end": 1560.38, "word": " وهذا", "probability": 0.811279296875}, {"start": 1560.38, "end": 1560.76, "word": " العدد", "probability": 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1587.83, "word": " على", "probability": 0.578125}, {"start": 1587.83, "end": 1588.23, "word": " square", "probability": 0.88671875}, {"start": 1588.23, "end": 1588.69, "word": " root", "probability": 0.91796875}, {"start": 1588.69, "end": 1588.91, "word": " ل", "probability": 0.9072265625}, {"start": 1588.91, "end": 1589.19, "word": " n", "probability": 0.39501953125}, {"start": 1589.19, "end": 1590.49, "word": " بساوي", "probability": 0.9053955078125}, {"start": 1590.49, "end": 1590.93, "word": " سفر", "probability": 0.97314453125}, {"start": 1590.93, "end": 1591.25, "word": " وهو", "probability": 0.764404296875}, {"start": 1591.25, "end": 1591.83, "word": " المطلوب", "probability": 0.9893798828125}, {"start": 1591.83, "end": 1593.29, "word": " طبعا", "probability": 0.8341064453125}], "temperature": 1.0}, {"id": 56, "seek": 162435, "start": 1597.55, "end": 1624.35, "text": "طبعاً في حال تاني أو في برهان تاني باستخدام الـ monotone convergence theorem إذا ال solution to use monotone convergence theorem ال sequence أنا عندي xn بساوي واحد على square root ل n", "tokens": [9566, 3555, 3615, 995, 14111, 8978, 11331, 6027, 6055, 7649, 1829, 34051, 8978, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 4724, 995, 14851, 9778, 3215, 10943, 2423, 39184, 1108, 310, 546, 32181, 20904, 11933, 15730, 2423, 3827, 281, 764, 1108, 310, 546, 32181, 20904, 2423, 8310, 41850, 18871, 16254, 2031, 77, 4724, 3794, 995, 45865, 36764, 24401, 15844, 3732, 5593, 5296, 297], "avg_logprob": -0.26041666761277216, "compression_ratio": 1.503030303030303, "no_speech_prob": 0.0, "words": [{"start": 1597.55, "end": 1597.97, "word": "طبعاً", "probability": 0.756982421875}, {"start": 1597.97, "end": 1598.07, "word": " في", "probability": 0.77978515625}, {"start": 1598.07, "end": 1598.43, "word": " حال", "probability": 0.7939453125}, {"start": 1598.43, "end": 1599.07, "word": " تاني", "probability": 0.8639322916666666}, {"start": 1599.07, "end": 1599.61, "word": " أو", "probability": 0.49462890625}, {"start": 1599.61, "end": 1599.79, "word": " في", "probability": 0.84228515625}, {"start": 1599.79, "end": 1600.15, "word": " برهان", "probability": 0.8060302734375}, {"start": 1600.15, "end": 1600.73, "word": " تاني", "probability": 0.9918619791666666}, {"start": 1600.73, "end": 1602.23, "word": " باستخدام", "probability": 0.9806315104166666}, {"start": 1602.23, "end": 1602.93, "word": " الـ", "probability": 0.777099609375}, {"start": 1602.93, "end": 1603.41, "word": " monotone", "probability": 0.77392578125}, {"start": 1603.41, "end": 1604.15, "word": " convergence", "probability": 0.943359375}, {"start": 1604.15, "end": 1604.75, "word": " theorem", "probability": 0.8994140625}, {"start": 1604.75, "end": 1608.77, "word": " إذا", "probability": 0.506591796875}, {"start": 1608.77, "end": 1609.21, "word": " ال", "probability": 0.35400390625}, {"start": 1609.21, "end": 1610.15, "word": " solution", "probability": 0.9072265625}, {"start": 1610.15, "end": 1611.23, "word": " to", "probability": 0.28662109375}, {"start": 1611.23, "end": 1614.35, "word": " use", "probability": 0.87744140625}, {"start": 1614.35, "end": 1615.87, "word": " monotone", "probability": 0.9811197916666666}, {"start": 1615.87, "end": 1617.57, "word": " convergence", "probability": 0.9326171875}, {"start": 1617.57, "end": 1618.21, "word": " theorem", "probability": 0.82373046875}, {"start": 1618.21, "end": 1620.05, "word": " ال", "probability": 0.69091796875}, {"start": 1620.05, "end": 1620.51, "word": " sequence", "probability": 0.849609375}, {"start": 1620.51, "end": 1620.75, "word": " أنا", "probability": 0.564453125}, {"start": 1620.75, "end": 1621.09, "word": " عندي", "probability": 0.9287109375}, {"start": 1621.09, "end": 1621.81, "word": " xn", "probability": 0.42919921875}, {"start": 1621.81, "end": 1622.45, "word": " بساوي", "probability": 0.65203857421875}, {"start": 1622.45, "end": 1622.93, "word": " واحد", "probability": 0.888671875}, {"start": 1622.93, "end": 1623.15, "word": " على", "probability": 0.69189453125}, {"start": 1623.15, "end": 1623.49, "word": " square", "probability": 0.82421875}, {"start": 1623.49, "end": 1623.89, "word": " root", "probability": 0.91162109375}, {"start": 1623.89, "end": 1624.11, "word": " ل", "probability": 0.8701171875}, {"start": 1624.11, "end": 1624.35, "word": " n", "probability": 0.405029296875}], "temperature": 1.0}, {"id": 57, "seek": 164855, "start": 1629.01, "end": 1648.55, "text": "هذا أكبر من أو ساوي واحد على square root ل n زايد واحد اللي هو xn زايد واحد وبالتالي ال sequence is decreasing صح؟", "tokens": [3224, 15730, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 36764, 24401, 15844, 3732, 5593, 5296, 297, 30767, 995, 25708, 36764, 24401, 13672, 1829, 31439, 2031, 77, 30767, 995, 25708, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 2423, 8310, 307, 23223, 20328, 5016, 22807], "avg_logprob": -0.1500868108537462, "compression_ratio": 1.323076923076923, "no_speech_prob": 0.0, "words": [{"start": 1629.0100000000002, "end": 1630.0700000000002, "word": "هذا", "probability": 0.893310546875}, {"start": 1630.0700000000002, "end": 1631.13, "word": " أكبر", "probability": 0.87109375}, {"start": 1631.13, "end": 1631.39, "word": " من", "probability": 0.98291015625}, {"start": 1631.39, "end": 1631.59, "word": " أو", "probability": 0.9501953125}, {"start": 1631.59, "end": 1632.11, "word": " ساوي", "probability": 0.7522786458333334}, {"start": 1632.11, "end": 1632.65, "word": " واحد", "probability": 0.944091796875}, {"start": 1632.65, "end": 1632.83, "word": " على", "probability": 0.70849609375}, {"start": 1632.83, "end": 1633.21, "word": " square", "probability": 0.73486328125}, {"start": 1633.21, "end": 1633.73, "word": " root", "probability": 0.91845703125}, {"start": 1633.73, "end": 1633.93, "word": " ل", "probability": 0.82177734375}, {"start": 1633.93, "end": 1634.15, "word": " n", "probability": 0.280517578125}, {"start": 1634.15, "end": 1634.59, "word": " زايد", "probability": 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"probability": 0.8836263020833334}], "temperature": 1.0}, {"id": 58, "seek": 167910, "start": 1649.8, "end": 1679.1, "text": "بعدين انا عندي absolute xn بساوي absolute واحد على جدر ال n أصغر من أو ساوي واحد لكل n لكل n فهذا بيقدي ان ال sequence xn is bounded اذا bounded و decrease in", "tokens": [3555, 22488, 9957, 1975, 8315, 18871, 16254, 8236, 2031, 77, 4724, 3794, 995, 45865, 8236, 36764, 24401, 15844, 10874, 3215, 2288, 2423, 297, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 36764, 24401, 5296, 28820, 297, 5296, 28820, 297, 6156, 3224, 15730, 4724, 1829, 4587, 16254, 16472, 2423, 8310, 2031, 77, 307, 37498, 1975, 15730, 37498, 4032, 11514, 294], "avg_logprob": -0.22989241314716027, "compression_ratio": 1.4675324675324675, "no_speech_prob": 0.0, "words": [{"start": 1649.8, "end": 1650.36, "word": "بعدين", "probability": 0.81298828125}, {"start": 1650.36, "end": 1650.48, "word": " انا", "probability": 0.5904541015625}, {"start": 1650.48, "end": 1650.78, "word": " عندي", "probability": 0.887451171875}, {"start": 1650.78, "end": 1651.24, "word": " absolute", "probability": 0.74853515625}, {"start": 1651.24, "end": 1652.22, "word": " xn", "probability": 0.4544677734375}, {"start": 1652.22, "end": 1652.9, "word": " بساوي", "probability": 0.78564453125}, {"start": 1652.9, "end": 1653.44, "word": " absolute", "probability": 0.71826171875}, {"start": 1653.44, "end": 1654.1, "word": " واحد", "probability": 0.938232421875}, {"start": 1654.1, "end": 1654.28, "word": " على", "probability": 0.67138671875}, {"start": 1654.28, "end": 1654.78, "word": " جدر", "probability": 0.8649088541666666}, {"start": 1654.78, "end": 1654.96, "word": " ال", "probability": 0.8935546875}, {"start": 1654.96, "end": 1655.26, "word": " n", "probability": 0.58837890625}, {"start": 1655.26, "end": 1656.8, "word": " أصغر", "probability": 0.8865966796875}, {"start": 1656.8, "end": 1656.98, "word": " من", "probability": 0.970703125}, {"start": 1656.98, "end": 1657.18, "word": " أو", "probability": 0.74072265625}, {"start": 1657.18, "end": 1657.68, "word": " ساوي", "probability": 0.93359375}, {"start": 1657.68, "end": 1658.3, "word": " واحد", "probability": 0.980224609375}, {"start": 1658.3, "end": 1660.74, "word": " لكل", "probability": 0.982666015625}, {"start": 1660.74, "end": 1661.08, "word": " n", "probability": 0.84228515625}, {"start": 1661.08, "end": 1667.12, "word": " لكل", "probability": 0.896240234375}, {"start": 1667.12, "end": 1667.54, "word": " n", "probability": 0.919921875}, {"start": 1667.54, "end": 1670.34, "word": " فهذا", "probability": 0.9306640625}, {"start": 1670.34, "end": 1670.86, "word": " بيقدي", "probability": 0.66094970703125}, {"start": 1670.86, "end": 1671.1, "word": " ان", "probability": 0.712890625}, {"start": 1671.1, "end": 1671.28, "word": " ال", "probability": 0.638671875}, {"start": 1671.28, "end": 1671.76, "word": " sequence", "probability": 0.9609375}, {"start": 1671.76, "end": 1672.74, "word": " xn", "probability": 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5296, 2031, 77, 11331, 1829, 12984, 297, 7251, 29399, 2304, 1829, 30731, 297, 6156, 10721, 12984, 3555, 2655, 8315, 14739, 2423, 7649, 5172, 2304, 10943, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288, 20666, 5172, 2288, 3615, 13672, 1829, 10874, 16758, 43761], "avg_logprob": -0.27581520875295, "compression_ratio": 1.3821656050955413, "no_speech_prob": 0.0, "words": [{"start": 1680.09, "end": 1680.43, "word": "إذا", "probability": 0.51727294921875}, {"start": 1680.43, "end": 1680.65, "word": " الـ", "probability": 0.686279296875}, {"start": 1680.65, "end": 1681.17, "word": " sequence", "probability": 0.732421875}, {"start": 1681.17, "end": 1682.49, "word": " xn", "probability": 0.4052734375}, {"start": 1682.49, "end": 1682.75, "word": " by", "probability": 0.57275390625}, {"start": 1682.75, "end": 1683.39, "word": " monotone", "probability": 0.8846028645833334}, {"start": 1683.39, "end": 1684.03, "word": " convergence", "probability": 0.9638671875}, {"start": 1684.03, "end": 1684.57, "word": " theorem", "probability": 0.9072265625}, {"start": 1684.57, "end": 1686.25, "word": " limit", "probability": 0.87890625}, {"start": 1686.25, "end": 1686.97, "word": " xn", "probability": 0.964599609375}, {"start": 1686.97, "end": 1687.69, "word": " بساوة", "probability": 0.624462890625}, {"start": 1687.69, "end": 1688.49, "word": " الانفمام", "probability": 0.65867919921875}, {"start": 1688.49, "end": 1691.31, "word": " ل", "probability": 0.9443359375}, {"start": 1691.31, "end": 1692.35, "word": " xn", "probability": 0.74658203125}, {"start": 1692.35, "end": 1692.89, "word": " حيث", "probability": 0.9827473958333334}, {"start": 1692.89, "end": 1693.19, "word": " n", "probability": 0.7314453125}, {"start": 1693.19, "end": 1694.01, "word": " ينتمي", "probability": 0.95654296875}, {"start": 1694.01, "end": 1694.23, "word": " إلى", "probability": 0.89990234375}, {"start": 1694.23, "end": 1694.59, "word": " n", "probability": 0.64453125}, {"start": 1694.59, "end": 1697.53, "word": " فأثبتنا", "probability": 0.7340494791666666}, {"start": 1697.53, "end": 1697.71, "word": " أن", "probability": 0.420166015625}, {"start": 1697.71, "end": 1698.33, "word": " الانفمام", "probability": 0.95625}, {"start": 1698.33, "end": 1698.85, "word": " بساوة", "probability": 0.941015625}, {"start": 1698.85, "end": 1699.25, "word": " سفر", "probability": 0.7820638020833334}, {"start": 1699.25, "end": 1701.47, "word": " بالفرع", "probability": 0.79779052734375}, {"start": 1701.47, "end": 1701.65, "word": " اللي", "probability": 0.93505859375}, {"start": 1701.65, "end": 1702.07, "word": " جابله", "probability": 0.8919270833333334}], "temperature": 1.0}, {"id": 60, "seek": 172640, "start": 1704.0, "end": 1726.4, "text": "فهذا برهان تاني لكن احنا ال .. ال monotone convergence الكلام مش داخله في الامتحان فممكن انكم يعني ال .. تستخدموا البرهان الأول يعني ممكن تستخدموا البرهان الأول يعني okay تمام طيب نكمل", "tokens": [5172, 3224, 15730, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 44381, 1975, 5016, 8315, 2423, 4386, 2423, 1108, 310, 546, 32181, 2423, 28820, 10943, 37893, 11778, 47283, 43761, 8978, 2423, 10943, 2655, 5016, 7649, 6156, 2304, 43020, 16472, 24793, 37495, 22653, 2423, 4386, 6055, 14851, 9778, 40448, 14407, 2423, 26890, 3224, 7649, 16247, 12610, 37495, 22653, 3714, 43020, 6055, 14851, 9778, 40448, 14407, 2423, 26890, 3224, 7649, 16247, 12610, 37495, 22653, 1392, 46811, 10943, 23032, 1829, 3555, 8717, 24793, 1211], "avg_logprob": -0.16763117578294542, "compression_ratio": 1.8176470588235294, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1704.0, "end": 1704.36, "word": "فهذا", "probability": 0.8424479166666666}, {"start": 1704.36, "end": 1704.76, "word": " برهان", "probability": 0.957275390625}, {"start": 1704.76, "end": 1705.12, "word": " تاني", "probability": 0.9134114583333334}, {"start": 1705.12, "end": 1705.36, "word": " لكن", "probability": 0.68115234375}, {"start": 1705.36, "end": 1705.8, "word": " احنا", "probability": 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0.751220703125}, {"start": 1712.84, "end": 1713.3, "word": " يعني", "probability": 0.952880859375}, {"start": 1713.3, "end": 1713.76, "word": " ال", "probability": 0.89453125}, {"start": 1713.76, "end": 1714.86, "word": " ..", "probability": 0.8505859375}, {"start": 1714.86, "end": 1715.92, "word": " تستخدموا", "probability": 0.97861328125}, {"start": 1715.92, "end": 1716.42, "word": " البرهان", "probability": 0.9447021484375}, {"start": 1716.42, "end": 1716.8, "word": " الأول", "probability": 0.82421875}, {"start": 1716.8, "end": 1717.12, "word": " يعني", "probability": 0.760986328125}, {"start": 1717.12, "end": 1717.54, "word": " ممكن", "probability": 0.78857421875}, {"start": 1717.54, "end": 1718.12, "word": " تستخدموا", "probability": 0.94560546875}, {"start": 1718.12, "end": 1718.58, "word": " البرهان", "probability": 0.978515625}, {"start": 1718.58, "end": 1718.98, "word": " الأول", "probability": 0.964111328125}, {"start": 1718.98, "end": 1719.3, "word": " يعني", "probability": 0.861083984375}, {"start": 1719.3, "end": 1722.36, "word": " okay", "probability": 0.56591796875}, {"start": 1722.36, "end": 1722.92, "word": " تمام", "probability": 0.94580078125}, {"start": 1722.92, "end": 1725.66, "word": " طيب", "probability": 0.8828125}, {"start": 1725.66, "end": 1726.4, "word": " نكمل", "probability": 0.896484375}], "temperature": 1.0}, {"id": 61, "seek": 177410, "start": 1749.04, "end": 1774.1, "text": "find the supremum الجزء C بإنه نوجد ال supremum لواحد سالب واحد على الجدر ال N حيث N ينتمي ل N فهذا حسب exercise أخدنا supremum A زائد 6 بساوي A زائد supremum", "tokens": [35072, 264, 23710, 449, 25724, 11622, 38207, 383, 4724, 28814, 1863, 3224, 8717, 29245, 3215, 2423, 23710, 449, 5296, 14407, 24401, 8608, 6027, 3555, 36764, 24401, 15844, 25724, 3215, 2288, 2423, 426, 11331, 1829, 12984, 426, 7251, 29399, 2304, 1829, 5296, 426, 6156, 3224, 15730, 11331, 35457, 5380, 5551, 9778, 3215, 8315, 23710, 449, 316, 30767, 16373, 3215, 1386, 4724, 3794, 995, 45865, 316, 30767, 16373, 3215, 23710, 449], "avg_logprob": -0.25468750170298987, "compression_ratio": 1.4743589743589745, "no_speech_prob": 0.0, "words": [{"start": 1749.04, "end": 1750.0, "word": "find", "probability": 0.37939453125}, {"start": 1750.0, "end": 1750.3, "word": " the", "probability": 0.22900390625}, {"start": 1750.3, "end": 1750.92, "word": " supremum", "probability": 0.8798828125}, {"start": 1750.92, "end": 1752.02, "word": " الجزء", "probability": 0.85498046875}, {"start": 1752.02, "end": 1752.4, "word": " C", "probability": 0.521484375}, {"start": 1752.4, "end": 1753.9, "word": " بإنه", "probability": 0.54803466796875}, {"start": 1753.9, "end": 1754.2, "word": " نوجد", "probability": 0.9217122395833334}, {"start": 1754.2, "end": 1754.4, "word": " ال", "probability": 0.439208984375}, {"start": 1754.4, "end": 1755.54, "word": " supremum", "probability": 0.8408203125}, {"start": 1755.54, "end": 1756.64, "word": " لواحد", "probability": 0.628173828125}, {"start": 1756.64, "end": 1757.14, "word": " سالب", "probability": 0.84716796875}, {"start": 1757.14, "end": 1757.56, "word": " واحد", "probability": 0.978515625}, {"start": 1757.56, "end": 1757.76, "word": " على", "probability": 0.8837890625}, {"start": 1757.76, "end": 1758.24, "word": " الجدر", "probability": 0.9065755208333334}, {"start": 1758.24, "end": 1758.4, "word": " ال", "probability": 0.42626953125}, {"start": 1758.4, "end": 1758.7, "word": " N", "probability": 0.73193359375}, {"start": 1758.7, "end": 1759.28, "word": " حيث", "probability": 0.9700520833333334}, {"start": 1759.28, "end": 1759.6, "word": " N", "probability": 0.3291015625}, {"start": 1759.6, "end": 1760.28, "word": " ينتمي", "probability": 0.9725341796875}, {"start": 1760.28, "end": 1760.48, "word": " ل", "probability": 0.438232421875}, {"start": 1760.48, "end": 1760.8, "word": " N", "probability": 0.72900390625}, {"start": 1760.8, "end": 1763.78, "word": " فهذا", "probability": 0.9415690104166666}, {"start": 1763.78, "end": 1764.28, "word": " حسب", "probability": 0.98291015625}, {"start": 1764.28, "end": 1766.44, "word": " exercise", "probability": 0.75048828125}, {"start": 1766.44, "end": 1766.98, "word": " أخدنا", "probability": 0.938232421875}, {"start": 1766.98, "end": 1768.7, "word": " supremum", "probability": 0.748291015625}, {"start": 1768.7, "end": 1769.08, "word": " A", "probability": 0.59375}, {"start": 1769.08, "end": 1769.7, "word": " زائد", "probability": 0.8634440104166666}, {"start": 1769.7, "end": 1770.22, "word": " 6", "probability": 0.81787109375}, {"start": 1770.22, "end": 1772.2, "word": " بساوي", "probability": 0.8326416015625}, {"start": 1772.2, "end": 1772.62, "word": " A", "probability": 0.8466796875}, {"start": 1772.62, "end": 1773.32, "word": " زائد", "probability": 0.9568684895833334}, {"start": 1773.32, "end": 1774.1, "word": " supremum", "probability": 0.893310546875}], "temperature": 1.0}, {"id": 62, "seek": 179664, "start": 1776.02, "end": 1796.64, "text": "الست اللي هي عناصرها سالب واحد على جدر ال N هيف N ينتمي الى N ويساوي واحد الان supremum سالب ست بيساوي سالب ال infimum لعناصر الست", "tokens": [6027, 14851, 13672, 1829, 39896, 18871, 33546, 2288, 11296, 8608, 6027, 3555, 36764, 24401, 15844, 10874, 3215, 2288, 2423, 426, 39896, 5172, 426, 7251, 29399, 2304, 1829, 2423, 7578, 426, 4032, 1829, 3794, 995, 45865, 36764, 24401, 2423, 7649, 23710, 449, 8608, 6027, 3555, 8608, 2655, 4724, 1829, 3794, 995, 45865, 8608, 6027, 3555, 2423, 1536, 332, 449, 5296, 3615, 1863, 33546, 2288, 2423, 14851], "avg_logprob": -0.29521781386751117, "compression_ratio": 1.588235294117647, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1776.02, "end": 1776.64, "word": "الست", "probability": 0.734130859375}, {"start": 1776.64, "end": 1776.86, "word": " اللي", "probability": 0.55908203125}, {"start": 1776.86, "end": 1777.0, "word": " هي", "probability": 0.828125}, {"start": 1777.0, "end": 1777.58, "word": " عناصرها", "probability": 0.9691162109375}, {"start": 1777.58, "end": 1778.0, "word": " سالب", "probability": 0.8474934895833334}, {"start": 1778.0, "end": 1778.44, "word": " واحد", "probability": 0.982666015625}, {"start": 1778.44, "end": 1778.64, "word": " على", "probability": 0.76513671875}, {"start": 1778.64, "end": 1779.2, "word": " جدر", "probability": 0.7694498697916666}, {"start": 1779.2, "end": 1779.38, "word": " ال", "probability": 0.94091796875}, {"start": 1779.38, "end": 1779.7, "word": " N", "probability": 0.364501953125}, {"start": 1779.7, "end": 1781.02, "word": " هيف", "probability": 0.34503173828125}, {"start": 1781.02, "end": 1781.24, "word": " N", "probability": 0.380859375}, {"start": 1781.24, "end": 1781.92, "word": " ينتمي", "probability": 0.968505859375}, {"start": 1781.92, "end": 1782.24, "word": " الى", "probability": 0.596435546875}, {"start": 1782.24, "end": 1782.52, "word": " N", "probability": 0.626953125}, {"start": 1782.52, "end": 1785.22, "word": " ويساوي", "probability": 0.791796875}, {"start": 1785.22, "end": 1788.18, "word": " واحد", "probability": 0.961181640625}, {"start": 1788.18, "end": 1788.8, "word": " الان", "probability": 0.5023193359375}, {"start": 1788.8, "end": 1789.56, "word": " supremum", "probability": 0.6082763671875}, {"start": 1789.56, "end": 1790.12, "word": " سالب", "probability": 0.9562174479166666}, {"start": 1790.12, "end": 1790.56, "word": " ست", "probability": 0.8544921875}, {"start": 1790.56, "end": 1792.36, "word": " بيساوي", "probability": 0.827294921875}, {"start": 1792.36, "end": 1792.94, "word": " سالب", "probability": 0.9593098958333334}, {"start": 1792.94, "end": 1793.14, "word": " ال", "probability": 0.841796875}, {"start": 1793.14, "end": 1793.9, "word": " infimum", "probability": 0.7963053385416666}, {"start": 1793.9, "end": 1796.14, "word": " لعناصر", "probability": 0.8015625}, {"start": 1796.14, "end": 1796.64, "word": " الست", "probability": 0.75537109375}], "temperature": 1.0}, {"id": 63, "seek": 183057, "start": 1805.05, "end": 1830.57, "text": "و احنا لسه مثبتين ان ال inform هذا بساوي سفر اذا ال suprem للست هذي بطلع واحد طيب فرقة تانية show that اثبتي انه limit cosine n على n او على جدر ال n", "tokens": [2407, 1975, 5016, 8315, 5296, 3794, 3224, 3714, 12984, 3555, 2655, 9957, 16472, 2423, 1356, 23758, 4724, 3794, 995, 45865, 8608, 5172, 2288, 1975, 15730, 2423, 23710, 24976, 14851, 8032, 8848, 1829, 4724, 9566, 1211, 3615, 36764, 24401, 23032, 1829, 3555, 6156, 2288, 28671, 6055, 7649, 10632, 855, 300, 1975, 12984, 3555, 31371, 16472, 3224, 4948, 23565, 297, 15844, 297, 1975, 2407, 15844, 10874, 3215, 2288, 2423, 297], "avg_logprob": -0.2461503657741823, "compression_ratio": 1.4171779141104295, "no_speech_prob": 0.0, "words": [{"start": 1805.05, "end": 1805.31, "word": "و", "probability": 0.5537109375}, {"start": 1805.31, "end": 1805.57, "word": " احنا", "probability": 0.7749837239583334}, {"start": 1805.57, "end": 1805.87, "word": " لسه", "probability": 0.9192708333333334}, {"start": 1805.87, "end": 1806.51, "word": " مثبتين", "probability": 0.88857421875}, {"start": 1806.51, "end": 1806.67, "word": " ان", "probability": 0.89404296875}, {"start": 1806.67, "end": 1806.83, "word": " ال", "probability": 0.8984375}, {"start": 1806.83, "end": 1807.13, "word": " inform", "probability": 0.2303466796875}, {"start": 1807.13, "end": 1807.59, "word": " هذا", "probability": 0.79052734375}, {"start": 1807.59, "end": 1808.21, "word": " بساوي", "probability": 0.787841796875}, {"start": 1808.21, "end": 1808.75, "word": " سفر", "probability": 0.8240559895833334}, {"start": 1808.75, "end": 1809.91, "word": " اذا", "probability": 0.51788330078125}, {"start": 1809.91, "end": 1810.13, "word": " ال", "probability": 0.82177734375}, {"start": 1810.13, "end": 1810.45, "word": " suprem", "probability": 0.57470703125}, {"start": 1810.45, "end": 1811.11, "word": " للست", "probability": 0.755615234375}, {"start": 1811.11, "end": 1811.35, "word": " هذي", "probability": 0.5591634114583334}, {"start": 1811.35, "end": 1811.73, "word": " 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N أصغر لو سوى واحد على جدر ال N لكل N في N", "tokens": [3615, 23942, 10874, 3215, 2288, 2423, 426, 6225, 3215, 3215, 3714, 29245, 3555, 6156, 1211, 2407, 48812, 25513, 2655, 9673, 2655, 3555, 995, 9957, 3660, 29538, 8978, 36764, 24401, 15844, 10874, 3215, 2288, 2423, 426, 6156, 3555, 9566, 1211, 3615, 8608, 6027, 3555, 36764, 24401, 15844, 10874, 3215, 2288, 2423, 426, 5551, 9381, 17082, 2288, 45164, 8608, 2407, 7578, 23565, 426, 15844, 10874, 3215, 2288, 2423, 426, 5551, 9381, 17082, 2288, 45164, 8608, 2407, 7578, 36764, 24401, 15844, 10874, 3215, 2288, 2423, 426, 5296, 28820, 426, 8978, 426], "avg_logprob": -0.1254389018155216, "compression_ratio": 2.007142857142857, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 1861.53, "end": 1861.93, "word": "على", "probability": 0.734375}, {"start": 1861.93, "end": 1862.45, "word": " جدر", "probability": 0.9046223958333334}, {"start": 1862.45, "end": 1862.59, "word": " ال", "probability": 0.91796875}, {"start": 1862.59, "end": 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" N", "probability": 0.986328125}, {"start": 1880.15, "end": 1880.89, "word": " لكل", "probability": 0.972900390625}, {"start": 1880.89, "end": 1881.25, "word": " N", "probability": 0.8359375}, {"start": 1881.25, "end": 1881.45, "word": " في", "probability": 0.93359375}, {"start": 1881.45, "end": 1881.69, "word": " N", "probability": 0.7412109375}], "temperature": 1.0}, {"id": 66, "seek": 190791, "start": 1884.53, "end": 1907.91, "text": "طيب انا عند ال sequence لسه مثبتين احنا هنا في الفرع b ان ال sequence هذه ال limit تبعتها as n tends to infinity بساوة سفر و ال sequence هذه ال limit تبعتها سالب limit واحد على الجدر ال n و سالب واحد بسفر بطلع سفر as n tends to infinity اذا by squeeze theorem", "tokens": [9566, 1829, 3555, 1975, 8315, 43242, 2423, 8310, 5296, 3794, 3224, 50113, 3555, 2655, 9957, 1975, 5016, 8315, 34105, 8978, 27188, 2288, 3615, 272, 16472, 2423, 8310, 29538, 2423, 4948, 6055, 3555, 34268, 11296, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 2407, 3660, 8608, 5172, 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{"start": 1901.75, "end": 1902.15, "word": " بسفر", "probability": 0.80413818359375}, {"start": 1902.15, "end": 1902.55, "word": " بطلع", "probability": 0.8868408203125}, {"start": 1902.55, "end": 1902.91, "word": " سفر", "probability": 0.9763997395833334}, {"start": 1902.91, "end": 1903.31, "word": " as", "probability": 0.94091796875}, {"start": 1903.31, "end": 1903.69, "word": " n", "probability": 0.90234375}, {"start": 1903.69, "end": 1904.01, "word": " tends", "probability": 0.8056640625}, {"start": 1904.01, "end": 1904.63, "word": " to", "probability": 0.96923828125}, {"start": 1904.63, "end": 1905.09, "word": " infinity", "probability": 0.91259765625}, {"start": 1905.09, "end": 1906.51, "word": " اذا", "probability": 0.781982421875}, {"start": 1906.51, "end": 1906.83, "word": " by", "probability": 0.8212890625}, {"start": 1906.83, "end": 1907.43, "word": " squeeze", "probability": 0.405517578125}, {"start": 1907.43, "end": 1907.91, "word": " theorem", "probability": 0.90771484375}], "temperature": 1.0}, {"id": 67, "seek": 193607, "start": 1911.91, "end": 1936.07, "text": "بسكويز تيريم أو sandwich theorem بطلع limit ال sequence اللي لحد تبعها محصور في النص اللي هو cosine n على square root ل n as n times infinity بساوي 7 تمام؟ إذن هذا برهان الجزء دي", "tokens": [3555, 3794, 4117, 45865, 11622, 6055, 13546, 32640, 34051, 11141, 20904, 4724, 9566, 1211, 3615, 4948, 2423, 8310, 13672, 1829, 5296, 24401, 6055, 3555, 3615, 11296, 3714, 5016, 9381, 13063, 8978, 28239, 9381, 13672, 1829, 31439, 23565, 297, 15844, 3732, 5593, 5296, 297, 382, 297, 1413, 13202, 4724, 3794, 995, 45865, 1614, 46811, 10943, 22807, 11933, 8848, 1863, 23758, 4724, 2288, 3224, 7649, 25724, 11622, 38207, 11778, 1829], "avg_logprob": -0.38971918907718384, "compression_ratio": 1.2964824120603016, "no_speech_prob": 0.0, "words": [{"start": 1911.91, "end": 1912.89, "word": "بسكويز", "probability": 0.59267578125}, {"start": 1912.89, "end": 1913.45, "word": " تيريم", "probability": 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2102.13, "end": 2103.75, "word": " infimum", "probability": 0.7586263020833334}, {"start": 2103.75, "end": 2106.73, "word": " لواحد", "probability": 0.739501953125}, {"start": 2106.73, "end": 2106.93, "word": " على", "probability": 0.826171875}, {"start": 2106.93, "end": 2107.49, "word": " جذر", "probability": 0.6029459635416666}, {"start": 2107.49, "end": 2107.79, "word": " n", "probability": 0.4970703125}, {"start": 2107.79, "end": 2108.35, "word": " سالب", "probability": 0.6568196614583334}, {"start": 2108.35, "end": 2108.81, "word": " سفر", "probability": 0.8489583333333334}], "temperature": 1.0}, {"id": 74, "seek": 213637, "start": 2118.07, "end": 2136.37, "text": "هذا الان from أثبتنا أنه بساوي سبر إذا حسب by nested intervals theorem", "tokens": [3224, 15730, 2423, 7649, 490, 5551, 12984, 3555, 2655, 8315, 14739, 3224, 4724, 3794, 995, 45865, 8608, 3555, 2288, 11933, 15730, 11331, 35457, 538, 15646, 292, 26651, 20904], "avg_logprob": -0.5312500164426607, "compression_ratio": 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"end": 2132.45, "word": " intervals", "probability": 0.8759765625}, {"start": 2132.45, "end": 2136.37, "word": " theorem", "probability": 0.7890625}], "temperature": 1.0}, {"id": 75, "seek": 216878, "start": 2139.78, "end": 2168.78, "text": "الـ intersection has unique element has unique element يعني في التقاطع مافيش أنصر وحيد طب احنا قلنا ان السفر ينتمي لتقاطع وتقاطع في أنصر واحد لذا لازم يساوي السفر in a sense", "tokens": [6027, 39184, 15236, 575, 3845, 4478, 575, 3845, 4478, 37495, 22653, 8978, 16712, 4587, 41193, 3615, 19446, 41185, 8592, 14739, 9381, 2288, 4032, 5016, 25708, 23032, 3555, 1975, 5016, 8315, 12174, 1211, 8315, 16472, 21136, 5172, 2288, 7251, 29399, 2304, 1829, 5296, 2655, 4587, 41193, 3615, 4032, 2655, 4587, 41193, 3615, 8978, 14739, 9381, 2288, 36764, 24401, 5296, 15730, 5296, 31377, 2304, 7251, 3794, 995, 45865, 21136, 5172, 2288, 294, 257, 2020], "avg_logprob": -0.2765410991564189, "compression_ratio": 1.593939393939394, "no_speech_prob": 0.0, "words": [{"start": 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{"start": 2154.54, "end": 2155.26, "word": " أنصر", "probability": 0.7730305989583334}, {"start": 2155.26, "end": 2155.66, "word": " وحيد", "probability": 0.698974609375}, {"start": 2155.66, "end": 2156.62, "word": " طب", "probability": 0.857666015625}, {"start": 2156.62, "end": 2156.82, "word": " احنا", "probability": 0.8943684895833334}, {"start": 2156.82, "end": 2157.16, "word": " قلنا", "probability": 0.8525390625}, {"start": 2157.16, "end": 2157.34, "word": " ان", "probability": 0.51806640625}, {"start": 2157.34, "end": 2157.84, "word": " السفر", "probability": 0.8160807291666666}, {"start": 2157.84, "end": 2158.78, "word": " ينتمي", "probability": 0.974609375}, {"start": 2158.78, "end": 2160.6, "word": " لتقاطع", "probability": 0.8025146484375}, {"start": 2160.6, "end": 2161.74, "word": " وتقاطع", "probability": 0.7427734375}, {"start": 2161.74, "end": 2161.9, "word": " في", "probability": 0.81103515625}, {"start": 2161.9, "end": 2162.36, "word": " أنصر", "probability": 0.9728190104166666}, {"start": 2162.36, "end": 2162.86, "word": " واحد", "probability": 0.9853515625}, {"start": 2162.86, "end": 2163.18, "word": " لذا", "probability": 0.4593505859375}, {"start": 2163.18, "end": 2163.5, "word": " لازم", "probability": 0.9430338541666666}, {"start": 2163.5, "end": 2163.82, "word": " يساوي", "probability": 0.8492431640625}, {"start": 2163.82, "end": 2164.3, "word": " السفر", "probability": 0.9728190104166666}, {"start": 2164.3, "end": 2167.96, "word": " in", "probability": 0.1402587890625}, {"start": 2167.96, "end": 2168.06, "word": " a", "probability": 0.83056640625}, {"start": 2168.06, "end": 2168.78, "word": " sense", "probability": 0.89013671875}], "temperature": 1.0}, {"id": 76, "seek": 219361, "start": 2173.33, "end": 2193.61, "text": "بما أن السفر ينتمي للتقاطع يجب أن يكون التقاطع من N بساوي واحد infinity ل I N بساوي بس single tone سفر", "tokens": [3555, 15042, 14739, 21136, 5172, 2288, 7251, 29399, 2304, 1829, 24976, 2655, 4587, 41193, 3615, 7251, 7435, 3555, 14739, 7251, 30544, 16712, 4587, 41193, 3615, 9154, 426, 4724, 3794, 995, 45865, 36764, 24401, 13202, 5296, 286, 426, 4724, 3794, 995, 45865, 4724, 3794, 2167, 8027, 8608, 5172, 2288], "avg_logprob": -0.31473214285714285, "compression_ratio": 1.3064516129032258, "no_speech_prob": 0.0, "words": [{"start": 2173.33, "end": 2174.35, "word": "بما", "probability": 0.873779296875}, {"start": 2174.35, "end": 2174.57, "word": " أن", "probability": 0.71044921875}, {"start": 2174.57, "end": 2175.07, "word": " السفر", "probability": 0.7891438802083334}, {"start": 2175.07, "end": 2175.67, "word": " ينتمي", "probability": 0.896484375}, {"start": 2175.67, "end": 2176.63, "word": " للتقاطع", "probability": 0.89775390625}, {"start": 2176.63, "end": 2181.21, "word": " يجب", "probability": 0.8070475260416666}, {"start": 2181.21, "end": 2185.21, "word": " أن", "probability": 0.64990234375}, {"start": 2185.21, "end": 2185.63, "word": " يكون", "probability": 0.870849609375}, {"start": 2185.63, "end": 2186.37, "word": " التقاطع", "probability": 0.9700927734375}, {"start": 2186.37, "end": 2186.55, "word": " من", "probability": 0.73974609375}, {"start": 2186.55, "end": 2186.83, "word": " N", "probability": 0.2003173828125}, {"start": 2186.83, "end": 2187.31, "word": " بساوي", "probability": 0.6513519287109375}, {"start": 2187.31, "end": 2187.75, "word": " واحد", "probability": 0.801513671875}, {"start": 2187.75, "end": 2188.33, "word": " infinity", "probability": 0.2408447265625}, {"start": 2188.33, "end": 2189.65, "word": " ل", "probability": 0.814453125}, {"start": 2189.65, "end": 2189.99, "word": " I", "probability": 0.556640625}, {"start": 2189.99, "end": 2190.37, "word": " N", "probability": 0.548828125}, {"start": 2190.37, "end": 2191.07, "word": " بساوي", "probability": 0.8807373046875}, {"start": 2191.07, "end": 2191.59, "word": " بس", "probability": 0.667236328125}, {"start": 2191.59, "end": 2192.77, "word": " single", "probability": 0.476806640625}, {"start": 2192.77, "end": 2193.15, "word": " tone", "probability": 0.28466796875}, {"start": 2193.15, "end": 2193.61, "word": " سفر", "probability": 0.8494466145833334}], "temperature": 1.0}, {"id": 77, "seek": 220611, "start": 2199.67, "end": 2206.11, "text": "طبعا ممكن اعطيتكم انا برهان زي هذا بس كان بدل 1 على جدر ال N 1 على N", "tokens": [9566, 3555, 3615, 995, 3714, 43020, 1975, 3615, 9566, 36081, 24793, 1975, 8315, 4724, 2288, 3224, 7649, 30767, 1829, 23758, 4724, 3794, 25961, 47525, 1211, 502, 15844, 10874, 3215, 2288, 2423, 426, 502, 15844, 426], "avg_logprob": -0.1705729237033261, "compression_ratio": 1.2777777777777777, "no_speech_prob": 0.0, "words": [{"start": 2199.67, "end": 2200.11, "word": "طبعا", "probability": 0.90478515625}, {"start": 2200.11, "end": 2200.47, "word": " ممكن", "probability": 0.953369140625}, {"start": 2200.47, "end": 2201.03, "word": " اعطيتكم", "probability": 0.8548828125}, {"start": 2201.03, "end": 2201.21, "word": " انا", "probability": 0.865234375}, {"start": 2201.21, "end": 2201.69, "word": " برهان", "probability": 0.9378662109375}, {"start": 2201.69, "end": 2201.99, "word": " زي", "probability": 0.932373046875}, {"start": 2201.99, "end": 2202.37, "word": " هذا", "probability": 0.93798828125}, {"start": 2202.37, "end": 2202.93, "word": " بس", "probability": 0.860595703125}, {"start": 2202.93, "end": 2203.11, "word": " كان", "probability": 0.96630859375}, {"start": 2203.11, "end": 2203.55, "word": " بدل", "probability": 0.95703125}, {"start": 2203.55, "end": 2204.49, "word": " 1", "probability": 0.440673828125}, {"start": 2204.49, "end": 2204.71, "word": " على", "probability": 0.54638671875}, {"start": 2204.71, "end": 2205.01, "word": " جدر", "probability": 0.92041015625}, {"start": 2205.01, "end": 2205.13, "word": " ال", "probability": 0.451416015625}, {"start": 2205.13, "end": 2205.25, "word": " N", "probability": 0.66650390625}, {"start": 2205.25, "end": 2205.51, "word": " 1", "probability": 0.69873046875}, {"start": 2205.51, "end": 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"end": 2226.6, "word": " بساوي", "probability": 0.76708984375}, {"start": 2226.6, "end": 2227.06, "word": " سفر", "probability": 0.8645833333333334}, {"start": 2227.06, "end": 2227.6, "word": " و", "probability": 0.958984375}, {"start": 2227.6, "end": 2227.88, "word": " عملنا", "probability": 0.9264322916666666}, {"start": 2227.88, "end": 2228.26, "word": " برهان", "probability": 0.94287109375}, {"start": 2228.26, "end": 2229.0, "word": " بالتناقض", "probability": 0.965234375}, {"start": 2229.0, "end": 2230.18, "word": " افرضي", "probability": 0.76025390625}, {"start": 2230.18, "end": 2230.28, "word": " إن", "probability": 0.39111328125}, {"start": 2230.28, "end": 2230.38, "word": " ال", "probability": 0.57568359375}, {"start": 2230.38, "end": 2230.54, "word": " x", "probability": 0.85205078125}, {"start": 2230.54, "end": 2231.1, "word": " مابسويش", "probability": 0.78857421875}, {"start": 2231.1, "end": 2231.48, "word": " سفر", "probability": 0.97314453125}, {"start": 2231.48, "end": 2232.34, "word": " إذا", "probability": 0.653076171875}, {"start": 2232.34, "end": 2232.8, "word": " أكبر", "probability": 0.9529622395833334}, {"start": 2232.8, "end": 2233.0, "word": " من", "probability": 0.99609375}, {"start": 2233.0, "end": 2233.38, "word": " سفر", "probability": 0.9807942708333334}, {"start": 2233.38, "end": 2233.52, "word": " و", "probability": 0.98876953125}, {"start": 2233.52, "end": 2233.88, "word": " وصلنا", "probability": 0.8182779947916666}, {"start": 2233.88, "end": 2234.1, "word": " إلى", "probability": 0.92626953125}, {"start": 2234.1, "end": 2234.66, "word": " تناقض", "probability": 0.9483642578125}, {"start": 2234.66, "end": 2234.9, "word": " by", "probability": 0.79541015625}, {"start": 2234.9, "end": 2235.62, "word": " Archimedean", "probability": 0.91552734375}, {"start": 2235.62, "end": 2236.1, "word": " property", "probability": 0.623046875}], "temperature": 1.0}, {"id": 79, "seek": 226550, "start": 2237.08, "end": 2265.5, "text": "إذا في برهان تاني لكن أنا حبيت أعطيكم البرهان هذا التاني اللي ما خناش زيه طبعا صح إذا أنا عندكم برهانين طيب هاي كمان الجزء بيه من السؤال هذا أنا في عندي sequence xn sequence mr", "tokens": [28814, 15730, 8978, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 44381, 41850, 11331, 21292, 2655, 5551, 3615, 9566, 1829, 24793, 2423, 26890, 3224, 7649, 23758, 16712, 7649, 1829, 13672, 1829, 19446, 16490, 1863, 33599, 30767, 1829, 3224, 23032, 3555, 3615, 995, 20328, 5016, 11933, 15730, 41850, 43242, 24793, 4724, 2288, 3224, 7649, 9957, 23032, 1829, 3555, 8032, 47302, 9122, 2304, 7649, 25724, 11622, 38207, 4724, 1829, 3224, 9154, 21136, 33604, 6027, 23758, 41850, 8978, 18871, 16254, 8310, 2031, 77, 8310, 33660], "avg_logprob": -0.25800304587294415, "compression_ratio": 1.7267441860465116, "no_speech_prob": 0.0, "words": [{"start": 2237.08, "end": 2237.32, "word": "إذا", "probability": 0.40179443359375}, {"start": 2237.32, "end": 2237.5, "word": " في", "probability": 0.5576171875}, {"start": 2237.5, "end": 2237.94, "word": " برهان", "probability": 0.926513671875}, {"start": 2237.94, "end": 2238.28, "word": " تاني", "probability": 0.9412434895833334}, {"start": 2238.28, "end": 2238.52, "word": " لكن", "probability": 0.76416015625}, {"start": 2238.52, "end": 2238.78, "word": " أنا", "probability": 0.56787109375}, {"start": 2238.78, "end": 2239.94, "word": " حبيت", "probability": 0.7328287760416666}, {"start": 2239.94, "end": 2240.74, "word": " أعطيكم", "probability": 0.82314453125}, {"start": 2240.74, "end": 2241.32, "word": " البرهان", "probability": 0.9107666015625}, {"start": 2241.32, "end": 2241.82, "word": " هذا", "probability": 0.74365234375}, {"start": 2241.82, "end": 2242.26, "word": " التاني", "probability": 0.9762369791666666}, {"start": 2242.26, "end": 2242.38, "word": " اللي", "probability": 0.883544921875}, {"start": 2242.38, "end": 2242.5, "word": " ما", "probability": 0.73974609375}, {"start": 2242.5, "end": 2242.92, "word": " خناش", "probability": 0.6118977864583334}, {"start": 2242.92, "end": 2243.44, "word": " زيه", "probability": 0.9853515625}, {"start": 2243.44, "end": 2246.84, "word": " طبعا", "probability": 0.83380126953125}, {"start": 2246.84, "end": 2250.92, "word": " صح", "probability": 0.83251953125}, {"start": 2250.92, "end": 2251.62, "word": " إذا", "probability": 0.5316162109375}, {"start": 2251.62, "end": 2251.78, "word": " أنا", "probability": 0.2198486328125}, {"start": 2251.78, "end": 2252.2, "word": " عندكم", "probability": 0.978271484375}, {"start": 2252.2, "end": 2253.48, "word": " برهانين", "probability": 0.95693359375}, {"start": 2253.48, "end": 2257.34, "word": " طيب", "probability": 0.8763020833333334}, {"start": 2257.34, "end": 2257.54, "word": " هاي", "probability": 0.640380859375}, {"start": 2257.54, "end": 2258.12, "word": " كمان", "probability": 0.9861653645833334}, {"start": 2258.12, "end": 2259.74, "word": " الجزء", "probability": 0.9186197916666666}, {"start": 2259.74, "end": 2260.06, "word": " بيه", "probability": 0.7696940104166666}, {"start": 2260.06, "end": 2260.18, "word": " من", "probability": 0.9755859375}, {"start": 2260.18, "end": 2260.66, "word": " السؤال", "probability": 0.9959309895833334}, {"start": 2260.66, "end": 2261.2, "word": " هذا", "probability": 0.94921875}, {"start": 2261.2, "end": 2262.2, "word": " أنا", "probability": 0.61279296875}, {"start": 2262.2, "end": 2262.38, "word": " في", "probability": 0.927734375}, {"start": 2262.38, "end": 2262.7, "word": " عندي", "probability": 0.843994140625}, {"start": 2262.7, "end": 2263.24, "word": " sequence", "probability": 0.955078125}, {"start": 2263.24, "end": 2264.02, "word": " xn", "probability": 0.4530029296875}, {"start": 2264.02, "end": 2264.8, "word": " sequence", "probability": 0.869140625}, {"start": 2264.8, "end": 2265.5, "word": " mr", "probability": 0.80419921875}], "temperature": 1.0}, {"id": 80, "seek": 228821, "start": 2266.72, "end": 2288.22, "text": "وعندي xn أصغر من أو ساوي سفر لكل n وبدنا نثبت إنه إذا كان limit xn بساوي x then ال x بتطلع أيضا أصغر من أو ساوي سفر", "tokens": [2407, 3615, 1863, 16254, 2031, 77, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 8608, 5172, 2288, 5296, 28820, 297, 4032, 44510, 8315, 8717, 12984, 3555, 2655, 36145, 3224, 11933, 15730, 25961, 4948, 2031, 77, 4724, 3794, 995, 45865, 2031, 550, 2423, 2031, 39894, 9566, 1211, 3615, 36632, 11242, 995, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.139282223302871, "compression_ratio": 1.4724409448818898, "no_speech_prob": 0.0, "words": [{"start": 2266.72, "end": 2267.56, "word": "وعندي", "probability": 0.755615234375}, {"start": 2267.56, "end": 2268.36, "word": " xn", "probability": 0.624755859375}, {"start": 2268.36, "end": 2269.04, "word": " أصغر", "probability": 0.959716796875}, {"start": 2269.04, "end": 2269.22, "word": " من", "probability": 0.982421875}, {"start": 2269.22, "end": 2269.42, "word": " أو", "probability": 0.951171875}, {"start": 2269.42, "end": 2269.82, "word": " ساوي", "probability": 0.849609375}, {"start": 2269.82, "end": 2270.3, "word": " سفر", "probability": 0.76953125}, {"start": 2270.3, "end": 2270.84, "word": " لكل", "probability": 0.982666015625}, {"start": 2270.84, "end": 2271.26, "word": " n", "probability": 0.67626953125}, {"start": 2271.26, "end": 2273.56, "word": " وبدنا", "probability": 0.74267578125}, {"start": 2273.56, "end": 2274.2, "word": " نثبت", "probability": 0.994873046875}, {"start": 2274.2, "end": 2275.88, "word": " إنه", "probability": 0.6751708984375}, {"start": 2275.88, "end": 2278.86, "word": " إذا", "probability": 0.883056640625}, {"start": 2278.86, "end": 2279.24, "word": " كان", "probability": 0.994140625}, {"start": 2279.24, "end": 2279.66, "word": " limit", "probability": 0.9375}, {"start": 2279.66, "end": 2281.44, "word": " xn", "probability": 0.94970703125}, {"start": 2281.44, "end": 2282.26, "word": " بساوي", "probability": 0.840576171875}, {"start": 2282.26, "end": 2282.72, "word": " x", "probability": 0.97412109375}, {"start": 2282.72, "end": 2283.14, "word": " then", "probability": 0.56201171875}, {"start": 2283.14, "end": 2285.36, "word": " ال", "probability": 0.8564453125}, {"start": 2285.36, "end": 2285.64, "word": " x", "probability": 0.73486328125}, {"start": 2285.64, "end": 2286.16, "word": " بتطلع", "probability": 0.96337890625}, {"start": 2286.16, "end": 2286.62, "word": " أيضا", "probability": 0.81689453125}, {"start": 2286.62, "end": 2287.14, "word": " أصغر", "probability": 0.9912109375}, {"start": 2287.14, "end": 2287.32, "word": " من", "probability": 0.9912109375}, {"start": 2287.32, "end": 2287.5, "word": " أو", "probability": 0.99169921875}, {"start": 2287.5, "end": 2287.86, "word": " ساوي", "probability": 0.95703125}, {"start": 2287.86, "end": 2288.22, "word": " سفر", "probability": 0.9654947916666666}], "temperature": 1.0}, {"id": 81, "seek": 231365, "start": 2289.55, "end": 2313.65, "text": "أنا في عندى sequence of real numbers كل حدودها غير ثالثة و ال sequence converge ل X بالدفعة ان ال limit ايضا بتطلع غير ثالثة فهي البرهان proof برهان بالتناقض assume on contrary", "tokens": [10721, 8315, 8978, 43242, 7578, 8310, 295, 957, 3547, 28242, 11331, 3215, 23328, 11296, 32771, 13546, 38637, 6027, 12984, 3660, 4032, 2423, 8310, 41881, 5296, 1783, 20666, 3215, 5172, 27884, 16472, 2423, 4948, 1975, 1829, 11242, 995, 39894, 9566, 1211, 3615, 32771, 13546, 38637, 6027, 12984, 3660, 6156, 3224, 1829, 2423, 26890, 3224, 7649, 8177, 4724, 2288, 3224, 7649, 20666, 2655, 8315, 4587, 11242, 6552, 322, 19506], "avg_logprob": -0.26378677215646296, "compression_ratio": 1.4382022471910112, "no_speech_prob": 0.0, "words": [{"start": 2289.55, "end": 2290.67, "word": "أنا", "probability": 0.5367431640625}, {"start": 2290.67, "end": 2290.81, "word": " في", "probability": 0.7666015625}, {"start": 2290.81, "end": 2291.07, "word": " عندى", "probability": 0.45361328125}, {"start": 2291.07, "end": 2291.39, "word": " sequence", "probability": 0.92333984375}, {"start": 2291.39, "end": 2291.73, "word": " of", "probability": 0.94580078125}, {"start": 2291.73, "end": 2291.95, "word": " real", "probability": 0.96923828125}, {"start": 2291.95, "end": 2292.53, "word": " numbers", "probability": 0.814453125}, {"start": 2292.53, "end": 2293.09, "word": " كل", "probability": 0.580078125}, {"start": 2293.09, "end": 2293.63, "word": " حدودها", "probability": 0.991455078125}, {"start": 2293.63, "end": 2293.91, "word": " غير", "probability": 0.99658203125}, {"start": 2293.91, "end": 2294.61, "word": " ثالثة", "probability": 0.7718505859375}, {"start": 2294.61, "end": 2295.23, "word": " و", "probability": 0.427490234375}, {"start": 2295.23, "end": 2295.35, "word": " ال", "probability": 0.7900390625}, {"start": 2295.35, "end": 2295.67, "word": " sequence", "probability": 0.97509765625}, {"start": 2295.67, "end": 2296.27, "word": " converge", "probability": 0.88916015625}, {"start": 2296.27, "end": 2296.43, "word": " ل", "probability": 0.8603515625}, {"start": 2296.43, "end": 2296.73, "word": " X", "probability": 0.58740234375}, {"start": 2296.73, "end": 2297.49, "word": " بالدفعة", "probability": 0.51190185546875}, {"start": 2297.49, "end": 2297.75, "word": " ان", "probability": 0.2449951171875}, {"start": 2297.75, "end": 2297.93, "word": " ال", "probability": 0.5029296875}, {"start": 2297.93, "end": 2298.33, "word": " limit", "probability": 0.95703125}, {"start": 2298.33, "end": 2299.17, "word": " ايضا", "probability": 0.875732421875}, {"start": 2299.17, "end": 2299.65, "word": " بتطلع", "probability": 0.9151611328125}, {"start": 2299.65, "end": 2300.01, "word": " غير", "probability": 0.99658203125}, {"start": 2300.01, "end": 2300.83, "word": " ثالثة", "probability": 0.978515625}, {"start": 2300.83, "end": 2301.93, "word": " فهي", "probability": 0.8346354166666666}, {"start": 2301.93, "end": 2302.55, "word": " البرهان", "probability": 0.909423828125}, {"start": 2302.55, "end": 2303.19, "word": " proof", "probability": 0.4658203125}, {"start": 2303.19, "end": 2307.85, "word": " برهان", "probability": 0.830078125}, {"start": 2307.85, "end": 2308.67, "word": " بالتناقض", "probability": 0.972265625}, {"start": 2308.67, "end": 2309.33, "word": " assume", "probability": 0.7353515625}, {"start": 2309.33, "end": 2312.93, "word": " on", "probability": 0.8955078125}, {"start": 2312.93, "end": 2313.65, "word": " contrary", "probability": 0.869140625}], "temperature": 1.0}, {"id": 82, "seek": 233778, "start": 2316.4, "end": 2337.78, "text": "إن X أكبر من سفر و بدنا نصل إلى تناقض افرض إن X أكبر من سفر طيب let epsilon بساوي X ع 2 هذا بطل عدل موجب", "tokens": [28814, 1863, 1783, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 4032, 47525, 8315, 8717, 36520, 30731, 6055, 8315, 4587, 11242, 1975, 5172, 43042, 36145, 1783, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 23032, 1829, 3555, 718, 17889, 4724, 3794, 995, 45865, 1783, 6225, 568, 23758, 4724, 9566, 1211, 6225, 3215, 1211, 3714, 29245, 3555], "avg_logprob": -0.28096591992811726, "compression_ratio": 1.3658536585365855, "no_speech_prob": 0.0, "words": [{"start": 2316.4, "end": 2316.7, "word": "إن", "probability": 0.6688232421875}, {"start": 2316.7, "end": 2317.12, "word": " X", "probability": 0.5810546875}, {"start": 2317.12, "end": 2317.58, "word": " أكبر", "probability": 0.9095052083333334}, {"start": 2317.58, "end": 2317.8, "word": " من", "probability": 0.99267578125}, {"start": 2317.8, "end": 2318.22, "word": " سفر", "probability": 0.73974609375}, {"start": 2318.22, "end": 2319.56, "word": " و", "probability": 0.533203125}, {"start": 2319.56, "end": 2320.12, "word": " بدنا", "probability": 0.6639404296875}, {"start": 2320.12, "end": 2320.56, "word": " نصل", "probability": 0.971435546875}, {"start": 2320.56, "end": 2320.92, "word": " إلى", "probability": 0.86474609375}, {"start": 2320.92, "end": 2322.42, "word": " تناقض", "probability": 0.9573974609375}, {"start": 2322.42, "end": 2325.86, "word": " افرض", "probability": 0.7652180989583334}, {"start": 2325.86, "end": 2326.1, "word": " إن", "probability": 0.220703125}, {"start": 2326.1, "end": 2326.5, "word": " X", "probability": 0.935546875}, {"start": 2326.5, "end": 2326.98, "word": " أكبر", "probability": 0.9703776041666666}, {"start": 2326.98, "end": 2327.22, "word": " من", "probability": 0.9970703125}, {"start": 2327.22, "end": 2328.04, "word": " سفر", "probability": 0.9676106770833334}, {"start": 2328.04, "end": 2329.9, "word": " طيب", "probability": 0.7284342447916666}, {"start": 2329.9, "end": 2332.58, "word": " let", "probability": 0.708984375}, {"start": 2332.58, "end": 2333.86, "word": " epsilon", "probability": 0.2449951171875}, {"start": 2333.86, "end": 2335.3, "word": " بساوي", "probability": 0.7298583984375}, {"start": 2335.3, "end": 2335.64, "word": " X", "probability": 0.845703125}, {"start": 2335.64, "end": 2335.78, "word": " ع", "probability": 0.6064453125}, {"start": 2335.78, "end": 2336.28, "word": " 2", "probability": 0.560546875}, {"start": 2336.28, "end": 2336.62, "word": " هذا", "probability": 0.54248046875}, {"start": 2336.62, "end": 2336.94, "word": " بطل", "probability": 0.7449544270833334}, {"start": 2336.94, "end": 2337.26, "word": " عدل", "probability": 0.7701822916666666}, {"start": 2337.26, "end": 2337.78, "word": " موجب", "probability": 0.9842122395833334}], "temperature": 1.0}, {"id": 83, "seek": 235046, "start": 2339.4, "end": 2350.46, "text": "بما أنه since .. since xn converges ل x إذا يوجد capital N يعتمد على إبسلون", "tokens": [3555, 15042, 14739, 3224, 1670, 4386, 1670, 2031, 77, 9652, 2880, 5296, 2031, 11933, 15730, 7251, 29245, 3215, 4238, 426, 7251, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536], "avg_logprob": -0.22442035617366915, "compression_ratio": 1.0098039215686274, "no_speech_prob": 0.0, "words": [{"start": 2339.4, "end": 2339.88, "word": "بما", "probability": 0.9853515625}, {"start": 2339.88, "end": 2340.32, "word": " أنه", "probability": 0.681640625}, {"start": 2340.32, 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0.460693359375}, {"start": 2446.97, "end": 2447.53, "word": " بتناقض", "probability": 0.7882080078125}, {"start": 2447.53, "end": 2447.71, "word": " مع", "probability": 0.990234375}, {"start": 2447.71, "end": 2448.19, "word": " الفرض", "probability": 0.943115234375}, {"start": 2448.19, "end": 2450.03, "word": " تمام؟", "probability": 0.6637369791666666}, {"start": 2450.03, "end": 2450.25, "word": " اذا", "probability": 0.7342529296875}, {"start": 2450.25, "end": 2450.59, "word": " هذا", "probability": 0.401123046875}, {"start": 2450.59, "end": 2450.91, "word": " سبب", "probability": 0.89697265625}, {"start": 2450.91, "end": 2451.57, "word": " التناقض", "probability": 0.9864501953125}, {"start": 2451.57, "end": 2452.09, "word": " هذا", "probability": 0.84521484375}, {"start": 2452.09, "end": 2452.91, "word": " هو", "probability": 0.9892578125}, {"start": 2452.91, "end": 2453.07, "word": " ال", "probability": 0.97802734375}, {"start": 2453.07, "end": 2453.69, "word": " assumption", "probability": 0.9375}, {"start": 2453.69, "end": 2454.53, "word": " تبعنا", "probability": 0.940185546875}, {"start": 2454.53, "end": 2454.97, "word": " ان", "probability": 0.892578125}, {"start": 2454.97, "end": 2455.55, "word": " x", "probability": 0.62744140625}, {"start": 2455.55, "end": 2456.29, "word": " أكبر", "probability": 0.82275390625}, {"start": 2456.29, "end": 2456.71, "word": " من", "probability": 0.994140625}, {"start": 2456.71, "end": 2457.43, "word": " السفر", "probability": 0.7254638671875}, {"start": 2457.43, "end": 2459.55, "word": " okay؟", "probability": 0.60595703125}, {"start": 2459.55, "end": 2459.87, "word": " اذا", "probability": 0.87841796875}, {"start": 2459.87, "end": 2460.65, "word": " الصح", "probability": 0.9716796875}, {"start": 2460.65, "end": 2461.05, "word": " ان", "probability": 0.86572265625}, {"start": 2461.05, "end": 2461.53, "word": " x", "probability": 0.9169921875}, {"start": 2461.53, "end": 2462.17, "word": " أصغر", "probability": 0.9373779296875}, {"start": 2462.17, "end": 2462.41, "word": " من", "probability": 0.96337890625}, {"start": 2462.41, "end": 2462.65, "word": " أو", "probability": 0.71826171875}, {"start": 2462.65, "end": 2463.03, "word": " ساوي", "probability": 0.7958984375}, {"start": 2463.03, "end": 2463.49, "word": " سفر", "probability": 0.9357096354166666}, {"start": 2463.49, "end": 2464.73, "word": " وهو", "probability": 0.81005859375}, {"start": 2464.73, "end": 2465.49, "word": " المطلوب", "probability": 0.9365234375}], "temperature": 1.0}, {"id": 88, "seek": 249583, "start": 2468.07, "end": 2495.83, "text": "Okay تمام اذا يعني هذه يعني بعض الأسئلة الاسئلة الرابعة هذا نظرية أخدناها و اللي بعدي أعتقد حلناها في المحاضرة فهواجف هنا و هيك بنكون يعني راجعنا تقريبا امتحان", "tokens": [8297, 46811, 10943, 1975, 15730, 37495, 22653, 29538, 37495, 22653, 45030, 11242, 16247, 3794, 19986, 37977, 2423, 32277, 19986, 37977, 34892, 16758, 27884, 23758, 8717, 19913, 2288, 10632, 5551, 9778, 3215, 8315, 11296, 4032, 13672, 1829, 45030, 16254, 5551, 34268, 28543, 11331, 1211, 8315, 11296, 8978, 9673, 5016, 46958, 25720, 6156, 3224, 2407, 26108, 5172, 34105, 4032, 39896, 4117, 44945, 30544, 37495, 22653, 12602, 26108, 3615, 8315, 6055, 4587, 16572, 3555, 995, 1975, 2304, 2655, 5016, 7649], "avg_logprob": -0.1512419879436493, "compression_ratio": 1.6923076923076923, "no_speech_prob": 0.0, "words": [{"start": 2468.07, "end": 2468.51, "word": "Okay", "probability": 0.92138671875}, {"start": 2468.51, "end": 2469.09, "word": " تمام", "probability": 0.8955078125}, {"start": 2469.09, "end": 2470.77, "word": " اذا", "probability": 0.57611083984375}, {"start": 2470.77, "end": 2471.51, "word": " يعني", "probability": 0.888427734375}, {"start": 2471.51, "end": 2471.95, "word": " هذه", "probability": 0.7392578125}, {"start": 2471.95, "end": 2472.45, "word": " يعني", "probability": 0.85205078125}, {"start": 2472.45, "end": 2472.97, "word": " بعض", "probability": 0.984130859375}, {"start": 2472.97, "end": 2473.73, "word": " الأسئلة", "probability": 0.9307861328125}, {"start": 2473.73, "end": 2476.71, "word": " الاسئلة", "probability": 0.74078369140625}, {"start": 2476.71, "end": 2477.17, "word": " الرابعة", "probability": 0.8849283854166666}, {"start": 2477.17, "end": 2477.45, "word": " هذا", "probability": 0.3642578125}, {"start": 2477.45, "end": 2478.17, "word": " نظرية", "probability": 0.9813232421875}, {"start": 2478.17, "end": 2480.01, "word": " أخدناها", "probability": 0.94638671875}, {"start": 2480.01, "end": 2481.61, "word": " و", "probability": 0.72314453125}, {"start": 2481.61, "end": 2481.89, "word": " اللي", "probability": 0.894287109375}, {"start": 2481.89, "end": 2482.43, "word": " بعدي", "probability": 0.72021484375}, {"start": 2482.43, "end": 2483.77, "word": " أعتقد", "probability": 0.8912760416666666}, {"start": 2483.77, "end": 2485.63, "word": " حلناها", "probability": 0.889892578125}, {"start": 2485.63, "end": 2485.73, "word": " في", "probability": 0.943359375}, {"start": 2485.73, "end": 2486.61, "word": " المحاضرة", "probability": 0.9901123046875}, {"start": 2486.61, "end": 2488.61, "word": " فهواجف", "probability": 0.8283203125}, {"start": 2488.61, "end": 2488.93, "word": " هنا", "probability": 0.99755859375}, {"start": 2488.93, "end": 2490.27, "word": " و", "probability": 0.755859375}, {"start": 2490.27, "end": 2491.19, "word": " هيك", "probability": 0.966552734375}, {"start": 2491.19, "end": 2491.63, "word": " بنكون", "probability": 0.873046875}, {"start": 2491.63, "end": 2491.99, "word": " يعني", "probability": 0.97998046875}, {"start": 2491.99, "end": 2494.49, "word": " راجعنا", "probability": 0.929443359375}, {"start": 2494.49, "end": 2495.19, "word": " تقريبا", "probability": 0.96240234375}, {"start": 2495.19, "end": 2495.83, "word": " امتحان", "probability": 0.90859375}], "temperature": 1.0}, {"id": 89, "seek": 250169, "start": 2496.57, "end": 2501.69, "text": "للمتحانة النصفة ونشوفكم ان شاء الله بقرا", "tokens": [1211, 19528, 2655, 5016, 7649, 3660, 28239, 9381, 5172, 3660, 4032, 1863, 8592, 38688, 24793, 16472, 13412, 16606, 21984, 4724, 4587, 23557], "avg_logprob": -0.2083899482436802, "compression_ratio": 1.0724637681159421, "no_speech_prob": 0.0, "words": [{"start": 2496.57, "end": 2497.39, "word": "للمتحانة", "probability": 0.7505696614583334}, {"start": 2497.39, "end": 2498.03, "word": " النصفة", "probability": 0.9246826171875}, {"start": 2498.03, "end": 2500.83, "word": " ونشوفكم", "probability": 0.8611328125}, {"start": 2500.83, "end": 2500.97, "word": " ان", "probability": 0.78955078125}, {"start": 2500.97, "end": 2501.15, "word": " شاء", "probability": 0.988037109375}, {"start": 2501.15, "end": 2501.29, "word": " الله", "probability": 0.9697265625}, {"start": 2501.29, "end": 2501.69, "word": " بقرا", "probability": 0.73291015625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2502.60025, "duration_after_vad": 2243.446249999994}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/a-utq7LmSIM_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/a-utq7LmSIM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..74b9d594f1d94dea11f0cad4816dec3809e6fc2e
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/a-utq7LmSIM_raw.srt
@@ -0,0 +1,1408 @@
+1
+00:00:21,060 --> 00:00:27,740
+اليوم ان شاء الله هنحاول نحل امتحان نصفي سابق
+
+2
+00:00:27,740 --> 00:00:32,680
+كمراجعة للامتحان النصفي الأول اللي هناخده ان شاء
+
+3
+00:00:32,680 --> 00:00:40,040
+الله غدا اول سؤال في الامتحان هذا عبارة عن سؤال
+
+4
+00:00:40,040 --> 00:00:45,120
+true or false اذا العبارة صح فبنعلم عليها صح او
+
+5
+00:00:45,120 --> 00:00:52,670
+true واذا خطأ بنعلم عليها خطأ فأول عبارةبتقول لإن
+
+6
+00:00:52,670 --> 00:00:58,250
+ال absolute value ل X سالب Y بساوي absolute X سالب
+
+7
+00:00:58,250 --> 00:01:05,030
+absolute Y وحتى
+
+8
+00:01:05,030 --> 00:01:09,810
+لو كانت تلاقي إشارة موجبة، لو سمحتوا ما تتكلميش
+
+9
+00:01:09,810 --> 00:01:16,710
+إلا لما ترفع إيدك و إعزالك هه هل هذا الكلام صحيح
+
+10
+00:01:16,710 --> 00:01:25,300
+لكلX وY أنتمي إلى R فالعبارة هذه false ليست صحيحة
+
+11
+00:01:25,300 --> 00:01:32,380
+ممكن نجيب أكتر من counter example صح العبارة
+
+12
+00:01:32,380 --> 00:01:36,300
+التانية لو كان S subset of the set of all real
+
+13
+00:01:36,300 --> 00:01:42,420
+numbers is finite لو
+
+14
+00:01:42,420 --> 00:01:48,840
+كانت ال set هذه finiteفهذا بيقدّي أن ال supremum S
+
+15
+00:01:48,840 --> 00:01:59,400
+و ال infimum S ينتمي لل set S هل هذا true؟ هل لما
+
+16
+00:01:59,400 --> 00:02:03,420
+تكون ال set finite ال supremum تبعها و ال infimum
+
+17
+00:02:03,420 --> 00:02:08,840
+تبعها ينتمي إلها؟ هذا صحيح كان تمرين exercise و
+
+18
+00:02:08,840 --> 00:02:11,380
+برهنة، إذن هذا true
+
+19
+00:02:15,490 --> 00:02:22,250
+تلاتة لو أعرفت ال set I of S على إنها the set of
+
+20
+00:02:22,250 --> 00:02:34,870
+all V حيث V بساوي infimum ال set S فال
+
+21
+00:02:34,870 --> 00:02:40,430
+set هذه contains contains
+
+22
+00:02:40,430 --> 00:02:44,010
+more than one element more than
+
+23
+00:02:46,860 --> 00:02:52,280
+one element المجموعة
+
+24
+00:02:52,280 --> 00:02:58,000
+هذه تحتوي على أكتر من عنصر، هل هذا صحيح؟ هل الست
+
+25
+00:02:58,000 --> 00:03:01,860
+ممكن يكون إلها أكتر من infamous؟ لأ، لأ، لأ، لأ،
+
+26
+00:03:01,860 --> 00:03:02,560
+لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ،
+
+27
+00:03:02,560 --> 00:03:07,020
+لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ،
+
+28
+00:03:07,020 --> 00:03:07,020
+لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ،
+
+29
+00:03:07,020 --> 00:03:07,100
+لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ، لأ،
+
+30
+00:03:07,100 --> 00:03:10,040
+لأ، لأ، لأ، لأ، لأ، لأ، ل
+
+31
+00:03:13,960 --> 00:03:19,580
+العبارة الرابعة every monotone sequence converges
+
+32
+00:03:19,580 --> 00:03:25,980
+if and only if it is bounded هذه
+
+33
+00:03:25,980 --> 00:03:28,780
+عبارة عن الـ monotone convergence theorem فهذه
+
+34
+00:03:28,780 --> 00:03:34,620
+true ال
+
+35
+00:03:34,620 --> 00:03:40,020
+sequence سالب
+
+36
+00:03:40,020 --> 00:03:55,840
+واحد أس N على Nإن ينتمي لإن is convergent هل
+
+37
+00:03:55,840 --> 00:03:56,760
+هذا صحيح؟
+
+38
+00:04:06,660 --> 00:04:12,140
+طيب ماشي المشكلة أن الأقلام السودة اللي عندي كلها
+
+39
+00:04:12,140 --> 00:04:22,160
+صارت فاتعة اه في واحد جبت انت و
+
+40
+00:04:22,160 --> 00:04:32,480
+الله انا مش كتير يعني
+
+41
+00:04:32,480 --> 00:04:36,020
+هدم يعني احسن نشوف يعني
+
+42
+00:04:42,570 --> 00:04:47,330
+فال sequence هذه convergent هل هذا صحيح ولا خطأ
+
+43
+00:04:47,330 --> 00:04:54,790
+هذا true و
+
+44
+00:04:54,790 --> 00:05:00,050
+لو بدنا نبرهن الكلام هذا و بالمناسبة ال sequence
+
+45
+00:05:00,050 --> 00:05:06,410
+هذه converge لصفر converge و ال limit تبعتها صفر
+
+46
+00:05:07,900 --> 00:05:14,940
+ليه؟ لأنه هاي المسافة بين ال inf term لحد انه ليه؟
+
+47
+00:05:14,940 --> 00:05:23,620
+سالب واحد أس ان على n سالب سفر ايش
+
+48
+00:05:23,620 --> 00:05:27,520
+هاد بالساوي؟
+
+49
+00:05:27,520 --> 00:05:28,320
+بتساوي
+
+50
+00:05:37,950 --> 00:05:45,590
+بتساوي واحد على ان صح؟ نصبوت؟ وهذا أصغر من اذا كان
+
+51
+00:05:45,590 --> 00:05:54,410
+واحد ضرب واحد على ان هذا عدد موجب واحد على ان تقول
+
+52
+00:05:54,410 --> 00:06:05,090
+للسفر إذن حسب نظرية سابقة رقمها
+
+53
+00:06:05,090 --> 00:06:11,610
+كان في النقص اتنين اربعةبطلع limit xn بساوي سفر
+
+54
+00:06:11,610 --> 00:06:15,190
+limit سالب
+
+55
+00:06:15,190 --> 00:06:20,750
+واحد بس n على n لما n تقوى ل infinity بساوي سفر
+
+56
+00:06:20,750 --> 00:06:25,230
+إذا
+
+57
+00:06:25,230 --> 00:06:28,650
+ال sequence هذي convergent إذا العبارة هذه true
+
+58
+00:06:28,650 --> 00:06:31,650
+طيب عبارة تانية
+
+59
+00:06:45,940 --> 00:06:55,320
+product of two divergent sequences
+
+60
+00:06:55,320 --> 00:07:03,700
+is divergent هل
+
+61
+00:07:03,700 --> 00:07:05,280
+هذا true ولا false؟
+
+62
+00:07:08,090 --> 00:07:10,990
+العبارة هادف false طب لما قلنا لكوا جيبوا counter
+
+63
+00:07:10,990 --> 00:07:17,310
+example ممكن تجيب سالب واحد اثنان وكمان ال
+
+64
+00:07:17,310 --> 00:07:23,290
+sequence التانية تكون سالب واحد اثنان او
+
+65
+00:07:23,290 --> 00:07:26,830
+ممكن تكون سالب واحد اثنان زائد واحد نفس الشغل
+
+66
+00:07:26,830 --> 00:07:32,390
+بتطلع convergent ال productهذه الـ divergent وهذه
+
+67
+00:07:32,390 --> 00:07:38,210
+الـ divergent لكن X in في Y in هساوي ال sequence
+
+68
+00:07:38,210 --> 00:07:44,650
+سلب واحد واست اتنين in لما
+
+69
+00:07:44,650 --> 00:07:49,370
+نضربهم في بعض فهذا بيعطيني ال sequence ثابت واحد
+
+70
+00:07:49,370 --> 00:07:55,730
+وهذه converge لواحد اذا هذه في عندي example of two
+
+71
+00:07:55,730 --> 00:08:01,090
+divergent sequences لكن حصل ضربهم بيطلعconvergent
+
+72
+00:08:01,090 --> 00:08:06,230
+وليس divergent طيب
+
+73
+00:08:06,230 --> 00:08:21,470
+لو كانت S bounded S bounded subset of R وS0
+
+74
+00:08:21,470 --> 00:08:28,510
+subset من S هل هذا بيقدّي ان انفمم
+
+75
+00:08:29,760 --> 00:08:42,480
+S0 subset أصغر من أو يساوي انفمه من ال S هل
+
+76
+00:08:42,480 --> 00:08:51,020
+هذه العبارة صحيحة؟ أيه
+
+77
+00:08:51,020 --> 00:08:59,320
+رأيكم؟ العبارة هذه falseلما مجموعة تصغر ال inform
+
+78
+00:08:59,320 --> 00:09:07,940
+تباعها بيكبر، لكن اللي بيكون صح أنه ال supremum لو
+
+79
+00:09:07,940 --> 00:09:12,600
+أخدت ال supremum للمجموعة الجزئية S0 فهذا بيطلع
+
+80
+00:09:12,600 --> 00:09:17,040
+أصغر من أو ساوي ال supremum للمجموع S، هذه العبارة
+
+81
+00:09:17,040 --> 00:09:17,320
+true
+
+82
+00:09:20,870 --> 00:09:25,710
+ال supremum لما المجموعه تكبر بيكبر لكن ال infimum
+
+83
+00:09:25,710 --> 00:09:38,110
+لما المجموعه تكبر بيصغر عكس بعض طيب
+
+84
+00:09:43,200 --> 00:09:50,100
+لو كانت x in sequence of positive real numbers و
+
+85
+00:09:50,100 --> 00:10:02,060
+limit x in plus one over x in بساوي واحد فهذا
+
+86
+00:10:02,060 --> 00:10:08,180
+بيقدي ان ال sequence x in diverges
+
+87
+00:10:08,180 --> 00:10:15,870
+هل هذه العبارة صح ولا خطأ؟العبارة هذه خطأ لأنه
+
+88
+00:10:15,870 --> 00:10:22,230
+احنا فينا في تمرين سبتاشر في section تلاتة اتنين
+
+89
+00:10:22,230 --> 00:10:26,230
+بقول اذا كانت ال sequence .. اذا كان ال limit
+
+90
+00:10:26,230 --> 00:10:29,630
+ratio هذا بساوي واحد فال sequence ممكن تكون
+
+91
+00:10:29,630 --> 00:10:33,890
+convergent او divergent وفي السؤال هذا اعطينا
+
+92
+00:10:33,890 --> 00:10:41,240
+مثلين واحد ل sequence convergentال ratio limit ال
+
+93
+00:10:41,240 --> 00:10:45,100
+ratio تبعها بيساوي واحد لكنها convergent ومثال
+
+94
+00:10:45,100 --> 00:10:50,300
+تاني ل sequence limit ال ratio تبعها أيضا بيساوي
+
+95
+00:10:50,300 --> 00:10:53,700
+واحد لكنها divergent إذا لو كان limit ال ratio
+
+96
+00:10:53,700 --> 00:10:57,100
+بيساوي واحد فال sequence إما converge أو divergent
+
+97
+00:10:57,100 --> 00:11:01,400
+مالضايش نجزم إنها convergent أو نجزم إنها
+
+98
+00:11:01,400 --> 00:11:07,020
+divergent okay إذا العبارة هذه خطأ أو false إذا
+
+99
+00:11:07,020 --> 00:11:13,950
+العبارة هذه falseلو بدلت divergent بconvergent
+
+100
+00:11:13,950 --> 00:11:21,530
+برضه false الصح انها may converge or may diverge
+
+101
+00:11:21,530 --> 00:11:26,810
+طيب
+
+102
+00:11:26,810 --> 00:11:43,910
+العبارة الأخيرة تسعةأي open interval
+
+103
+00:11:43,910 --> 00:11:47,950
+a وb تحتوي
+
+104
+00:11:47,950 --> 00:11:49,970
+rational number R
+
+105
+00:11:54,840 --> 00:11:59,220
+في rational number محصور من a وb يعني ال open
+
+106
+00:11:59,220 --> 00:12:03,920
+interval تحتوي ال R كذلك في نتيجة ال density
+
+107
+00:12:03,920 --> 00:12:08,300
+theorem أي open interval زي هذه تحتوي ال rational
+
+108
+00:12:08,300 --> 00:12:13,780
+إذن هذا الكلام صحيح
+
+109
+00:12:13,780 --> 00:12:18,340
+إذن هذا أول سؤال صح وخطأ السؤال التاني
+
+110
+00:12:33,670 --> 00:12:50,070
+Question 2 اذا
+
+111
+00:12:50,070 --> 00:12:55,770
+احنا
+
+112
+00:12:55,770 --> 00:13:01,210
+بنحل هذا امتحان هذا امتحان Med
+
+113
+00:13:03,740 --> 00:13:11,660
+mid term one التاريخ
+
+114
+00:13:11,660 --> 00:13:24,780
+تبعه اتناش تلاتة الفين و تلاتاش السؤال
+
+115
+00:13:24,780 --> 00:13:32,630
+او الفرع B من السؤال الأولإذا هذا السؤال الأول
+
+116
+00:13:32,630 --> 00:13:37,290
+الفرق بيه
+
+117
+00:13:37,290 --> 00:13:44,130
+X و Y ينتموا إلى R أعداد حقيقية Such that absolute
+
+118
+00:13:44,130 --> 00:13:53,190
+X minus Y أصغر من واحد على N لكل N في N لما نثبت
+
+119
+00:13:53,190 --> 00:13:55,470
+أن هذا بقدر X بساوي Y
+
+120
+00:14:00,570 --> 00:14:09,470
+البرهان هنستخدم الـ Archimedean property assume on
+
+121
+00:14:09,470 --> 00:14:13,930
+contrary على
+
+122
+00:14:13,930 --> 00:14:23,710
+المقيد ان X لا يساوي Y هذا
+
+123
+00:14:23,710 --> 00:14:24,410
+بيقدّي
+
+124
+00:14:28,910 --> 00:14:44,930
+أكبر من سفر لأي
+
+125
+00:14:44,930 --> 00:14:53,850
+عدد موجب يوجد in zero عدد طبيعيبحيث ان واحد على ن
+
+126
+00:14:53,850 --> 00:15:05,390
+زيرو أصغر من absolute X سالب Y هذا
+
+127
+00:15:05,390 --> 00:15:13,530
+من وين من ال Archimedean property طيب
+
+128
+00:15:13,530 --> 00:15:16,610
+انا عندي من الفرض
+
+129
+00:15:19,930 --> 00:15:36,910
+من الفرض by hypothesis هذا أصغر من واحد على n لكل
+
+130
+00:15:36,910 --> 00:15:39,770
+n وبالتالي هذا صحيح ل n zero
+
+131
+00:15:48,170 --> 00:15:53,370
+إذن بيطلع عندى 1 على N0 أصغر من 1 على N0 إذن هذا
+
+132
+00:15:53,370 --> 00:16:00,590
+بيقدي إن 1 على N0 أصغر من 1 على N0 وهذا تناقض
+
+133
+00:16:00,590 --> 00:16:07,610
+contradiction تناقض إذن هذا التناقض بيقول إن ال X
+
+134
+00:16:07,610 --> 00:16:14,490
+لازم تساوي ال Y كما هو مطلوب okay تمامإذن هذا
+
+135
+00:16:14,490 --> 00:16:22,890
+برهان الجزء التاني من السؤال الأول في عند
+
+136
+00:16:22,890 --> 00:16:35,710
+هنا السؤال التاني question
+
+137
+00:16:35,710 --> 00:16:48,690
+2 الفرع a show S&Tإن الـ infimum لست واحد على
+
+138
+00:16:48,690 --> 00:17:03,630
+الجذر ال N حيث N عدد طوابيعي بساوي سفر نشوف
+
+139
+00:17:03,630 --> 00:17:08,610
+مع بعض واضح
+
+140
+00:17:08,610 --> 00:17:14,940
+إن سفر أصغر من أو ساوي واحد على الجذر ال Nلكل n
+
+141
+00:17:14,940 --> 00:17:24,180
+عدد طبيعي صح؟ في حد عنده شك في ذلك؟ وبالتالي
+
+142
+00:17:24,180 --> 00:17:31,040
+so سفر is zero
+
+143
+00:17:31,040 --> 00:17:41,460
+is a lower bound a lower bound of ال set S اللي هي
+
+144
+00:17:43,280 --> 00:17:48,720
+بنعرفها لأنها مجموعة العناصر واحد على square root
+
+145
+00:17:48,720 --> 00:17:56,140
+of N حيث N ينتمي ل N طيب
+
+146
+00:17:56,140 --> 00:18:01,600
+to show أن
+
+147
+00:18:01,600 --> 00:18:07,160
+السفر هو ال minimum ل S أو هو ال greatest lower
+
+148
+00:18:07,160 --> 00:18:28,050
+bound لمجموعة S فبناخد let Wbe any lower bound of
+
+149
+00:18:28,050 --> 00:18:37,130
+S فبالنسبة
+
+150
+00:18:37,130 --> 00:18:46,710
+لـ claimبنثبت أن السفر أصغر من أو ساوي W عفوا W
+
+151
+00:18:46,710 --> 00:18:53,150
+أصغر من أو ساوي 0 وبالتالي هيك يكون السفر أكبر من
+
+152
+00:18:53,150 --> 00:18:57,150
+أو ساوي أي lower bound يعني السفر هو ال greatest
+
+153
+00:18:57,150 --> 00:19:04,590
+lower bound صح فلبرهان ذلك assume بنعمل برهان
+
+154
+00:19:04,590 --> 00:19:13,920
+بالتناقض on contraryأفرضه على النقيد أن w أكبر من
+
+155
+00:19:13,920 --> 00:19:18,420
+سفر then
+
+156
+00:19:18,420 --> 00:19:21,540
+by
+
+157
+00:19:21,540 --> 00:19:28,160
+Archimedean property .. by Archimedean property
+
+158
+00:19:28,160 --> 00:19:37,150
+لأي عدد موجب زي هذايوجد عدد طبيعي N0 ينتمي إلى N
+
+159
+00:19:37,150 --> 00:19:43,130
+بحيث انه مقلوب N0
+
+160
+00:19:43,130 --> 00:19:51,250
+أصغر من العدد الموجب W تربية طبعا W تربية إذا W
+
+161
+00:19:51,250 --> 00:19:57,960
+عدد موجب فW تربية بالتأكيد عدد موجبفانا بطبّق الـ
+
+162
+00:19:57,960 --> 00:20:01,360
+Archimedean property مش على w وعلى w تربية العدد
+
+163
+00:20:01,360 --> 00:20:06,060
+الموجب w square فبقدر ألاقي by Archimedean
+
+164
+00:20:06,060 --> 00:20:11,140
+property natural number n0 مقلوب وأصغر من w تربية
+
+165
+00:20:11,140 --> 00:20:21,940
+طبعا هذا بيقدّي أن واحد على جذر n0 أصغر من w ومن
+
+166
+00:20:21,940 --> 00:20:22,240
+ال
+
+167
+00:20:26,300 --> 00:20:34,080
+من الفرض الـ W هذا lower bound للست S وبالتالي هذا
+
+168
+00:20:34,080 --> 00:20:41,320
+أصغر من أو سوى واحد على الجدر التربيهي ل N0 لأن
+
+169
+00:20:41,320 --> 00:20:46,440
+هذا ينتمي ل S، هذا عنصر في S، صح؟ واحنا فرضين أن
+
+170
+00:20:46,440 --> 00:20:53,370
+الـ W أشمل lower boundللمجموع S وهذا answer في
+
+171
+00:20:53,370 --> 00:20:58,730
+المجموع S إذا ال W كونه lower bound ل S أصغر من أو
+
+172
+00:20:58,730 --> 00:21:04,310
+ساوي واحد على square root ل N0 فطبعا هذا بيدي ان
+
+173
+00:21:04,310 --> 00:21:09,150
+واحد على square root ل N0 أصغر من واحد على square
+
+174
+00:21:09,150 --> 00:21:18,330
+root ل N0 وهذا مدينة تناقضكيف عدد بيطلع أصغر من
+
+175
+00:21:18,330 --> 00:21:21,730
+نفسه هذا تناقض لأن هذا التناقض بيقول لإن ال
+
+176
+00:21:21,730 --> 00:21:26,010
+assumption تبعنا ان ال W أكبر من السفر خطأ
+
+177
+00:21:26,010 --> 00:21:33,210
+وبالتالي ال W لازم يكون أصغر بدل ما هو أكبر من
+
+178
+00:21:33,210 --> 00:21:37,970
+السفر يطلع أصغر من أو يساوي سفر وبالتالي ال W
+
+179
+00:21:37,970 --> 00:21:43,830
+السفر هو أكبر lower bound okay إذا هذا بيكمل
+
+180
+00:21:43,830 --> 00:21:45,750
+البرهان تمام
+
+181
+00:22:04,850 --> 00:22:17,070
+انجاب على الفرق بيه من السؤال التاني show
+
+182
+00:22:17,070 --> 00:22:26,770
+ان ال limit لواحد على جذر ال N as N times infinity
+
+183
+00:22:26,770 --> 00:22:28,370
+بساوي سفر
+
+184
+00:22:40,270 --> 00:22:45,910
+Proof بإن احنا في النهاية بيستخدم تعريف طبعا
+
+185
+00:22:45,910 --> 00:22:51,790
+epsilon capital N للنهايات ففي نهاية الأمر بدنا ال
+
+186
+00:22:51,790 --> 00:22:55,190
+absolute value ل 1 على square root of N minus 0
+
+187
+00:22:55,190 --> 00:23:02,270
+بدنا ده يطلع أصغر من أي epsilon صح؟ طب ما هذا
+
+188
+00:23:02,270 --> 00:23:05,070
+بيساوي 1 على square root of N
+
+189
+00:23:09,000 --> 00:23:16,500
+متى هذا بيكون أصغر من epsilon فانا
+
+190
+00:23:16,500 --> 00:23:24,040
+هاخد هذا
+
+191
+00:23:24,040 --> 00:23:28,080
+لما واحد على n أصغر من epsilon تربية لو ربعت
+
+192
+00:23:28,080 --> 00:23:33,920
+الطرفين فمتى
+
+193
+00:23:33,920 --> 00:23:35,920
+هذا بيكون أصغر من epsilon تربية
+
+194
+00:23:38,730 --> 00:23:45,210
+Okay إذا هنا هاخد انا واحد على capital N أصغر من
+
+195
+00:23:45,210 --> 00:23:54,110
+epsilon تربية إذا
+
+196
+00:23:54,110 --> 00:23:58,570
+نستخدم ال Archimedean property هذا epsilon تربية
+
+197
+00:23:58,570 --> 00:24:02,690
+عدد موجب By Archimedean property بقدر ألاقي عدد
+
+198
+00:24:02,690 --> 00:24:07,550
+طبيعي capital N مقلوب وأصغر من epsilon تربية ناشي
+
+199
+00:24:08,920 --> 00:24:15,320
+بدا هنا given epsilon
+
+200
+00:24:15,320 --> 00:24:27,320
+أكبر من السفر use الarchimedean property to choose
+
+201
+00:24:27,320 --> 00:24:36,760
+n عدد طبيعي بحيث أن واحد على n أصغر من epsilon
+
+202
+00:24:39,750 --> 00:24:54,590
+الان لو خدت small n أكبر من أو ساوي capital N فهذا
+
+203
+00:24:54,590 --> 00:24:59,410
+بيقدي أن واحد على small n أصغر لو ساوي واحد على
+
+204
+00:24:59,410 --> 00:25:00,190
+capital N
+
+205
+00:25:05,570 --> 00:25:09,570
+بيقدي واحد على square root ل N أصغر من لو يساوي
+
+206
+00:25:09,570 --> 00:25:16,450
+واحد على square root ل capital N وهذا بيقدي ان ال
+
+207
+00:25:16,450 --> 00:25:23,650
+absolute value لواحد على square root ل N minus صفر
+
+208
+00:25:23,650 --> 00:25:27,790
+بيساوي واحد على square root ل N وهذا أصغر من لو
+
+209
+00:25:27,790 --> 00:25:34,250
+يساوي واحد على square root ل capital N وهذا من هنا
+
+210
+00:25:36,260 --> 00:25:40,760
+لو سمينا الـ inequality هذه الـ star إذا by star
+
+211
+00:25:40,760 --> 00:25:48,620
+واحد على square root ل N أصغر من epsilon إذا
+
+212
+00:25:48,620 --> 00:25:53,440
+هاي نحققنا تعريف epsilon capital N للنهايات for
+
+213
+00:25:53,440 --> 00:25:58,600
+any given epsilon أثبتنا إنه يوجد capital N عدد
+
+214
+00:25:58,600 --> 00:26:02,760
+طبيعي وهذا العدد الطبيعي يعتمد على epsilon هاي
+
+215
+00:26:02,760 --> 00:26:08,910
+مرتبط بepsilonبحيث لكل n أكبر من أو ساوي capital N
+
+216
+00:26:08,910 --> 00:26:16,010
+طلع المسافة بين xn و x اللي هي سفر أصغر من epsilon
+
+217
+00:26:16,010 --> 00:26:23,530
+اذا by definition by
+
+218
+00:26:23,530 --> 00:26:28,910
+definition بطلع عندي limit واحد على square root ل
+
+219
+00:26:28,910 --> 00:26:33,290
+n بساوي سفر وهو المطلوب طبعا
+
+220
+00:26:37,550 --> 00:26:42,930
+طبعاً في حال تاني أو في برهان تاني باستخدام الـ
+
+221
+00:26:42,930 --> 00:26:48,770
+monotone convergence theorem إذا
+
+222
+00:26:48,770 --> 00:26:54,350
+ال solution to use
+
+223
+00:26:54,350 --> 00:27:01,090
+monotone convergence theorem ال sequence أنا عندي
+
+224
+00:27:01,090 --> 00:27:04,350
+xn بساوي واحد على square root ل n
+
+225
+00:27:09,010 --> 00:27:14,590
+هذا أكبر من أو ساوي واحد على square root ل n زايد
+
+226
+00:27:14,590 --> 00:27:26,090
+واحد اللي هو xn زايد واحد وبالتالي ال sequence is
+
+227
+00:27:26,090 --> 00:27:32,900
+decreasing صح؟بعدين انا عندي absolute xn بساوي
+
+228
+00:27:32,900 --> 00:27:38,300
+absolute واحد على جدر ال n أصغر من أو ساوي واحد
+
+229
+00:27:38,300 --> 00:27:47,120
+لكل n لكل
+
+230
+00:27:47,120 --> 00:27:57,520
+n فهذا بيقدي ان ال sequence xn is bounded اذا
+
+231
+00:27:57,520 --> 00:28:02,750
+bounded و decrease inإذا الـ sequence xn by
+
+232
+00:28:02,750 --> 00:28:07,690
+monotone convergence theorem limit xn بساوة
+
+233
+00:28:07,690 --> 00:28:17,710
+الانفمام ل xn حيث n ينتمي إلى n فأثبتنا أن
+
+234
+00:28:17,710 --> 00:28:24,760
+الانفمام بساوة سفر بالفرع اللي جابلهفهذا برهان
+
+235
+00:28:24,760 --> 00:28:28,540
+تاني لكن احنا ال .. ال monotone convergence الكلام
+
+236
+00:28:28,540 --> 00:28:34,860
+مش داخله في الامتحان فممكن انكم يعني ال ..
+
+237
+00:28:34,860 --> 00:28:38,580
+تستخدموا البرهان الأول يعني ممكن تستخدموا البرهان
+
+238
+00:28:38,580 --> 00:28:42,360
+الأول يعني okay
+
+239
+00:28:42,360 --> 00:28:46,400
+تمام طيب نكمل
+
+240
+00:29:09,040 --> 00:29:15,540
+find the supremum الجزء C بإنه نوجد ال supremum
+
+241
+00:29:15,540 --> 00:29:20,800
+لواحد سالب واحد على الجدر ال N حيث N ينتمي ل N
+
+242
+00:29:20,800 --> 00:29:32,620
+فهذا حسب exercise أخدنا supremum A زائد 6 بساوي A
+
+243
+00:29:32,620 --> 00:29:38,640
+زائد supremumالست اللي هي عناصرها سالب واحد على
+
+244
+00:29:38,640 --> 00:29:48,800
+جدر ال N هيف N ينتمي الى N ويساوي واحد الان
+
+245
+00:29:48,800 --> 00:29:56,140
+supremum سالب ست بيساوي سالب ال infimum لعناصر
+
+246
+00:29:56,140 --> 00:29:56,640
+الست
+
+247
+00:30:05,050 --> 00:30:09,910
+و احنا لسه مثبتين ان ال inform هذا بساوي سفر اذا
+
+248
+00:30:09,910 --> 00:30:15,210
+ال suprem للست هذي بطلع واحد طيب
+
+249
+00:30:15,210 --> 00:30:25,970
+فرقة تانية show that اثبتي انه limit cosine
+
+250
+00:30:25,970 --> 00:30:34,290
+n على n او على جدر ال nas n tends to infinity
+
+251
+00:30:34,290 --> 00:30:38,650
+بساوي سفر proof
+
+252
+00:30:38,650 --> 00:30:43,290
+أنا
+
+253
+00:30:43,290 --> 00:30:50,010
+عندي cosine n أكبر من أو أصغر من أو ساوي واحد أكبر
+
+254
+00:30:50,010 --> 00:30:59,630
+من أو ساوي سالب واحد لكل n في n وواحد
+
+255
+00:31:01,530 --> 00:31:07,070
+على جدر ال N عدد موجب فلو ضربت المتباينة هذه في
+
+256
+00:31:07,070 --> 00:31:11,550
+واحد على جدر ال N فبطلع سالب واحد على جدر ال N
+
+257
+00:31:11,550 --> 00:31:18,130
+أصغر لو سوى cosine N على جدر ال N أصغر لو سوى واحد
+
+258
+00:31:18,130 --> 00:31:21,690
+على جدر ال N لكل N في N
+
+259
+00:31:24,530 --> 00:31:28,310
+طيب انا عند ال sequence لسه مثبتين احنا هنا في
+
+260
+00:31:28,310 --> 00:31:32,510
+الفرع b ان ال sequence هذه ال limit تبعتها as n
+
+261
+00:31:32,510 --> 00:31:35,970
+tends to infinity بساوة سفر و ال sequence هذه ال
+
+262
+00:31:35,970 --> 00:31:41,430
+limit تبعتها سالب limit واحد على الجدر ال n و سالب
+
+263
+00:31:41,430 --> 00:31:46,830
+واحد بسفر بطلع سفر as n tends to infinity اذا by
+
+264
+00:31:46,830 --> 00:31:47,910
+squeeze theorem
+
+265
+00:31:51,910 --> 00:31:57,090
+بسكويز تيريم أو sandwich theorem بطلع limit ال
+
+266
+00:31:57,090 --> 00:32:02,250
+sequence اللي لحد تبعها محصور في النص اللي هو
+
+267
+00:32:02,250 --> 00:32:06,590
+cosine n على square root ل n as n times infinity
+
+268
+00:32:06,590 --> 00:32:16,070
+بساوي 7 تمام؟ إذن هذا برهان الجزء دي
+
+269
+00:32:23,640 --> 00:32:30,920
+واضحة للحلول question
+
+270
+00:32:30,920 --> 00:32:38,680
+تلاتة الفرع a انا عندي in بالساوية closed interval
+
+271
+00:32:38,680 --> 00:32:44,480
+من سفر لواحد على جدر ال n prove
+
+272
+00:32:46,630 --> 00:32:51,890
+إنه ال intersection from n equals one to infinity
+
+273
+00:32:51,890 --> 00:33:02,970
+ل I n بساوي single twin zipper فممكن
+
+274
+00:33:02,970 --> 00:33:11,570
+نستخدم ال nested interval property proof
+
+275
+00:33:11,570 --> 00:33:23,930
+أول شي واضحواضح ان سفر ينتمي ل I N لكل N في N هذا
+
+276
+00:33:23,930 --> 00:33:28,690
+بيقدي ان السفر ينتمي
+
+277
+00:33:28,690 --> 00:33:35,210
+لتقاطعه صح
+
+278
+00:33:35,210 --> 00:33:40,310
+الفترة المغلقة هذه السفر دائما ينتمي لها لكل عدد
+
+279
+00:33:40,310 --> 00:33:45,330
+طبيعيالان انا في عندي sequence of intervals و
+
+280
+00:33:45,330 --> 00:33:49,970
+السفر ينتمي لكل عنصر في ال set إذا السفر ينتمي
+
+281
+00:33:49,970 --> 00:34:00,410
+لتقاطه كل المجموعات طيب الفترة I in is closed صح
+
+282
+00:34:00,410 --> 00:34:06,850
+and bounded لكل
+
+283
+00:34:06,850 --> 00:34:16,080
+inو بعدين واضح ان I N contains I N زاد واحد لكل N
+
+284
+00:34:16,080 --> 00:34:23,340
+في N صح؟ يعني ال sequence هذه nested يعني ال
+
+285
+00:34:23,340 --> 00:34:29,200
+sequence I N nested
+
+286
+00:34:29,200 --> 00:34:32,300
+اذا
+
+287
+00:34:32,300 --> 00:34:35,940
+by
+
+288
+00:34:35,940 --> 00:34:38,000
+nested
+
+289
+00:34:44,870 --> 00:34:50,710
+intervals theorem استقاط
+
+290
+00:34:50,710 --> 00:34:59,510
+وهذا لازم يكون في unique element لأنه انا عندي طيب
+
+291
+00:34:59,510 --> 00:35:07,490
+جبل ما نطبق نستد انا عندي ال infimum لواحد على جذر
+
+292
+00:35:07,490 --> 00:35:08,810
+n سالب سفر
+
+293
+00:35:18,070 --> 00:35:26,910
+هذا الان from أثبتنا أنه بساوي سبر إذا حسب by
+
+294
+00:35:26,910 --> 00:35:31,470
+nested
+
+295
+00:35:31,470 --> 00:35:36,370
+intervals theorem
+
+296
+00:35:39,780 --> 00:35:47,720
+الـ intersection has unique element has
+
+297
+00:35:47,720 --> 00:35:52,920
+unique element
+
+298
+00:35:52,920 --> 00:35:57,340
+يعني في التقاطع مافيش أنصر وحيد طب احنا قلنا ان
+
+299
+00:35:57,340 --> 00:36:03,500
+السفر ينتمي لتقاطع وتقاطع في أنصر واحد لذا لازم
+
+300
+00:36:03,500 --> 00:36:07,960
+يساوي السفر in
+
+301
+00:36:07,960 --> 00:36:08,780
+a sense
+
+302
+00:36:13,330 --> 00:36:21,210
+بما أن السفر ينتمي للتقاطع يجب
+
+303
+00:36:21,210 --> 00:36:25,210
+أن
+
+304
+00:36:25,210 --> 00:36:31,070
+يكون التقاطع من N بساوي واحد infinity ل I N بساوي
+
+305
+00:36:31,070 --> 00:36:33,610
+بس single tone سفر
+
+306
+00:36:39,670 --> 00:36:44,490
+طبعا ممكن اعطيتكم انا برهان زي هذا بس كان بدل 1
+
+307
+00:36:44,490 --> 00:36:46,110
+على جدر ال N 1 على N
+
+308
+00:36:50,700 --> 00:36:57,680
+و أثبتنا إن ال set هذى طبعا واضح من هنا إن هذى
+
+309
+00:36:57,680 --> 00:37:04,260
+دايما صحيحة و أثبتنا العكس و قلنا إنه لو أخدت أي x
+
+310
+00:37:04,260 --> 00:37:07,880
+تتقاطع فبنا نثبت إن هذا ال x بساوي سفر و عملنا
+
+311
+00:37:07,880 --> 00:37:12,800
+برهان بالتناقض افرضي إن ال x مابسويش سفر إذا أكبر
+
+312
+00:37:12,800 --> 00:37:16,100
+من سفر و وصلنا إلى تناقض by Archimedean property
+
+313
+00:37:17,080 --> 00:37:21,820
+إذا في برهان تاني لكن أنا حبيت أعطيكم البرهان هذا
+
+314
+00:37:21,820 --> 00:37:26,840
+التاني اللي ما خناش زيه طبعا
+
+315
+00:37:26,840 --> 00:37:30,920
+صح
+
+316
+00:37:30,920 --> 00:37:37,340
+إذا أنا عندكم برهانين طيب
+
+317
+00:37:37,340 --> 00:37:42,700
+هاي كمان الجزء بيه من السؤال هذا أنا في عندي
+
+318
+00:37:42,700 --> 00:37:49,820
+sequence xn sequence mrوعندي xn أصغر من أو ساوي
+
+319
+00:37:49,820 --> 00:38:02,720
+سفر لكل n وبدنا نثبت إنه إذا كان limit xn بساوي x
+
+320
+00:38:02,720 --> 00:38:10,810
+then ال x بتطلع أيضا أصغر من أو ساوي سفرأنا في
+
+321
+00:38:10,810 --> 00:38:14,610
+عندى sequence of real numbers كل حدودها غير ثالثة
+
+322
+00:38:14,610 --> 00:38:18,330
+و ال sequence converge ل X بالدفعة ان ال limit
+
+323
+00:38:18,330 --> 00:38:27,850
+ايضا بتطلع غير ثالثة فهي البرهان proof برهان
+
+324
+00:38:27,850 --> 00:38:32,930
+بالتناقض assume on
+
+325
+00:38:32,930 --> 00:38:33,650
+contrary
+
+326
+00:38:36,400 --> 00:38:45,860
+إن X أكبر من سفر و بدنا نصل إلى تناقض افرض
+
+327
+00:38:45,860 --> 00:38:56,620
+إن X أكبر من سفر طيب let epsilon بساوي X ع 2 هذا
+
+328
+00:38:56,620 --> 00:39:05,960
+بطل عدل موجببما أنه since .. since xn converges ل
+
+329
+00:39:05,960 --> 00:39:13,010
+x إذا يوجد capital N يعتمد على إبسلونعدد طبيعي
+
+330
+00:39:13,010 --> 00:39:19,190
+بحيث أنه لو كان n أكبر من أو ساوي capital N فهذا
+
+331
+00:39:19,190 --> 00:39:26,110
+بقدر أن absolute xn minus x أصغر من epsilon طيب ال
+
+332
+00:39:26,110 --> 00:39:30,910
+epsilon هذا أخدناها بساوي x ع 2 إذا أنا عندي هي xn
+
+333
+00:39:30,910 --> 00:39:36,830
+minus x أصغر من x ع 2 أكبر
+
+334
+00:39:36,830 --> 00:39:38,550
+من سالب x ع 2
+
+335
+00:39:42,440 --> 00:39:49,560
+طيب ما هيك بطلع عندي x in ضيفي x لكل الأطراف فبطلع
+
+336
+00:39:49,560 --> 00:40:01,200
+x in أصغر من تلاتة x عتنين أكبر من x عتنين طيب
+
+337
+00:40:01,200 --> 00:40:06,000
+هذا
+
+338
+00:40:06,000 --> 00:40:12,140
+معناه وهذا الكلام صحيح لكل inأكبر من و ساوي
+
+339
+00:40:12,140 --> 00:40:18,740
+capital N فلو أخدت .. إذا بطلع من هنا X capital N
+
+340
+00:40:18,740 --> 00:40:22,660
+لو أخدت small n هذه بساوي capital N فبطلع X
+
+341
+00:40:22,660 --> 00:40:29,740
+capital N أكبر من X ع 2 و أنا عندي X ع 2 عدد موجب
+
+342
+00:40:29,740 --> 00:40:35,100
+أكبر من 0 إذا أنا بطلع عندي X capital N أكبر من 0
+
+343
+00:40:35,100 --> 00:40:41,550
+وهذا تناقضلأن انا فرض ان كل حدود ال sequence كلهم
+
+344
+00:40:41,550 --> 00:40:46,770
+اعداد غير سالبة فكيف طلع هالحد رقم capital N موجب
+
+345
+00:40:46,770 --> 00:40:52,090
+هذا بتناقض مع الفرض تمام؟ اذا هذا سبب التناقض هذا
+
+346
+00:40:52,090 --> 00:40:59,550
+هو ال assumption تبعنا ان x أكبر من السفر okay؟
+
+347
+00:40:59,550 --> 00:41:05,490
+اذا الصح ان x أصغر من أو ساوي سفر وهو المطلوب
+
+348
+00:41:08,070 --> 00:41:16,710
+Okay تمام اذا يعني هذه يعني بعض الأسئلة الاسئلة
+
+349
+00:41:16,710 --> 00:41:25,630
+الرابعة هذا نظرية أخدناها و اللي بعدي أعتقد حلناها
+
+350
+00:41:25,630 --> 00:41:34,490
+في المحاضرة فهواجف هنا و هيك بنكون يعني راجعنا
+
+351
+00:41:34,490 --> 00:41:41,290
+تقريبا امتحانللمتحانة النصفة ونشوفكم ان شاء الله
+
+352
+00:41:41,290 --> 00:41:41,690
+بقرا
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d77e4185aef670ad10d471bea973a5b20c6e3ca4
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y.srt
@@ -0,0 +1,1955 @@
+1
+00:00:21,750 --> 00:00:27,870
+بسم الله الرحمن الرحيم في محاضرة اليوم هنكمل ما 
+
+2
+00:00:27,870 --> 00:00:32,170
+بدأنا حاول 
+
+3
+00:00:32,170 --> 00:00:37,890
+موضوع ال infinite limits المرة الفاتت عرفنا ما معناه 
+
+4
+00:00:37,890 --> 00:00:42,370
+أن ال limit ل function و cluster point بالساوية 
+
+5
+00:00:42,370 --> 00:00:49,050
+plus أو minus infinity وشوفنا نظرية أو نظرية 
+
+6
+00:00:49,050 --> 00:00:54,800
+برهنها خاصة بهذا النوع من ال limits كانت النظرية 
+
+7
+00:00:54,800 --> 00:01:05,340
+التالية خلينا نكتبها let 
+
+8
+00:01:05,340 --> 00:01:14,420
+f و g be functions from a to r و c ال cluster 
+
+9
+00:01:14,420 --> 00:01:18,320
+point 
+
+10
+00:01:20,530 --> 00:01:28,690
+of the set A such that f 
+
+11
+00:01:28,690 --> 00:01:34,590
+of x less than or equal g of x for every x تنتمي 
+
+12
+00:01:34,590 --> 00:01:42,630
+إلى a different from c فشوفنا أنه لو كان ال limit 
+
+13
+00:01:42,630 --> 00:01:45,950
+لـ f 
+
+14
+00:01:45,950 --> 00:01:48,190
+of x as x tends 
+
+15
+00:02:05,090 --> 00:02:07,390
+وكذلك لو كانت ال limit 
+
+16
+00:02:10,530 --> 00:02:17,930
+لـ g of x as x tends to c بساوي negative infinity 
+
+17
+00:02:17,930 --> 00:02:26,810
+فهذا بتضمن أن limit ل f of x as x tends to c بساوي 
+
+18
+00:02:26,810 --> 00:02:35,730
+negative infinity طيب 
+
+19
+00:02:35,730 --> 00:02:36,350
+الـ ..
+
+20
+00:02:41,050 --> 00:02:47,350
+اليوم هنعرف ما معناه أن ال limit و c من اليمين 
+
+21
+00:02:47,350 --> 00:02:52,030
+بالساوي infinity أو ال limit و c من اليسار 
+
+22
+00:02:52,030 --> 00:02:55,990
+بالساوي infinity وكذلك نفس الشيء ال one sided 
+
+23
+00:02:55,990 --> 00:03:02,030
+limit و c ما معناه أنها ساوي سالب infinity لأن 
+
+24
+00:03:02,030 --> 00:03:07,010
+هذه كانت two sided limit المرة الأخيرة اتعرفنا ما 
+
+25
+00:03:07,010 --> 00:03:10,410
+معناه أن ال two sided limit تكون infinite أو ساوي 
+
+26
+00:03:10,410 --> 00:03:14,430
+infinity أو plus أو minus infinity اليوم ما معناه 
+
+27
+00:03:14,430 --> 00:03:18,370
+أن ال one sided limit تكون infinity أو negative 
+
+28
+00:03:18,370 --> 00:03:23,970
+infinity فناخد التعريف مشابه 
+
+29
+00:03:23,970 --> 00:03:29,050
+للتعريف الـ one sided limit تكون بتساوي 
+
+30
+00:03:29,050 --> 00:03:33,590
+real number فـ
+
+31
+00:03:33,590 --> 00:03:44,120
+let f be function from a to r و 
+
+32
+00:03:44,120 --> 00:03:49,240
+c cluster point 
+
+33
+00:03:49,240 --> 00:03:58,760
+of الـ set اللي هي a تقاطع الفترة المفتوحة من c لما 
+
+34
+00:03:58,760 --> 00:03:59,640
+إلى نهاية
+
+35
+00:04:04,430 --> 00:04:10,290
+يقول إن الـ limit لـ 
+
+36
+00:04:10,290 --> 00:04:15,530
+function f of x as x tends to c from the right 
+
+37
+00:04:15,530 --> 00:04:23,110
+بالساوي infinity respectively 
+
+38
+00:04:23,110 --> 00:04:30,370
+على التوالي بنقول إن ال limit ل f of x لما x تقول 
+
+39
+00:04:30,370 --> 00:04:39,170
+إلى c من اليمين بتساوي negative infinity إذا 
+
+40
+00:04:39,170 --> 00:04:44,790
+تحقق الشرط التالي لأي 
+
+41
+00:04:44,790 --> 00:04:51,950
+Alpha for any Alpha 
+
+42
+00:04:51,950 --> 00:04:57,450
+belonging to R نقدر 
+
+43
+00:04:57,450 --> 00:05:07,810
+نلاقي Delta تعتمد على Alpha  على دلتا بحيث أنه لكل 
+
+44
+00:05:07,810 --> 00:05:17,050
+x ينتمي إلى a و ال x على يمين ال c و المسافة بينها 
+
+45
+00:05:17,050 --> 00:05:22,730
+و بين ال c أصغر من دلتا فلازم هذا يضمن أن f of x 
+
+46
+00:05:22,730 --> 00:05:32,310
+أكبر من alpha أو على التواري respectively ال f of 
+
+47
+00:05:32,310 --> 00:05:37,330
+x هتكون في حالة ال limit بالساول سالب infinity 
+
+48
+00:05:37,330 --> 00:05:42,550
+عايزينها تكون أصغر من ال alpha أصغر من ال given 
+
+49
+00:05:42,550 --> 00:05:46,330
+alpha okay 
+
+50
+00:05:46,330 --> 00:05:53,550
+إذا أنا هنا عندي limit ال function و c من اليمين 
+
+51
+00:05:53,550 --> 00:05:57,570
+بالساول infinity معناته لأي real number alpha بقدر 
+
+52
+00:05:57,570 --> 00:06:05,620
+أخلي f of x أكبر من Alpha لكل X على يمين الـC لأن X 
+
+53
+00:06:05,620 --> 00:06:08,920
+تقوى للـC من اليمين فX على يمين الـC يعني X أكبر 
+
+54
+00:06:08,920 --> 00:06:13,160
+من الـC يعني X ناقص C أكبر من صفر والمسافة بين 
+
+55
+00:06:13,160 --> 00:06:18,140
+الـC والـX أو الـX والـC أصغر من الـD فلكل الـX 
+
+56
+00:06:18,140 --> 00:06:22,140
+اللي زيها دي بدي أخلي F of X أكبر من Alpha عشان 
+
+57
+00:06:22,140 --> 00:06:26,280
+أقدر أقول أن ال limit لF of X tends to infinity 
+
+58
+00:06:28,130 --> 00:06:31,870
+بالمثل ما معنى أن limit f of x عن c من اليمين 
+
+59
+00:06:31,870 --> 00:06:38,050
+بالساوى سالب infinity معناه بقدر أخلي لكل x زي ما 
+
+60
+00:06:38,050 --> 00:06:44,350
+شوفنا أو لأي عدد real number alpha يوجد delta بحيث 
+
+61
+00:06:44,350 --> 00:06:48,670
+لكل x على يمين ال C والمسافة بينها و بين ال C أصغر 
+
+62
+00:06:48,670 --> 00:06:52,310
+من ال delta لازم صورتها تكون أصغر من ال given 
+
+63
+00:06:52,310 --> 00:07:00,470
+alpha okay تمام طيب خلينا الآن كتير من النظريات 
+
+64
+00:07:00,470 --> 00:07:06,970
+اللي أخدناها for two sided limit زي هذه مثلاً بتكون 
+
+65
+00:07:06,970 --> 00:07:12,210
+صحيحة لل right limit و لل left limit طبعاً ممكن 
+
+66
+00:07:12,210 --> 00:07:18,570
+كمان نعرف بنفس الطريقة ال limit from the left أو 
+
+67
+00:07:18,570 --> 00:07:22,690
+ال left hand limit مايعني أن ال left hand limit 
+
+68
+00:07:22,690 --> 00:07:24,410
+تساوي 
+
+69
+00:07:25,850 --> 00:07:31,370
+Infinity أو سالب Infinity إذا 
+
+70
+00:07:31,370 --> 00:07:36,690
+لو أنا بدي أعدل أعرف ما معنى أن ال limit ل F عن C 
+
+71
+00:07:36,690 --> 00:07:40,750
+من اليسار بالساوية Infinity أو ما معنى أن ال limit 
+
+72
+00:07:40,750 --> 00:07:46,150
+ل F عن C من اليسار بالساوية سالب Infinity فباخد 
+
+73
+00:07:46,150 --> 00:07:53,340
+let F be هكذا و C cluster point هتصير لـ a تقاطع 
+
+74
+00:07:53,340 --> 00:08:00,220
+الفترة من سالب infinity إلى C فبنقول 
+
+75
+00:08:00,220 --> 00:08:04,680
+إن ال limit لما X تقول إلى C من اليسار بالساوي 
+
+76
+00:08:04,680 --> 00:08:10,260
+infinity أو ال limit لما X تقول إلى C من اليسار 
+
+77
+00:08:10,260 --> 00:08:15,860
+بالساوي السالب infinity إذا كان لأي Alpha يوجد 
+
+78
+00:08:15,860 --> 00:08:21,200
+Delta تعتمد على Alpha الآن ال X هتكون على يسار ال C 
+
+79
+00:08:21,200 --> 00:08:31,200
+وبالتالي هذا هنستبدله بـ C ناقص X أكبر 
+
+80
+00:08:31,200 --> 00:08:38,680
+من صفر أصغر من دلتر فلكل X زي هذه أنا عايز أن تكون 
+
+81
+00:08:38,680 --> 00:08:43,620
+F of X أكبر من Alpha أو في الحالة هذه F of X أصغر 
+
+82
+00:08:43,620 --> 00:08:49,710
+من Alpha هنا ذيك نكون عرفنا الـ left limit عن c ما 
+
+83
+00:08:49,710 --> 00:08:56,170
+معنى أنها ساوي plus أو minus infinity إذن قلنا إن 
+
+84
+00:08:56,170 --> 00:09:00,310
+كل النظريات اللي برهنها for two sided limits هتكون 
+
+85
+00:09:00,310 --> 00:09:08,730
+صحيحة لل left limit و لل right limit من ضمنهم 
+
+86
+00:09:08,730 --> 00:09:14,010
+النظرية السابقة طيب 
+
+87
+00:09:14,010 --> 00:09:15,130
+لو بدي أنا يعني 
+
+88
+00:09:18,090 --> 00:09:24,870
+آخد أمثلة كيف نستخدم التعريف هذا فيه إثبات أن الـ 
+
+89
+00:09:24,870 --> 00:09:32,230
+limits تطلع plus أو minus infinity فناخد أول مثال 
+
+90
+00:09:32,230 --> 00:09:40,250
+let f of x بساوي واحد 
+
+91
+00:09:40,250 --> 00:09:45,170
+على x حيث x لا يساوي صفر سبق و show 
+
+92
+00:09:49,780 --> 00:09:58,060
+عايزين نفدت واحد أن ال limit لواحد على x أو f of x 
+
+93
+00:09:58,060 --> 00:10:05,820
+هنا لما x تقول إلى صفر من اليمين بساوي ال infinity 
+
+94
+00:10:05,820 --> 00:10:09,960
+و 2 limit 
+
+95
+00:10:11,300 --> 00:10:19,200
+ل f of X لما X تقول إلى 0 من اليسار يساوي سالب 
+
+96
+00:10:19,200 --> 00:10:23,880
+infinity okay فلو 
+
+97
+00:10:23,880 --> 00:10:29,540
+بدنا نبرم الجزء الأول مثلاً to show 
+
+98
+00:10:32,410 --> 00:10:38,710
+المقاومة ل f of x as x tends to 0 from the right 
+
+99
+00:10:38,710 --> 00:10:45,670
+بساوي infinity فبدي ابدا بـ alpha تنتمي ل R فبقول 
+
+100
+00:10:45,670 --> 00:10:57,490
+let alpha belonging to R be given و 
+
+101
+00:10:57,490 --> 00:11:02,790
+بدي ارد عليها بDelta بدي أرد على ال alpha دي الـ 
+
+102
+00:11:02,790 --> 00:11:08,630
+delta عدد موجب ويعتمد على ال alpha ف choose delta 
+
+103
+00:11:08,630 --> 00:11:15,890
+بتساوي واحد على absolute alpha زائد واحد بالتأكيد 
+
+104
+00:11:15,890 --> 00:11:21,910
+هذا عدد موجب لأن absolute ال alpha دي عدد حقيقي 
+
+105
+00:11:21,910 --> 00:11:28,450
+القيمة المطلقة له عدد غير سالب ممكن يساوي صفر إذا 
+
+106
+00:11:28,450 --> 00:11:32,980
+كانت alpha بالساوية صفر لكن زائد واحد بصير موجب 
+
+107
+00:11:32,980 --> 00:11:37,720
+المقام موجب إذا أنا بضيف واحد ليه عشان أضمن أن 
+
+108
+00:11:37,720 --> 00:11:41,880
+المقام ما يسويش صفر لأن في احتمال أن ال alpha ساوي 
+
+109
+00:11:41,880 --> 00:11:45,640
+صفر فبصير عندي مشكلة عشان أتخلص من المشكلة هذه 
+
+110
+00:11:45,640 --> 00:11:51,140
+بجسم على absolute alpha زائد واحد الآن هذا عدد 
+
+111
+00:11:51,140 --> 00:11:57,010
+موجب و يعتمد على alpha هي ال delta هي مرتبطة معرفة 
+
+112
+00:11:57,010 --> 00:12:02,350
+بدلالة alpha هي معناه أنها تعتمد على alpha إذا لأي 
+
+113
+00:12:02,350 --> 00:12:09,410
+alpha ينتمي ل R خد ال delta اللي بتنظرها هي واحد 
+
+114
+00:12:09,410 --> 00:12:13,740
+على absolute alpha زائد واحد هذا أكيد عدد موجب 
+
+115
+00:12:13,740 --> 00:12:20,440
+then من مراتبة الآن أن كل x في المجال تبع الدالة 
+
+116
+00:12:20,440 --> 00:12:24,380
+اللي هو كل الأعداد الحقيقية مع عدد صفر و ال x على 
+
+117
+00:12:24,380 --> 00:12:30,420
+يمين ال c اللي هو الصفر و من هنا x ينتمي إلى a 
+
+118
+00:12:30,420 --> 00:12:35,560
+اللي هي R المجال تبع الدالة كل الأعداد الحقيقية مع 
+
+119
+00:12:35,560 --> 00:12:42,940
+عدد صفر و X ناقص صفر ال C هنا ال cluster point هي 
+
+120
+00:12:42,940 --> 00:12:47,100
+الصفر الآن ال X على يمين الصفر يعني X ناقص صفر 
+
+121
+00:12:47,100 --> 00:12:54,040
+أكبر من صفر فإذا كانت ال X هذه أصغر من Delta فهذا 
+
+122
+00:12:54,040 --> 00:13:03,900
+هيعطيني أن ال 1 على X أكبر من واحد على دلتا 
+
+123
+00:13:03,900 --> 00:13:07,560
+وبالتالي 
+
+124
+00:13:07,560 --> 00:13:14,720
+هذا بيقدي أن f of x اللي هي بالساوي واحد على x 
+
+125
+00:13:14,720 --> 00:13:20,940
+أكبر من واحد على دلتا اللي هي بالساوي واحد مقلوب 
+
+126
+00:13:20,940 --> 00:13:28,280
+الدلتا بيطلع absolute alpha زائد واحد وهذه أكبر من 
+
+127
+00:13:28,280 --> 00:13:32,300
+absolute ال alpha absolute alpha زائد واحد أكبر من 
+
+128
+00:13:32,300 --> 00:13:36,880
+absolute alpha و absolute alpha أكبر من أو يساوي 
+
+129
+00:13:36,880 --> 00:13:41,920
+alpha أي عدد حقيقي القيمة المطلقة تبعته أكبر من أو 
+
+130
+00:13:41,920 --> 00:13:48,220
+يساوي نفسه فالنهاية أثبتنا أن f of x أكبر من ال 
+
+131
+00:13:48,220 --> 00:13:48,920
+given alpha 
+
+132
+00:13:53,520 --> 00:13:58,440
+Okay تمام بما أن ال alpha دي كانت arbitrarily 
+
+133
+00:13:58,440 --> 00:14:06,640
+since alpha belong to R was arbitrarily إذا هن 
+
+134
+00:14:06,640 --> 00:14:12,180
+أثبتنا إذا معناه هذا الكلام هذا أن لكل alpha فيه 
+
+135
+00:14:12,180 --> 00:14:17,280
+delta تعتمد عليها بتخلي f of x أكبر من alpha لكل x 
+
+136
+00:14:17,280 --> 00:14:23,440
+قريبة من الصفر within مسافة delta إن هذا معناه حسب 
+
+137
+00:14:23,440 --> 00:14:29,340
+التعريف إن ال limit ل f of x لما x تقول إلى صفر من 
+
+138
+00:14:29,340 --> 00:14:36,780
+اليمين بساوي infinity برهانه 
+
+139
+00:14:36,780 --> 00:14:38,320
+جزء الثاني مشابه 
+
+140
+00:14:47,840 --> 00:14:54,400
+is similar مشابه لل part للجزء الأول يعني فهسيبكم 
+
+141
+00:14:54,400 --> 00:14:59,860
+انتم تكتبوا برهان مشابه مع التعديلات اللازمة و
+
+142
+00:14:59,860 --> 00:15:05,660
+أيه طبعًا التعريف تبع ال limit from the left موجود
+
+143
+00:15:05,660 --> 00:15:13,280
+okay تمام اللي هو بالأزرق تمام مثال
+
+144
+00:15:13,280 --> 00:15:33,190
+ثاني ممكن برضه ناخد مثال ثاني 
+
+145
+00:15:33,190 --> 00:15:41,050
+show limit for function e to one على x as x tends
+
+146
+00:15:41,050 --> 00:15:45,070
+to zero from the right  بساوي infinity
+
+147
+00:15:55,920 --> 00:16:02,700
+أنا عندي ال function تبعتي f of x بيسمي E أس واحد
+
+148
+00:16:02,700 --> 00:16:07,840
+على X طبعًا ال X هنا ال function مش معرفة عند الصفر
+
+149
+00:16:07,840 --> 00:16:12,900
+مجال الدالة هذه كل الأعداد الحقيقية ما عدا الصفر
+
+150
+00:16:12,900 --> 00:16:20,860
+المثال هذا أخذناه المرة اللي فاتت we
+
+151
+00:16:20,860 --> 00:16:21,440
+have
+
+152
+00:16:25,650 --> 00:16:34,410
+from previous example من
+
+153
+00:16:34,410 --> 00:16:44,110
+المثال السابق فانا هادرس سابق أنه واحد
+
+154
+00:16:44,110 --> 00:16:51,370
+على اكس أكبر من صفر وأصغر من واحد على اكس لكل
+
+155
+00:16:51,370 --> 00:17:00,340
+اكس أكبر من صفر لكل x على يمين الصفر كان في عندي T
+
+156
+00:17:00,340 --> 00:17:09,980
+أصغر من E of T لكل T عدد موجب طيب
+
+157
+00:17:09,980 --> 00:17:18,000
+احنا لسه بتعامل since ال
+
+158
+00:17:18,000 --> 00:17:24,340
+limit لواحد على x لما x تقول إلى الصفر من اليمين
+
+159
+00:17:26,540 --> 00:17:31,760
+بساوي infinity فممكن
+
+160
+00:17:31,760 --> 00:17:37,160
+نطبق ال comparison test هذا فهي عندي f of x أصغر
+
+161
+00:17:37,160 --> 00:17:45,800
+من g of x يعني خليني أسمي هذه g of x  كلمشي مع النظرية
+
+162
+00:17:45,800 --> 00:17:52,020
+يعني وخلني f of x بساوي واحد على x فهي عندي f of x
+
+163
+00:17:52,020 --> 00:18:00,560
+أصغر من g of x لكل x في R أو لكل X لا يساوي صفر لكل
+
+164
+00:18:00,560 --> 00:18:07,820
+X موجبة بقى أو على يمين الصفر لكل
+
+165
+00:18:07,820 --> 00:18:16,880
+X في R تقاطع صفر إلى ما لا نهاية okay فإذا
+
+166
+00:18:16,880 --> 00:18:22,320
+قلنا النظرية هذه صحيحة لل right limit باستخدام
+
+167
+00:18:22,320 --> 00:18:33,480
+النظرية by above theorem by above theorem for right
+
+168
+00:18:33,480 --> 00:18:39,680
+limits للنهايات
+
+169
+00:18:39,680 --> 00:18:49,990
+من اليمين we have نحصل على انه ال limitلقيت واحد
+
+170
+00:18:49,990 --> 00:18:55,370
+على اكس لما اكس تقول إلى صفر من اليمين بساوي plus
+
+171
+00:18:55,370 --> 00:19:04,770
+infinity وهذا اللي بدنا إياه هي مظبوط صح؟ تمام؟ إذن
+
+172
+00:19:04,770 --> 00:19:08,850
+ممكن نطبق النظرية هذه لإثبات أن ال limit ل ال
+
+173
+00:19:08,850 --> 00:19:13,690
+function E to 1 ل X من X أو ل 0 من اليمين بساوي
+
+174
+00:19:13,690 --> 00:19:22,130
+infinity برضه ممكن نطبق التعريف يعني ممكن أعطي
+
+175
+00:19:22,130 --> 00:19:28,530
+برهان ثاني و أقول بما أن هذه المتباينة صحيحة لكل X
+
+176
+00:19:28,530 --> 00:19:34,480
+موجبة وبما انه ال limit هذه لو أخذنا ال function
+
+177
+00:19:34,480 --> 00:19:38,080
+واحد على X عن صفر من اليمين بيساوي infinity
+
+178
+00:19:38,080 --> 00:19:41,560
+معناته انا بقدر أخلي واحد ال function واحد على X
+
+179
+00:19:41,560 --> 00:19:46,940
+هذه أكبر من Alpha لأي real number Alpha صح؟
+
+180
+00:19:48,400 --> 00:19:52,300
+وبالتالي بقدر أخلي أي T واحد على اكس أكبر من اي
+
+181
+00:19:52,300 --> 00:19:59,600
+real number Alpha لكل X طبعًا على يمين الصفر وعلى
+
+182
+00:19:59,600 --> 00:20:06,660
+مسافة أصغر من Delta نقدر نجيب طبعًا Delta لكل Alpha
+
+183
+00:20:06,660 --> 00:20:13,800
+فممكن برضه استخدم التعريف لإثبات ان ال limit لإي ت
+
+184
+00:20:13,800 --> 00:20:16,800
+واحد على اكس لما اكسه تؤول ل صفر من اليمين بيساوي
+
+185
+00:20:16,800 --> 00:20:21,100
+infinity Okay تمام ان انا ممكن استخدم التعريف أو
+
+186
+00:20:21,100 --> 00:20:27,860
+استخدم ال comparison test اللي فوق واضح في اي سؤال
+
+187
+00:20:27,860 --> 00:20:37,380
+طب
+
+188
+00:20:37,380 --> 00:20:45,280
+احنا يعني لاحظوا في ال chapter هذا اتعرضنا ل ال ..
+
+189
+00:20:47,310 --> 00:20:53,690
+لتعريف النهايات للدوال and cluster point للمجال
+
+190
+00:20:53,690 --> 00:20:58,390
+تبعها أو and cluster point لتقاطع مجالها مع فترة
+
+191
+00:20:58,390 --> 00:21:02,470
+مفتوحة زي هذه أو فترة مفتوحة زي هذه في حالة ال
+
+192
+00:21:02,470 --> 00:21:06,870
+infinite limits وفي كل ال limits هذه دائما ال X
+
+193
+00:21:06,870 --> 00:21:12,290
+كانت تؤول ل C لعدد ل cluster point سواء من اليمين
+
+194
+00:21:12,290 --> 00:21:15,910
+أو من اليسار لكن أحيانًا
+
+195
+00:21:17,840 --> 00:21:29,720
+بتصادفنا نهايات أحيانًا
+
+196
+00:21:29,720 --> 00:21:37,020
+نتعرض لمواقف زي هذه أنه كيف أنا بدي .. يعني ممكن
+
+197
+00:21:37,020 --> 00:21:42,840
+يكون عندي limit ل a for x بدل ما x تؤول ل cluster
+
+198
+00:21:42,840 --> 00:21:45,920
+point c x تؤول ل infinity
+
+199
+00:21:49,360 --> 00:21:52,820
+ما معنى ان ال limit ل f of x لما x تؤول ال
+
+200
+00:21:52,820 --> 00:22:00,720
+infinity بيساوي عدد L أو ما معنى ان ال limit ل ال
+
+201
+00:22:00,720 --> 00:22:05,080
+function f لما x تؤول ل سالب infinity بيساوي أيضًا
+
+202
+00:22:05,080 --> 00:22:12,060
+عدد L هذا ما تعرضنا إليه فبنلاقي تعريف نشوف كيف
+
+203
+00:22:12,060 --> 00:22:20,300
+التعريف تبع ال limits هذه بيكونمثلًا أنا عندي ال
+
+204
+00:22:20,300 --> 00:22:28,380
+limit نرجع لل function واحد على X فانا
+
+205
+00:22:28,380 --> 00:22:33,980
+عندي ال limit يعني Y بيساوي واحد على X فانا عندي ال
+
+206
+00:22:33,980 --> 00:22:40,980
+limit واحد على X لما X تؤول infinity واضح انها
+
+207
+00:22:40,980 --> 00:22:47,330
+بيساوي عدد L صفر وبرضه كمان لو كانت x تؤول ل سالب
+
+208
+00:22:47,330 --> 00:22:53,250
+infinity برضه ال limit بيساوي صفر، إذا كيف اثبت
+
+209
+00:22:53,250 --> 00:22:59,750
+أو كيف أعرف ان ال limit عند ال infinity بيساوي
+
+210
+00:22:59,750 --> 00:23:04,150
+عدد أو عند السالب infinity بيساوي عدد ما؟
+
+211
+00:23:04,790 --> 00:23:10,050
+التعريفات هذه ما مرت لسه علينا فنحتاج ان احنا نعرف
+
+212
+00:23:10,050 --> 00:23:18,950
+أو ناخد هذه التعريفات إذا دلوقت نقول التعريف هذا
+
+213
+00:23:18,950 --> 00:23:28,570
+ناخد 
+
+214
+00:23:28,570 --> 00:23:29,070
+definition
+
+215
+00:23:48,740 --> 00:24:03,120
+فالتعريف let f be function from A to R and
+
+216
+00:24:03,120 --> 00:24:10,780
+الفترة من A إلى ما لا نهاية تكون داخل المجموعة A
+
+217
+00:24:10,780 --> 00:24:13,640
+for some A ينتمي إلى R
+
+218
+00:24:20,670 --> 00:24:32,430
+فبنعرف ونقول ان ال ينتمي لار is
+
+219
+00:24:32,430 --> 00:24:36,710
+a limit of
+
+220
+00:24:36,710 --> 00:24:46,950
+ال function f as x tends to infinity and right و
+
+221
+00:24:46,950 --> 00:24:52,540
+بنكتب في الحالة هذه ان ال limitلـ f of x as x
+
+222
+00:24:52,540 --> 00:24:58,820
+tends to infinity بيساوي لعدد L إذا تحقق الشرط
+
+223
+00:24:58,820 --> 00:25:06,960
+التالي لكل إبسلون for any إبسلون أكبر من 0 نقدر
+
+224
+00:25:06,960 --> 00:25:14,660
+نلاقي capital K عدد حقيقي يعتمد على إبسلون وهذا
+
+225
+00:25:14,660 --> 00:25:23,260
+العدد أكبر من العدد A اللي هو عدد حقيقي معين بحيث
+
+226
+00:25:23,260 --> 00:25:33,980
+أنه لكل لو كان ال X أكبر من ال K فهذا بتضمن أنه
+
+227
+00:25:33,980 --> 00:25:39,780
+absolute F of X minus L أصغر من ال given epsilon
+
+228
+00:25:39,780 --> 00:25:44,280
+تمام؟
+
+229
+00:25:44,280 --> 00:25:46,120
+بالمثل ممكن أعرف
+
+230
+00:25:52,400 --> 00:25:58,400
+العرف ما معناه ان ال limit لل function f لما x
+
+231
+00:25:58,400 --> 00:26:04,120
+تؤول ل سالب infinity بيساوي عدد L في الحالة هذه
+
+232
+00:26:04,120 --> 00:26:13,200
+باشترط ان المجموعة المجال يحتوي على فترة زي هذه
+
+233
+00:26:13,200 --> 00:26:17,840
+بدأت
+
+234
+00:26:17,840 --> 00:26:19,300
+الفترة هذه فترة زي هذه
+
+235
+00:26:22,770 --> 00:26:30,990
+هنا قلنا بدل infinity نبدلها بال-infinity وهنا بال
+
+236
+00:26:30,990 --> 00:26:37,030
+-infinity ونقول
+
+237
+00:26:37,030 --> 00:26:46,190
+أنه يوجد K المرة هذه بدل أكبر من A أصغر من A وهذه
+
+238
+00:26:46,190 --> 00:26:48,150
+تتغير لكل X
+
+239
+00:26:53,500 --> 00:26:59,960
+أصغر من K أصغر
+
+240
+00:26:59,960 --> 00:27:03,760
+من Y أصغر
+
+241
+00:27:03,760 --> 00:27:06,920
+من K أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+242
+00:27:06,920 --> 00:27:07,020
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+243
+00:27:07,020 --> 00:27:07,760
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+244
+00:27:07,760 --> 00:27:12,040
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أص
+
+245
+00:27:12,440 --> 00:27:18,680
+Okay طيب إذا أنا لأن في عندي تعريفات جديدة كمان
+
+246
+00:27:18,680 --> 00:27:25,880
+مرة كل النظريات اللي أثبتناها سابقًا بس
+
+247
+00:27:25,880 --> 00:27:31,500
+بدل ما X تؤول ل C بيصير X تؤول ل infinity ال
+
+248
+00:27:31,500 --> 00:27:37,040
+sequence أول شيء إذا كانت ال limit هذه أو هذه
+
+249
+00:27:37,040 --> 00:27:42,630
+موجودة سواء عدد Lفممكن إثباتي انها unique زيها زي
+
+250
+00:27:42,630 --> 00:27:49,210
+أي two sided limit أو زيها زي أي limit أخرى كذلك
+
+251
+00:27:49,210 --> 00:27:53,370
+ممكن نثبت sequential criterion لل limits زي هدول
+
+252
+00:27:53,370 --> 00:27:59,230
+وممكن النظرية زي هذه تكون صحيحة لهذا النوع من ال
+
+253
+00:27:59,230 --> 00:28:06,130
+limits okay إذا معظم النظريات معظم النظريات اللي
+
+254
+00:28:06,130 --> 00:28:13,240
+أثبتناها تكون صحيحة لهذا النوع الجديد من الـ
+
+255
+00:28:13,240 --> 00:28:17,660
+infinite نسميها infinite limits at infinity هذه
+
+256
+00:28:17,660 --> 00:28:21,980
+limits at infinity أو سالب infinity هذه كانت
+
+257
+00:28:21,980 --> 00:28:27,520
+نسميها infinite limits فمثلا
+
+258
+00:28:27,520 --> 00:28:31,900
+على سبيل المثال وليس الحصر ممكن ان احنا نكتب
+
+259
+00:28:31,900 --> 00:28:35,060
+sequential criterion لهذا النوع من ال limits
+
+260
+00:28:42,580 --> 00:28:57,860
+هي sequential theorem sequential
+
+261
+00:28:57,860 --> 00:29:03,680
+.. sequential
+
+262
+00:29:03,680 --> 00:29:07,500
+criterion
+
+263
+00:29:07,500 --> 00:29:18,320
+.. sequential criterion for limits for limits at
+
+264
+00:29:18,320 --> 00:29:23,580
+infinity the
+
+265
+00:29:23,580 --> 00:29:29,480
+following statements are equivalent are equivalent
+
+266
+00:29:29,480 --> 00:29:40,830
+واحد limit F of X as X tends to infinity بيساوي عدد
+
+267
+00:29:40,830 --> 00:29:47,970
+M اثنين for every
+
+268
+00:29:47,970 --> 00:29:48,910
+sequence
+
+269
+00:29:51,330 --> 00:29:57,750
+x in contained in a تقابل فترة مفتوحة من a إلى ما
+
+270
+00:29:57,750 --> 00:30:05,030
+للإلهية such that limit x in as n tends to
+
+271
+00:30:05,030 --> 00:30:14,410
+infinity بيساوي ال infinity لازم
+
+272
+00:30:14,410 --> 00:30:19,300
+يطلع عندي limit ال image بيساوي العدد L لسيكوانس Xn
+
+273
+00:30:19,300 --> 00:30:25,380
+as N تؤول ل Infinity بيساوي العدد L لذا هذه
+
+274
+00:30:25,380 --> 00:30:34,580
+Sequential criterion for limits at infinity وممكن
+
+275
+00:30:34,580 --> 00:30:39,860
+نثبت النظرية هذه زي ما أثبتنا Sequential criterion
+
+276
+00:30:39,860 --> 00:30:46,300
+for finite two-sided limits أو for finite one
+
+277
+00:30:46,300 --> 00:30:51,520
+-sided limits مثلًا لو أريد أن أثبت واحد implies
+
+278
+00:30:51,520 --> 00:30:55,660
+اثنين فبقول
+
+279
+00:30:55,660 --> 00:31:05,080
+assume أنه one holds هذا
+
+280
+00:31:05,080 --> 00:31:12,440
+معناه أن ال limit لf of x as x tends to infinity
+
+281
+00:31:12,440 --> 00:31:17,120
+بسوي عدد الـ L طيب to prove
+
+282
+00:31:32,110 --> 00:31:38,430
+to prove two holds فابد
+
+283
+00:31:38,430 --> 00:31:46,430
+أثبت لأي sequence لأي sequence بالمواصفات هذه
+
+284
+00:31:46,430 --> 00:31:57,370
+limit صورتها بساوي الـ L فببدأ بقول let let XM contain
+
+285
+00:31:57,370 --> 00:32:08,400
+بالـ A قاطع الفترة هذه بيجيبن
+
+286
+00:32:08,400 --> 00:32:11,560
+بحيث
+
+287
+00:32:11,560 --> 00:32:20,660
+أن ال limit لسيكوينس xn هذه بساوي
+
+288
+00:32:20,660 --> 00:32:24,660
+infinity و
+
+289
+00:32:24,660 --> 00:32:27,380
+بالثبات أن ال limit صورتها بساوي الـ
+
+290
+00:32:30,930 --> 00:32:34,970
+عشان أثبت أنه two holds، بدي أثبت أنه الـ limit
+
+291
+00:32:34,970 --> 00:32:45,370
+لصورة الـ xn as n times infinity بساوي L لأن هذه
+
+292
+00:32:45,370 --> 00:32:48,930
+عبارة عن sequence، بدي أثبت limit sequence بالساوي
+
+293
+00:32:48,930 --> 00:32:54,030
+عدد، بستخدم تعريف Y capital N لل limit of a
+
+294
+00:32:54,030 --> 00:33:00,120
+sequence، صح؟ إذا أنا بقول let epsilon أكبر من
+
+295
+00:33:00,120 --> 00:33:07,240
+الصفر be given طيب،
+
+296
+00:33:07,240 --> 00:33:16,140
+أنا عندي فارض since ال limit لـ F of X as X tends
+
+297
+00:33:16,140 --> 00:33:26,080
+to infinity بتساوي العدد L إذا من تعريف ال limit of
+
+298
+00:33:26,080 --> 00:33:31,180
+infinity اللي زيها دي هي التعريف هي تحت تقول إنه
+
+299
+00:33:31,180 --> 00:33:41,300
+for any given epsilon يوجد عدد حقيقي K يعتمد على
+
+300
+00:33:41,300 --> 00:33:47,220
+epsilon وهذا أكبر من A بحيث
+
+301
+00:33:47,220 --> 00:33:57,390
+إنه لو كان X أكبر من الـ K بيقدي أنه absolute f of
+
+302
+00:33:57,390 --> 00:34:03,750
+x minus الـ L أصغر من إبسلون أسمي ال implication هذه
+
+303
+00:34:03,750 --> 00:34:08,830
+star طيب
+
+304
+00:34:08,830 --> 00:34:13,750
+أنا برضه عندي أنا فارد أن ال sequence هذه ال given
+
+305
+00:34:13,750 --> 00:34:15,590
+sequence ال limit تبعتها
+
+306
+00:34:21,580 --> 00:34:26,380
+بساوي infinity ومن
+
+307
+00:34:26,380 --> 00:34:30,540
+تعريف أن تكون ال sequence limit تبعتها infinite
+
+308
+00:34:30,540 --> 00:34:37,180
+هذا معناه أن مقدر أخلي x in أكبر من أي عدد حقيقي
+
+309
+00:34:37,180 --> 00:34:41,440
+alpha فاخد alpha هنا بساوي K مش هذا عدد حقيقي
+
+310
+00:34:41,440 --> 00:34:48,520
+محترم فاخد هوبما عنده limit لسيكوينس Xn بالساوي
+
+311
+00:34:48,520 --> 00:34:56,920
+infinity then for any real number يوجد
+
+312
+00:34:56,920 --> 00:35:06,700
+capital N عدد طبيعي يعتمد على K هذا أشمله عدد طبيعي
+
+313
+00:35:06,700 --> 00:35:14,800
+بحيث أنه لكل n أكبر من أو يساوي capital N هذا بتضمن
+
+314
+00:35:14,800 --> 00:35:22,980
+أن ال Xn أكبر من العدد K العدد الحقيقي K نسمي ال
+
+315
+00:35:22,980 --> 00:35:25,160
+implication هذه double star
+
+316
+00:35:39,650 --> 00:35:45,850
+تمام هيك إذا هذا ناخده من كوننا أن احنا فرضين أن
+
+317
+00:35:45,850 --> 00:35:49,350
+limit ال sequence xn بالساوي infinity ومن تعريف
+
+318
+00:35:49,350 --> 00:35:56,710
+ال infinite limit لل sequence الآن الآن في تحول في
+
+319
+00:35:56,710 --> 00:36:05,270
+البرهان now star and double star yield
+
+320
+00:36:09,010 --> 00:36:17,430
+بيعطوني التالي لو كانت n أكبر من أو يساوي capital N
+
+321
+00:36:17,430 --> 00:36:28,590
+فهذا بيؤدي إنه xn أكبر من k هذا موجود أخدناه من
+
+322
+00:36:28,590 --> 00:36:33,930
+double star لكل n أكبر من أو يساوي capital N بيطلع
+
+323
+00:36:33,930 --> 00:36:40,980
+xn أكبر من k طيب و من ال star و هذا بيقدي باستخدام
+
+324
+00:36:40,980 --> 00:36:47,640
+ال star ال star بيقول لي لكل x لو كانت ال x أو ال
+
+325
+00:36:47,640 --> 00:36:56,860
+xn أكبر من capital K هذا بيقدي أن صورتها المسافة
+
+326
+00:36:56,860 --> 00:37:01,840
+بينها و بين الـ L أصغر من إبسلون صح؟
+
+327
+00:37:06,200 --> 00:37:10,360
+إذا نجي نلخص كمان مرة، إيه اللي عملناها؟ أنا إيش
+
+328
+00:37:10,360 --> 00:37:14,560
+بدأنا في بتلقى limited sequence F of X N لما N تقول
+
+329
+00:37:14,560 --> 00:37:18,460
+الـ infinity بالساوي عدد L فهذه البديات بإبسلون
+
+330
+00:37:18,460 --> 00:37:22,940
+أكبر من الصفر given أثبتت أن يوجد capital N عدد
+
+331
+00:37:22,940 --> 00:37:27,680
+طبيعي يعتمد على الـ K والـ K تعتمد على إبسلون، إذا
+
+332
+00:37:27,680 --> 00:37:33,810
+الـ N هذه تعتمد على الـ given إبسلون و هذه ال N لكل
+
+333
+00:37:33,810 --> 00:37:38,050
+small n أكبر من أو يساوي ال capital N هذه طلع عند
+
+334
+00:37:38,050 --> 00:37:41,350
+المسافة بين الحد النوني لل sequence و L أصغر من
+
+335
+00:37:41,350 --> 00:37:47,750
+Epsilon بما أن Epsilon was arbitrary since Epsilon
+
+336
+00:37:47,750 --> 00:37:54,250
+أكبر من الصفر was arbitrary إذا by epsilon capital
+
+337
+00:37:54,250 --> 00:37:58,650
+N definition of limit of sequence بنكون هيك حسب
+
+338
+00:37:58,650 --> 00:38:04,570
+التعريف أثبتنا أن ال limit لسيكوينس F of X N as N
+
+339
+00:38:04,570 --> 00:38:09,390
+tends to infinity بالساوي لعدد L وبالتالي هيك
+
+340
+00:38:09,390 --> 00:38:16,950
+بنكون أثبتنا إذا العبارة 2 holds وهيك بنكون أثبتنا
+
+341
+00:38:16,950 --> 00:38:23,230
+أن العبارة statement 1 implies statement 2 Okay
+
+342
+00:38:23,230 --> 00:38:30,010
+تمام إذا نحن ممكن نبرهن ال sequential criterion لل
+
+343
+00:38:30,010 --> 00:38:35,730
+infinite limit و لل one sided limit و لكل أنواع ال
+
+344
+00:38:35,730 --> 00:38:47,310
+limit و هاي أثبتنا جزء برهان الجزء الثاني مماثل ال
+
+345
+00:38:47,310 --> 00:38:58,010
+proof of 2 implies 1 is similar to
+
+346
+00:38:58,010 --> 00:39:03,410
+original
+
+347
+00:39:03,410 --> 00:39:08,690
+proof or proof of
+
+348
+00:39:08,690 --> 00:39:14,810
+original original
+
+349
+00:39:14,810 --> 00:39:21,470
+sequential criterion original sequential criterion
+
+350
+00:39:21,470 --> 00:39:26,530
+مع عمل التعديلات اللازمة فأنا بقول لكم أنكم ترجعوا ل
+
+351
+00:39:26,530 --> 00:39:30,910
+sequential criterion الأساسية تقرأوا البرهان تبعها
+
+352
+00:39:30,910 --> 00:39:36,050
+كيف أنا برهان اتنين ده أو احدو تعملوا التعديلات ..
+
+353
+00:39:36,050 --> 00:39:41,070
+إزاي نبرهن واحد بدل اتنين .. okay تمام .. إذا أن
+
+354
+00:39:41,070 --> 00:39:44,910
+هذه تعتبر sequential criterion لل limits at
+
+355
+00:39:44,910 --> 00:39:50,710
+infinity بالمثل ممكن أن احنا نحصل على sequential
+
+356
+00:39:50,710 --> 00:39:57,810
+criterion لل limits at negative infinity يعني ال
+
+357
+00:39:57,810 --> 00:40:04,930
+.. يعني هذه ممكن تبدلها ب negative infinity وهذه
+
+358
+00:40:04,930 --> 00:40:10,570
+ممكن تبدلها ب negative infinity وهذه ممكن تبدلها ب
+
+359
+00:40:10,570 --> 00:40:15,670
+سالب infinity إلى a وهذه ممكن تبدلها ب negative
+
+360
+00:40:15,670 --> 00:40:19,790
+infinity وهذه
+
+361
+00:40:19,790 --> 00:40:23,550
+ممكن تبدلها بسالب
+
+362
+00:40:23,550 --> 00:40:24,030
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+363
+00:40:24,030 --> 00:40:24,130
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+364
+00:40:24,130 --> 00:40:24,750
+تبدلها بسالب infinity إلى a وهذه ممكن تبدلها بسالب
+
+365
+00:40:24,750 --> 00:40:24,830
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+366
+00:40:24,830 --> 00:40:25,250
+تبدلها بسالب infinity إلى a وهذه ممكن تبدلها بسالب
+
+367
+00:40:25,250 --> 00:40:26,690
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+368
+00:40:26,690 --> 00:40:27,830
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+369
+00:40:27,830 --> 00:40:38,590
+تبدلها بسالب… البرنامج طبعا مشابه للنظرية السابقة ناخد
+
+370
+00:40:38,590 --> 00:40:44,150
+أمثلة طبعا
+
+371
+00:40:44,150 --> 00:40:52,230
+في نظريات كتيرة صحيحة لنوع هذا من ال limits فلو
+
+372
+00:40:52,230 --> 00:40:58,940
+احتجنا زي ال squeeze theorem زي ال comparison test
+
+373
+00:40:58,940 --> 00:41:04,240
+أو الآخر فمثلا
+
+374
+00:41:04,240 --> 00:41:23,600
+ناخد بعض الأمثلة مثلا
+
+375
+00:41:23,600 --> 00:41:25,080
+ناخد examples
+
+376
+00:41:41,280 --> 00:41:50,920
+F of X يساوي واحد على X و X لا يساوي صفر Show
+
+377
+00:41:50,920 --> 00:41:54,720
+that limit
+
+378
+00:41:54,720 --> 00:42:03,660
+F of X as X tends to infinity يساوي صفر و كذلك
+
+379
+00:42:03,660 --> 00:42:11,410
+limit F of x as x tends to negative infinity بساوي
+
+380
+00:42:11,410 --> 00:42:16,950
+صفر المظبوط
+
+381
+00:42:16,950 --> 00:42:24,370
+ال function واحد على x ثانية فلما
+
+382
+00:42:24,370 --> 00:42:29,070
+x تقول infinity واحد على x بتقول صفر لما x
+
+383
+00:42:29,070 --> 00:42:32,690
+تقولها سالب infinity برضه ال function واحد على x
+
+384
+00:42:32,690 --> 00:42:39,550
+بتقول صفر فلو بدي اثبات الجزء الأول فممكن استخدم
+
+385
+00:42:39,550 --> 00:42:45,590
+التعريف أو استخدم ال sequential criterion فمثلا
+
+386
+00:42:45,590 --> 00:42:56,850
+to use a definition لو بدي استخدم التعريف مثلا
+
+387
+00:42:58,750 --> 00:43:02,270
+limit f of x من x سواء و لا انا كنت بتساوي صفر
+
+388
+00:43:02,270 --> 00:43:10,950
+فبابدأ حسب التعريف بابدأ بإبسلون أكبر من الصفر ل
+
+389
+00:43:10,950 --> 00:43:18,510
+إبسلون أكبر من الصفر بكلمة وبعدين بدأ أثبت أنه في
+
+390
+00:43:18,510 --> 00:43:26,420
+K يعتمد على إبسلون ف choose K بطريقة الكلى انه واحد
+
+391
+00:43:26,420 --> 00:43:30,840
+على ابسلان فهذا تطلع عدد موجب ويعتمد على ابسلان
+
+392
+00:43:30,840 --> 00:43:36,780
+و ال L طبعا هنا في السؤال هذا هي الصفر يعني هنا ال
+
+393
+00:43:36,780 --> 00:43:40,680
+domain تبعت لكل العداد الحقيقية مع ده الصفر فممكن
+
+394
+00:43:40,680 --> 00:43:46,440
+اخد الصفر الفقرة هذه contained in ال domain تبع
+
+395
+00:43:46,440 --> 00:43:51,100
+الـ a اللي هو كل العداد الحقيقية من عدد الصفر هنا
+
+396
+00:43:51,100 --> 00:43:52,620
+لأي X أكبر من K أكبر من أكبر من أكبر من أكبر من أكبر من
+
+397
+00:43:52,620 --> 00:43:53,060
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+398
+00:43:53,060 --> 00:43:53,980
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+399
+00:43:53,980 --> 00:43:57,560
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+400
+00:43:57,560 --> 00:44:03,280
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+401
+00:44:03,280 --> 00:44:06,340
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+402
+00:44:06,340 --> 00:44:08,320
+أكبر من أكبر من أكبر
+
+403
+00:44:09,640 --> 00:44:19,540
+أن واحد على X أصغر من واحد على K و هذا بدوره
+
+404
+00:44:19,540 --> 00:44:25,560
+بقدر أن absolute of f of x minus الصفر الـ L هنا
+
+405
+00:44:25,560 --> 00:44:31,480
+هو الصفر فهذا بيطلع بساوي absolute واحد على X
+
+406
+00:44:31,480 --> 00:44:34,420
+فهذا عبارة عن واحد على X
+
+407
+00:44:45,380 --> 00:44:48,660
+و هذا أقل من واحد على K و واحد على K أقل من
+
+408
+00:44:48,660 --> 00:44:54,400
+إبسلو و هذا أصغر طبعا من واحد على K طبعا عندي ال
+
+409
+00:44:54,400 --> 00:44:59,520
+X هنا أكبر من K لحظة X أكبر من K و ال K موجة بقى
+
+410
+00:45:03,660 --> 00:45:08,560
+القيمة المطلقة لـ 1 على X هي 1 على X اطرح صفر
+
+411
+00:45:08,560 --> 00:45:14,760
+ما بتعملش حاجة هذا أصغر من 1 على K ومن هنا 1 على K
+
+412
+00:45:14,760 --> 00:45:15,580
+بساوي L
+
+413
+00:45:18,310 --> 00:45:22,630
+إذن هاني أثبتت لي أي epsilon أكبر من صفر يوجد K
+
+414
+00:45:22,630 --> 00:45:27,590
+عدد حقيقي يعتمد على epsilon بحيث لكل x أكبر من K
+
+415
+00:45:27,590 --> 00:45:33,190
+يطلع absolute f of x minus L أصغر من epsilon okay
+
+416
+00:45:33,190 --> 00:45:39,430
+إذن هذا معناه أن ال limit حسب التعريف limit واحد
+
+417
+00:45:39,430 --> 00:45:46,030
+على x لما x تقول إلى infinity بساوي صفر
+
+418
+00:45:49,020 --> 00:45:53,960
+بالمثل ممكن نثبت الجزء الثاني نفس البرهان مع
+
+419
+00:45:53,960 --> 00:46:02,520
+التعديل في تعريف limit at سالب infinity ممكن
+
+420
+00:46:02,520 --> 00:46:10,180
+برضه نستخدم sequential criterion لو بدأت 
+
+421
+00:46:10,180 --> 00:46:15,900
+استخدم sequential criterion لإثبات
+
+422
+00:46:15,900 --> 00:46:26,470
+limit بأخذ بقول إن هنا let xn be a sequence contained
+
+423
+00:46:26,470 --> 00:46:35,250
+in 0 و infinity بحيث إن limit xn تساوي infinity
+
+424
+00:46:35,250 --> 00:46:39,570
+إذا 
+
+425
+00:46:39,570 --> 00:46:48,340
+limit f of xn has n times infinity طبعا هذا بيقود
+
+426
+00:46:48,340 --> 00:46:51,500
+هذا
+
+427
+00:46:51,500 --> 00:46:58,320
+بيقود إن limit واحد على xn  بساوي صفر exercise
+
+428
+00:46:58,320 --> 00:47:02,480
+أخذناها أخذنا إن limit sequence xn بساوي infinity
+
+429
+00:47:02,480 --> 00:47:06,990
+if and only if limit مقلوب السيكوانس بساوي صفر الآن
+
+430
+00:47:06,990 --> 00:47:13,670
+limit f of xn بساوي limit واحد على xn as n tends
+
+431
+00:47:13,670 --> 00:47:21,150
+to infinity وهذا بيساوي 6 لأي
+
+432
+00:47:21,150 --> 00:47:26,510
+sequence نهايتها infinity نهاية صورتها بيساوي
+
+433
+00:47:26,510 --> 00:47:27,330
+العدد L
+
+434
+00:47:35,720 --> 00:47:42,000
+بنطلع ال limit ل ال function f of x as x tends to 
+
+435
+00:47:42,000 --> 00:47:47,200
+infinity بساوية 0 إذا هذا برهان ثاني using
+
+436
+00:47:47,200 --> 00:47:55,620
+sequential criterion okay واضح مفهوم مثال
+
+437
+00:47:55,620 --> 00:47:56,160
+ثاني
+
+438
+00:48:05,300 --> 00:48:10,520
+بناخد g of x بساوي
+
+439
+00:48:10,520 --> 00:48:15,820
+واحد على x تربيع g of x لا يساوي صغير فبدنا نثبت أن
+
+440
+00:48:15,820 --> 00:48:21,660
+ال limit ل g of x لما x تقول ل infinity ولما x
+
+441
+00:48:21,660 --> 00:48:27,280
+تقول ل سالب infinity بساوي صغير برضه ممكن نستخدم
+
+442
+00:48:27,280 --> 00:48:32,440
+sequential criterion لثبات الجزء الأول أو الثاني
+
+443
+00:48:37,280 --> 00:48:42,020
+هذا كان limit xn بالساوية infinity فlimit 1 على xn
+
+444
+00:48:42,020 --> 00:48:46,300
+بالساوية infinity بساوي صفر وبالتالي limit 1 على
+
+445
+00:48:46,300 --> 00:48:57,500
+xn تربيع يعني هذا بيقود وهذا
+
+446
+00:48:57,500 --> 00:49:05,370
+بيقود إن limit 1 على xn تربيع لما n تنتقل ل infinity
+
+447
+00:49:05,370 --> 00:49:16,510
+بساوي limit 1 على xn ضرب limit 1 على xn وهذا بساوي 
+
+448
+00:49:16,510 --> 00:49:27,310
+0 ضرب 0 بساوي 0 و limit g ل xn as n tends to
+
+449
+00:49:27,310 --> 00:49:33,810
+infinity بساوي limit 1 على x n تربيع يعني بالساعة
+
+450
+00:49:33,810 --> 00:49:40,990
+صفر صح إذا by sequential criterion
+
+451
+00:49:40,990 --> 00:49:44,030
+أثبتت
+
+452
+00:49:44,030 --> 00:49:49,130
+إن لأي sequence x حدودها موجبة أو نهايتها
+
+453
+00:49:49,130 --> 00:49:57,970
+infinity ف limit صورتها بالساعة صفر هذا معناه أن
+
+454
+00:49:57,970 --> 00:50:06,540
+ال limitلـ function g of x لما x تقول infinity
+
+455
+00:50:06,540 --> 00:50:10,980
+بساوي صفر هنا نريد أن نكون أخرجنا الجزء الأول
+
+456
+00:50:10,980 --> 00:50:14,800
+باستخدام sequential criterion بالمثل وكنا نستخدم 
+
+457
+00:50:14,800 --> 00:50:18,280
+sequential criterion اللي أتبعت الجزء التالي بس
+
+458
+00:50:18,280 --> 00:50:25,580
+هنا هناخد x الموجودة في الفترة هذه وهكذا ونفس
+
+459
+00:50:25,580 --> 00:50:29,860
+النظرية اللي أخذناها في القصة السابقة بالكون صحيحة
+
+460
+00:50:29,860 --> 00:50:33,760
+هذا بقى يقود المقلوب ال sequence إذا كانت limit ال
+
+461
+00:50:33,760 --> 00:50:37,320
+sequence infinity فlimit المقلوب صفر وبالتالي
+
+462
+00:50:37,320 --> 00:50:43,420
+limit المقلوب المربع بساوي صفر ممكن كمان نستخدم
+
+463
+00:50:43,420 --> 00:50:48,440
+squeeze theorem ممكن نستخدم squeeze theorem فمثلا
+
+464
+00:50:48,440 --> 00:50:56,100
+لو بدنا نبرهن كمان واحد بطريقة ثانية فممكن إن احنا
+
+465
+00:50:56,100 --> 00:51:05,360
+note that for x أكبر من 1 لما أخدت x أكبر من
+
+466
+00:51:05,360 --> 00:51:11,700
+1 بطلع عندي دائما x تربيع أكبر من أو يساوي x
+
+467
+00:51:13,190 --> 00:51:18,610
+وبالتالي هذا بيقود إن 1 على x تربيع أكبر من 
+
+468
+00:51:18,610 --> 00:51:23,930
+أو يساوي 1 على x وطبعا أكبر من الصفر لكل x
+
+469
+00:51:23,930 --> 00:51:29,310
+أكبر من 1 طب احنا لسه مثلنا في الجدول شوية إن limit
+
+470
+00:51:29,310 --> 00:51:33,810
+ال function 1 على x لما x تقول إلى infinity
+
+471
+00:51:33,810 --> 00:51:40,930
+بساوي صفر صح؟ لسه 130 في المثال الأول هذا المثال
+
+472
+00:51:40,930 --> 00:51:44,630
+الثاني في المثال السابق قصدنا إن limit ال function
+
+473
+00:51:44,630 --> 00:51:48,790
+1 على x لما x تقول infinity تساوي صفر وlimit
+
+474
+00:51:48,790 --> 00:51:54,090
+الدالة ثابت صفر لما x تقول infinity تبقى صفر إذا 
+
+475
+00:51:54,090 --> 00:52:03,790
+by squeeze theorem by
+
+476
+00:52:03,790 --> 00:52:09,590
+squeeze theorem for limits at infinity Limited دالة
+
+477
+00:52:09,590 --> 00:52:19,130
+المحصورة اللي هي 1 على x تربيع as X tends to
+
+478
+00:52:19,130 --> 00:52:24,310
+infinity بساوي صفر
+
+479
+00:52:24,310 --> 00:52:32,490
+okay واضح وبالمثل ممكن نعطي براهين زي هذا أو زي
+
+480
+00:52:32,490 --> 00:52:36,290
+هذا إما باستخدام squeeze theorem أو sequential
+
+481
+00:52:36,290 --> 00:52:43,790
+criterion أو حتى definition okay واضح في أي سؤال
+
+482
+00:52:43,790 --> 00:52:49,050
+في أي استفسار okay هنوقف إذن هنا والمرة الجاية في
+
+483
+00:52:49,050 --> 00:52:53,190
+بعض أنواع ال infinite limits هنتكلم عنهم يعني
+
+484
+00:52:53,190 --> 00:52:59,640
+باختصار وبعدين نحاول نجمّل section أربعة تلاتة 
+
+485
+00:52:59,640 --> 00:53:04,100
+وبالتالي ننهي ال chapter اللي هو chapter أربعة و
+
+486
+00:53:04,100 --> 00:53:08,620
+بعدين نبدأ في chapter خمسة اللي هو آخر chapter في
+
+487
+00:53:08,620 --> 00:53:15,240
+المقرر وهو أهم chapter طبعا في حد عنده أي سؤال أو
+
+488
+00:53:15,240 --> 00:53:19,640
+استفسار؟ شكرا لاصغائكم ونشوف إن شاء الله المرة
+
+489
+00:53:19,640 --> 00:53:20,040
+القادمة
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b659d86b900a995dc4cbc5b68ce6adcf9a7d4a8f
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_postprocess.srt
@@ -0,0 +1,1956 @@
+1
+00:00:21,750 --> 00:00:27,870
+بسم الله الرحمن الرحيم في محاضرة اليوم هنكمل ما
+
+2
+00:00:27,870 --> 00:00:32,170
+بدأنا حاول
+
+3
+00:00:32,170 --> 00:00:37,890
+موضوع ال infinite limits المرة تاعة عرفنا ما معناه
+
+4
+00:00:37,890 --> 00:00:42,370
+ان ال limit ل function and cluster point بالساوية
+
+5
+00:00:42,370 --> 00:00:49,050
+plus او minus infinity وشوفنا نظرية اخر نظرية
+
+6
+00:00:49,050 --> 00:00:54,800
+برهنهاخاصة بهذا النوع من ال limits كانت النظرية
+
+7
+00:00:54,800 --> 00:01:05,340
+التالية خلينا نكتبها let
+
+8
+00:01:05,340 --> 00:01:14,420
+f و g be functions from a to r و c ال cluster
+
+9
+00:01:14,420 --> 00:01:18,320
+point
+
+10
+00:01:20,530 --> 00:01:28,690
+of the set A such that f
+
+11
+00:01:28,690 --> 00:01:34,590
+of x less than or equal g of x for every x تمتمي
+
+12
+00:01:34,590 --> 00:01:42,630
+إلى a different from c فشوفنا أنه لو كان ال limit
+
+13
+00:01:42,630 --> 00:01:45,950
+فf
+
+14
+00:01:45,950 --> 00:01:48,190
+of x as x tends
+
+15
+00:02:05,090 --> 00:02:07,390
+وكذلك لو كانت ال limit
+
+16
+00:02:10,530 --> 00:02:17,930
+لـ u of x as x tends to c بساوي negative infinity
+
+17
+00:02:17,930 --> 00:02:26,810
+فهذا بتضمن ان limit ل f of x as x tends to c بساوي
+
+18
+00:02:26,810 --> 00:02:35,730
+negative infinity طيب
+
+19
+00:02:35,730 --> 00:02:36,350
+ال ..
+
+20
+00:02:41,050 --> 00:02:47,350
+اليوم هنعرف ما معناه ان ال limit and c من اليمين
+
+21
+00:02:47,350 --> 00:02:52,030
+بالساوي infinity او ال limit and c من اليسار
+
+22
+00:02:52,030 --> 00:02:55,990
+بالساوي infinity و كذلك نفس الشيء ال one sided
+
+23
+00:02:55,990 --> 00:03:02,030
+limit and c ما معناه أنها ساوي سالب infinity لأن
+
+24
+00:03:02,030 --> 00:03:07,010
+هذه كانت two sided limit المرة الأخيرة اتعرفناما
+
+25
+00:03:07,010 --> 00:03:10,410
+معناه ان ال two sided limit تكون infinite او ساوي
+
+26
+00:03:10,410 --> 00:03:14,430
+infinity او plus او minus infinity اليوم ما معناه
+
+27
+00:03:14,430 --> 00:03:18,370
+ان ال one sided limit تكون infinity او negative
+
+28
+00:03:18,370 --> 00:03:23,970
+infinity فناخد التعريف مشابه
+
+29
+00:03:23,970 --> 00:03:29,050
+للتعريف التعريف ال one sided limit تكون بتساوي
+
+30
+00:03:29,050 --> 00:03:33,590
+real number ف
+
+31
+00:03:33,590 --> 00:03:44,120
+letfd function from a to r و
+
+32
+00:03:44,120 --> 00:03:49,240
+c cluster point
+
+33
+00:03:49,240 --> 00:03:58,760
+of الست اللي هي a تقاطع الفترة المفتوحة من c لما
+
+34
+00:03:58,760 --> 00:03:59,640
+إلى نهاية
+
+35
+00:04:04,430 --> 00:04:10,290
+يقول إن الـ limit لـ
+
+36
+00:04:10,290 --> 00:04:15,530
+function f of x as x tends to c from the right
+
+37
+00:04:15,530 --> 00:04:23,110
+بالساوي infinity respectively
+
+38
+00:04:23,110 --> 00:04:30,370
+على التوالي بنقول إن ال limit ل f of x لما x تقول
+
+39
+00:04:30,370 --> 00:04:39,170
+إلى c من اليمينبتساوي negative infinity إذا
+
+40
+00:04:39,170 --> 00:04:44,790
+تحقق الشرط التالي لأي
+
+41
+00:04:44,790 --> 00:04:51,950
+Alpha for any Alpha
+
+42
+00:04:51,950 --> 00:04:57,450
+belonging to R نقدر
+
+43
+00:04:57,450 --> 00:05:07,810
+نلاقي Delta تعتمد على Alphaعلى دموجة بحيث انه لكل
+
+44
+00:05:07,810 --> 00:05:17,050
+x ينتمي إلى a و ال x على يمين ال c و المسافة بينها
+
+45
+00:05:17,050 --> 00:05:22,730
+و بين ال c أصغر من دلتا فلازم هذا يضمن انه f of x
+
+46
+00:05:22,730 --> 00:05:32,310
+أكبر من alpha او على التواري respectively ال f of
+
+47
+00:05:32,310 --> 00:05:37,330
+xهتكون في حالة ال limit بالساول سالب infinity
+
+48
+00:05:37,330 --> 00:05:42,550
+عايزينها تكون أصغر من ال alpha أصغر من ال given
+
+49
+00:05:42,550 --> 00:05:46,330
+alpha okay
+
+50
+00:05:46,330 --> 00:05:53,550
+إذا أنا هنا عندي limit ال function and c من اليمين
+
+51
+00:05:53,550 --> 00:05:57,570
+بالساول infinity معناته لأي real number alpha بقدر
+
+52
+00:05:57,570 --> 00:06:05,620
+أخلي f of xأكبر من Alpha لكل X على يمين الـC لأن X
+
+53
+00:06:05,620 --> 00:06:08,920
+تقوى للـC من اليمين فX على يمين الـC يعني X أكبر
+
+54
+00:06:08,920 --> 00:06:13,160
+من الـC يعني X ثالث C أكبر من الثالث والمسافة بين
+
+55
+00:06:13,160 --> 00:06:18,140
+الـC والـX أو الـX والـC أصغر من الـD فلكل الـX
+
+56
+00:06:18,140 --> 00:06:22,140
+اللي زيها دي بدي أخلي F of X أكبر من Alpha عشان
+
+57
+00:06:22,140 --> 00:06:26,280
+أقدر أقول أن ال limit لF of X tends to infinity
+
+58
+00:06:28,130 --> 00:06:31,870
+بالمثل ما معنى انه limit f of x عن c من اليمين
+
+59
+00:06:31,870 --> 00:06:38,050
+بالساوى سالب infinity معناه بقدر اخلى لكل x زى ما
+
+60
+00:06:38,050 --> 00:06:44,350
+شوفنا او لأى عدد real number alpha يوجد delta بحيث
+
+61
+00:06:44,350 --> 00:06:48,670
+لكل x على يمين ال C والمسافة بينها و بين ال C أصغر
+
+62
+00:06:48,670 --> 00:06:52,310
+من ال delta لازم صورتها تكون أصغر من ال given
+
+63
+00:06:52,310 --> 00:07:00,470
+alpha okay تمامطيب خلّينا الان كتير من النظريات
+
+64
+00:07:00,470 --> 00:07:06,970
+اللي أخدناها for two sided limit زي هذه مثلا بتكون
+
+65
+00:07:06,970 --> 00:07:12,210
+صحيحة لل right limit و لل left limit طبعا ممكن
+
+66
+00:07:12,210 --> 00:07:18,570
+كمان نعرف بنفس الطريقة ال limit from the left او
+
+67
+00:07:18,570 --> 00:07:22,690
+ال left hand limit مايعني ان ال left hand limit
+
+68
+00:07:22,690 --> 00:07:24,410
+تساوي
+
+69
+00:07:25,850 --> 00:07:31,370
+Infinity او سالب Infinity اذا
+
+70
+00:07:31,370 --> 00:07:36,690
+لو انا بدى اعدل اعرف مامعنى ان ال limit ل F عن C
+
+71
+00:07:36,690 --> 00:07:40,750
+من اليسار بالساوية Infinity او مامعنى ان ال limit
+
+72
+00:07:40,750 --> 00:07:46,150
+ل F عن C من اليسار بالساوية سالب Infinity فباخد
+
+73
+00:07:46,150 --> 00:07:53,340
+let F be هكذاو C cluster point هتصير لإيه تقاطع
+
+74
+00:07:53,340 --> 00:08:00,220
+الفترة من سالب infinity إلى C فبنقول
+
+75
+00:08:00,220 --> 00:08:04,680
+إن ال limit لما X تقول إلى C من اليسار بالساوي
+
+76
+00:08:04,680 --> 00:08:10,260
+infinity أو ال limit لما X تقول إلى C من اليسار
+
+77
+00:08:10,260 --> 00:08:15,860
+بالساوي السالب infinity إذا كان لأي Alpha يوجد
+
+78
+00:08:15,860 --> 00:08:21,200
+Delta تعتمد على Alphaالان ال X هتكون على يسار ال C
+
+79
+00:08:21,200 --> 00:08:31,200
+وبالتالي هذا هنستبدله بC سالب X أكبر
+
+80
+00:08:31,200 --> 00:08:38,680
+من سفر أصغر من دلتر فلكل X زي هذه انا عايز ان تكون
+
+81
+00:08:38,680 --> 00:08:43,620
+F of X أكبر من Alpha او في الحالة هذه F of X أصغر
+
+82
+00:08:43,620 --> 00:08:49,710
+من Alpha هنا ذيك نكون عرفناالـ left limit عن c ما
+
+83
+00:08:49,710 --> 00:08:56,170
+معنى أنها ساوي plus أو minus infinity إذن قلنا إن
+
+84
+00:08:56,170 --> 00:09:00,310
+كل النظريات اللي برهنها for two sided limits هتكون
+
+85
+00:09:00,310 --> 00:09:08,730
+صحيحة لل left limit و لل right limit من ضمنهم
+
+86
+00:09:08,730 --> 00:09:14,010
+النظرية السابقة طيب
+
+87
+00:09:14,010 --> 00:09:15,130
+لو بدي أنا يعني
+
+88
+00:09:18,090 --> 00:09:24,870
+أخد أمثلة كيف نستخدم التعريف هذا فيه إثبات إن ال
+
+89
+00:09:24,870 --> 00:09:32,230
+limits تطلع plus أو minus infinity فناخد أول مثال
+
+90
+00:09:32,230 --> 00:09:40,250
+let f of x بسوي واحد
+
+91
+00:09:40,250 --> 00:09:45,170
+على x حيث x لا يساوي سبق و show
+
+92
+00:09:49,780 --> 00:09:58,060
+عايزين نفدت واحد ان ال limit لواحد على x او f of x
+
+93
+00:09:58,060 --> 00:10:05,820
+هنا لما x تقول الى ستر من اليمين بساوي ال infinity
+
+94
+00:10:05,820 --> 00:10:09,960
+و 2 limit
+
+95
+00:10:11,300 --> 00:10:19,200
+لف of X لما X تقول ال 0 من اليسار يساوي سالب
+
+96
+00:10:19,200 --> 00:10:23,880
+infinity okay فلو
+
+97
+00:10:23,880 --> 00:10:29,540
+بدنا نبرم الجزء الأول مثلا to show
+
+98
+00:10:32,410 --> 00:10:38,710
+المقاومة لـ f of x as x tends to 0 from the right
+
+99
+00:10:38,710 --> 00:10:45,670
+بساوي infinity فبدي ابدا بـ alpha تنتمي ل R فبقول
+
+100
+00:10:45,670 --> 00:10:57,490
+let alpha belonging to R be given و
+
+101
+00:10:57,490 --> 00:11:02,790
+بدي ارد عليها بDeltaبدي ارد على ال alpha دي ال
+
+102
+00:11:02,790 --> 00:11:08,630
+delta عدد موجب ويعتمد على ال alpha ف choose delta
+
+103
+00:11:08,630 --> 00:11:15,890
+بتساوي واحد على absolute alpha زاد واحد بالتأكيد
+
+104
+00:11:15,890 --> 00:11:21,910
+هذا عدد موجب لأن absolute ال alpha دي عدد حقيقي
+
+105
+00:11:21,910 --> 00:11:28,450
+القيمة المطلقة له عدد غير سالم ممكن يساوي سفر إذا
+
+106
+00:11:28,450 --> 00:11:32,980
+كانت alpha بالساوية سفرلكن زائد واحد بصير موجب
+
+107
+00:11:32,980 --> 00:11:37,720
+المقام موجب اذا انا بضيف واحد ليه عشان اضمن ان
+
+108
+00:11:37,720 --> 00:11:41,880
+المقام مايسويش سفر لان في احتمال ان ال alpha ساوي
+
+109
+00:11:41,880 --> 00:11:45,640
+سفر فبصير عندى مشكلة عشان اتخلص من المشكلة هذه
+
+110
+00:11:45,640 --> 00:11:51,140
+بجسم على absolute alpha زائد واحد الان هذا عدد
+
+111
+00:11:51,140 --> 00:11:57,010
+موجبو يعتمد على alpha هي ال delta هي مرتبطة معرفة
+
+112
+00:11:57,010 --> 00:12:02,350
+بدلالة alpha هي معناه أنها تعتمد على alpha إذا لأي
+
+113
+00:12:02,350 --> 00:12:09,410
+alpha ينتمي ل R خد ال delta اللي بتنظرها هي واحد
+
+114
+00:12:09,410 --> 00:12:13,740
+على absolute alpha الذات واحدةهذا اكيد عدد موجة
+
+115
+00:12:13,740 --> 00:12:20,440
+then من مراتبة الان ان كل x في المجال تبع الدالة
+
+116
+00:12:20,440 --> 00:12:24,380
+اللى هو كل العداد الحقيقية مع عدد صفر و ال x على
+
+117
+00:12:24,380 --> 00:12:30,420
+يمين ال c اللى هو الصفر و من هنا x ينتمي الى a
+
+118
+00:12:30,420 --> 00:12:35,560
+اللى هى R المجال تبع الدالة كل العداد الحقيقية مع
+
+119
+00:12:35,560 --> 00:12:42,940
+عدد صفرو X سالب سفر ال C هنا ال cluster point هي
+
+120
+00:12:42,940 --> 00:12:47,100
+السفر الآن ال X على يمين السفر يعني X minus سفر
+
+121
+00:12:47,100 --> 00:12:54,040
+أكبر من سفر فإذا كانت ال X هذه أصغر من Delta فهذا
+
+122
+00:12:54,040 --> 00:13:03,900
+هيعطيني أن ال 1 على Xأكبر من واحد على دلتا
+
+123
+00:13:03,900 --> 00:13:07,560
+وبالتالي
+
+124
+00:13:07,560 --> 00:13:14,720
+هذا بيقدي انه f of x اللي هي بالساوي واحد على اكس
+
+125
+00:13:14,720 --> 00:13:20,940
+أكبر من واحد على دلتا اللي هي بالساوي واحد مقلوب
+
+126
+00:13:20,940 --> 00:13:28,280
+الدلتا بيطلع absolute alpha زائد واحد وهذه أكبر من
+
+127
+00:13:28,280 --> 00:13:32,300
+absolute ال alphaabsolute alpha زاد واحد أكبر من
+
+128
+00:13:32,300 --> 00:13:36,880
+absolute alpha وabsolute alpha أكبر من أو يساوي
+
+129
+00:13:36,880 --> 00:13:41,920
+alpha أي عدد حقيقي القيمة المطلقة تبعته أكبر من أو
+
+130
+00:13:41,920 --> 00:13:48,220
+يساوي نفسه فالنهاية أثبتنا أن f of x أكبر من ال
+
+131
+00:13:48,220 --> 00:13:48,920
+given alpha
+
+132
+00:13:53,520 --> 00:13:58,440
+Okay تمام بما ان ال alpha دي كانت arbitrarily
+
+133
+00:13:58,440 --> 00:14:06,640
+since alpha belong to R was arbitrarily اذا هين
+
+134
+00:14:06,640 --> 00:14:12,180
+اثبتنا اذا معناه هذا الكلام هذا انه لكل alpha فيه
+
+135
+00:14:12,180 --> 00:14:17,280
+delta تعتمد عليها بتخلي f of x اكبر من alpha لكل x
+
+136
+00:14:17,280 --> 00:14:23,440
+قريبة من السفر within مسافة deltaإن هذا معناه حسب
+
+137
+00:14:23,440 --> 00:14:29,340
+التعريف إن ال limit ل f of x لما x تقول إلى سفر من
+
+138
+00:14:29,340 --> 00:14:36,780
+اليمين بساوي بورهانش
+
+139
+00:14:36,780 --> 00:14:38,320
+جزء التاني مشابه
+
+140
+00:14:47,840 --> 00:14:54,400
+is similar مشابه لل part للجزء الأول يعني فهسيبكم
+
+141
+00:14:54,400 --> 00:14:59,860
+انتوا تكتبوا برهان مشابه مع التعديلات اللازمة و
+
+142
+00:14:59,860 --> 00:15:05,660
+أيه طبعا التعريف تبع ال limit from the left موجود
+
+143
+00:15:05,660 --> 00:15:13,280
+okay تمام اللي هو بالأزرق تمام مثال
+
+144
+00:15:13,280 --> 00:15:33,190
+تاني ممكن برضهناخد مثال تاني
+
+145
+00:15:33,190 --> 00:15:41,050
+show limit for function e to one على x as x tends
+
+146
+00:15:41,050 --> 00:15:45,070
+to zero from the right بساوي infinity
+
+147
+00:15:55,920 --> 00:16:02,700
+أنا عندي ال function تبعتي f of x بيسمي E أس واحد
+
+148
+00:16:02,700 --> 00:16:07,840
+على X طبعا ال X هنا ال function مش معرفة عند السفر
+
+149
+00:16:07,840 --> 00:16:12,900
+المجال الدالي هذه كل الأعداد الحقيقية مع ده السفر
+
+150
+00:16:12,900 --> 00:16:20,860
+المثل هذا أخدناه المرة اللي فاتت we
+
+151
+00:16:20,860 --> 00:16:21,440
+have
+
+152
+00:16:25,650 --> 00:16:34,410
+from previous example من
+
+153
+00:16:34,410 --> 00:16:44,110
+المثال السابق فانا هادرس سابق انه واحد
+
+154
+00:16:44,110 --> 00:16:51,370
+على اكس اكبر من سفر اصغر من اكس واحد على اكس لكل
+
+155
+00:16:51,370 --> 00:17:00,340
+اكس اكبر من سفرلكل x على يمين السفر كان في ندي T
+
+156
+00:17:00,340 --> 00:17:09,980
+أزرق من E of T لكل T عدد مؤجد طيب
+
+157
+00:17:09,980 --> 00:17:18,000
+احنا لسه بتبتيل since ال
+
+158
+00:17:18,000 --> 00:17:24,340
+limit لواحد على x لما x تقول إلى السفر من اليمين
+
+159
+00:17:26,540 --> 00:17:31,760
+بساوي infinity فممكن
+
+160
+00:17:31,760 --> 00:17:37,160
+نطبخ ال comparison test هذا فهي عندي f of x أصغر
+
+161
+00:17:37,160 --> 00:17:45,800
+من g of x يعني خليني أسمي هذه g of x كمشي مع نظري
+
+162
+00:17:45,800 --> 00:17:52,020
+يعني وخلني f of x بساوي واحد على x فهي عندي f of x
+
+163
+00:17:52,020 --> 00:18:00,560
+أصغر من g of x لكل x في Rأو لكل X لا يساوي سفر لكل
+
+164
+00:18:00,560 --> 00:18:07,820
+X موجة بقى أو على يمين السفر لكل
+
+165
+00:18:07,820 --> 00:18:16,880
+X في R تقاطع سفر إلى ملا نهاية okay فإذا
+
+166
+00:18:16,880 --> 00:18:22,320
+قلنا النظرية هذه صحيحة لل right limit باستخدام
+
+167
+00:18:22,320 --> 00:18:33,480
+النظريةby above theorem by above theorem for right
+
+168
+00:18:33,480 --> 00:18:39,680
+limits للنهايات
+
+169
+00:18:39,680 --> 00:18:49,990
+من اليمين we have نحصل على انه ال limitلقيت واحد
+
+170
+00:18:49,990 --> 00:18:55,370
+على اكس لما اكس تقول إلى سفر من اليمين بساوي plus
+
+171
+00:18:55,370 --> 00:19:04,770
+infinity وهذا اللي بدناه هي مظبوط صح؟ تمام؟إذن
+
+172
+00:19:04,770 --> 00:19:08,850
+ممكن نطبق النظرية هذه لإثبات أن ال limit ل ال
+
+173
+00:19:08,850 --> 00:19:13,690
+function E to 1 ل X من X أو ل 0 من اليمين بساوي
+
+174
+00:19:13,690 --> 00:19:22,130
+infinity برضه ممكن نطبق التعريف يعني ممكن أعطي
+
+175
+00:19:22,130 --> 00:19:28,530
+برهان تاني و أقول بما أن هذه المتباينة صحيحة لكل X
+
+176
+00:19:28,530 --> 00:19:34,480
+موجبةو بما انه ال limit هذه لو احلى ال function
+
+177
+00:19:34,480 --> 00:19:38,080
+واحد على X عن سفر من اليانين بالساوي infinity
+
+178
+00:19:38,080 --> 00:19:41,560
+معناته انا بقدر اخلي واحد ال function واحد على X
+
+179
+00:19:41,560 --> 00:19:46,940
+هذه اكبر من Alpha لأي real number Alpha صح؟
+
+180
+00:19:48,400 --> 00:19:52,300
+وبالتالي بقدر اخل اي ت واحد على اكس اكبر من اي
+
+181
+00:19:52,300 --> 00:19:59,600
+real number Alpha لكل X طبعا على يمين السفر وعلى
+
+182
+00:19:59,600 --> 00:20:06,660
+مسافة اصغر من Delta نقدر نجيب طبعا Delta لكل Alpha
+
+183
+00:20:06,660 --> 00:20:13,800
+فممكن برضه استخدم التعريف لاثبات ان ال limit لإي ت
+
+184
+00:20:13,800 --> 00:20:16,800
+واحد على اكس على ما اكسه ولا سفر من اليمين بالساوي
+
+185
+00:20:16,800 --> 00:20:21,100
+infinityOkay تمام ان انا ممكن استخدم التعريف او
+
+186
+00:20:21,100 --> 00:20:27,860
+استخدم ال comparison test اللي فوق واضح في اي سؤال
+
+187
+00:20:27,860 --> 00:20:37,380
+طب
+
+188
+00:20:37,380 --> 00:20:45,280
+احنا يعني لاحظوا في ال chapter هذا اتعرضنا ل ال ..
+
+189
+00:20:47,310 --> 00:20:53,690
+لتعريف النهايات للدول and cluster point للمجال
+
+190
+00:20:53,690 --> 00:20:58,390
+تبعها او and cluster point لتقاطع مجالها مع فترة
+
+191
+00:20:58,390 --> 00:21:02,470
+مفتوحة زي هذه او فترة مفتوحة زي هذه في حالة ال
+
+192
+00:21:02,470 --> 00:21:06,870
+infinite limits وفي كل ال limits هذه دائما ال X
+
+193
+00:21:06,870 --> 00:21:12,290
+كانت تقول ل C لعدد ل cluster point سواء من اليمين
+
+194
+00:21:12,290 --> 00:21:15,910
+او من اليسار لكن احيانا
+
+195
+00:21:17,840 --> 00:21:29,720
+بتصادفنا نهايات احيانا
+
+196
+00:21:29,720 --> 00:21:37,020
+نتعرض لمواقف زي هذه انه كيف انا بدي .. يعني ممكن
+
+197
+00:21:37,020 --> 00:21:42,840
+يكون عندي limit ل a for x بدل ما x تقول ل cluster
+
+198
+00:21:42,840 --> 00:21:45,920
+point c x تقول ل infinity
+
+199
+00:21:49,360 --> 00:21:52,820
+ما معنى ان ال limit ل f of x لما x تقول ال
+
+200
+00:21:52,820 --> 00:22:00,720
+infinity بساوي عدد L او ما معنى ان ال limit ل ال
+
+201
+00:22:00,720 --> 00:22:05,080
+function f لما x تقول ال سالب infinity بساوي ايضا
+
+202
+00:22:05,080 --> 00:22:12,060
+عدد L هذا ما اتعرضنا اليه فبنلا تعريف نشوف كيف
+
+203
+00:22:12,060 --> 00:22:20,300
+التعريف تبع ال limits هذه بيكونمثلًا انا عندي ال
+
+204
+00:22:20,300 --> 00:22:28,380
+limit نرجع لل function واحد على X فانا
+
+205
+00:22:28,380 --> 00:22:33,980
+عندي ال limit يعني Y بساوي واحد على X فانا عندي ال
+
+206
+00:22:33,980 --> 00:22:40,980
+limit واحد على X لما X تقول infinity واضح انها
+
+207
+00:22:40,980 --> 00:22:47,330
+بساوي عدد L صفروبرضه كمان لو كانت x تقولنا سالب
+
+208
+00:22:47,330 --> 00:22:53,250
+infinity برضه ال limit بالساوية سفر، اذا كيف اثبت
+
+209
+00:22:53,250 --> 00:22:59,750
+او كيف اعرف ان ال limit عند ال infinity بالساوية
+
+210
+00:22:59,750 --> 00:23:04,150
+عدد او عند السالب infinity بالساوية عدد ما؟
+
+211
+00:23:04,790 --> 00:23:10,050
+التعريفات هذه ما مرت لسه علينا فنحتاج ان احنا نعرف
+
+212
+00:23:10,050 --> 00:23:18,950
+او ناخد هذه التعريفات اذا دلوقت نقصح التعريف هذا
+
+213
+00:23:18,950 --> 00:23:28,570
+ناخد
+
+214
+00:23:28,570 --> 00:23:29,070
+definition
+
+215
+00:23:48,740 --> 00:24:03,120
+فالتعريف let f be function from A to R and
+
+216
+00:24:03,120 --> 00:24:10,780
+الفترة من A إلى ماله نهاية تكون داخل المجموعة A
+
+217
+00:24:10,780 --> 00:24:13,640
+for some A ينتمي إلى R
+
+218
+00:24:20,670 --> 00:24:32,430
+فبنعرف و نقول ان ال ينتمي لار is
+
+219
+00:24:32,430 --> 00:24:36,710
+a limit of
+
+220
+00:24:36,710 --> 00:24:46,950
+ال function f as x tends to infinity and right و
+
+221
+00:24:46,950 --> 00:24:52,540
+بنكتب في الحالة هذه ان ال limitلـ f of x as x
+
+222
+00:24:52,540 --> 00:24:58,820
+tends to infinity بالساوية لعدد L إذا تحقق الشرط
+
+223
+00:24:58,820 --> 00:25:06,960
+التالي لكل إبسلون for any إبسلون أكبر من 0 نقدر
+
+224
+00:25:06,960 --> 00:25:14,660
+نلاقي capital K عدد حقيقي يعتمد على إبسلون وهذا
+
+225
+00:25:14,660 --> 00:25:23,260
+العدد أكبر من العدد A اللي هو عدد حقيقيمعين بحيث
+
+226
+00:25:23,260 --> 00:25:33,980
+أنه لكل لو كان ال X أكبر من ال K فهذا بتضمن أنه
+
+227
+00:25:33,980 --> 00:25:39,780
+absolute F of X minus L أصغر من ال given epsilon
+
+228
+00:25:39,780 --> 00:25:44,280
+تمام؟
+
+229
+00:25:44,280 --> 00:25:46,120
+بالمثل ممكن أعرف
+
+230
+00:25:52,400 --> 00:25:58,400
+العرف ما معناه ان ال limit لل function f لما x
+
+231
+00:25:58,400 --> 00:26:04,120
+تقول لسالب infinity بالساوي عدد L في الحالة هذه
+
+232
+00:26:04,120 --> 00:26:13,200
+باشترط ان المجموع المجال يحتوي على فترة زي هذه
+
+233
+00:26:13,200 --> 00:26:17,840
+بدأت
+
+234
+00:26:17,840 --> 00:26:19,300
+فترة هذه فترة زي هذه
+
+235
+00:26:22,770 --> 00:26:30,990
+هنا قلنا بدل infinity نبدلها بال-infinity وهنا بال
+
+236
+00:26:30,990 --> 00:26:37,030
+-infinity ونقول
+
+237
+00:26:37,030 --> 00:26:46,190
+إنه يوجد K المرة هذه بدل أكبر من A أصغر من A وهذه
+
+238
+00:26:46,190 --> 00:26:48,150
+تتغير لكل X
+
+239
+00:26:53,500 --> 00:26:59,960
+أصغر من K أصغر
+
+240
+00:26:59,960 --> 00:27:03,760
+من Y أصغر
+
+241
+00:27:03,760 --> 00:27:06,920
+من K أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+242
+00:27:06,920 --> 00:27:07,020
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+243
+00:27:07,020 --> 00:27:07,760
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+244
+00:27:07,760 --> 00:27:12,040
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أص
+
+245
+00:27:12,440 --> 00:27:18,680
+Okay طيب اذا انا لان في عندي تعريفات جديدة كمان
+
+246
+00:27:18,680 --> 00:27:25,880
+مرة كل نظريات اللي أثبتناها سابقا بس
+
+247
+00:27:25,880 --> 00:27:31,500
+بدل ما X أول ل C بيصير X أول ل infinity ال
+
+248
+00:27:31,500 --> 00:27:37,040
+sequence أول شيء إذا كانت ال limit هذه أو هذه
+
+249
+00:27:37,040 --> 00:27:42,630
+موجودة بسوء عدد Lفممكن اثباتي انها unique زيها زي
+
+250
+00:27:42,630 --> 00:27:49,210
+اي two sided limit او زيها زي اي limit اخرى كذلك
+
+251
+00:27:49,210 --> 00:27:53,370
+ممكن نثبت sequential criterion لل limits زي هدول
+
+252
+00:27:53,370 --> 00:27:59,230
+وممكن النظرية زي هذه تكون صحيحة لهذا النوع من ال
+
+253
+00:27:59,230 --> 00:28:06,130
+limits okay اذا معظم النظريات معظم النظريات اللي
+
+254
+00:28:06,130 --> 00:28:13,240
+اثبتناها تكون صحيحة لهذا النوع الجديد من الـ
+
+255
+00:28:13,240 --> 00:28:17,660
+infinite نسميها infinite limits at infinity هذه
+
+256
+00:28:17,660 --> 00:28:21,980
+limits at infinity أو سالم infinity هذه كانت
+
+257
+00:28:21,980 --> 00:28:27,520
+نسميها infinite limits فمثلا
+
+258
+00:28:27,520 --> 00:28:31,900
+على سبيل المثال وليس الحصر ممكن ان احنا نكتب
+
+259
+00:28:31,900 --> 00:28:35,060
+sequential criterion لهذا النوع من ال limits
+
+260
+00:28:42,580 --> 00:28:57,860
+هي sequential theorem sequential
+
+261
+00:28:57,860 --> 00:29:03,680
+.. sequential
+
+262
+00:29:03,680 --> 00:29:07,500
+criterion
+
+263
+00:29:07,500 --> 00:29:18,320
+.. sequential criterionfor limits for limits at
+
+264
+00:29:18,320 --> 00:29:23,580
+infinity the
+
+265
+00:29:23,580 --> 00:29:29,480
+following statements are equivalent are equivalent
+
+266
+00:29:29,480 --> 00:29:40,830
+واحد limitF of X as X tends to infinity بساوي عدد
+
+267
+00:29:40,830 --> 00:29:47,970
+M اتنين for every
+
+268
+00:29:47,970 --> 00:29:48,910
+sequence
+
+269
+00:29:51,330 --> 00:29:57,750
+x in contained in a تقابل فترة مفتوحة من a إلى م
+
+270
+00:29:57,750 --> 00:30:05,030
+للإلهية such that limit x in as n tends to
+
+271
+00:30:05,030 --> 00:30:14,410
+infinity بساوي ال infinity لازم
+
+272
+00:30:14,410 --> 00:30:19,300
+يطلع عندى limit ال imageبساوي العدد L لسيكوانس Xn
+
+273
+00:30:19,300 --> 00:30:25,380
+as N times Infinity بساوي العدد L لذا هذه
+
+274
+00:30:25,380 --> 00:30:34,580
+Sequential criterion for limits at infinity وممكن
+
+275
+00:30:34,580 --> 00:30:39,860
+نثبت النظرية هذه زي ما أثبتنا Sequential criterion
+
+276
+00:30:39,860 --> 00:30:46,300
+for finite two-sided limits أو for finite one
+
+277
+00:30:46,300 --> 00:30:51,520
+-sided limitsمثلًا لو أريد أن أثبت واحد implies
+
+278
+00:30:51,520 --> 00:30:55,660
+اتنين فبقول
+
+279
+00:30:55,660 --> 00:31:05,080
+assume أنه one holds هذا
+
+280
+00:31:05,080 --> 00:31:12,440
+معناه أن ال limit لf of x as x tends to infinity
+
+281
+00:31:12,440 --> 00:31:17,120
+بسوى عدد L طيب to prove
+
+282
+00:31:32,110 --> 00:31:38,430
+to prove two holes فابد
+
+283
+00:31:38,430 --> 00:31:46,430
+أثبت لأي sequence لأي sequence بالمواصفات هذه
+
+284
+00:31:46,430 --> 00:31:57,370
+limit صورتها بساوي L فببدأ بقول let let XM contain
+
+285
+00:31:57,370 --> 00:32:08,400
+بالـ A قاطعالفترة هذه بيجيبن
+
+286
+00:32:08,400 --> 00:32:11,560
+بحيث
+
+287
+00:32:11,560 --> 00:32:20,660
+ان ال limit لسيكوينس xn هذه بساوي
+
+288
+00:32:20,660 --> 00:32:24,660
+infinity و
+
+289
+00:32:24,660 --> 00:32:27,380
+بالثبات ان ال limit صورتها بساوي ال
+
+290
+00:32:30,930 --> 00:32:34,970
+عشان أثبت أنه two holds، بدي أثبت أنه الـ limit
+
+291
+00:32:34,970 --> 00:32:45,370
+لصورة الـ xn as n times infinity بساوي n لأن هذه
+
+292
+00:32:45,370 --> 00:32:48,930
+عبارة عن sequence، بدي أثبت limit sequence بالساوي
+
+293
+00:32:48,930 --> 00:32:54,030
+عدد، بستخدم تعريف Y capital N لل limit of a
+
+294
+00:32:54,030 --> 00:33:00,120
+sequence، صح؟إذا أنا بقول let epsilon أكبر من
+
+295
+00:33:00,120 --> 00:33:07,240
+السفر be given طيب،
+
+296
+00:33:07,240 --> 00:33:16,140
+أنا عندي فارض since ال limit ل F of X as X tends
+
+297
+00:33:16,140 --> 00:33:26,080
+to infinity بتساوي العدد Lإذا من تعريف ال limit of
+
+298
+00:33:26,080 --> 00:33:31,180
+infinity اللي زيها دي هي التعريف هي تحت تقول إنه
+
+299
+00:33:31,180 --> 00:33:41,300
+for any given epsilon يوجد عدد حقيقي K يعتمد على
+
+300
+00:33:41,300 --> 00:33:47,220
+epsilon وهذا أكبر من A بحيث
+
+301
+00:33:47,220 --> 00:33:57,390
+إنه لو كان Xأكبر من الـ K بيقدي أنه absolute f of
+
+302
+00:33:57,390 --> 00:34:03,750
+x minus ال L أصغر من إبسل أسمي ال implication هذه
+
+303
+00:34:03,750 --> 00:34:08,830
+star طيب
+
+304
+00:34:08,830 --> 00:34:13,750
+أنا برضه عندي أنا فارد أن ال sequence هذه ال given
+
+305
+00:34:13,750 --> 00:34:15,590
+sequence ال limit تبعتها
+
+306
+00:34:21,580 --> 00:34:26,380
+بساوي infinity ومن
+
+307
+00:34:26,380 --> 00:34:30,540
+تعريف ان تكون ال sequence limit تبعتها infinite
+
+308
+00:34:30,540 --> 00:34:37,180
+هذا معناه ان مقدر اخلى x in اكبر من اي عدد حقيقي
+
+309
+00:34:37,180 --> 00:34:41,440
+alpha فاخد alpha هنا بساوي k مش هذا عدد حقيقي
+
+310
+00:34:41,440 --> 00:34:48,520
+محترم فاخد هوبما عنده limit لسيكوينس Xn بالساوي
+
+311
+00:34:48,520 --> 00:34:56,920
+infinity then for any real number يوجد
+
+312
+00:34:56,920 --> 00:35:06,700
+capital N عدد طبيعي يعتمد على Kهذا أشمله عدد طبيعي
+
+313
+00:35:06,700 --> 00:35:14,800
+بحيث أنه لكل n أكبر من أو ساوي capital N هذا بتضمن
+
+314
+00:35:14,800 --> 00:35:22,980
+أن ال Xn أكبر من العدد K العدد الحقيقي K نسمي ال
+
+315
+00:35:22,980 --> 00:35:25,160
+implication هذه double star
+
+316
+00:35:39,650 --> 00:35:45,850
+تمام هيك إذا هذا ناخده من كوننا ان احنا فرضين ان
+
+317
+00:35:45,850 --> 00:35:49,350
+limit ال sequence xn بالساوية infinity ومن تعريف
+
+318
+00:35:49,350 --> 00:35:56,710
+ال infinite limit لل sequence الان الان في تحول في
+
+319
+00:35:56,710 --> 00:36:05,270
+البرهار now star and double star yield
+
+320
+00:36:09,010 --> 00:36:17,430
+بيعطوني التالي لو كانت n أكبر من أو ساوى capital N
+
+321
+00:36:17,430 --> 00:36:28,590
+فهذا بيؤدي إنه xn أكبر من k هذا موجود أخدناه من
+
+322
+00:36:28,590 --> 00:36:33,930
+double star لكل n أكبر من أو ساوى capital N بيطلع
+
+323
+00:36:33,930 --> 00:36:40,980
+xn أكبر من kطيب و من ال star و هذا بيقدي باستخدام
+
+324
+00:36:40,980 --> 00:36:47,640
+ال star ال star بيقوللي لكل x لو كانت ال x او ال
+
+325
+00:36:47,640 --> 00:36:56,860
+xn أكبر من capital K هذا بيقدي ان صورتها المسافة
+
+326
+00:36:56,860 --> 00:37:01,840
+بينها و بين ال L أصغر من إبسل صح؟
+
+327
+00:37:06,200 --> 00:37:10,360
+إذا نجي نلخص كمان مرة، إيه اللي عملناها؟ أنا إيش
+
+328
+00:37:10,360 --> 00:37:14,560
+بدأ في بتلقى limited sequence F of X N لما N تقول
+
+329
+00:37:14,560 --> 00:37:18,460
+الـinfinity بالساوي عدد L فهذه البديات بإبسلون
+
+330
+00:37:18,460 --> 00:37:22,940
+أكبر من السفر given أثبتت إن يوجد capital N عدد
+
+331
+00:37:22,940 --> 00:37:27,680
+طبيعي يعتمد على الـK والـK تعتمد على إبسلون، إذا
+
+332
+00:37:27,680 --> 00:37:33,810
+الـN هذه تعتمد على الـgiven إبسلونو هذه ال N لكل
+
+333
+00:37:33,810 --> 00:37:38,050
+small n أكبر من أو ساوي ال capital N هذه طلع عند
+
+334
+00:37:38,050 --> 00:37:41,350
+المسافة بين الحد النوني لل sequence و L أصغر من
+
+335
+00:37:41,350 --> 00:37:47,750
+Epsilon بما أن Epsilon was arbitrary since Epsilon
+
+336
+00:37:47,750 --> 00:37:54,250
+أكبر من السفر was arbitraryإذا by epsilon capital
+
+337
+00:37:54,250 --> 00:37:58,650
+N definition of limit of sequence بنكون هيك حسب
+
+338
+00:37:58,650 --> 00:38:04,570
+التعريف أثبتنا أنه limit لسيكوينس F of X N as N
+
+339
+00:38:04,570 --> 00:38:09,390
+tends to infinity بالساوي لعدد N وبالتالي هيك
+
+340
+00:38:09,390 --> 00:38:16,950
+بنكون أثبتنا إذا العبارة 2 holds وهيك بنكون أثبتنا
+
+341
+00:38:16,950 --> 00:38:23,230
+أن العبارة statement 1 implies statement 2Okay
+
+342
+00:38:23,230 --> 00:38:30,010
+تمام إذا نحن ممكن نبرهن ال sequential criterion لل
+
+343
+00:38:30,010 --> 00:38:35,730
+infinite limit و لل one sided limit و لكل أنواع ال
+
+344
+00:38:35,730 --> 00:38:47,310
+limit و هاي أثبتنا جزء برهان الجزء التاني مماثل ال
+
+345
+00:38:47,310 --> 00:38:58,010
+proof of 2 implies 1is similar to
+
+346
+00:38:58,010 --> 00:39:03,410
+original
+
+347
+00:39:03,410 --> 00:39:08,690
+proof or proof of
+
+348
+00:39:08,690 --> 00:39:14,810
+original original
+
+349
+00:39:14,810 --> 00:39:21,470
+sequential criterion original sequential criterion
+
+350
+00:39:21,470 --> 00:39:26,530
+مععمل التعديلات اللازمة فانا بقولكم انكم ترجعوا ل
+
+351
+00:39:26,530 --> 00:39:30,910
+sequential criterion الأساسية تقرأوا البرهان تبعها
+
+352
+00:39:30,910 --> 00:39:36,050
+كيف انا برهان اتنين ده او احدو تعملوا التعديلات ..
+
+353
+00:39:36,050 --> 00:39:41,070
+ازاي نبران واحد بدي لاتنين .. okay تمام .. اذا ان
+
+354
+00:39:41,070 --> 00:39:44,910
+هذه تعتبر sequential criterion لل limits at
+
+355
+00:39:44,910 --> 00:39:50,710
+infinity بالمثل ممكن ان احنا نحصل على sequential
+
+356
+00:39:50,710 --> 00:39:57,810
+criterion لل limits at negative infinity يعني ال
+
+357
+00:39:57,810 --> 00:40:04,930
+..يعني هذه ممكن تبدلها ب negative infinity وهذه
+
+358
+00:40:04,930 --> 00:40:10,570
+ممكن تبدلها ب negative infinity وهذه ممكن تبدلها ب
+
+359
+00:40:10,570 --> 00:40:15,670
+سالب infinity إلى a وهذه ممكن تبدلها ب negative
+
+360
+00:40:15,670 --> 00:40:19,790
+infinity وهذه
+
+361
+00:40:19,790 --> 00:40:23,550
+ممكن تبدلها بسالب
+
+362
+00:40:23,550 --> 00:40:24,030
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+363
+00:40:24,030 --> 00:40:24,130
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+364
+00:40:24,130 --> 00:40:24,750
+تبدلها بسالب infinity إلى a وهذه ممكن تبدلها بسالب
+
+365
+00:40:24,750 --> 00:40:24,830
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+366
+00:40:24,830 --> 00:40:25,250
+تبدلها بسالب infinity إلى a وهذه ممكن تبدلها بسالب
+
+367
+00:40:25,250 --> 00:40:26,690
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+368
+00:40:26,690 --> 00:40:27,830
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+369
+00:40:27,830 --> 00:40:38,590
+تبدلها بسوالبرنامج طبعا مشابه للنظرية السابقة ناخد
+
+370
+00:40:38,590 --> 00:40:44,150
+أمثلة طبعا
+
+371
+00:40:44,150 --> 00:40:52,230
+في نظريات كتيرة صحيحة لنوع هذا من ال limits فلو
+
+372
+00:40:52,230 --> 00:40:58,940
+احتجنا زي ال squeeze theorem زي ال comparisontest
+
+373
+00:40:58,940 --> 00:41:04,240
+او الاخر فمثلا
+
+374
+00:41:04,240 --> 00:41:23,600
+ناخد بعض الأمثلة مثلا
+
+375
+00:41:23,600 --> 00:41:25,080
+ناخد examples
+
+376
+00:41:41,280 --> 00:41:50,920
+F of X يساوي واحد على X و X لا يساوي ساقر Show
+
+377
+00:41:50,920 --> 00:41:54,720
+that limit
+
+378
+00:41:54,720 --> 00:42:03,660
+F of X as X tends to infinity يساوي ساقر و كذلك
+
+379
+00:42:03,660 --> 00:42:11,410
+limitلف of x as x tends to negative infinity بساوة
+
+380
+00:42:11,410 --> 00:42:16,950
+ستة المظبوط
+
+381
+00:42:16,950 --> 00:42:24,370
+ال function واحد على x ثانية فلما
+
+382
+00:42:24,370 --> 00:42:29,070
+x تقول infinity واحد على x بتقولها ستة لما x
+
+383
+00:42:29,070 --> 00:42:32,690
+تقولها سالب infinity برضه ال function واحد على x
+
+384
+00:42:32,690 --> 00:42:39,550
+تقول إلى ستةفلو بدي اثبات الجزء الأول فممكن استخدم
+
+385
+00:42:39,550 --> 00:42:45,590
+التعريف او استخدم الـ sequential criterion فمثلا
+
+386
+00:42:45,590 --> 00:42:56,850
+to use a definition لو بدي استخدم التعريف مثلا
+
+387
+00:42:58,750 --> 00:43:02,270
+Limit f of x من x سواء و لا انا كنت بتساوي سفر
+
+388
+00:43:02,270 --> 00:43:10,950
+فبابدأ حسب التعريف بابدأ بإبسلون أكبر من السفر لت
+
+389
+00:43:10,950 --> 00:43:18,510
+إبسلون أكبر من السفر بكلمة و بعدين بدأ أثبت أنه في
+
+390
+00:43:18,510 --> 00:43:26,420
+ك يعتمد على إبسلون ف choose كارتة الكعلى انه واحد
+
+391
+00:43:26,420 --> 00:43:30,840
+على ابسلان فهذا تطلع عدد موجب ويعتمد على ابسلان
+
+392
+00:43:30,840 --> 00:43:36,780
+وال ا طبعا هنا في السؤال هذا هي السفر يعني هنا ال
+
+393
+00:43:36,780 --> 00:43:40,680
+domain تبعد لكل العداد الحقيقية مع ده السفرفممكن
+
+394
+00:43:40,680 --> 00:43:46,440
+اخد السفر الفقرة هذه contained in ال domain تبع
+
+395
+00:43:46,440 --> 00:43:51,100
+الـ a اللي هو كل العداد الحقيقية من عدد السفر هنا
+
+396
+00:43:51,100 --> 00:43:52,620
+لأي أكبر من أكبر من أكبر من أكبر من أكبر من أكبر
+
+397
+00:43:52,620 --> 00:43:53,060
+من أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+398
+00:43:53,060 --> 00:43:53,980
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+399
+00:43:53,980 --> 00:43:57,560
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+400
+00:43:57,560 --> 00:44:03,280
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+401
+00:44:03,280 --> 00:44:06,340
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+402
+00:44:06,340 --> 00:44:08,320
+أكبر من أكبر من أكبر
+
+403
+00:44:09,640 --> 00:44:19,540
+إن واحد على اكس أصغر من واحد على كم و هذا بدوره
+
+404
+00:44:19,540 --> 00:44:25,560
+بقدر ان absolute of f of x minus الصفر ال ال هنا
+
+405
+00:44:25,560 --> 00:44:31,480
+هو الصفر فهذا بيطلع بساوي absolute واحد على اكس
+
+406
+00:44:31,480 --> 00:44:34,420
+فهذا عبارة عن واحد على اكس
+
+407
+00:44:45,380 --> 00:44:48,660
+و هذا أقل من واحد على كي و واحد على كي أقل من
+
+408
+00:44:48,660 --> 00:44:54,400
+إبسلو و هذا أصغر طبعا من واحد على كي طبعا عندي ال
+
+409
+00:44:54,400 --> 00:44:59,520
+X هنا أكبر من كي لحظة X أكبر من كي و ال K موجة بقى
+
+410
+00:45:03,660 --> 00:45:08,560
+القيمة المطلقة ل 1 على X هي 1 على X إطرح سفر
+
+411
+00:45:08,560 --> 00:45:14,760
+مابتعملش حاجة هذا أصغر من 1 على K ومن هنا 1 على K
+
+412
+00:45:14,760 --> 00:45:15,580
+بساوي Y
+
+413
+00:45:18,310 --> 00:45:22,630
+إذن هاني أثبتت لي أي epsilon أكبر من سفر يوجد k
+
+414
+00:45:22,630 --> 00:45:27,590
+عدد حقيقي يعتمد على epsilon بحيث لكل x أكبر من k
+
+415
+00:45:27,590 --> 00:45:33,190
+طلع absolute f of x minus L أصغر من epsilon okay
+
+416
+00:45:33,190 --> 00:45:39,430
+إذن هذا معناه إن ال limit حسب التعريف limit واحد
+
+417
+00:45:39,430 --> 00:45:46,030
+على x لما x تقول إلى infinity بساوي سفر
+
+418
+00:45:49,020 --> 00:45:53,960
+بالمثل ممكن نثبت الجزء التاني نفس البرهان مع
+
+419
+00:45:53,960 --> 00:46:02,520
+التعديل في تعريف limit at سالب infinity ممكن
+
+420
+00:46:02,520 --> 00:46:10,180
+برضه نستخدم sequential criterion لو بدأت
+
+421
+00:46:10,180 --> 00:46:15,900
+استخدم sequential criterion لإثبات
+
+422
+00:46:15,900 --> 00:46:26,470
+limitبأخد بقول ان هنا let xn be sequence contained
+
+423
+00:46:26,470 --> 00:46:35,250
+in 0 و infinity بحيث انه limit xn تساوي infinity
+
+424
+00:46:35,250 --> 00:46:39,570
+اذا
+
+425
+00:46:39,570 --> 00:46:48,340
+limit f of xn has n times infinityطبعا هذا بيقدي
+
+426
+00:46:48,340 --> 00:46:51,500
+هذا
+
+427
+00:46:51,500 --> 00:46:58,320
+بيقدي انه limit واحد على xn بساوي سفر exercise
+
+428
+00:46:58,320 --> 00:47:02,480
+أخدناها أخدنا انه limit sequence xn بساوي infinity
+
+429
+00:47:02,480 --> 00:47:06,990
+if and only if limit مقلوب السيكوانس بساوي سفرالان
+
+430
+00:47:06,990 --> 00:47:13,670
+limit f of xn بساوي limit واحد على xn as n tends
+
+431
+00:47:13,670 --> 00:47:21,150
+to infinity وهذا بيساوي ستة لأي
+
+432
+00:47:21,150 --> 00:47:26,510
+sequence نهايتها infinity نهاية سورتها بيساوي
+
+433
+00:47:26,510 --> 00:47:27,330
+العدد L
+
+434
+00:47:35,720 --> 00:47:42,000
+بنطلع ال limit ل ال function f of x as x tends to
+
+435
+00:47:42,000 --> 00:47:47,200
+infinity بساوية 0 إذا هذا برهان تاني using
+
+436
+00:47:47,200 --> 00:47:55,620
+sequential criterion okay واضح مفهوم مثال
+
+437
+00:47:55,620 --> 00:47:56,160
+تاني
+
+438
+00:48:05,300 --> 00:48:10,520
+بناخد g of x بساوي
+
+439
+00:48:10,520 --> 00:48:15,820
+واحد على extra g of x لا يساوي صغير فبدنا نثبت ان
+
+440
+00:48:15,820 --> 00:48:21,660
+ال limit ل g of x لما x تقول ل infinity و لما x
+
+441
+00:48:21,660 --> 00:48:27,280
+تقول ل سالب infinity بساوي صغير برضه ممكن نستخدم
+
+442
+00:48:27,280 --> 00:48:32,440
+sequential criterion لثبات الجزء الأول أو التاني
+
+443
+00:48:37,280 --> 00:48:42,020
+هذا كان limit xn بالساوية infinity فlimit 1 على xn
+
+444
+00:48:42,020 --> 00:48:46,300
+بالساوية infinity بساوية سفر وبالتالي limit 1 على
+
+445
+00:48:46,300 --> 00:48:57,500
+xn تربية يعني هذا بيقدر وهذا
+
+446
+00:48:57,500 --> 00:49:05,370
+بيقدر ان limit1 على xn ترجية لما انتقل ل infinity
+
+447
+00:49:05,370 --> 00:49:16,510
+بساوي limit 1 على xn ضرب limit 1 على xn وهذا بساوي
+
+448
+00:49:16,510 --> 00:49:27,310
+0 ضرب 0 بساوي 0 و limit g ل xn as n tends to
+
+449
+00:49:27,310 --> 00:49:33,810
+infinity بساوي limit1 على x in third يعني بالساعة
+
+450
+00:49:33,810 --> 00:49:40,990
+سفر صح إذا by sequential criterion
+
+451
+00:49:40,990 --> 00:49:44,030
+أثبتت
+
+452
+00:49:44,030 --> 00:49:49,130
+أنه لأي sequence x in حدودها موجب أو نهايتها
+
+453
+00:49:49,130 --> 00:49:57,970
+infinity ف limit صورتها بالساعة سفر هذا معناه أن
+
+454
+00:49:57,970 --> 00:50:06,540
+ال limitلـ function g of x لما x تقول انفينيتي
+
+455
+00:50:06,540 --> 00:50:10,980
+بساوي ستة هنا نريد أن نكون أخرجنا الجزء الأول
+
+456
+00:50:10,980 --> 00:50:14,800
+باستخدام sequential criterion بالمثل و كنا نستخدم
+
+457
+00:50:14,800 --> 00:50:18,280
+sequential criterion اللي اتبعت الجزء التالي بس
+
+458
+00:50:18,280 --> 00:50:25,580
+هنا هناخد x الموجودة في الفترة هذه و هكذاو نفس
+
+459
+00:50:25,580 --> 00:50:29,860
+النظرية اللى خدناها في القصة السابقة بالكون صحيحة
+
+460
+00:50:29,860 --> 00:50:33,760
+هذا بقى قدر المقلوب ال sequence إذا كانت limit ال
+
+461
+00:50:33,760 --> 00:50:37,320
+sequence infinity فlimit المقلوب سفر وبالتالي
+
+462
+00:50:37,320 --> 00:50:43,420
+limit المقلوب المربع بساوة سفر ممكن كمان نستخدم
+
+463
+00:50:43,420 --> 00:50:48,440
+squeeze theorem ممكن نستخدم squeeze theoremفمثلا
+
+464
+00:50:48,440 --> 00:50:56,100
+لو بدنا نبره كمان واحد بطريقة تانية فممكن ان احنا
+
+465
+00:50:56,100 --> 00:51:05,360
+note that for x أكبر من واحد لما خدت x أكبر من
+
+466
+00:51:05,360 --> 00:51:11,700
+واحد بطلع عندي دايما x تربيه أكبر من أو ساوي x
+
+467
+00:51:13,190 --> 00:51:18,610
+وبالتالي هذا بيقدي ان واحد على اكس تربيه اكبر من
+
+468
+00:51:18,610 --> 00:51:23,930
+او ساوي واحد على اكس وطبعا اكبر من السفر لكل اكس
+
+469
+00:51:23,930 --> 00:51:29,310
+اكبر من واحد طب احنا لسه مثل في الجد شوية ان limit
+
+470
+00:51:29,310 --> 00:51:33,810
+ال function واحد على اكس لما اكس تقول الى infinity
+
+471
+00:51:33,810 --> 00:51:40,930
+بساوي سفر صح؟لسه 130 في المثال الأول هذا المثال
+
+472
+00:51:40,930 --> 00:51:44,630
+الثاني في المثال السابق قصدنا ان limit ال function
+
+473
+00:51:44,630 --> 00:51:48,790
+واحد على x لما x تقول infinity تساوي سفر و limit
+
+474
+00:51:48,790 --> 00:51:54,090
+الدالة ثابت سفر لما x تقول infinity تبقى سفر اذا
+
+475
+00:51:54,090 --> 00:52:03,790
+by squeeze theorem by
+
+476
+00:52:03,790 --> 00:52:09,590
+squeeze theorem for limits at infinityLimited دالة
+
+477
+00:52:09,590 --> 00:52:19,130
+المحصورة اللي هي واحد على اكس تربيه as X tends to
+
+478
+00:52:19,130 --> 00:52:24,310
+infinity بساوي تمام
+
+479
+00:52:24,310 --> 00:52:32,490
+okay واضحو بالمثل ممكن نعطي براهين زي هذا او زي
+
+480
+00:52:32,490 --> 00:52:36,290
+هذا اما باستخدام squeeze theorem او sequential
+
+481
+00:52:36,290 --> 00:52:43,790
+criterion او حتى definition okay واضح في اي سؤال
+
+482
+00:52:43,790 --> 00:52:49,050
+في اي استفسار okay هنوقف اذا هنا و المرة الجاية في
+
+483
+00:52:49,050 --> 00:52:53,190
+بعض انواع ال infinite limits هنتكلم عنهم يعني
+
+484
+00:52:53,190 --> 00:52:59,640
+باختصارو بعدين نحاول نجمل section اربعة تلاتة
+
+485
+00:52:59,640 --> 00:53:04,100
+وبالتالي ننهي ال chapter اللي هو chapter اربعة و
+
+486
+00:53:04,100 --> 00:53:08,620
+بعدين نبدأ في chapter خمسة اللي هو اخر chapter في
+
+487
+00:53:08,620 --> 00:53:15,240
+المخرج هو اهم chapter طبعا في حد عنده اي سؤال او
+
+488
+00:53:15,240 --> 00:53:19,640
+استفسار؟ شكرا لاصغاكم و نشوف ان شاء الله المرة
+
+489
+00:53:19,640 --> 00:53:20,040
+القادمة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..4e826a184bab1c5e9cc359cce7e3e53d2bc681de
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 4973, "start": 21.75, "end": 49.73, "text": "بسم الله الرحمن الرحيم في محاضرة اليوم هنكمل ما بدأنا حاول موضوع ال infinite limits المرة تاعة عرفنا ما معناه ان ال limit ل function and cluster point بالساوية plus او minus infinity وشوفنا نظرية اخر نظرية برهنها", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 8978, 3714, 5016, 46958, 25720, 45595, 20498, 8032, 1863, 24793, 1211, 19446, 47525, 10721, 8315, 11331, 995, 12610, 3714, 2407, 11242, 45367, 2423, 13785, 10406, 9673, 25720, 6055, 995, 27884, 6225, 28480, 8315, 19446, 20449, 8315, 3224, 16472, 2423, 4948, 5296, 2445, 293, 13630, 935, 20666, 3794, 995, 2407, 10632, 1804, 1975, 2407, 3175, 13202, 4032, 8592, 38688, 8315, 8717, 19913, 2288, 10632, 1975, 34740, 8717, 19913, 2288, 10632, 4724, 2288, 3224, 1863, 11296], "avg_logprob": -0.15904017644269125, "compression_ratio": 1.6237623762376239, "no_speech_prob": 0.0, "words": [{"start": 21.75, "end": 22.07, "word": "بسم", "probability": 0.878173828125}, {"start": 22.07, "end": 22.31, "word": " الله", "probability": 0.95703125}, {"start": 22.31, "end": 22.67, "word": " الرحمن", "probability": 0.9698893229166666}, {"start": 22.67, "end": 23.39, "word": " الرحيم", "probability": 0.9939778645833334}, {"start": 23.39, "end": 24.79, "word": " في", "probability": 0.822265625}, {"start": 24.79, "end": 25.33, "word": " محاضرة", "probability": 0.9713134765625}, {"start": 25.33, "end": 25.91, "word": " اليوم", "probability": 0.961669921875}, {"start": 25.91, "end": 27.63, "word": " هنكمل", "probability": 0.93798828125}, {"start": 27.63, "end": 27.87, "word": " ما", "probability": 0.92724609375}, {"start": 27.87, "end": 28.61, "word": " بدأنا", "probability": 0.9908854166666666}, {"start": 28.61, "end": 32.17, "word": " حاول", "probability": 0.7467447916666666}, {"start": 32.17, "end": 32.65, "word": " موضوع", "probability": 0.93603515625}, {"start": 32.65, "end": 32.81, "word": " ال", "probability": 0.84130859375}, {"start": 32.81, "end": 33.19, "word": " infinite", "probability": 0.88330078125}, {"start": 33.19, "end": 33.83, "word": " limits", "probability": 0.943359375}, {"start": 33.83, "end": 35.45, "word": " المرة", "probability": 0.95068359375}, {"start": 35.45, "end": 35.83, "word": " تاعة", "probability": 0.4022623697916667}, {"start": 35.83, "end": 37.11, "word": " عرفنا", "probability": 0.90576171875}, {"start": 37.11, "end": 37.37, "word": " ما", "probability": 0.890625}, {"start": 37.37, "end": 37.89, "word": " معناه", "probability": 0.6995442708333334}, {"start": 37.89, "end": 38.09, "word": " ان", "probability": 0.55517578125}, {"start": 38.09, "end": 38.29, "word": " ال", "probability": 0.54443359375}, {"start": 38.29, "end": 38.71, "word": " limit", "probability": 0.953125}, {"start": 38.71, "end": 38.93, "word": " ل", "probability": 0.6689453125}, {"start": 38.93, "end": 39.57, "word": " function", "probability": 0.7744140625}, {"start": 39.57, "end": 40.69, 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{"start": 53.04, "end": 53.42, "word": " limits", "probability": 0.88623046875}, {"start": 53.42, "end": 54.02, "word": " كانت", "probability": 0.967529296875}, {"start": 54.02, "end": 54.8, "word": " النظرية", "probability": 0.974365234375}, {"start": 54.8, "end": 55.7, "word": " التالية", "probability": 0.9913736979166666}, {"start": 55.7, "end": 58.3, "word": " خلينا", "probability": 0.748291015625}, {"start": 58.3, "end": 60.02, "word": " نكتبها", "probability": 0.9904296875}, {"start": 60.02, "end": 65.34, "word": " let", "probability": 0.56640625}, {"start": 65.34, "end": 65.82, "word": " f", "probability": 0.763671875}, {"start": 65.82, "end": 66.16, "word": " و", "probability": 0.708984375}, {"start": 66.16, "end": 66.54, "word": " g", "probability": 0.87646484375}, {"start": 66.54, "end": 66.94, "word": " be", "probability": 0.7353515625}, {"start": 66.94, "end": 67.82, "word": " functions", "probability": 0.875}, {"start": 67.82, "end": 68.34, "word": " from", "probability": 0.89013671875}, {"start": 68.34, "end": 69.02, "word": " a", "probability": 0.74560546875}, {"start": 69.02, "end": 69.32, "word": " to", "probability": 0.9716796875}, {"start": 69.32, "end": 70.14, "word": " r", "probability": 0.93896484375}, {"start": 70.14, "end": 72.6, "word": " و", "probability": 0.890625}, {"start": 72.6, "end": 73.36, "word": " c", "probability": 0.7705078125}, {"start": 73.36, "end": 73.9, "word": " ال", "probability": 0.431640625}, {"start": 73.9, "end": 74.42, "word": " cluster", "probability": 0.96728515625}, {"start": 74.42, "end": 78.32, "word": " point", "probability": 0.96826171875}], "temperature": 1.0}, {"id": 3, "seek": 10819, "start": 80.53, "end": 108.19, "text": "of the set A such that f of x less than or equal g of x for every x تمتمي إلى a different from c فشوفنا أنه لو كان ال limit فf of x as x tends", "tokens": [2670, 264, 992, 316, 1270, 300, 283, 295, 2031, 1570, 813, 420, 2681, 290, 295, 2031, 337, 633, 2031, 46811, 39237, 1829, 30731, 257, 819, 490, 269, 6156, 8592, 38688, 8315, 14739, 3224, 45164, 25961, 2423, 4948, 6156, 69, 295, 2031, 382, 2031, 12258], "avg_logprob": -0.2843749947018094, "compression_ratio": 1.210144927536232, "no_speech_prob": 0.0, "words": [{"start": 80.53, "end": 81.07, "word": "of", "probability": 0.126953125}, {"start": 81.07, "end": 81.43, "word": " the", "probability": 0.673828125}, {"start": 81.43, "end": 81.73, "word": " set", "probability": 0.95849609375}, {"start": 81.73, "end": 82.27, "word": " A", "probability": 0.796875}, {"start": 82.27, "end": 84.49, "word": " such", "probability": 0.8330078125}, {"start": 84.49, "end": 85.49, "word": " that", "probability": 0.95751953125}, {"start": 85.49, "end": 88.69, "word": " f", "probability": 0.57666015625}, {"start": 88.69, "end": 88.97, "word": " of", "probability": 0.489501953125}, {"start": 88.97, "end": 89.37, "word": " x", "probability": 0.85693359375}, {"start": 89.37, "end": 89.71, "word": " less", "probability": 0.77978515625}, {"start": 89.71, "end": 90.03, "word": " than", "probability": 0.94873046875}, {"start": 90.03, "end": 90.17, "word": " or", "probability": 0.91357421875}, {"start": 90.17, "end": 90.39, "word": " equal", "probability": 0.9111328125}, {"start": 90.39, "end": 90.77, "word": " g", "probability": 0.44677734375}, {"start": 90.77, "end": 91.03, "word": " of", "probability": 0.95751953125}, {"start": 91.03, "end": 91.47, "word": " x", "probability": 0.98388671875}, {"start": 91.47, "end": 93.07, "word": " for", "probability": 0.94970703125}, {"start": 93.07, "end": 93.57, "word": " every", "probability": 0.79541015625}, {"start": 93.57, "end": 93.91, "word": " x", "probability": 0.96435546875}, {"start": 93.91, "end": 94.59, "word": " تمتمي", "probability": 0.6083984375}, {"start": 94.59, "end": 94.85, "word": " إلى", "probability": 0.71484375}, {"start": 94.85, "end": 95.33, "word": " a", "probability": 0.58251953125}, {"start": 95.33, "end": 96.85, "word": " different", "probability": 0.89892578125}, {"start": 96.85, "end": 97.63, "word": " from", "probability": 0.919921875}, {"start": 97.63, "end": 98.15, "word": " c", "probability": 0.76220703125}, {"start": 98.15, "end": 100.95, "word": " فشوفنا", "probability": 0.94287109375}, {"start": 100.95, "end": 101.45, "word": " أنه", "probability": 0.68701171875}, {"start": 101.45, "end": 101.63, "word": " لو", "probability": 0.98486328125}, {"start": 101.63, "end": 102.07, "word": " كان", "probability": 0.9970703125}, {"start": 102.07, "end": 102.25, "word": " ال", "probability": 0.7353515625}, {"start": 102.25, "end": 102.63, "word": " limit", "probability": 0.84814453125}, {"start": 102.63, "end": 105.95, "word": " فf", "probability": 0.39776611328125}, {"start": 105.95, "end": 106.25, "word": " of", "probability": 0.85302734375}, {"start": 106.25, "end": 106.73, "word": " x", "probability": 0.9921875}, {"start": 106.73, "end": 107.29, "word": " as", "probability": 0.87255859375}, {"start": 107.29, "end": 107.75, "word": " x", "probability": 0.71875}, {"start": 107.75, "end": 108.19, "word": " tends", "probability": 0.8779296875}], "temperature": 1.0}, {"id": 4, "seek": 12739, "start": 125.09, "end": 127.39, "text": "وكذلك لو كانت ال limit", "tokens": [2407, 4117, 8848, 23275, 45164, 25961, 2655, 2423, 4948], "avg_logprob": -0.784375011920929, "compression_ratio": 0.875, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 125.09, "end": 125.85, "word": "وكذلك", "probability": 0.505950927734375}, {"start": 125.85, "end": 126.23, "word": " لو", "probability": 0.693359375}, {"start": 126.23, "end": 126.91, "word": " كانت", "probability": 0.92138671875}, {"start": 126.91, "end": 127.03, "word": " ال", "probability": 0.312255859375}, {"start": 127.03, "end": 127.39, "word": " limit", "probability": 0.66552734375}], "temperature": 1.0}, {"id": 5, "seek": 15635, "start": 130.53, "end": 156.35, "text": "لـ u of x as x tends to c بساوي negative infinity فهذا بتضمن ان limit ل f of x as x tends to c بساوي negative infinity طيب ال ..", "tokens": [1211, 39184, 344, 295, 2031, 382, 2031, 12258, 281, 269, 4724, 3794, 995, 45865, 3671, 13202, 6156, 3224, 15730, 39894, 11242, 27842, 16472, 4948, 5296, 283, 295, 2031, 382, 2031, 12258, 281, 269, 4724, 3794, 995, 45865, 3671, 13202, 23032, 1829, 3555, 2423, 4386], "avg_logprob": -0.2690972169240316, "compression_ratio": 1.4272727272727272, "no_speech_prob": 0.0, "words": [{"start": 130.53, "end": 131.53, "word": "لـ", "probability": 0.33795166015625}, {"start": 131.53, "end": 132.53, "word": " u", "probability": 0.12396240234375}, {"start": 132.53, "end": 132.83, "word": " of", "probability": 0.254150390625}, {"start": 132.83, "end": 133.23, "word": " x", "probability": 0.93896484375}, {"start": 133.23, "end": 133.73, "word": " as", "probability": 0.8671875}, {"start": 133.73, "end": 134.19, "word": " x", "probability": 0.9521484375}, {"start": 134.19, "end": 134.51, "word": " tends", "probability": 0.393798828125}, {"start": 134.51, 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"probability": 0.9892578125}, {"start": 143.75, "end": 144.87, "word": " as", "probability": 0.96728515625}, {"start": 144.87, "end": 145.27, "word": " x", "probability": 0.9814453125}, {"start": 145.27, "end": 145.61, "word": " tends", "probability": 0.74267578125}, {"start": 145.61, "end": 145.81, "word": " to", "probability": 0.984375}, {"start": 145.81, "end": 146.11, "word": " c", "probability": 0.95556640625}, {"start": 146.11, "end": 146.81, "word": " بساوي", "probability": 0.9552001953125}, {"start": 146.81, "end": 147.35, "word": " negative", "probability": 0.9716796875}, {"start": 147.35, "end": 148.09, "word": " infinity", "probability": 0.92578125}, {"start": 148.09, "end": 155.73, "word": " طيب", "probability": 0.9679361979166666}, {"start": 155.73, "end": 156.03, "word": " ال", "probability": 0.63525390625}, {"start": 156.03, "end": 156.35, "word": " ..", "probability": 0.9169921875}], "temperature": 1.0}, {"id": 6, "seek": 18621, "start": 161.05, "end": 186.21, "text": "اليوم هنعرف ما معناه ان ال limit and c من اليمين بالساوي infinity او ال limit and c من اليسار بالساوي infinity و كذلك نفس الشيء ال one sided limit and c ما معناه أنها ساوي سالب infinity لأن هذه كانت two sided limit المرة الأخيرة اتعرفنا", "tokens": [6027, 1829, 20498, 8032, 1863, 3615, 28480, 19446, 20449, 8315, 3224, 16472, 2423, 4948, 293, 269, 9154, 45595, 2304, 9957, 20666, 3794, 995, 45865, 13202, 1975, 2407, 2423, 4948, 293, 269, 9154, 45595, 3794, 9640, 20666, 3794, 995, 45865, 13202, 4032, 9122, 8848, 23275, 8717, 36178, 25124, 1829, 38207, 2423, 472, 41651, 4948, 293, 269, 19446, 20449, 8315, 3224, 14739, 11296, 8608, 995, 45865, 8608, 6027, 3555, 13202, 5296, 33456, 29538, 25961, 2655, 732, 41651, 4948, 9673, 25720, 16247, 9778, 48923, 1975, 2655, 3615, 28480, 8315], "avg_logprob": -0.20510057676797624, "compression_ratio": 1.9135135135135135, "no_speech_prob": 0.0, "words": [{"start": 161.05, "end": 161.63, "word": "اليوم", "probability": 0.91796875}, {"start": 161.63, 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او", "probability": 0.84033203125}, {"start": 170.07, "end": 170.23, "word": " ال", "probability": 0.7421875}, {"start": 170.23, "end": 170.51, "word": " limit", "probability": 0.931640625}, {"start": 170.51, "end": 170.85, "word": " and", "probability": 0.9501953125}, {"start": 170.85, "end": 171.25, "word": " c", "probability": 0.9560546875}, {"start": 171.25, "end": 171.47, "word": " من", "probability": 0.99365234375}, {"start": 171.47, "end": 172.03, "word": " اليسار", "probability": 0.9905598958333334}, {"start": 172.03, "end": 172.49, "word": " بالساوي", "probability": 0.9774169921875}, {"start": 172.49, "end": 172.93, "word": " infinity", "probability": 0.86572265625}, {"start": 172.93, "end": 173.15, "word": " و", "probability": 0.81201171875}, {"start": 173.15, "end": 173.47, "word": " كذلك", "probability": 0.8273111979166666}, {"start": 173.47, "end": 173.79, "word": " نفس", "probability": 0.99365234375}, {"start": 173.79, "end": 174.99, "word": " الشيء", "probability": 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"end": 182.03, "word": " لأن", "probability": 0.4893798828125}, {"start": 182.03, "end": 182.27, "word": " هذه", "probability": 0.67578125}, {"start": 182.27, "end": 182.81, "word": " كانت", "probability": 0.972412109375}, {"start": 182.81, "end": 183.23, "word": " two", "probability": 0.8974609375}, {"start": 183.23, "end": 183.65, "word": " sided", "probability": 0.85986328125}, {"start": 183.65, "end": 184.19, "word": " limit", "probability": 0.978515625}, {"start": 184.19, "end": 185.25, "word": " المرة", "probability": 0.6536865234375}, {"start": 185.25, "end": 185.59, "word": " الأخيرة", "probability": 0.5233561197916666}, {"start": 185.59, "end": 186.21, "word": " اتعرفنا", "probability": 0.868896484375}], "temperature": 1.0}, {"id": 7, "seek": 21399, "start": 186.77, "end": 213.99, "text": "ما معناه ان ال two sided limit تكون infinite او ساوي infinity او plus او minus infinity اليوم ما معناه ان ال one sided limit تكون infinity او negative infinity فناخد التعريف مشابه للتعريف التعريف ال one sided limit تكون بتساوي real number ف let", "tokens": [15042, 20449, 8315, 3224, 16472, 2423, 732, 41651, 4948, 6055, 30544, 13785, 1975, 2407, 8608, 995, 45865, 13202, 1975, 2407, 1804, 1975, 2407, 3175, 13202, 45595, 20498, 19446, 20449, 8315, 3224, 16472, 2423, 472, 41651, 4948, 6055, 30544, 13202, 1975, 2407, 3671, 13202, 6156, 1863, 47283, 3215, 16712, 3615, 16572, 5172, 37893, 16758, 3224, 24976, 2655, 3615, 16572, 5172, 16712, 3615, 16572, 5172, 2423, 472, 41651, 4948, 6055, 30544, 39894, 3794, 995, 45865, 957, 1230, 6156, 718], "avg_logprob": -0.20733173688252768, "compression_ratio": 1.994047619047619, "no_speech_prob": 0.0, "words": [{"start": 186.77, "end": 187.01, "word": "ما", "probability": 0.67333984375}, {"start": 187.01, "end": 187.39, "word": " معناه", "probability": 0.6494140625}, {"start": 187.39, "end": 187.57, "word": " ان", "probability": 0.654296875}, {"start": 187.57, "end": 187.75, "word": " ال", "probability": 0.8251953125}, {"start": 187.75, 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"end": 438.83, "word": " ال", "probability": 0.67919921875}, {"start": 438.83, "end": 439.19, "word": " left", "probability": 0.931640625}, {"start": 439.19, "end": 439.53, "word": " hand", "probability": 0.80126953125}, {"start": 439.53, "end": 440.01, "word": " limit", "probability": 0.9833984375}, {"start": 440.01, "end": 441.15, "word": " مايعني", "probability": 0.5310872395833334}, {"start": 441.15, "end": 441.41, "word": " ان", "probability": 0.6982421875}, {"start": 441.41, "end": 441.59, "word": " ال", "probability": 0.9169921875}, {"start": 441.59, "end": 441.85, "word": " left", "probability": 0.95166015625}, {"start": 441.85, "end": 442.21, "word": " hand", "probability": 0.91796875}, {"start": 442.21, "end": 442.69, "word": " limit", "probability": 0.9833984375}, {"start": 442.69, "end": 444.41, "word": " تساوي", "probability": 0.8896484375}], "temperature": 1.0}, {"id": 16, "seek": 46793, "start": 445.85, "end": 467.93, "text": "Infinity او سالب Infinity اذا لو انا بدى اعدل اعرف مامعنى ان ال limit ل F عن C من اليسار بالساوية Infinity او مامعنى ان ال limit ل F عن C من اليسار بالساوية سالب Infinity فباخد let F be هكذا", "tokens": [4575, 5194, 507, 1975, 2407, 8608, 6027, 3555, 34762, 1975, 15730, 45164, 1975, 8315, 47525, 7578, 1975, 22488, 1211, 1975, 3615, 28480, 19446, 2304, 3615, 1863, 7578, 16472, 2423, 4948, 5296, 479, 18871, 383, 9154, 45595, 3794, 9640, 20666, 3794, 995, 2407, 10632, 34762, 1975, 2407, 19446, 2304, 3615, 1863, 7578, 16472, 2423, 4948, 5296, 479, 18871, 383, 9154, 45595, 3794, 9640, 20666, 3794, 995, 2407, 10632, 8608, 6027, 3555, 34762, 6156, 3555, 47283, 3215, 718, 479, 312, 8032, 4117, 15730], "avg_logprob": -0.2749619011471911, "compression_ratio": 1.986206896551724, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 445.85, "end": 446.61, "word": "Infinity", "probability": 0.6205851236979166}, {"start": 446.61, "end": 446.93, "word": " او", "probability": 0.64013671875}, {"start": 446.93, "end": 447.47, "word": " 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"text": "الـ left limit عن c ما معنى أنها ساوي plus أو minus infinity إذن قلنا إن كل النظريات اللي برهنها for two sided limits هتكون صحيحة لل left limit و لل right limit من ضمنهم النظرية السابقة طيب لو بدي أنا يعني", "tokens": [6027, 39184, 1411, 4948, 18871, 269, 19446, 20449, 1863, 7578, 14739, 11296, 8608, 995, 45865, 1804, 34051, 3175, 13202, 11933, 8848, 1863, 12174, 1211, 8315, 36145, 28242, 28239, 19913, 16572, 9307, 13672, 1829, 4724, 2288, 3224, 1863, 11296, 337, 732, 41651, 10406, 8032, 2655, 30544, 20328, 5016, 1829, 5016, 3660, 24976, 1411, 4948, 4032, 24976, 558, 4948, 9154, 48812, 27842, 16095, 28239, 19913, 2288, 10632, 21136, 16758, 28671, 23032, 1829, 3555, 45164, 4724, 16254, 41850, 37495, 22653], "avg_logprob": -0.26782852029189086, "compression_ratio": 1.5532994923857868, "no_speech_prob": 0.0, "words": [{"start": 526.99, "end": 527.41, "word": "الـ", "probability": 0.52783203125}, {"start": 527.41, "end": 527.63, "word": " left", "probability": 0.55908203125}, 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"end": 535.97, "word": " قلنا", "probability": 0.79638671875}, {"start": 535.97, "end": 536.17, "word": " إن", "probability": 0.544921875}, {"start": 536.17, "end": 536.73, "word": " كل", "probability": 0.80712890625}, {"start": 536.73, "end": 537.49, "word": " النظريات", "probability": 0.84857177734375}, {"start": 537.49, "end": 537.67, "word": " اللي", "probability": 0.734619140625}, {"start": 537.67, "end": 538.21, "word": " برهنها", "probability": 0.83984375}, {"start": 538.21, "end": 538.45, "word": " for", "probability": 0.83837890625}, {"start": 538.45, "end": 538.71, "word": " two", "probability": 0.919921875}, {"start": 538.71, "end": 539.01, "word": " sided", "probability": 0.51123046875}, {"start": 539.01, "end": 539.53, "word": " limits", "probability": 0.9619140625}, {"start": 539.53, "end": 540.31, "word": " هتكون", "probability": 0.8701171875}, {"start": 540.31, "end": 540.81, "word": " صحيحة", "probability": 0.9525390625}, {"start": 540.81, "end": 540.99, "word": " لل", "probability": 0.90625}, {"start": 540.99, "end": 541.23, "word": " left", "probability": 0.46728515625}, {"start": 541.23, "end": 541.71, "word": " limit", "probability": 0.98291015625}, {"start": 541.71, "end": 542.45, "word": " و", "probability": 0.5908203125}, {"start": 542.45, "end": 542.59, "word": " لل", "probability": 0.91650390625}, {"start": 542.59, "end": 542.85, "word": " right", "probability": 0.93896484375}, {"start": 542.85, "end": 545.57, "word": " limit", "probability": 0.98779296875}, {"start": 545.57, "end": 548.27, "word": " من", "probability": 0.96435546875}, {"start": 548.27, "end": 548.73, "word": " ضمنهم", "probability": 0.9912109375}, {"start": 548.73, "end": 549.37, "word": " النظرية", "probability": 0.9693603515625}, {"start": 549.37, "end": 550.23, "word": " السابقة", "probability": 0.7181803385416666}, {"start": 550.23, "end": 554.01, "word": " طيب", "probability": 0.96923828125}, {"start": 554.01, "end": 554.17, "word": " لو", "probability": 0.95654296875}, {"start": 554.17, "end": 554.51, "word": " بدي", "probability": 0.6556396484375}, {"start": 554.51, "end": 554.71, "word": " أنا", "probability": 0.6298828125}, {"start": 554.71, "end": 555.13, "word": " يعني", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 20, "seek": 58516, "start": 558.09, "end": 585.17, "text": "أخد أمثلة كيف نستخدم التعريف هذا فيه إثبات إن ال limits تطلع plus أو minus infinity فناخد أول مثال let f of x بسوي واحد على x حيث x لا يساوي سبق و show", "tokens": [10721, 9778, 3215, 5551, 2304, 12984, 37977, 9122, 33911, 8717, 14851, 9778, 40448, 16712, 3615, 16572, 5172, 23758, 8978, 3224, 11933, 12984, 3555, 9307, 36145, 2423, 10406, 6055, 9566, 1211, 3615, 1804, 34051, 3175, 13202, 6156, 1863, 47283, 3215, 5551, 12610, 50113, 6027, 718, 283, 295, 2031, 4724, 3794, 45865, 36764, 24401, 15844, 2031, 11331, 1829, 12984, 2031, 20193, 7251, 3794, 995, 45865, 8608, 3555, 4587, 4032, 855], "avg_logprob": -0.2744565338328265, "compression_ratio": 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"probability": 0.8955078125}, {"start": 566.33, "end": 566.85, "word": " plus", "probability": 0.80615234375}, {"start": 566.85, "end": 567.11, "word": " أو", "probability": 0.8037109375}, {"start": 567.11, "end": 567.57, "word": " minus", "probability": 0.98828125}, {"start": 567.57, "end": 569.49, "word": " infinity", "probability": 0.8232421875}, {"start": 569.49, "end": 571.39, "word": " فناخد", "probability": 0.8212890625}, {"start": 571.39, "end": 571.63, "word": " أول", "probability": 0.9482421875}, {"start": 571.63, "end": 572.23, "word": " مثال", "probability": 0.81298828125}, {"start": 572.23, "end": 572.65, "word": " let", "probability": 0.70654296875}, {"start": 572.65, "end": 575.17, "word": " f", "probability": 0.73291015625}, {"start": 575.17, "end": 575.41, "word": " of", "probability": 0.33203125}, {"start": 575.41, "end": 575.85, "word": " x", "probability": 0.83251953125}, {"start": 575.85, "end": 576.89, "word": " بسوي", "probability": 0.6021321614583334}, {"start": 576.89, "end": 580.25, "word": " واحد", "probability": 0.843505859375}, {"start": 580.25, "end": 580.43, "word": " على", "probability": 0.541015625}, {"start": 580.43, "end": 580.81, "word": " x", "probability": 0.5625}, {"start": 580.81, "end": 581.17, "word": " حيث", "probability": 0.5448404947916666}, {"start": 581.17, "end": 581.47, "word": " x", "probability": 0.91455078125}, {"start": 581.47, "end": 581.71, "word": " لا", "probability": 0.830078125}, {"start": 581.71, "end": 582.17, "word": " يساوي", "probability": 0.8507080078125}, {"start": 582.17, "end": 582.69, "word": " سبق", "probability": 0.6231689453125}, {"start": 582.69, "end": 584.57, "word": " و", "probability": 0.9052734375}, {"start": 584.57, "end": 585.17, "word": " show", "probability": 0.5498046875}], "temperature": 1.0}, {"id": 21, "seek": 60996, "start": 589.78, "end": 609.96, "text": "عايزين نفدت واحد ان ال limit لواحد على x او f of x هنا لما x تقول الى ستر من اليمين بساوي ال infinity و 2 limit", "tokens": 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"word": " على", "probability": 0.72607421875}, {"start": 595.72, "end": 596.16, "word": " x", "probability": 0.39599609375}, {"start": 596.16, "end": 597.32, "word": " او", "probability": 0.83544921875}, {"start": 597.32, "end": 597.58, "word": " f", "probability": 0.78662109375}, {"start": 597.58, "end": 597.76, "word": " of", "probability": 0.416259765625}, {"start": 597.76, "end": 598.06, "word": " x", "probability": 0.9619140625}, {"start": 598.06, "end": 598.34, "word": " هنا", "probability": 0.66259765625}, {"start": 598.34, "end": 601.16, "word": " لما", "probability": 0.85546875}, {"start": 601.16, "end": 601.44, "word": " x", "probability": 0.890625}, {"start": 601.44, "end": 601.94, "word": " تقول", "probability": 0.881103515625}, {"start": 601.94, "end": 602.3, "word": " الى", "probability": 0.18939208984375}, {"start": 602.3, "end": 602.88, "word": " ستر", "probability": 0.7814127604166666}, {"start": 602.88, "end": 603.12, "word": " من", "probability": 0.98388671875}, {"start": 603.12, "end": 603.86, "word": " اليمين", "probability": 0.8487955729166666}, {"start": 603.86, "end": 605.24, "word": " بساوي", "probability": 0.65948486328125}, {"start": 605.24, "end": 605.38, "word": " ال", "probability": 0.417724609375}, {"start": 605.38, "end": 605.82, "word": " infinity", "probability": 0.87646484375}, {"start": 605.82, "end": 606.56, "word": " و", "probability": 0.7060546875}, {"start": 606.56, "end": 608.04, "word": " 2", "probability": 0.1600341796875}, {"start": 608.04, "end": 609.96, "word": " limit", "probability": 0.3330078125}], "temperature": 1.0}, {"id": 22, "seek": 62954, "start": 611.3, "end": 629.54, "text": "لف of X لما X تقول ال 0 من اليسار يساوي سالب infinity okay فلو بدنا نبرم الجزء الأول مثلا to show", "tokens": [1211, 5172, 295, 1783, 5296, 15042, 1783, 6055, 39648, 2423, 1958, 9154, 45595, 3794, 9640, 7251, 3794, 995, 45865, 8608, 6027, 3555, 13202, 1392, 6156, 1211, 2407, 47525, 8315, 8717, 26890, 2304, 25724, 11622, 38207, 16247, 12610, 50113, 15040, 281, 855], "avg_logprob": -0.30338541099003385, "compression_ratio": 1.2295081967213115, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 611.3, "end": 611.86, "word": "لف", "probability": 0.3897705078125}, {"start": 611.86, "end": 612.14, "word": " of", "probability": 0.642578125}, {"start": 612.14, "end": 612.52, "word": " X", "probability": 0.50390625}, {"start": 612.52, "end": 613.16, "word": " لما", "probability": 0.718994140625}, {"start": 613.16, "end": 613.48, "word": " X", "probability": 0.95654296875}, {"start": 613.48, "end": 614.12, "word": " تقول", "probability": 0.884521484375}, {"start": 614.12, "end": 615.36, "word": " ال", "probability": 0.338134765625}, {"start": 615.36, "end": 615.98, "word": " 0", "probability": 0.293701171875}, {"start": 615.98, "end": 616.3, "word": " من", "probability": 0.99462890625}, {"start": 616.3, "end": 617.12, "word": " اليسار", "probability": 0.97509765625}, {"start": 617.12, "end": 618.68, "word": " يساوي", "probability": 0.764617919921875}, {"start": 618.68, "end": 619.2, "word": " سالب", "probability": 0.86865234375}, {"start": 619.2, "end": 619.74, "word": " infinity", "probability": 0.456298828125}, {"start": 619.74, "end": 620.52, "word": " okay", "probability": 0.2373046875}, {"start": 620.52, "end": 623.88, "word": " فلو", "probability": 0.88720703125}, {"start": 623.88, "end": 624.16, "word": " بدنا", "probability": 0.961181640625}, {"start": 624.16, "end": 624.72, "word": " نبرم", "probability": 0.734375}, {"start": 624.72, "end": 626.36, "word": " الجزء", "probability": 0.873046875}, {"start": 626.36, "end": 626.76, "word": " الأول", "probability": 0.8603515625}, {"start": 626.76, "end": 627.28, "word": " مثلا", "probability": 0.982177734375}, {"start": 627.28, "end": 629.02, "word": " to", "probability": 0.92333984375}, {"start": 629.02, "end": 629.54, "word": " show", "probability": 0.96875}], "temperature": 1.0}, {"id": 23, "seek": 65921, "start": 632.41, "end": 659.21, "text": "المقاومة لـ f of x as x tends to 0 from the right بساوي infinity فبدي ابدا بـ alpha تنتمي ل R فبقول let alpha belonging to R be given و بدي ارد عليها بDelta", "tokens": [45340, 4587, 995, 20498, 3660, 5296, 39184, 283, 295, 2031, 382, 2031, 12258, 281, 1958, 490, 264, 558, 4724, 3794, 995, 45865, 13202, 6156, 3555, 16254, 48127, 28259, 4724, 39184, 8961, 6055, 29399, 2304, 1829, 5296, 497, 6156, 3555, 39648, 718, 8961, 22957, 281, 497, 312, 2212, 4032, 4724, 16254, 1975, 2288, 3215, 25894, 11296, 4724, 40848, 1328], "avg_logprob": -0.4541843280953876, "compression_ratio": 1.2654320987654322, "no_speech_prob": 0.0, "words": [{"start": 632.4100000000001, "end": 633.33, "word": "المقاومة", "probability": 0.4727783203125}, {"start": 633.33, "end": 634.25, "word": " لـ", "probability": 0.4246826171875}, {"start": 634.25, "end": 634.43, "word": " f", "probability": 0.2445068359375}, {"start": 634.43, "end": 634.67, "word": " of", "probability": 0.227783203125}, {"start": 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" ابدا", "probability": 0.624755859375}, {"start": 642.29, "end": 642.57, "word": " بـ", "probability": 0.4725341796875}, {"start": 642.57, "end": 642.87, "word": " alpha", "probability": 0.4013671875}, {"start": 642.87, "end": 644.59, "word": " تنتمي", "probability": 0.9407958984375}, {"start": 644.59, "end": 644.67, "word": " ل", "probability": 0.87255859375}, {"start": 644.67, "end": 644.99, "word": " R", "probability": 0.26953125}, {"start": 644.99, "end": 645.67, "word": " فبقول", "probability": 0.9192708333333334}, {"start": 645.67, "end": 646.11, "word": " let", "probability": 0.75830078125}, {"start": 646.11, "end": 648.41, "word": " alpha", "probability": 0.76806640625}, {"start": 648.41, "end": 648.99, "word": " belonging", "probability": 0.69482421875}, {"start": 648.99, "end": 649.33, "word": " to", "probability": 0.98486328125}, {"start": 649.33, "end": 649.87, "word": " R", "probability": 0.98876953125}, {"start": 649.87, "end": 651.55, "word": " be", "probability": 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"word": " alpha", "probability": 0.350830078125}, {"start": 662.15, "end": 662.61, "word": " دي", "probability": 0.62255859375}, {"start": 662.61, "end": 662.79, "word": " ال", "probability": 0.313232421875}, {"start": 662.79, "end": 663.25, "word": " delta", "probability": 0.7783203125}, {"start": 663.25, "end": 664.61, "word": " عدد", "probability": 0.9554036458333334}, {"start": 664.61, "end": 665.21, "word": " موجب", "probability": 0.98779296875}, {"start": 665.21, "end": 666.05, "word": " ويعتمد", "probability": 0.817626953125}, {"start": 666.05, "end": 666.25, "word": " على", "probability": 0.84033203125}, {"start": 666.25, "end": 666.43, "word": " ال", "probability": 0.95703125}, {"start": 666.43, "end": 666.85, "word": " alpha", "probability": 0.92822265625}, {"start": 666.85, "end": 667.21, "word": " ف", "probability": 0.625}, {"start": 667.21, "end": 667.93, "word": " choose", "probability": 0.405029296875}, {"start": 667.93, "end": 668.63, "word": " delta", "probability": 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{"start": 679.61, "end": 680.35, "word": " absolute", "probability": 0.450439453125}, {"start": 680.35, "end": 680.53, "word": " ال", "probability": 0.7705078125}, {"start": 680.53, "end": 680.79, "word": " alpha", "probability": 0.90966796875}, {"start": 680.79, "end": 681.01, "word": " دي", "probability": 0.96533203125}, {"start": 681.01, "end": 681.29, "word": " عدد", "probability": 0.9943033854166666}, {"start": 681.29, "end": 681.91, "word": " حقيقي", "probability": 0.98486328125}, {"start": 681.91, "end": 682.49, "word": " القيمة", "probability": 0.9560546875}, {"start": 682.49, "end": 682.93, "word": " المطلقة", "probability": 0.9854736328125}, {"start": 682.93, "end": 683.27, "word": " له", "probability": 0.1678466796875}, {"start": 683.27, "end": 684.49, "word": " عدد", "probability": 0.99169921875}, {"start": 684.49, "end": 684.97, "word": " غير", "probability": 0.9970703125}, {"start": 684.97, "end": 685.49, "word": " سالم", "probability": 0.7333984375}, {"start": 685.49, "end": 687.55, "word": " ممكن", "probability": 0.933349609375}, {"start": 687.55, "end": 687.93, "word": " يساوي", "probability": 0.9610595703125}, {"start": 687.93, "end": 688.27, "word": " سفر", "probability": 0.8527018229166666}, {"start": 688.27, "end": 688.45, "word": " إذا", "probability": 0.6920166015625}, {"start": 688.45, "end": 688.81, "word": " كانت", "probability": 0.9228515625}, {"start": 688.81, "end": 688.99, "word": " alpha", "probability": 0.46728515625}, {"start": 688.99, "end": 689.47, "word": " بالساوية", "probability": 0.613330078125}, {"start": 689.47, "end": 689.81, "word": " سفر", "probability": 0.9397786458333334}], "temperature": 1.0}, {"id": 25, "seek": 71164, "start": 690.62, "end": 711.64, "text": "لكن زائد واحد بصير موجب المقام موجب اذا انا بضيف واحد ليه عشان اضمن ان المقام مايسويش سفر لان في احتمال ان ال alpha ساوي سفر فبصير عندى مشكلة عشان اتخلص من المشكلة هذه بجسم على absolute alpha زائد واحد الان هذا عدد موجب", "tokens": [1211, 19452, 30767, 16373, 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{"start": 697.72, "end": 698.22, "word": " المقام", "probability": 0.9832356770833334}, {"start": 698.22, "end": 698.82, "word": " مايسويش", "probability": 0.81611328125}, {"start": 698.82, "end": 699.18, "word": " سفر", "probability": 0.9099934895833334}, {"start": 699.18, "end": 700.58, "word": " لان", "probability": 0.86376953125}, {"start": 700.58, "end": 700.74, "word": " في", "probability": 0.8369140625}, {"start": 700.74, "end": 701.16, "word": " احتمال", "probability": 0.9490966796875}, {"start": 701.16, "end": 701.28, "word": " ان", "probability": 0.9462890625}, {"start": 701.28, "end": 701.38, "word": " ال", "probability": 0.599609375}, {"start": 701.38, "end": 701.56, "word": " alpha", "probability": 0.40869140625}, {"start": 701.56, "end": 701.88, "word": " ساوي", "probability": 0.6222330729166666}, {"start": 701.88, "end": 702.36, "word": " سفر", "probability": 0.9698893229166666}, {"start": 702.36, "end": 703.46, "word": " فبصير", "probability": 0.808349609375}, {"start": 703.46, "end": 703.72, "word": " عندى", "probability": 0.72216796875}, {"start": 703.72, "end": 704.1, "word": " مشكلة", "probability": 0.9793294270833334}, {"start": 704.1, "end": 704.3, "word": " عشان", "probability": 0.9765625}, {"start": 704.3, "end": 704.74, "word": " اتخلص", "probability": 0.8982421875}, {"start": 704.74, "end": 704.86, "word": " من", "probability": 0.96826171875}, {"start": 704.86, "end": 705.32, "word": " المشكلة", "probability": 0.9893798828125}, {"start": 705.32, "end": 705.64, "word": " هذه", "probability": 0.363037109375}, {"start": 705.64, "end": 707.1, "word": " بجسم", "probability": 0.8880208333333334}, {"start": 707.1, "end": 707.34, "word": " على", "probability": 0.91015625}, {"start": 707.34, "end": 707.66, "word": " absolute", "probability": 0.6123046875}, {"start": 707.66, "end": 708.06, "word": " alpha", "probability": 0.7509765625}, {"start": 708.06, "end": 708.5, "word": " زائد", "probability": 0.8883463541666666}, {"start": 708.5, "end": 709.62, "word": " واحد", "probability": 0.9736328125}, {"start": 709.62, "end": 710.5, "word": " الان", "probability": 0.6707763671875}, {"start": 710.5, "end": 710.78, "word": " هذا", "probability": 0.7158203125}, {"start": 710.78, "end": 711.14, "word": " عدد", "probability": 0.98681640625}, {"start": 711.14, "end": 711.64, "word": " موجب", "probability": 0.9928385416666666}], "temperature": 1.0}, {"id": 26, "seek": 73133, "start": 713.15, "end": 731.33, "text": "و يعتمد على alpha هي ال delta هي مرتبطة معرفة بدلالة alpha هي معناه أنها تعتمد على alpha إذا لأي alpha ينتمي ل R خد ال delta اللي بتنظرها هي واحد على absolute alpha الذات واحدة", "tokens": [2407, 7251, 34268, 2304, 3215, 15844, 8961, 39896, 2423, 8289, 39896, 3714, 43500, 3555, 9566, 3660, 20449, 28480, 3660, 47525, 1211, 6027, 3660, 8961, 39896, 20449, 8315, 3224, 14739, 11296, 6055, 34268, 2304, 3215, 15844, 8961, 11933, 15730, 5296, 10721, 1829, 8961, 7251, 29399, 2304, 1829, 5296, 497, 16490, 3215, 2423, 8289, 13672, 1829, 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"probability": 0.94140625}, {"start": 717.01, "end": 717.73, "word": " بدلالة", "probability": 0.97607421875}, {"start": 717.73, "end": 718.07, "word": " alpha", "probability": 0.79443359375}, {"start": 718.07, "end": 718.81, "word": " هي", "probability": 0.425048828125}, {"start": 718.81, "end": 719.15, "word": " معناه", "probability": 0.8631184895833334}, {"start": 719.15, "end": 719.33, "word": " أنها", "probability": 0.68994140625}, {"start": 719.33, "end": 720.01, "word": " تعتمد", "probability": 0.97314453125}, {"start": 720.01, "end": 720.61, "word": " على", "probability": 0.9150390625}, {"start": 720.61, "end": 720.95, "word": " alpha", "probability": 0.6865234375}, {"start": 720.95, "end": 721.93, "word": " إذا", "probability": 0.6036376953125}, {"start": 721.93, "end": 722.35, "word": " لأي", "probability": 0.8388671875}, {"start": 722.35, "end": 722.67, "word": " alpha", "probability": 0.88134765625}, {"start": 722.67, "end": 723.17, "word": " ينتمي", "probability": 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{"start": 846.64, "end": 847.3, "word": " اثبتنا", "probability": 0.91455078125}, {"start": 847.3, "end": 847.5, "word": " اذا", "probability": 0.74462890625}, {"start": 847.5, "end": 847.94, "word": " معناه", "probability": 0.71875}, {"start": 847.94, "end": 848.22, "word": " هذا", "probability": 0.44482421875}, {"start": 848.22, "end": 849.56, "word": " الكلام", "probability": 0.7676595052083334}, {"start": 849.56, "end": 850.08, "word": " هذا", "probability": 0.4404296875}, {"start": 850.08, "end": 851.0, "word": " انه", "probability": 0.642578125}, {"start": 851.0, "end": 851.38, "word": " لكل", "probability": 0.9365234375}, {"start": 851.38, "end": 851.78, "word": " alpha", "probability": 0.7822265625}, {"start": 851.78, "end": 852.18, "word": " فيه", "probability": 0.722412109375}, {"start": 852.18, "end": 852.56, "word": " delta", "probability": 0.61181640625}, {"start": 852.56, "end": 853.48, "word": " تعتمد", "probability": 0.9041748046875}, {"start": 853.48, "end": 854.02, "word": " عليها", "probability": 0.988525390625}, {"start": 854.02, "end": 855.02, "word": " بتخلي", "probability": 0.8694661458333334}, {"start": 855.02, "end": 855.22, "word": " f", "probability": 0.55029296875}, {"start": 855.22, "end": 855.4, "word": " of", "probability": 0.73193359375}, {"start": 855.4, "end": 855.64, "word": " x", "probability": 0.70556640625}, {"start": 855.64, "end": 856.0, "word": " اكبر", "probability": 0.8702799479166666}, {"start": 856.0, "end": 856.14, "word": " من", "probability": 0.44287109375}, {"start": 856.14, "end": 856.38, "word": " alpha", "probability": 0.63134765625}, {"start": 856.38, "end": 856.9, "word": " لكل", "probability": 0.923828125}, {"start": 856.9, "end": 857.28, "word": " x", "probability": 0.8740234375}, {"start": 857.28, "end": 858.38, "word": " قريبة", "probability": 0.9552408854166666}, {"start": 858.38, "end": 858.84, "word": " من", "probability": 0.990234375}, {"start": 858.84, "end": 859.64, "word": " السفر", "probability": 0.8484700520833334}, {"start": 859.64, "end": 860.58, "word": " within", "probability": 0.67529296875}, {"start": 860.58, "end": 861.3, "word": " مسافة", "probability": 0.9879557291666666}, {"start": 861.3, "end": 861.58, "word": " delta", "probability": 0.61865234375}], "temperature": 1.0}, {"id": 32, "seek": 87832, "start": 862.34, "end": 878.32, "text": "إن هذا معناه حسب التعريف إن ال limit ل f of x لما x تقول إلى سفر من اليمين بساوي بورهانش جزء التاني مشابه", "tokens": [28814, 1863, 23758, 20449, 8315, 3224, 11331, 35457, 16712, 3615, 16572, 5172, 36145, 2423, 4948, 5296, 283, 295, 2031, 5296, 15042, 2031, 6055, 39648, 30731, 8608, 5172, 2288, 9154, 2423, 32640, 9957, 4724, 3794, 995, 45865, 4724, 13063, 3224, 7649, 8592, 10874, 11622, 38207, 16712, 7649, 1829, 37893, 16758, 3224], "avg_logprob": -0.3547794094272688, "compression_ratio": 1.3409090909090908, "no_speech_prob": 0.0, "words": [{"start": 862.34, "end": 862.58, "word": "إن", "probability": 0.4678955078125}, {"start": 862.58, "end": 862.8, "word": " هذا", "probability": 0.6748046875}, {"start": 862.8, "end": 863.14, "word": " معناه", "probability": 0.8264973958333334}, {"start": 863.14, "end": 863.44, "word": " حسب", "probability": 0.80859375}, {"start": 863.44, "end": 864.06, "word": " التعريف", "probability": 0.9466552734375}, {"start": 864.06, "end": 864.28, "word": " إن", "probability": 0.14599609375}, {"start": 864.28, "end": 864.5, "word": " ال", "probability": 0.39306640625}, {"start": 864.5, "end": 864.94, "word": " limit", "probability": 0.6943359375}, {"start": 864.94, "end": 866.0, "word": " ل", "probability": 0.7529296875}, {"start": 866.0, "end": 866.26, "word": " f", "probability": 0.3125}, {"start": 866.26, "end": 866.52, "word": " of", "probability": 0.496826171875}, {"start": 866.52, "end": 866.86, "word": " x", "probability": 0.89501953125}, {"start": 866.86, "end": 867.22, "word": " لما", "probability": 0.734619140625}, {"start": 867.22, "end": 867.54, "word": " x", "probability": 0.8798828125}, {"start": 867.54, "end": 868.14, "word": " تقول", "probability": 0.922607421875}, {"start": 868.14, "end": 868.5, "word": " إلى", "probability": 0.84228515625}, {"start": 868.5, "end": 869.06, "word": " سفر", "probability": 0.74853515625}, {"start": 869.06, "end": 869.34, "word": " من", "probability": 0.99072265625}, {"start": 869.34, "end": 870.88, "word": " اليمين", "probability": 0.8517252604166666}, {"start": 870.88, "end": 872.18, "word": " بساوي", "probability": 0.8380126953125}, {"start": 872.18, "end": 876.78, "word": " بورهانش", "probability": 0.675146484375}, {"start": 876.78, "end": 877.1, "word": " جزء", "probability": 0.9716796875}, {"start": 877.1, "end": 877.58, "word": " التاني", "probability": 0.8349609375}, {"start": 877.58, "end": 878.32, "word": " مشابه", "probability": 0.9583333333333334}], "temperature": 1.0}, {"id": 33, "seek": 91488, "start": 887.84, "end": 914.88, "text": "is similar مشابه لل part للجزء الأول يعني فهسيبكم انتوا تكتبوا برهان مشابه مع التعديلات اللازمة و أيه طبعا التعريف تبع ال limit from the left موجود okay تمام اللي هو بالأزرق تمام مثال تاني ممكن برضه", "tokens": [271, 2531, 37893, 16758, 3224, 24976, 644, 24976, 7435, 11622, 38207, 16247, 12610, 37495, 22653, 6156, 3224, 3794, 1829, 3555, 24793, 16472, 2655, 14407, 6055, 4117, 2655, 3555, 14407, 4724, 2288, 3224, 7649, 37893, 16758, 3224, 20449, 16712, 3615, 16254, 1211, 9307, 13672, 31377, 46007, 4032, 36632, 3224, 23032, 3555, 3615, 995, 16712, 3615, 16572, 5172, 6055, 3555, 3615, 2423, 4948, 490, 264, 1411, 3714, 29245, 23328, 1392, 46811, 10943, 13672, 1829, 31439, 20666, 10721, 11622, 2288, 4587, 46811, 10943, 50113, 6027, 6055, 7649, 1829, 3714, 43020, 4724, 43042, 3224], "avg_logprob": -0.16595123397124992, "compression_ratio": 1.5721153846153846, "no_speech_prob": 0.0, "words": [{"start": 887.84, "end": 888.28, "word": "is", "probability": 0.5439453125}, {"start": 888.28, "end": 888.92, "word": " similar", "probability": 0.935546875}, {"start": 888.92, "end": 889.86, "word": " مشابه", "probability": 0.87548828125}, {"start": 889.86, "end": 890.66, "word": " لل", "probability": 0.41650390625}, {"start": 890.66, "end": 891.18, "word": " part", "probability": 0.52197265625}, {"start": 891.18, "end": 892.1, "word": " للجزء", "probability": 0.81396484375}, {"start": 892.1, "end": 892.58, "word": " الأول", "probability": 0.78125}, {"start": 892.58, "end": 893.5, "word": " يعني", "probability": 0.6240234375}, {"start": 893.5, "end": 894.4, "word": " فهسيبكم", "probability": 0.86474609375}, {"start": 894.4, "end": 894.98, "word": " انتوا", "probability": 0.6875}, {"start": 894.98, "end": 896.1, "word": " تكتبوا", "probability": 0.98896484375}, {"start": 896.1, "end": 896.62, "word": " برهان", "probability": 0.9307861328125}, {"start": 896.62, "end": 897.26, "word": " مشابه", "probability": 0.9903971354166666}, {"start": 897.26, "end": 897.58, "word": " مع", "probability": 0.9794921875}, {"start": 897.58, "end": 898.34, "word": " التعديلات", "probability": 0.92568359375}, {"start": 898.34, "end": 899.04, "word": " اللازمة", "probability": 0.99462890625}, {"start": 899.04, "end": 899.86, "word": " و", "probability": 0.736328125}, {"start": 899.86, "end": 900.08, "word": " أيه", "probability": 0.681640625}, {"start": 900.08, "end": 900.4, "word": " طبعا", "probability": 0.9664306640625}, {"start": 900.4, "end": 901.14, "word": " التعريف", "probability": 0.8988037109375}, {"start": 901.14, "end": 901.66, "word": " تبع", "probability": 0.7163899739583334}, {"start": 901.66, "end": 902.76, "word": " ال", "probability": 0.59375}, {"start": 902.76, "end": 903.14, "word": " limit", "probability": 0.92138671875}, {"start": 903.14, "end": 903.82, "word": " from", "probability": 0.90673828125}, {"start": 903.82, "end": 904.08, "word": " the", "probability": 0.90283203125}, {"start": 904.08, "end": 904.48, "word": " left", "probability": 0.96826171875}, {"start": 904.48, "end": 905.66, "word": " موجود", "probability": 0.96337890625}, {"start": 905.66, "end": 906.34, "word": " okay", "probability": 0.5126953125}, {"start": 906.34, "end": 906.78, "word": " تمام", "probability": 0.854736328125}, {"start": 906.78, "end": 906.9, "word": " اللي", "probability": 0.881103515625}, {"start": 906.9, "end": 907.04, "word": " هو", "probability": 0.99609375}, {"start": 907.04, "end": 907.96, "word": " بالأزرق", "probability": 0.8189453125}, {"start": 907.96, "end": 909.68, "word": " تمام", "probability": 0.95654296875}, {"start": 909.68, "end": 913.28, "word": " مثال", "probability": 0.92431640625}, {"start": 913.28, "end": 913.84, "word": " تاني", "probability": 0.94580078125}, {"start": 913.84, "end": 914.36, "word": " ممكن", "probability": 0.987060546875}, {"start": 914.36, "end": 914.88, "word": " برضه", "probability": 0.9905598958333334}], "temperature": 1.0}, {"id": 34, "seek": 94507, "start": 916.89, "end": 945.07, "text": "ناخد مثال تاني show limit for function e to one على x as x tends to zero from the right بساوي infinity", "tokens": [1863, 47283, 3215, 50113, 6027, 6055, 7649, 1829, 855, 4948, 337, 2445, 308, 281, 472, 15844, 2031, 382, 2031, 12258, 281, 4018, 490, 264, 558, 4724, 3794, 995, 45865, 13202], "avg_logprob": -0.3245967818844703, "compression_ratio": 1.0701754385964912, "no_speech_prob": 0.0, "words": [{"start": 916.89, "end": 917.77, "word": "ناخد", "probability": 0.6730143229166666}, {"start": 917.77, "end": 918.65, "word": " مثال", "probability": 0.944091796875}, {"start": 918.65, "end": 933.19, "word": " تاني", "probability": 0.8844401041666666}, {"start": 933.19, "end": 934.71, "word": " show", "probability": 0.15283203125}, {"start": 934.71, "end": 935.37, "word": " limit", "probability": 0.96142578125}, {"start": 935.37, "end": 936.75, "word": " for", "probability": 0.39501953125}, {"start": 936.75, "end": 937.35, "word": " function", "probability": 0.697265625}, {"start": 937.35, "end": 937.67, "word": " e", "probability": 0.404541015625}, {"start": 937.67, "end": 937.89, "word": " to", "probability": 0.73388671875}, {"start": 937.89, "end": 938.17, "word": " one", "probability": 0.4072265625}, {"start": 938.17, "end": 938.49, "word": " على", "probability": 0.60986328125}, {"start": 938.49, "end": 938.87, "word": " x", "probability": 0.8544921875}, {"start": 938.87, "end": 940.21, "word": " as", "probability": 0.92333984375}, {"start": 940.21, "end": 940.65, "word": " x", "probability": 0.96875}, {"start": 940.65, "end": 941.05, "word": " tends", "probability": 0.78564453125}, {"start": 941.05, "end": 941.27, "word": " to", "probability": 0.98583984375}, {"start": 941.27, "end": 941.65, "word": " zero", "probability": 0.875}, {"start": 941.65, "end": 941.95, "word": " from", "probability": 0.89013671875}, {"start": 941.95, "end": 942.21, "word": " the", "probability": 0.87109375}, {"start": 942.21, "end": 942.65, "word": " right", "probability": 0.93603515625}, {"start": 942.65, "end": 944.49, "word": " بساوي", "probability": 0.74365234375}, {"start": 944.49, "end": 945.07, "word": " infinity", "probability": 0.69287109375}], "temperature": 1.0}, {"id": 35, "seek": 98144, "start": 955.92, "end": 981.44, "text": "أنا عندي ال function تبعتي f of x بيسمي E أس واحد على X طبعا ال X هنا ال function مش معرفة عند السفر المجال الدالي هذه كل الأعداد الحقيقية مع ده السفر المثل هذا أخدناه المرة اللي فاتت we have", "tokens": [10721, 8315, 18871, 16254, 2423, 2445, 6055, 3555, 3615, 31371, 283, 295, 2031, 4724, 1829, 38251, 1829, 462, 5551, 3794, 36764, 24401, 15844, 1783, 23032, 3555, 3615, 995, 2423, 1783, 34105, 2423, 2445, 37893, 20449, 28480, 3660, 43242, 21136, 5172, 2288, 9673, 7435, 6027, 32748, 6027, 1829, 29538, 28242, 16247, 22488, 18513, 21542, 38436, 4587, 10632, 20449, 11778, 3224, 21136, 5172, 2288, 9673, 12984, 1211, 23758, 5551, 9778, 3215, 8315, 3224, 9673, 25720, 13672, 1829, 6156, 9307, 2655, 321, 362], "avg_logprob": -0.28838734567901236, "compression_ratio": 1.6302083333333333, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 955.92, "end": 956.4, "word": "أنا", "probability": 0.59698486328125}, {"start": 956.4, "end": 956.84, "word": " عندي", "probability": 0.75927734375}, {"start": 956.84, "end": 957.28, "word": " ال", "probability": 0.92041015625}, {"start": 957.28, "end": 957.9, "word": " function", "probability": 0.61328125}, {"start": 957.9, "end": 958.66, "word": " تبعتي", "probability": 0.8887939453125}, {"start": 958.66, "end": 958.98, "word": " f", "probability": 0.5791015625}, {"start": 958.98, "end": 959.2, "word": " of", "probability": 0.5146484375}, {"start": 959.2, "end": 959.68, "word": " x", "probability": 0.84423828125}, {"start": 959.68, "end": 961.76, "word": " بيسمي", "probability": 0.6204833984375}, {"start": 961.76, "end": 962.02, "word": " E", "probability": 0.10101318359375}, {"start": 962.02, "end": 962.3, "word": " أس", "probability": 0.717041015625}, {"start": 962.3, "end": 962.7, "word": " واحد", "probability": 0.962158203125}, {"start": 962.7, "end": 962.88, "word": " على", "probability": 0.708984375}, {"start": 962.88, "end": 963.18, "word": " X", "probability": 0.677734375}, {"start": 963.18, "end": 963.5, "word": " طبعا", "probability": 0.8570556640625}, {"start": 963.5, "end": 963.72, "word": " ال", "probability": 0.64599609375}, {"start": 963.72, "end": 964.08, "word": " X", "probability": 0.82666015625}, {"start": 964.08, "end": 964.42, "word": " هنا", "probability": 0.93994140625}, {"start": 964.42, "end": 965.76, "word": " ال", "probability": 0.6396484375}, {"start": 965.76, "end": 966.18, "word": " function", "probability": 0.931640625}, {"start": 966.18, "end": 966.48, "word": " مش", "probability": 0.98876953125}, {"start": 966.48, "end": 967.1, "word": " معرفة", "probability": 0.9249674479166666}, {"start": 967.1, "end": 967.32, "word": " عند", "probability": 0.955078125}, {"start": 967.32, "end": 967.84, "word": " السفر", "probability": 0.7864583333333334}, {"start": 967.84, "end": 970.08, "word": " المجال", "probability": 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0.953857421875}, {"start": 975.36, "end": 975.56, "word": " اللي", "probability": 0.93310546875}, {"start": 975.56, "end": 977.14, "word": " فاتت", "probability": 0.9910481770833334}, {"start": 977.14, "end": 980.86, "word": " we", "probability": 0.65185546875}, {"start": 980.86, "end": 981.44, "word": " have", "probability": 0.96484375}], "temperature": 1.0}, {"id": 36, "seek": 101319, "start": 985.65, "end": 1013.19, "text": "from previous example من المثال السابق فانا هادرس سابق انه واحد على اكس اكبر من سفر اصغر من اكس واحد على اكس لكل اكس اكبر من سفر", "tokens": [20579, 3894, 1365, 9154, 9673, 12984, 6027, 21136, 16758, 4587, 6156, 7649, 995, 8032, 18513, 2288, 3794, 8608, 16758, 4587, 16472, 3224, 36764, 24401, 15844, 1975, 4117, 3794, 1975, 4117, 26890, 9154, 8608, 5172, 2288, 1975, 9381, 17082, 2288, 9154, 1975, 4117, 3794, 36764, 24401, 15844, 1975, 4117, 3794, 5296, 28820, 1975, 4117, 3794, 1975, 4117, 26890, 9154, 8608, 5172, 2288], "avg_logprob": -0.275705631702177, 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limits للنهايات من اليمين we have نحصل على انه ال limit", "tokens": [2322, 3673, 20904, 538, 3673, 20904, 337, 558, 10406, 24976, 1863, 11296, 1829, 9307, 9154, 45595, 2304, 9957, 321, 362, 8717, 5016, 36520, 15844, 16472, 3224, 2423, 4948], "avg_logprob": -0.3089978530489165, "compression_ratio": 1.2574257425742574, "no_speech_prob": 0.0, "words": [{"start": 1103.96, "end": 1104.6, "word": "by", "probability": 0.1580810546875}, {"start": 1104.6, "end": 1105.74, "word": " above", "probability": 0.9013671875}, {"start": 1105.74, "end": 1106.36, "word": " theorem", "probability": 0.83251953125}, {"start": 1106.36, "end": 1108.92, "word": " by", "probability": 0.4755859375}, {"start": 1108.92, "end": 1109.5, "word": " above", "probability": 0.96923828125}, {"start": 1109.5, "end": 1111.02, "word": " theorem", "probability": 0.88623046875}, {"start": 1111.02, "end": 1112.88, "word": " for", "probability": 0.7626953125}, {"start": 1112.88, "end": 1113.48, "word": " right", "probability": 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"end": 1143.33, "text": "لقيت واحد على اكس لما اكس تقول إلى سفر من اليمين بساوي plus infinity وهذا اللي بدناه هي مظبوط صح؟ تمام؟", "tokens": [1211, 4587, 36081, 36764, 24401, 15844, 1975, 4117, 3794, 5296, 15042, 1975, 4117, 3794, 6055, 39648, 30731, 8608, 5172, 2288, 9154, 2423, 32640, 9957, 4724, 3794, 995, 45865, 1804, 13202, 37037, 15730, 13672, 1829, 47525, 8315, 3224, 39896, 3714, 19913, 3555, 2407, 9566, 20328, 5016, 22807, 46811, 10943, 22807], "avg_logprob": -0.31421875715255737, "compression_ratio": 1.288888888888889, "no_speech_prob": 0.0, "words": [{"start": 1128.73, "end": 1129.47, "word": "لقيت", "probability": 0.6780598958333334}, {"start": 1129.47, "end": 1129.99, "word": " واحد", "probability": 0.7183837890625}, {"start": 1129.99, "end": 1130.19, "word": " على", "probability": 0.55615234375}, {"start": 1130.19, "end": 1130.59, "word": " اكس", "probability": 0.7411295572916666}, {"start": 1130.59, "end": 1130.87, "word": " لما", "probability": 0.78564453125}, {"start": 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اتنين فبقول assume أنه one holds هذا معناه أن ال limit لf of x as x tends to infinity بسوى عدد L طيب to prove", "tokens": [2304, 12984, 1211, 14111, 995, 45164, 5551, 16572, 3215, 14739, 5551, 12984, 3555, 2655, 36764, 24401, 18779, 1975, 2655, 1863, 9957, 6156, 3555, 39648, 6552, 14739, 3224, 472, 9190, 23758, 20449, 8315, 3224, 14739, 2423, 4948, 5296, 69, 295, 2031, 382, 2031, 12258, 281, 13202, 4724, 3794, 2407, 7578, 6225, 3215, 3215, 441, 23032, 1829, 3555, 281, 7081], "avg_logprob": -0.37261653350571455, "compression_ratio": 1.2331288343558282, "no_speech_prob": 0.0, "words": [{"start": 1848.54, "end": 1849.28, "word": "مثلًا", "probability": 0.700732421875}, {"start": 1849.28, "end": 1849.4, "word": " لو", "probability": 0.43701171875}, {"start": 1849.4, "end": 1849.72, "word": " أريد", "probability": 0.66943359375}, {"start": 1849.72, "end": 1849.82, "word": " أن", "probability": 0.6357421875}, {"start": 1849.82, "end": 1850.22, "word": " أثبت", "probability": 0.9775390625}, 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"word": " صورتها", "probability": 0.9842529296875}, {"start": 2212.26, "end": 2216.86, "word": " المسافة", "probability": 0.9661865234375}, {"start": 2216.86, "end": 2217.24, "word": " بينها", "probability": 0.987548828125}, {"start": 2217.24, "end": 2217.4, "word": " و", "probability": 0.533203125}, {"start": 2217.4, "end": 2217.64, "word": " بين", "probability": 0.95166015625}, {"start": 2217.64, "end": 2217.86, "word": " ال", "probability": 0.9736328125}, {"start": 2217.86, "end": 2218.16, "word": " L", "probability": 0.7978515625}, {"start": 2218.16, "end": 2218.84, "word": " أصغر", "probability": 0.9822998046875}, {"start": 2218.84, "end": 2219.1, "word": " من", "probability": 0.9921875}, {"start": 2219.1, "end": 2219.64, "word": " إبسل", "probability": 0.64105224609375}, {"start": 2219.64, "end": 2221.84, "word": " صح؟", "probability": 0.8937174479166666}], "temperature": 1.0}, {"id": 80, "seek": 225006, "start": 2226.2, "end": 2250.06, "text": "إذا نجي نلخص كمان مرة، إيه اللي عملناها؟ أنا إيش بدأ في بتلقى limited sequence F of X N لما N تقول الـinfinity بالساوي عدد L فهذه البديات بإبسلون أكبر من السفر given أثبتت إن يوجد capital N عدد طبيعي يعتمد على الـK والـK تعتمد على إبسلون، إذا الـN هذه تعتمد على الـgiven إبسلون", "tokens": [28814, 15730, 8717, 7435, 1829, 8717, 1211, 9778, 9381, 9122, 2304, 7649, 3714, 25720, 12399, 11933, 1829, 3224, 13672, 1829, 6225, 42213, 8315, 11296, 22807, 41850, 11933, 1829, 8592, 47525, 10721, 8978, 39894, 1211, 4587, 7578, 5567, 8310, 479, 295, 1783, 426, 5296, 15042, 426, 6055, 39648, 2423, 39184, 259, 5194, 507, 20666, 3794, 995, 45865, 6225, 3215, 3215, 441, 6156, 3224, 24192, 29739, 16254, 9307, 4724, 28814, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 2212, 5551, 12984, 3555, 2655, 2655, 36145, 7251, 29245, 3215, 4238, 426, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 7251, 34268, 2304, 3215, 15844, 2423, 39184, 42, 16070, 39184, 42, 6055, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536, 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{"start": 2234.28, "end": 2234.56, "word": " تقول", "probability": 0.93359375}, {"start": 2234.56, "end": 2235.12, "word": " الـinfinity", "probability": 0.6229248046875}, {"start": 2235.12, "end": 2235.58, "word": " بالساوي", "probability": 0.66375732421875}, {"start": 2235.58, "end": 2235.94, "word": " عدد", "probability": 0.9811197916666666}, {"start": 2235.94, "end": 2236.26, "word": " L", "probability": 0.89013671875}, {"start": 2236.26, "end": 2237.34, "word": " فهذه", "probability": 0.7020670572916666}, {"start": 2237.34, "end": 2237.84, "word": " البديات", "probability": 0.7394205729166666}, {"start": 2237.84, "end": 2238.46, "word": " بإبسلون", "probability": 0.7781168619791666}, {"start": 2238.46, "end": 2238.88, "word": " أكبر", "probability": 0.98046875}, {"start": 2238.88, "end": 2239.08, "word": " من", "probability": 0.9921875}, {"start": 2239.08, "end": 2239.54, "word": " السفر", "probability": 0.8863932291666666}, {"start": 2239.54, "end": 2239.94, "word": " given", "probability": 0.71826171875}, {"start": 2239.94, "end": 2241.18, "word": " أثبتت", "probability": 0.98173828125}, {"start": 2241.18, "end": 2241.32, "word": " إن", "probability": 0.67626953125}, {"start": 2241.32, "end": 2241.76, "word": " يوجد", "probability": 0.962890625}, {"start": 2241.76, "end": 2242.24, "word": " capital", "probability": 0.7705078125}, {"start": 2242.24, "end": 2242.54, "word": " N", "probability": 0.96728515625}, {"start": 2242.54, "end": 2242.94, "word": " عدد", "probability": 0.9803059895833334}, {"start": 2242.94, "end": 2243.96, "word": " طبيعي", "probability": 0.9864501953125}, {"start": 2243.96, "end": 2244.6, "word": " يعتمد", "probability": 0.9061279296875}, {"start": 2244.6, "end": 2244.74, "word": " على", "probability": 0.837890625}, {"start": 2244.74, "end": 2245.08, "word": " الـK", "probability": 0.6607259114583334}, {"start": 2245.08, "end": 2245.58, "word": " والـK", "probability": 0.6874186197916666}, {"start": 2245.58, "end": 2246.16, "word": " تعتمد", "probability": 0.9195556640625}, {"start": 2246.16, "end": 2246.32, "word": " على", "probability": 0.92578125}, {"start": 2246.32, "end": 2247.46, "word": " إبسلون،", "probability": 0.9150390625}, {"start": 2247.46, "end": 2247.68, "word": " إذا", "probability": 0.811767578125}, {"start": 2247.68, "end": 2248.14, "word": " الـN", "probability": 0.9104817708333334}, {"start": 2248.14, "end": 2248.42, "word": " هذه", "probability": 0.431640625}, {"start": 2248.42, "end": 2249.0, "word": " تعتمد", "probability": 0.973876953125}, {"start": 2249.0, "end": 2249.14, "word": " على", "probability": 0.91259765625}, {"start": 2249.14, "end": 2249.56, "word": " الـgiven", "probability": 0.7470703125}, {"start": 2249.56, "end": 2250.06, "word": " إبسلون", "probability": 0.9482421875}], "temperature": 1.0}, {"id": 81, "seek": 227127, "start": 2251.55, "end": 2271.27, "text": "و هذه ال N لكل small n أكبر من أو ساوي ال capital N هذه طلع عند المسافة بين الحد النوني لل sequence و L أصغر من Epsilon بما أن Epsilon was arbitrary since Epsilon أكبر من السفر was arbitrary", "tokens": [2407, 29538, 2423, 426, 5296, 28820, 1359, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2423, 4238, 426, 29538, 23032, 1211, 3615, 43242, 9673, 3794, 31845, 3660, 49374, 21542, 3215, 28239, 11536, 1829, 24976, 8310, 4032, 441, 5551, 9381, 17082, 2288, 9154, 9970, 82, 15754, 4724, 15042, 14739, 9970, 82, 15754, 390, 23211, 1670, 9970, 82, 15754, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 390, 23211], "avg_logprob": -0.2851562561357723, "compression_ratio": 1.5084745762711864, "no_speech_prob": 0.0, "words": [{"start": 2251.55, "end": 2251.87, "word": "و", "probability": 0.82177734375}, {"start": 2251.87, "end": 2252.37, "word": " هذه", "probability": 0.1966552734375}, {"start": 2252.37, "end": 2252.55, "word": " ال", "probability": 0.767578125}, {"start": 2252.55, "end": 2252.83, "word": " N", "probability": 0.401123046875}, {"start": 2252.83, "end": 2253.81, "word": " لكل", 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"probability": 0.638671875}, {"start": 2258.05, "end": 2258.63, "word": " المسافة", "probability": 0.9591064453125}, {"start": 2258.63, "end": 2258.87, "word": " بين", "probability": 0.9345703125}, {"start": 2258.87, "end": 2259.31, "word": " الحد", "probability": 0.947998046875}, {"start": 2259.31, "end": 2259.83, "word": " النوني", "probability": 0.7820638020833334}, {"start": 2259.83, "end": 2259.99, "word": " لل", "probability": 0.8544921875}, {"start": 2259.99, "end": 2260.49, "word": " sequence", "probability": 0.8388671875}, {"start": 2260.49, "end": 2260.73, "word": " و", "probability": 0.3193359375}, {"start": 2260.73, "end": 2260.87, "word": " L", "probability": 0.69140625}, {"start": 2260.87, "end": 2261.17, "word": " أصغر", "probability": 0.9696044921875}, {"start": 2261.17, "end": 2261.35, "word": " من", "probability": 0.99462890625}, {"start": 2261.35, "end": 2261.79, "word": " Epsilon", "probability": 0.6659749348958334}, {"start": 2261.79, "end": 2263.11, "word": " بما", "probability": 0.88232421875}, {"start": 2263.11, "end": 2263.27, "word": " أن", "probability": 0.7939453125}, {"start": 2263.27, "end": 2263.71, "word": " Epsilon", "probability": 0.8956705729166666}, {"start": 2263.71, "end": 2263.97, "word": " was", "probability": 0.74169921875}, {"start": 2263.97, "end": 2264.53, "word": " arbitrary", "probability": 0.814453125}, {"start": 2264.53, "end": 2265.35, "word": " since", "probability": 0.7109375}, {"start": 2265.35, "end": 2267.75, "word": " Epsilon", "probability": 0.7762044270833334}, {"start": 2267.75, "end": 2268.25, "word": " أكبر", "probability": 0.9720052083333334}, {"start": 2268.25, "end": 2268.47, "word": " من", "probability": 0.99609375}, {"start": 2268.47, "end": 2269.13, "word": " السفر", "probability": 0.754150390625}, {"start": 2269.13, "end": 2269.99, "word": " was", "probability": 0.890625}, {"start": 2269.99, "end": 2271.27, "word": " arbitrary", "probability": 0.6259765625}], "temperature": 1.0}, {"id": 82, "seek": 230083, "start": 2272.23, "end": 2300.83, "text": "إذا by epsilon capital N definition of limit of sequence بنكون هيك حسب التعريف أثبتنا أنه limit لسيكوينس F of X N as N tends to infinity بالساوي لعدد N وبالتالي هيك بنكون أثبتنا إذا العبارة 2 holds وهيك بنكون أثبتنا أن العبارة statement 1 implies statement 2", "tokens": [28814, 15730, 538, 17889, 4238, 426, 7123, 295, 4948, 295, 8310, 44945, 30544, 39896, 4117, 11331, 35457, 16712, 3615, 16572, 5172, 5551, 12984, 3555, 2655, 8315, 14739, 3224, 4948, 5296, 3794, 1829, 4117, 2407, 9957, 3794, 479, 295, 1783, 426, 382, 426, 12258, 281, 13202, 20666, 3794, 995, 45865, 5296, 22488, 3215, 426, 46599, 6027, 2655, 6027, 1829, 39896, 4117, 44945, 30544, 5551, 12984, 3555, 2655, 8315, 11933, 15730, 18863, 3555, 9640, 3660, 568, 9190, 37037, 1829, 4117, 44945, 30544, 5551, 12984, 3555, 2655, 8315, 14739, 18863, 3555, 9640, 3660, 5629, 502, 18779, 5629, 568], "avg_logprob": -0.2547200536355376, "compression_ratio": 1.635135135135135, 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original original sequential criterion original sequential criterion مع", "tokens": [271, 2531, 281, 3380, 8177, 420, 8177, 295, 3380, 3380, 42881, 46691, 3380, 42881, 46691, 20449], "avg_logprob": -0.35500919117647056, "compression_ratio": 1.6521739130434783, "no_speech_prob": 0.0, "words": [{"start": 2334.03, "end": 2334.37, "word": "is", "probability": 0.42626953125}, {"start": 2334.37, "end": 2334.95, "word": " similar", "probability": 0.904296875}, {"start": 2334.95, "end": 2338.01, "word": " to", "probability": 0.94921875}, {"start": 2338.01, "end": 2343.41, "word": " original", "probability": 0.80517578125}, {"start": 2343.41, "end": 2344.03, "word": " proof", "probability": 0.91552734375}, {"start": 2344.03, "end": 2344.75, "word": " or", "probability": 0.1600341796875}, {"start": 2344.75, "end": 2345.35, "word": " proof", "probability": 0.63525390625}, {"start": 2345.35, "end": 2348.69, "word": " of", "probability": 0.9541015625}, {"start": 2348.69, "end": 2350.05, "word": " 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"probability": 0.82421875}, {"start": 2732.29, "end": 2733.19, "word": " okay", "probability": 0.2271728515625}, {"start": 2733.19, "end": 2734.21, "word": " إذن", "probability": 0.7869466145833334}, {"start": 2734.21, "end": 2734.51, "word": " هذا", "probability": 0.8671875}, {"start": 2734.51, "end": 2735.11, "word": " معناه", "probability": 0.990234375}, {"start": 2735.11, "end": 2735.31, "word": " إن", "probability": 0.58349609375}, {"start": 2735.31, "end": 2735.49, "word": " ال", "probability": 0.486328125}, {"start": 2735.49, "end": 2735.85, "word": " limit", "probability": 0.947265625}, {"start": 2735.85, "end": 2737.25, "word": " حسب", "probability": 0.9814453125}, {"start": 2737.25, "end": 2737.97, "word": " التعريف", "probability": 0.992919921875}, {"start": 2737.97, "end": 2738.83, "word": " limit", "probability": 0.9697265625}, {"start": 2738.83, "end": 2739.43, "word": " واحد", "probability": 0.7178955078125}, {"start": 2739.43, "end": 2739.63, "word": " على", "probability": 0.84765625}, {"start": 2739.63, "end": 2740.05, "word": " x", "probability": 0.93310546875}, {"start": 2740.05, "end": 2740.39, "word": " لما", "probability": 0.89990234375}, {"start": 2740.39, "end": 2740.69, "word": " x", "probability": 0.97119140625}, {"start": 2740.69, "end": 2741.23, "word": " تقول", "probability": 0.6727294921875}, {"start": 2741.23, "end": 2742.63, "word": " إلى", "probability": 0.82666015625}, {"start": 2742.63, "end": 2743.41, "word": " infinity", "probability": 0.86181640625}, {"start": 2743.41, "end": 2745.47, "word": " بساوي", "probability": 0.83349609375}, {"start": 2745.47, "end": 2746.03, "word": " سفر", "probability": 0.8912760416666666}], "temperature": 1.0}, {"id": 100, "seek": 277628, "start": 2749.02, "end": 2776.28, "text": "بالمثل ممكن نثبت الجزء التاني نفس البرهان مع التعديل في تعريف limit at سالب infinity ممكن برضه نستخدم sequential criterion لو بدأت استخدم sequential criterion لإثبات limit", "tokens": [3555, 45340, 12984, 1211, 3714, 43020, 8717, 12984, 3555, 2655, 25724, 11622, 38207, 16712, 7649, 1829, 8717, 36178, 2423, 26890, 3224, 7649, 20449, 16712, 3615, 16254, 1211, 8978, 37279, 16572, 5172, 4948, 412, 8608, 6027, 3555, 13202, 3714, 43020, 4724, 43042, 3224, 8717, 14851, 9778, 40448, 42881, 46691, 45164, 47525, 10721, 2655, 44713, 9778, 40448, 42881, 46691, 5296, 28814, 12984, 3555, 9307, 4948], "avg_logprob": -0.19372558034956455, "compression_ratio": 1.5542168674698795, "no_speech_prob": 0.0, "words": [{"start": 2749.02, "end": 2749.68, "word": "بالمثل", "probability": 0.8616943359375}, {"start": 2749.68, "end": 2750.04, "word": " ممكن", "probability": 0.8330078125}, {"start": 2750.04, "end": 2750.62, "word": " نثبت", "probability": 0.9652099609375}, {"start": 2750.62, "end": 2751.32, "word": " الجزء", "probability": 0.9586588541666666}, {"start": 2751.32, "end": 2752.12, "word": " التاني", "probability": 0.9186197916666666}, {"start": 2752.12, "end": 2753.12, "word": " نفس", "probability": 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limit f of xn has n times infinity", "tokens": [3555, 10721, 9778, 3215, 4724, 39648, 16472, 34105, 718, 2031, 77, 312, 8310, 16212, 294, 1958, 4032, 13202, 4724, 5016, 1829, 12984, 16472, 3224, 4948, 2031, 77, 6055, 3794, 995, 45865, 13202, 1975, 15730, 4948, 283, 295, 2031, 77, 575, 297, 1413, 13202], "avg_logprob": -0.3469460206952962, "compression_ratio": 1.2366412213740459, "no_speech_prob": 0.0, "words": [{"start": 2778.67, "end": 2779.81, "word": "بأخد", "probability": 0.68646240234375}, {"start": 2779.81, "end": 2781.03, "word": " بقول", "probability": 0.70458984375}, {"start": 2781.03, "end": 2781.67, "word": " ان", "probability": 0.1461181640625}, {"start": 2781.67, "end": 2782.01, "word": " هنا", "probability": 0.9453125}, {"start": 2782.01, "end": 2783.49, "word": " let", "probability": 0.69091796875}, {"start": 2783.49, "end": 2784.31, "word": " xn", "probability": 0.492919921875}, {"start": 2784.31, "end": 2784.93, "word": " be", "probability": 0.3876953125}, {"start": 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{"start": 2795.25, "end": 2799.57, "word": " اذا", "probability": 0.54168701171875}, {"start": 2799.57, "end": 2800.15, "word": " limit", "probability": 0.8896484375}, {"start": 2800.15, "end": 2801.91, "word": " f", "probability": 0.82080078125}, {"start": 2801.91, "end": 2802.27, "word": " of", "probability": 0.5205078125}, {"start": 2802.27, "end": 2803.29, "word": " xn", "probability": 0.960693359375}, {"start": 2803.29, "end": 2804.59, "word": " has", "probability": 0.77490234375}, {"start": 2804.59, "end": 2804.93, "word": " n", "probability": 0.76025390625}, {"start": 2804.93, "end": 2805.21, "word": " times", "probability": 0.4267578125}, {"start": 2805.21, "end": 2805.87, "word": " infinity", "probability": 0.91796875}], "temperature": 1.0}, {"id": 102, "seek": 282622, "start": 2807.04, "end": 2826.22, "text": "طبعا هذا بيقدي هذا بيقدي انه limit واحد على xn بساوي سفر exercise أخدناها أخدنا انه limit sequence xn بساوي infinity if and only if limit مقلوب السيكوانس بساوي سفر", 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"end": 2812.2, "word": " انه", "probability": 0.764404296875}, {"start": 2812.2, "end": 2812.62, "word": " limit", "probability": 0.94873046875}, {"start": 2812.62, "end": 2814.38, "word": " واحد", "probability": 0.68017578125}, {"start": 2814.38, "end": 2814.6, "word": " على", "probability": 0.56201171875}, {"start": 2814.6, "end": 2815.46, "word": " xn", "probability": 0.455322265625}, {"start": 2815.46, "end": 2816.24, "word": " بساوي", "probability": 0.8157958984375}, {"start": 2816.24, "end": 2816.84, "word": " سفر", "probability": 0.8741861979166666}, {"start": 2816.84, "end": 2818.32, "word": " exercise", "probability": 0.328125}, {"start": 2818.32, "end": 2819.14, "word": " أخدناها", "probability": 0.815625}, {"start": 2819.14, "end": 2820.4, "word": " أخدنا", "probability": 0.797607421875}, {"start": 2820.4, "end": 2820.64, "word": " انه", "probability": 0.794189453125}, {"start": 2820.64, "end": 2820.82, "word": " limit", "probability": 0.974609375}, {"start": 2820.82, "end": 2821.26, "word": " sequence", "probability": 0.9736328125}, {"start": 2821.26, "end": 2821.68, "word": " xn", "probability": 0.951171875}, {"start": 2821.68, "end": 2822.1, "word": " بساوي", "probability": 0.8575439453125}, {"start": 2822.1, "end": 2822.48, "word": " infinity", "probability": 0.8466796875}, {"start": 2822.48, "end": 2822.8, "word": " if", "probability": 0.86181640625}, {"start": 2822.8, "end": 2823.02, "word": " and", "probability": 0.943359375}, {"start": 2823.02, "end": 2823.26, "word": " only", "probability": 0.9150390625}, {"start": 2823.26, "end": 2823.54, "word": " if", "probability": 0.984375}, {"start": 2823.54, "end": 2824.26, "word": " limit", "probability": 0.9599609375}, {"start": 2824.26, "end": 2824.74, "word": " مقلوب", "probability": 0.9410400390625}, {"start": 2824.74, "end": 2825.42, "word": " السيكوانس", "probability": 0.7608235677083334}, {"start": 2825.42, "end": 2825.84, "word": " بساوي", "probability": 0.931884765625}, {"start": 2825.84, "end": 2826.22, "word": " سفر", "probability": 0.9822591145833334}], "temperature": 1.0}, {"id": 103, "seek": 284733, "start": 2826.51, "end": 2847.33, "text": "الان limit f of xn بساوي limit واحد على xn as n tends to infinity وهذا بيساوي ستة لأي sequence نهايتها infinity نهاية سورتها بيساوي العدد L", "tokens": [6027, 7649, 4948, 283, 295, 2031, 77, 4724, 3794, 995, 45865, 4948, 36764, 24401, 15844, 2031, 77, 382, 297, 12258, 281, 13202, 37037, 15730, 4724, 1829, 3794, 995, 45865, 8608, 2655, 3660, 5296, 10721, 1829, 8310, 8717, 11296, 36081, 11296, 13202, 8717, 11296, 10632, 8608, 13063, 2655, 11296, 4724, 1829, 3794, 995, 45865, 18863, 3215, 3215, 441], "avg_logprob": -0.35183190169005557, "compression_ratio": 1.3986013986013985, "no_speech_prob": 0.0, "words": [{"start": 2826.51, "end": 2826.99, "word": "الان", "probability": 0.52880859375}, {"start": 2826.99, "end": 2827.31, "word": " limit", "probability": 0.79345703125}, {"start": 2827.31, "end": 2827.57, "word": " f", "probability": 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{"start": 2834.27, "end": 2835.95, "word": " وهذا", "probability": 0.85986328125}, {"start": 2835.95, "end": 2836.33, "word": " بيساوي", "probability": 0.843115234375}, {"start": 2836.33, "end": 2836.73, "word": " ستة", "probability": 0.44482421875}, {"start": 2836.73, "end": 2841.15, "word": " لأي", "probability": 0.4368896484375}, {"start": 2841.15, "end": 2841.85, "word": " sequence", "probability": 0.92724609375}, {"start": 2841.85, "end": 2842.77, "word": " نهايتها", "probability": 0.869140625}, {"start": 2842.77, "end": 2843.35, "word": " infinity", "probability": 0.8564453125}, {"start": 2843.35, "end": 2844.91, "word": " نهاية", "probability": 0.6475423177083334}, {"start": 2844.91, "end": 2845.59, "word": " سورتها", "probability": 0.845947265625}, {"start": 2845.59, "end": 2846.51, "word": " بيساوي", "probability": 0.9099609375}, {"start": 2846.51, "end": 2846.99, "word": " العدد", "probability": 0.9417317708333334}, {"start": 2846.99, "end": 2847.33, "word": " L", "probability": 0.6552734375}], "temperature": 1.0}, {"id": 104, "seek": 287616, "start": 2855.72, "end": 2876.16, "text": "بنطلع ال limit ل ال function f of x as x tends to infinity بساوية 0 إذا هذا برهان تاني using sequential criterion okay واضح مفهوم مثال تاني", "tokens": [3555, 1863, 9566, 1211, 3615, 2423, 4948, 5296, 2423, 2445, 283, 295, 2031, 382, 2031, 12258, 281, 13202, 4724, 3794, 995, 2407, 10632, 1958, 11933, 15730, 23758, 4724, 2288, 3224, 7649, 6055, 7649, 1829, 1228, 42881, 46691, 1392, 4032, 46958, 5016, 3714, 5172, 3224, 20498, 50113, 6027, 6055, 7649, 1829], "avg_logprob": -0.43351714517555984, "compression_ratio": 1.230263157894737, "no_speech_prob": 0.0, "words": [{"start": 2855.7200000000003, "end": 2856.8, "word": "بنطلع", "probability": 0.55787353515625}, {"start": 2856.8, "end": 2856.94, "word": " ال", "probability": 0.277587890625}, {"start": 2856.94, "end": 2857.3, "word": " limit", "probability": 0.728515625}, {"start": 2857.3, "end": 2858.44, "word": " ل", "probability": 0.70263671875}, {"start": 2858.44, "end": 2858.76, "word": " ال", "probability": 0.1319580078125}, {"start": 2858.76, "end": 2859.24, "word": " function", "probability": 0.7578125}, {"start": 2859.24, "end": 2859.62, "word": " f", "probability": 0.7490234375}, {"start": 2859.62, "end": 2859.9, "word": " of", "probability": 0.330810546875}, {"start": 2859.9, "end": 2860.28, "word": " x", "probability": 0.84130859375}, {"start": 2860.28, "end": 2861.18, "word": " as", "probability": 0.6123046875}, {"start": 2861.18, "end": 2861.54, "word": " x", "probability": 0.86328125}, {"start": 2861.54, "end": 2861.88, "word": " tends", "probability": 0.6298828125}, {"start": 2861.88, "end": 2862.0, "word": " to", "probability": 0.8759765625}, {"start": 2862.0, "end": 2862.62, "word": " infinity", "probability": 0.86279296875}, {"start": 2862.62, "end": 2863.52, "word": " بساوية", "probability": 0.68056640625}, {"start": 2863.52, "end": 2864.06, "word": " 0", "probability": 0.294921875}, {"start": 2864.06, "end": 2864.98, "word": " إذا", "probability": 0.345672607421875}, {"start": 2864.98, "end": 2865.24, "word": " هذا", "probability": 0.233154296875}, {"start": 2865.24, "end": 2865.7, "word": " برهان", "probability": 0.93212890625}, {"start": 2865.7, "end": 2866.28, "word": " تاني", "probability": 0.96435546875}, {"start": 2866.28, "end": 2867.2, "word": " using", "probability": 0.86376953125}, {"start": 2867.2, "end": 2868.04, "word": " sequential", "probability": 0.9462890625}, {"start": 2868.04, "end": 2868.86, "word": " criterion", "probability": 0.7314453125}, {"start": 2868.86, "end": 2869.72, "word": " okay", "probability": 0.2235107421875}, {"start": 2869.72, "end": 2870.26, "word": " واضح", "probability": 0.97900390625}, {"start": 2870.26, "end": 2871.54, "word": " مفهوم", "probability": 0.883056640625}, {"start": 2871.54, "end": 2875.62, "word": " مثال", "probability": 0.8408203125}, {"start": 2875.62, "end": 2876.16, "word": " تاني", 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0.87158203125}, {"start": 2898.56, "end": 2898.78, "word": " x", "probability": 0.5986328125}, {"start": 2898.78, "end": 2899.02, "word": " تقول", "probability": 0.6388346354166666}, {"start": 2899.02, "end": 2899.16, "word": " ل", "probability": 0.474365234375}, {"start": 2899.16, "end": 2899.62, "word": " infinity", "probability": 0.576171875}, {"start": 2899.62, "end": 2901.04, "word": " و", "probability": 0.4970703125}, {"start": 2901.04, "end": 2901.38, "word": " لما", "probability": 0.82275390625}, {"start": 2901.38, "end": 2901.66, "word": " x", "probability": 0.96337890625}, {"start": 2901.66, "end": 2901.92, "word": " تقول", "probability": 0.7779947916666666}, {"start": 2901.92, "end": 2902.06, "word": " ل", "probability": 0.96728515625}, {"start": 2902.06, "end": 2902.42, "word": " سالب", "probability": 0.635498046875}, {"start": 2902.42, "end": 2902.94, "word": " infinity", "probability": 0.908203125}, {"start": 2902.94, "end": 2903.86, "word": " بساوي", "probability": 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"start": 2917.28, "end": 2939.02, "text": "هذا كان limit xn بالساوية infinity فlimit 1 على xn بالساوية infinity بساوية سفر وبالتالي limit 1 على xn تربية يعني هذا بيقدر وهذا بيقدر ان limit", "tokens": [3224, 15730, 25961, 4948, 2031, 77, 20666, 3794, 995, 2407, 10632, 13202, 6156, 4197, 270, 502, 15844, 2031, 77, 20666, 3794, 995, 2407, 10632, 13202, 4724, 3794, 995, 2407, 10632, 8608, 5172, 2288, 46599, 6027, 2655, 6027, 1829, 4948, 502, 15844, 2031, 77, 6055, 25513, 10632, 37495, 22653, 23758, 4724, 1829, 28543, 2288, 37037, 15730, 4724, 1829, 28543, 2288, 16472, 4948], "avg_logprob": -0.3019153365204411, "compression_ratio": 1.8016528925619835, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2917.28, "end": 2918.28, "word": "هذا", "probability": 0.42205810546875}, {"start": 2918.28, "end": 2918.62, "word": " كان", "probability": 0.58740234375}, {"start": 2918.62, "end": 2918.9, "word": " limit", "probability": 0.7783203125}, {"start": 2918.9, "end": 2919.56, "word": " 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"probability": 0.96435546875}], "temperature": 1.0}, {"id": 115, "seek": 317529, "start": 3147.03, "end": 3175.29, "text": "و بالمثل ممكن نعطي براهين زي هذا او زي هذا اما باستخدام squeeze theorem او sequential criterion او حتى definition okay واضح في اي سؤال في اي استفسار okay هنوقف اذا هنا و المرة الجاية في بعض انواع ال infinite limits هنتكلم عنهم يعني باختصار", "tokens": [2407, 20666, 2304, 12984, 1211, 3714, 43020, 8717, 3615, 9566, 1829, 4724, 23557, 3224, 9957, 30767, 1829, 23758, 1975, 2407, 30767, 1829, 23758, 1975, 15042, 4724, 995, 14851, 9778, 3215, 10943, 13578, 20904, 1975, 2407, 42881, 46691, 1975, 2407, 11331, 49975, 7123, 1392, 4032, 46958, 5016, 8978, 1975, 1829, 8608, 33604, 6027, 8978, 1975, 1829, 44713, 36178, 9640, 1392, 8032, 1863, 30543, 5172, 1975, 15730, 34105, 4032, 9673, 25720, 25724, 995, 10632, 8978, 45030, 11242, 16472, 14407, 3615, 2423, 13785, 10406, 8032, 29399, 4117, 19528, 18871, 16095, 37495, 22653, 4724, 47283, 2655, 9381, 9640], "avg_logprob": -0.14917763314749066, "compression_ratio": 1.5905172413793103, "no_speech_prob": 0.0, "words": [{"start": 3147.03, "end": 3147.39, "word": "و", "probability": 0.59814453125}, {"start": 3147.39, "end": 3147.99, "word": " بالمثل", "probability": 0.9517822265625}, {"start": 3147.99, "end": 3148.33, "word": " ممكن", "probability": 0.988525390625}, {"start": 3148.33, "end": 3148.71, "word": " نعطي", "probability": 0.9705810546875}, {"start": 3148.71, "end": 3149.35, "word": " براهين", "probability": 0.879638671875}, {"start": 3149.35, "end": 3150.53, "word": " زي", "probability": 0.834228515625}, {"start": 3150.53, "end": 3150.99, "word": " هذا", "probability": 0.67529296875}, {"start": 3150.99, "end": 3152.25, "word": " او", "probability": 0.77099609375}, {"start": 3152.25, "end": 3152.49, "word": " زي", "probability": 0.940673828125}, {"start": 3152.49, "end": 3152.93, "word": " هذا", "probability": 0.9404296875}, {"start": 3152.93, "end": 3153.51, "word": " اما", "probability": 0.867919921875}, {"start": 3153.51, "end": 3154.07, "word": " باستخدام", "probability": 0.9816080729166666}, {"start": 3154.07, "end": 3155.03, "word": " squeeze", "probability": 0.43212890625}, {"start": 3155.03, "end": 3155.47, "word": " theorem", "probability": 0.31689453125}, {"start": 3155.47, "end": 3155.83, "word": " او", "probability": 0.954345703125}, {"start": 3155.83, "end": 3156.29, "word": " sequential", "probability": 0.93017578125}, {"start": 3156.29, "end": 3156.77, "word": " criterion", "probability": 0.92236328125}, {"start": 3156.77, "end": 3157.75, "word": " او", "probability": 0.967529296875}, {"start": 3157.75, "end": 3158.01, "word": " حتى", "probability": 0.9453125}, {"start": 3158.01, "end": 3158.51, "word": " definition", "probability": 0.79443359375}, {"start": 3158.51, "end": 3161.47, "word": " okay", "probability": 0.53759765625}, {"start": 3161.47, "end": 3162.03, "word": " واضح", "probability": 0.9397786458333334}, {"start": 3162.03, "end": 3163.25, "word": " في", "probability": 0.38330078125}, {"start": 3163.25, "end": 3163.39, "word": " اي", "probability": 0.7017822265625}, {"start": 3163.39, "end": 3163.79, "word": " سؤال", "probability": 0.9352213541666666}, {"start": 3163.79, "end": 3163.99, "word": " في", "probability": 0.83251953125}, {"start": 3163.99, "end": 3164.15, "word": " اي", "probability": 0.898193359375}, {"start": 3164.15, "end": 3164.71, "word": " استفسار", "probability": 0.92431640625}, {"start": 3164.71, "end": 3165.81, "word": " okay", "probability": 0.845703125}, {"start": 3165.81, "end": 3166.47, "word": " هنوقف", "probability": 0.79254150390625}, {"start": 3166.47, "end": 3166.71, "word": " اذا", "probability": 0.849365234375}, {"start": 3166.71, "end": 3167.25, "word": " هنا", "probability": 0.990234375}, {"start": 3167.25, "end": 3168.11, "word": " و", "probability": 0.84326171875}, {"start": 3168.11, "end": 3168.47, "word": " المرة", "probability": 0.834716796875}, {"start": 3168.47, "end": 3168.83, "word": " الجاية", "probability": 0.8597005208333334}, {"start": 3168.83, "end": 3169.05, "word": " في", "probability": 0.888671875}, {"start": 3169.05, "end": 3169.37, "word": " بعض", "probability": 0.961181640625}, {"start": 3169.37, "end": 3169.75, "word": " انواع", "probability": 0.8929036458333334}, {"start": 3169.75, "end": 3169.97, "word": " ال", "probability": 0.7060546875}, {"start": 3169.97, "end": 3170.31, "word": " infinite", "probability": 0.876953125}, {"start": 3170.31, "end": 3170.81, "word": " limits", "probability": 0.9208984375}, {"start": 3170.81, "end": 3172.11, "word": " هنتكلم", "probability": 0.9569091796875}, {"start": 3172.11, "end": 3172.59, "word": " عنهم", "probability": 0.972900390625}, {"start": 3172.59, "end": 3173.19, "word": " يعني", "probability": 0.90625}, {"start": 3173.19, "end": 3175.29, "word": " باختصار", "probability": 0.98837890625}], "temperature": 1.0}, {"id": 116, "seek": 320004, "start": 3175.96, "end": 3200.04, "text": "و بعدين نحاول نجمل section اربعة تلاتة وبالتالي ننهي ال chapter اللي هو chapter اربعة و بعدين نبدأ في chapter خمسة اللي هو اخر chapter في المخرج هو اهم chapter طبعا في حد عنده اي سؤال او استفسار؟ شكرا لاصغاكم و نشوف ان شاء الله المرة القادمة", "tokens": [2407, 39182, 9957, 8717, 5016, 995, 12610, 8717, 7435, 42213, 3541, 1975, 25513, 27884, 6055, 1211, 9307, 3660, 46599, 6027, 2655, 6027, 1829, 8717, 1863, 3224, 1829, 2423, 7187, 13672, 1829, 31439, 7187, 1975, 25513, 27884, 4032, 39182, 9957, 8717, 44510, 10721, 8978, 7187, 16490, 2304, 3794, 3660, 13672, 1829, 31439, 1975, 34740, 7187, 8978, 9673, 34740, 7435, 31439, 1975, 16095, 7187, 23032, 3555, 3615, 995, 8978, 11331, 3215, 43242, 3224, 1975, 1829, 8608, 33604, 6027, 1975, 2407, 44713, 36178, 9640, 22807, 13412, 4117, 23557, 5296, 33546, 17082, 995, 24793, 4032, 8717, 8592, 38688, 16472, 13412, 16606, 21984, 9673, 25720, 25062, 18513, 46007], "avg_logprob": -0.17518029533899748, "compression_ratio": 1.841121495327103, "no_speech_prob": 2.2649765014648438e-06, "words": [{"start": 3175.96, "end": 3176.18, "word": "و", "probability": 0.7724609375}, {"start": 3176.18, "end": 3176.54, "word": " بعدين", "probability": 0.952880859375}, {"start": 3176.54, "end": 3177.0, "word": " نحاول", "probability": 0.9254150390625}, {"start": 3177.0, "end": 3177.5, "word": " نجمل", "probability": 0.9759114583333334}, {"start": 3177.5, "end": 3178.78, "word": " section", "probability": 0.8095703125}, {"start": 3178.78, "end": 3179.16, "word": " اربعة", "probability": 0.7078450520833334}, {"start": 3179.16, "end": 3179.64, "word": " تلاتة", "probability": 0.9029541015625}, {"start": 3179.64, "end": 3180.16, "word": " وبالتالي", "probability": 0.864453125}, {"start": 3180.16, "end": 3180.52, "word": " ننهي", "probability": 0.9715576171875}, {"start": 3180.52, "end": 3180.64, "word": " ال", "probability": 0.66064453125}, {"start": 3180.64, "end": 3180.98, "word": " chapter", "probability": 0.97021484375}, {"start": 3180.98, "end": 3182.34, "word": " اللي", "probability": 0.848388671875}, {"start": 3182.34, "end": 3182.62, "word": " هو", "probability": 0.98974609375}, {"start": 3182.62, "end": 3183.16, "word": " chapter", "probability": 0.93115234375}, {"start": 3183.16, "end": 3183.72, "word": " اربعة", "probability": 0.8424479166666666}, {"start": 3183.72, "end": 3184.1, "word": " و", "probability": 0.82373046875}, {"start": 3184.1, "end": 3184.5, "word": " بعدين", "probability": 0.931640625}, {"start": 3184.5, "end": 3184.98, "word": " نبدأ", "probability": 0.9661458333333334}, {"start": 3184.98, "end": 3185.16, "word": " في", "probability": 0.86767578125}, {"start": 3185.16, "end": 3185.62, "word": " chapter", "probability": 0.94482421875}, {"start": 3185.62, "end": 3186.7, "word": " خمسة", "probability": 0.8272705078125}, {"start": 3186.7, "end": 3186.94, "word": " اللي", "probability": 0.98779296875}, {"start": 3186.94, "end": 3187.16, "word": " هو", "probability": 0.990234375}, {"start": 3187.16, "end": 3187.54, "word": " اخر", "probability": 0.930419921875}, {"start": 3187.54, "end": 3188.04, "word": " chapter", "probability": 0.953125}, {"start": 3188.04, "end": 3188.62, "word": " في", "probability": 0.90185546875}, {"start": 3188.62, "end": 3189.26, "word": " المخرج", "probability": 0.84912109375}, {"start": 3189.26, "end": 3190.42, "word": " هو", "probability": 0.73974609375}, {"start": 3190.42, "end": 3190.84, "word": " اهم", "probability": 0.943359375}, {"start": 3190.84, "end": 3191.28, "word": " chapter", "probability": 0.9501953125}, {"start": 3191.28, "end": 3192.18, "word": " طبعا", "probability": 0.997802734375}, {"start": 3192.18, "end": 3194.16, "word": " في", "probability": 0.38330078125}, {"start": 3194.16, "end": 3194.42, "word": " حد", "probability": 0.989501953125}, {"start": 3194.42, "end": 3194.66, "word": " عنده", "probability": 0.93896484375}, {"start": 3194.66, "end": 3194.82, "word": " اي", "probability": 0.91259765625}, {"start": 3194.82, "end": 3195.1, "word": " سؤال", "probability": 0.94287109375}, {"start": 3195.1, "end": 3195.24, "word": " او", "probability": 0.941162109375}, {"start": 3195.24, "end": 3196.14, "word": " استفسار؟", "probability": 0.77703857421875}, {"start": 3196.14, "end": 3197.42, "word": " شكرا", "probability": 0.9606119791666666}, {"start": 3197.42, "end": 3198.22, "word": " لاصغاكم", "probability": 0.66826171875}, {"start": 3198.22, "end": 3198.48, "word": " و", "probability": 0.98583984375}, {"start": 3198.48, "end": 3198.84, "word": " نشوف", "probability": 0.91455078125}, {"start": 3198.84, "end": 3199.04, "word": " ان", "probability": 0.708984375}, {"start": 3199.04, "end": 3199.3, "word": " شاء", "probability": 0.978515625}, {"start": 3199.3, "end": 3199.34, "word": " الله", "probability": 0.9619140625}, {"start": 3199.34, "end": 3199.64, "word": " المرة", "probability": 0.32098388671875}, {"start": 3199.64, "end": 3200.04, "word": " القادمة", "probability": 0.7115071614583334}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3201.846, "duration_after_vad": 2939.860624999994}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e392169aa8fbf148207c1fee4d6f95d22657cd3d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/aBB9DzMBL4Y_raw.srt
@@ -0,0 +1,1976 @@
+1
+00:00:21,750 --> 00:00:27,870
+بسم الله الرحمن الرحيم في محاضرة اليوم هنكمل ما
+
+2
+00:00:27,870 --> 00:00:32,170
+بدأنا حاول
+
+3
+00:00:32,170 --> 00:00:37,890
+موضوع ال infinite limits المرة تاعة عرفنا ما معناه
+
+4
+00:00:37,890 --> 00:00:42,370
+ان ال limit ل function and cluster point بالساوية
+
+5
+00:00:42,370 --> 00:00:49,050
+plus او minus infinity وشوفنا نظرية اخر نظرية
+
+6
+00:00:49,050 --> 00:00:54,800
+برهنهاخاصة بهذا النوع من ال limits كانت النظرية
+
+7
+00:00:54,800 --> 00:01:05,340
+التالية خلينا نكتبها let
+
+8
+00:01:05,340 --> 00:01:14,420
+f و g be functions from a to r و c ال cluster
+
+9
+00:01:14,420 --> 00:01:18,320
+point
+
+10
+00:01:20,530 --> 00:01:28,690
+of the set A such that f
+
+11
+00:01:28,690 --> 00:01:34,590
+of x less than or equal g of x for every x تمتمي
+
+12
+00:01:34,590 --> 00:01:42,630
+إلى a different from c فشوفنا أنه لو كان ال limit
+
+13
+00:01:42,630 --> 00:01:45,950
+فf
+
+14
+00:01:45,950 --> 00:01:48,190
+of x as x tends
+
+15
+00:02:05,090 --> 00:02:07,390
+وكذلك لو كانت ال limit
+
+16
+00:02:10,530 --> 00:02:17,930
+لـ u of x as x tends to c بساوي negative infinity
+
+17
+00:02:17,930 --> 00:02:26,810
+فهذا بتضمن ان limit ل f of x as x tends to c بساوي
+
+18
+00:02:26,810 --> 00:02:35,730
+negative infinity طيب
+
+19
+00:02:35,730 --> 00:02:36,350
+ال ..
+
+20
+00:02:41,050 --> 00:02:47,350
+اليوم هنعرف ما معناه ان ال limit and c من اليمين
+
+21
+00:02:47,350 --> 00:02:52,030
+بالساوي infinity او ال limit and c من اليسار
+
+22
+00:02:52,030 --> 00:02:55,990
+بالساوي infinity و كذلك نفس الشيء ال one sided
+
+23
+00:02:55,990 --> 00:03:02,030
+limit and c ما معناه أنها ساوي سالب infinity لأن
+
+24
+00:03:02,030 --> 00:03:07,010
+هذه كانت two sided limit المرة الأخيرة اتعرفناما
+
+25
+00:03:07,010 --> 00:03:10,410
+معناه ان ال two sided limit تكون infinite او ساوي
+
+26
+00:03:10,410 --> 00:03:14,430
+infinity او plus او minus infinity اليوم ما معناه
+
+27
+00:03:14,430 --> 00:03:18,370
+ان ال one sided limit تكون infinity او negative
+
+28
+00:03:18,370 --> 00:03:23,970
+infinity فناخد التعريف مشابه
+
+29
+00:03:23,970 --> 00:03:29,050
+للتعريف التعريف ال one sided limit تكون بتساوي
+
+30
+00:03:29,050 --> 00:03:33,590
+real number ف
+
+31
+00:03:33,590 --> 00:03:44,120
+letfd function from a to r و
+
+32
+00:03:44,120 --> 00:03:49,240
+c cluster point
+
+33
+00:03:49,240 --> 00:03:58,760
+of الست اللي هي a تقاطع الفترة المفتوحة من c لما
+
+34
+00:03:58,760 --> 00:03:59,640
+إلى نهاية
+
+35
+00:04:04,430 --> 00:04:10,290
+يقول إن الـ limit لـ
+
+36
+00:04:10,290 --> 00:04:15,530
+function f of x as x tends to c from the right
+
+37
+00:04:15,530 --> 00:04:23,110
+بالساوي infinity respectively
+
+38
+00:04:23,110 --> 00:04:30,370
+على التوالي بنقول إن ال limit ل f of x لما x تقول
+
+39
+00:04:30,370 --> 00:04:39,170
+إلى c من اليمينبتساوي negative infinity إذا
+
+40
+00:04:39,170 --> 00:04:44,790
+تحقق الشرط التالي لأي
+
+41
+00:04:44,790 --> 00:04:51,950
+Alpha for any Alpha
+
+42
+00:04:51,950 --> 00:04:57,450
+belonging to R نقدر
+
+43
+00:04:57,450 --> 00:05:07,810
+نلاقي Delta تعتمد على Alphaعلى دموجة بحيث انه لكل
+
+44
+00:05:07,810 --> 00:05:17,050
+x ينتمي إلى a و ال x على يمين ال c و المسافة بينها
+
+45
+00:05:17,050 --> 00:05:22,730
+و بين ال c أصغر من دلتا فلازم هذا يضمن انه f of x
+
+46
+00:05:22,730 --> 00:05:32,310
+أكبر من alpha او على التواري respectively ال f of
+
+47
+00:05:32,310 --> 00:05:37,330
+xهتكون في حالة ال limit بالساول سالب infinity
+
+48
+00:05:37,330 --> 00:05:42,550
+عايزينها تكون أصغر من ال alpha أصغر من ال given
+
+49
+00:05:42,550 --> 00:05:46,330
+alpha okay
+
+50
+00:05:46,330 --> 00:05:53,550
+إذا أنا هنا عندي limit ال function and c من اليمين
+
+51
+00:05:53,550 --> 00:05:57,570
+بالساول infinity معناته لأي real number alpha بقدر
+
+52
+00:05:57,570 --> 00:06:05,620
+أخلي f of xأكبر من Alpha لكل X على يمين الـC لأن X
+
+53
+00:06:05,620 --> 00:06:08,920
+تقوى للـC من اليمين فX على يمين الـC يعني X أكبر
+
+54
+00:06:08,920 --> 00:06:13,160
+من الـC يعني X ثالث C أكبر من الثالث والمسافة بين
+
+55
+00:06:13,160 --> 00:06:18,140
+الـC والـX أو الـX والـC أصغر من الـD فلكل الـX
+
+56
+00:06:18,140 --> 00:06:22,140
+اللي زيها دي بدي أخلي F of X أكبر من Alpha عشان
+
+57
+00:06:22,140 --> 00:06:26,280
+أقدر أقول أن ال limit لF of X tends to infinity
+
+58
+00:06:28,130 --> 00:06:31,870
+بالمثل ما معنى انه limit f of x عن c من اليمين
+
+59
+00:06:31,870 --> 00:06:38,050
+بالساوى سالب infinity معناه بقدر اخلى لكل x زى ما
+
+60
+00:06:38,050 --> 00:06:44,350
+شوفنا او لأى عدد real number alpha يوجد delta بحيث
+
+61
+00:06:44,350 --> 00:06:48,670
+لكل x على يمين ال C والمسافة بينها و بين ال C أصغر
+
+62
+00:06:48,670 --> 00:06:52,310
+من ال delta لازم صورتها تكون أصغر من ال given
+
+63
+00:06:52,310 --> 00:07:00,470
+alpha okay تمامطيب خلّينا الان كتير من النظريات
+
+64
+00:07:00,470 --> 00:07:06,970
+اللي أخدناها for two sided limit زي هذه مثلا بتكون
+
+65
+00:07:06,970 --> 00:07:12,210
+صحيحة لل right limit و لل left limit طبعا ممكن
+
+66
+00:07:12,210 --> 00:07:18,570
+كمان نعرف بنفس الطريقة ال limit from the left او
+
+67
+00:07:18,570 --> 00:07:22,690
+ال left hand limit مايعني ان ال left hand limit
+
+68
+00:07:22,690 --> 00:07:24,410
+تساوي
+
+69
+00:07:25,850 --> 00:07:31,370
+Infinity او سالب Infinity اذا
+
+70
+00:07:31,370 --> 00:07:36,690
+لو انا بدى اعدل اعرف مامعنى ان ال limit ل F عن C
+
+71
+00:07:36,690 --> 00:07:40,750
+من اليسار بالساوية Infinity او مامعنى ان ال limit
+
+72
+00:07:40,750 --> 00:07:46,150
+ل F عن C من اليسار بالساوية سالب Infinity فباخد
+
+73
+00:07:46,150 --> 00:07:53,340
+let F be هكذاو C cluster point هتصير لإيه تقاطع
+
+74
+00:07:53,340 --> 00:08:00,220
+الفترة من سالب infinity إلى C فبنقول
+
+75
+00:08:00,220 --> 00:08:04,680
+إن ال limit لما X تقول إلى C من اليسار بالساوي
+
+76
+00:08:04,680 --> 00:08:10,260
+infinity أو ال limit لما X تقول إلى C من اليسار
+
+77
+00:08:10,260 --> 00:08:15,860
+بالساوي السالب infinity إذا كان لأي Alpha يوجد
+
+78
+00:08:15,860 --> 00:08:21,200
+Delta تعتمد على Alphaالان ال X هتكون على يسار ال C
+
+79
+00:08:21,200 --> 00:08:31,200
+وبالتالي هذا هنستبدله بC سالب X أكبر
+
+80
+00:08:31,200 --> 00:08:38,680
+من سفر أصغر من دلتر فلكل X زي هذه انا عايز ان تكون
+
+81
+00:08:38,680 --> 00:08:43,620
+F of X أكبر من Alpha او في الحالة هذه F of X أصغر
+
+82
+00:08:43,620 --> 00:08:49,710
+من Alpha هنا ذيك نكون عرفناالـ left limit عن c ما
+
+83
+00:08:49,710 --> 00:08:56,170
+معنى أنها ساوي plus أو minus infinity إذن قلنا إن
+
+84
+00:08:56,170 --> 00:09:00,310
+كل النظريات اللي برهنها for two sided limits هتكون
+
+85
+00:09:00,310 --> 00:09:08,730
+صحيحة لل left limit و لل right limit من ضمنهم
+
+86
+00:09:08,730 --> 00:09:14,010
+النظرية السابقة طيب
+
+87
+00:09:14,010 --> 00:09:15,130
+لو بدي أنا يعني
+
+88
+00:09:18,090 --> 00:09:24,870
+أخد أمثلة كيف نستخدم التعريف هذا فيه إثبات إن ال
+
+89
+00:09:24,870 --> 00:09:32,230
+limits تطلع plus أو minus infinity فناخد أول مثال
+
+90
+00:09:32,230 --> 00:09:40,250
+let f of x بسوي واحد
+
+91
+00:09:40,250 --> 00:09:45,170
+على x حيث x لا يساوي سبق و show
+
+92
+00:09:49,780 --> 00:09:58,060
+عايزين نفدت واحد ان ال limit لواحد على x او f of x
+
+93
+00:09:58,060 --> 00:10:05,820
+هنا لما x تقول الى ستر من اليمين بساوي ال infinity
+
+94
+00:10:05,820 --> 00:10:09,960
+و 2 limit
+
+95
+00:10:11,300 --> 00:10:19,200
+لف of X لما X تقول ال 0 من اليسار يساوي سالب
+
+96
+00:10:19,200 --> 00:10:23,880
+infinity okay فلو
+
+97
+00:10:23,880 --> 00:10:29,540
+بدنا نبرم الجزء الأول مثلا to show
+
+98
+00:10:32,410 --> 00:10:38,710
+المقاومة لـ f of x as x tends to 0 from the right
+
+99
+00:10:38,710 --> 00:10:45,670
+بساوي infinity فبدي ابدا بـ alpha تنتمي ل R فبقول
+
+100
+00:10:45,670 --> 00:10:57,490
+let alpha belonging to R be given و
+
+101
+00:10:57,490 --> 00:11:02,790
+بدي ارد عليها بDeltaبدي ارد على ال alpha دي ال
+
+102
+00:11:02,790 --> 00:11:08,630
+delta عدد موجب ويعتمد على ال alpha ف choose delta
+
+103
+00:11:08,630 --> 00:11:15,890
+بتساوي واحد على absolute alpha زاد واحد بالتأكيد
+
+104
+00:11:15,890 --> 00:11:21,910
+هذا عدد موجب لأن absolute ال alpha دي عدد حقيقي
+
+105
+00:11:21,910 --> 00:11:28,450
+القيمة المطلقة له عدد غير سالم ممكن يساوي سفر إذا
+
+106
+00:11:28,450 --> 00:11:32,980
+كانت alpha بالساوية سفرلكن زائد واحد بصير موجب
+
+107
+00:11:32,980 --> 00:11:37,720
+المقام موجب اذا انا بضيف واحد ليه عشان اضمن ان
+
+108
+00:11:37,720 --> 00:11:41,880
+المقام مايسويش سفر لان في احتمال ان ال alpha ساوي
+
+109
+00:11:41,880 --> 00:11:45,640
+سفر فبصير عندى مشكلة عشان اتخلص من المشكلة هذه
+
+110
+00:11:45,640 --> 00:11:51,140
+بجسم على absolute alpha زائد واحد الان هذا عدد
+
+111
+00:11:51,140 --> 00:11:57,010
+موجبو يعتمد على alpha هي ال delta هي مرتبطة معرفة
+
+112
+00:11:57,010 --> 00:12:02,350
+بدلالة alpha هي معناه أنها تعتمد على alpha إذا لأي
+
+113
+00:12:02,350 --> 00:12:09,410
+alpha ينتمي ل R خد ال delta اللي بتنظرها هي واحد
+
+114
+00:12:09,410 --> 00:12:13,740
+على absolute alpha الذات واحدةهذا اكيد عدد موجة
+
+115
+00:12:13,740 --> 00:12:20,440
+then من مراتبة الان ان كل x في المجال تبع الدالة
+
+116
+00:12:20,440 --> 00:12:24,380
+اللى هو كل العداد الحقيقية مع عدد صفر و ال x على
+
+117
+00:12:24,380 --> 00:12:30,420
+يمين ال c اللى هو الصفر و من هنا x ينتمي الى a
+
+118
+00:12:30,420 --> 00:12:35,560
+اللى هى R المجال تبع الدالة كل العداد الحقيقية مع
+
+119
+00:12:35,560 --> 00:12:42,940
+عدد صفرو X سالب سفر ال C هنا ال cluster point هي
+
+120
+00:12:42,940 --> 00:12:47,100
+السفر الآن ال X على يمين السفر يعني X minus سفر
+
+121
+00:12:47,100 --> 00:12:54,040
+أكبر من سفر فإذا كانت ال X هذه أصغر من Delta فهذا
+
+122
+00:12:54,040 --> 00:13:03,900
+هيعطيني أن ال 1 على Xأكبر من واحد على دلتا
+
+123
+00:13:03,900 --> 00:13:07,560
+وبالتالي
+
+124
+00:13:07,560 --> 00:13:14,720
+هذا بيقدي انه f of x اللي هي بالساوي واحد على اكس
+
+125
+00:13:14,720 --> 00:13:20,940
+أكبر من واحد على دلتا اللي هي بالساوي واحد مقلوب
+
+126
+00:13:20,940 --> 00:13:28,280
+الدلتا بيطلع absolute alpha زائد واحد وهذه أكبر من
+
+127
+00:13:28,280 --> 00:13:32,300
+absolute ال alphaabsolute alpha زاد واحد أكبر من
+
+128
+00:13:32,300 --> 00:13:36,880
+absolute alpha وabsolute alpha أكبر من أو يساوي
+
+129
+00:13:36,880 --> 00:13:41,920
+alpha أي عدد حقيقي القيمة المطلقة تبعته أكبر من أو
+
+130
+00:13:41,920 --> 00:13:48,220
+يساوي نفسه فالنهاية أثبتنا أن f of x أكبر من ال
+
+131
+00:13:48,220 --> 00:13:48,920
+given alpha
+
+132
+00:13:53,520 --> 00:13:58,440
+Okay تمام بما ان ال alpha دي كانت arbitrarily
+
+133
+00:13:58,440 --> 00:14:06,640
+since alpha belong to R was arbitrarily اذا هين
+
+134
+00:14:06,640 --> 00:14:12,180
+اثبتنا اذا معناه هذا الكلام هذا انه لكل alpha فيه
+
+135
+00:14:12,180 --> 00:14:17,280
+delta تعتمد عليها بتخلي f of x اكبر من alpha لكل x
+
+136
+00:14:17,280 --> 00:14:23,440
+قريبة من السفر within مسافة deltaإن هذا معناه حسب
+
+137
+00:14:23,440 --> 00:14:29,340
+التعريف إن ال limit ل f of x لما x تقول إلى سفر من
+
+138
+00:14:29,340 --> 00:14:36,780
+اليمين بساوي بورهانش
+
+139
+00:14:36,780 --> 00:14:38,320
+جزء التاني مشابه
+
+140
+00:14:47,840 --> 00:14:54,400
+is similar مشابه لل part للجزء الأول يعني فهسيبكم
+
+141
+00:14:54,400 --> 00:14:59,860
+انتوا تكتبوا برهان مشابه مع التعديلات اللازمة و
+
+142
+00:14:59,860 --> 00:15:05,660
+أيه طبعا التعريف تبع ال limit from the left موجود
+
+143
+00:15:05,660 --> 00:15:13,280
+okay تمام اللي هو بالأزرق تمام مثال
+
+144
+00:15:13,280 --> 00:15:33,190
+تاني ممكن برضهناخد مثال تاني
+
+145
+00:15:33,190 --> 00:15:41,050
+show limit for function e to one على x as x tends
+
+146
+00:15:41,050 --> 00:15:45,070
+to zero from the right بساوي infinity
+
+147
+00:15:55,920 --> 00:16:02,700
+أنا عندي ال function تبعتي f of x بيسمي E أس واحد
+
+148
+00:16:02,700 --> 00:16:07,840
+على X طبعا ال X هنا ال function مش معرفة عند السفر
+
+149
+00:16:07,840 --> 00:16:12,900
+المجال الدالي هذه كل الأعداد الحقيقية مع ده السفر
+
+150
+00:16:12,900 --> 00:16:20,860
+المثل هذا أخدناه المرة اللي فاتت we
+
+151
+00:16:20,860 --> 00:16:21,440
+have
+
+152
+00:16:25,650 --> 00:16:34,410
+from previous example من
+
+153
+00:16:34,410 --> 00:16:44,110
+المثال السابق فانا هادرس سابق انه واحد
+
+154
+00:16:44,110 --> 00:16:51,370
+على اكس اكبر من سفر اصغر من اكس واحد على اكس لكل
+
+155
+00:16:51,370 --> 00:17:00,340
+اكس اكبر من سفرلكل x على يمين السفر كان في ندي T
+
+156
+00:17:00,340 --> 00:17:09,980
+أزرق من E of T لكل T عدد مؤجد طيب
+
+157
+00:17:09,980 --> 00:17:18,000
+احنا لسه بتبتيل since ال
+
+158
+00:17:18,000 --> 00:17:24,340
+limit لواحد على x لما x تقول إلى السفر من اليمين
+
+159
+00:17:26,540 --> 00:17:31,760
+بساوي infinity فممكن
+
+160
+00:17:31,760 --> 00:17:37,160
+نطبخ ال comparison test هذا فهي عندي f of x أصغر
+
+161
+00:17:37,160 --> 00:17:45,800
+من g of x يعني خليني أسمي هذه g of x كمشي مع نظري
+
+162
+00:17:45,800 --> 00:17:52,020
+يعني وخلني f of x بساوي واحد على x فهي عندي f of x
+
+163
+00:17:52,020 --> 00:18:00,560
+أصغر من g of x لكل x في Rأو لكل X لا يساوي سفر لكل
+
+164
+00:18:00,560 --> 00:18:07,820
+X موجة بقى أو على يمين السفر لكل
+
+165
+00:18:07,820 --> 00:18:16,880
+X في R تقاطع سفر إلى ملا نهاية okay فإذا
+
+166
+00:18:16,880 --> 00:18:22,320
+قلنا النظرية هذه صحيحة لل right limit باستخدام
+
+167
+00:18:22,320 --> 00:18:33,480
+النظريةby above theorem by above theorem for right
+
+168
+00:18:33,480 --> 00:18:39,680
+limits للنهايات
+
+169
+00:18:39,680 --> 00:18:49,990
+من اليمين we have نحصل على انه ال limitلقيت واحد
+
+170
+00:18:49,990 --> 00:18:55,370
+على اكس لما اكس تقول إلى سفر من اليمين بساوي plus
+
+171
+00:18:55,370 --> 00:19:04,770
+infinity وهذا اللي بدناه هي مظبوط صح؟ تمام؟إذن
+
+172
+00:19:04,770 --> 00:19:08,850
+ممكن نطبق النظرية هذه لإثبات أن ال limit ل ال
+
+173
+00:19:08,850 --> 00:19:13,690
+function E to 1 ل X من X أو ل 0 من اليمين بساوي
+
+174
+00:19:13,690 --> 00:19:22,130
+infinity برضه ممكن نطبق التعريف يعني ممكن أعطي
+
+175
+00:19:22,130 --> 00:19:28,530
+برهان تاني و أقول بما أن هذه المتباينة صحيحة لكل X
+
+176
+00:19:28,530 --> 00:19:34,480
+موجبةو بما انه ال limit هذه لو احلى ال function
+
+177
+00:19:34,480 --> 00:19:38,080
+واحد على X عن سفر من اليانين بالساوي infinity
+
+178
+00:19:38,080 --> 00:19:41,560
+معناته انا بقدر اخلي واحد ال function واحد على X
+
+179
+00:19:41,560 --> 00:19:46,940
+هذه اكبر من Alpha لأي real number Alpha صح؟
+
+180
+00:19:48,400 --> 00:19:52,300
+وبالتالي بقدر اخل اي ت واحد على اكس اكبر من اي
+
+181
+00:19:52,300 --> 00:19:59,600
+real number Alpha لكل X طبعا على يمين السفر وعلى
+
+182
+00:19:59,600 --> 00:20:06,660
+مسافة اصغر من Delta نقدر نجيب طبعا Delta لكل Alpha
+
+183
+00:20:06,660 --> 00:20:13,800
+فممكن برضه استخدم التعريف لاثبات ان ال limit لإي ت
+
+184
+00:20:13,800 --> 00:20:16,800
+واحد على اكس على ما اكسه ولا سفر من اليمين بالساوي
+
+185
+00:20:16,800 --> 00:20:21,100
+infinityOkay تمام ان انا ممكن استخدم التعريف او
+
+186
+00:20:21,100 --> 00:20:27,860
+استخدم ال comparison test اللي فوق واضح في اي سؤال
+
+187
+00:20:27,860 --> 00:20:37,380
+طب
+
+188
+00:20:37,380 --> 00:20:45,280
+احنا يعني لاحظوا في ال chapter هذا اتعرضنا ل ال ..
+
+189
+00:20:47,310 --> 00:20:53,690
+لتعريف النهايات للدول and cluster point للمجال
+
+190
+00:20:53,690 --> 00:20:58,390
+تبعها او and cluster point لتقاطع مجالها مع فترة
+
+191
+00:20:58,390 --> 00:21:02,470
+مفتوحة زي هذه او فترة مفتوحة زي هذه في حالة ال
+
+192
+00:21:02,470 --> 00:21:06,870
+infinite limits وفي كل ال limits هذه دائما ال X
+
+193
+00:21:06,870 --> 00:21:12,290
+كانت تقول ل C لعدد ل cluster point سواء من اليمين
+
+194
+00:21:12,290 --> 00:21:15,910
+او من اليسار لكن احيانا
+
+195
+00:21:17,840 --> 00:21:29,720
+بتصادفنا نهايات احيانا
+
+196
+00:21:29,720 --> 00:21:37,020
+نتعرض لمواقف زي هذه انه كيف انا بدي .. يعني ممكن
+
+197
+00:21:37,020 --> 00:21:42,840
+يكون عندي limit ل a for x بدل ما x تقول ل cluster
+
+198
+00:21:42,840 --> 00:21:45,920
+point c x تقول ل infinity
+
+199
+00:21:49,360 --> 00:21:52,820
+ما معنى ان ال limit ل f of x لما x تقول ال
+
+200
+00:21:52,820 --> 00:22:00,720
+infinity بساوي عدد L او ما معنى ان ال limit ل ال
+
+201
+00:22:00,720 --> 00:22:05,080
+function f لما x تقول ال سالب infinity بساوي ايضا
+
+202
+00:22:05,080 --> 00:22:12,060
+عدد L هذا ما اتعرضنا اليه فبنلا تعريف نشوف كيف
+
+203
+00:22:12,060 --> 00:22:20,300
+التعريف تبع ال limits هذه بيكونمثلًا انا عندي ال
+
+204
+00:22:20,300 --> 00:22:28,380
+limit نرجع لل function واحد على X فانا
+
+205
+00:22:28,380 --> 00:22:33,980
+عندي ال limit يعني Y بساوي واحد على X فانا عندي ال
+
+206
+00:22:33,980 --> 00:22:40,980
+limit واحد على X لما X تقول infinity واضح انها
+
+207
+00:22:40,980 --> 00:22:47,330
+بساوي عدد L صفروبرضه كمان لو كانت x تقولنا سالب
+
+208
+00:22:47,330 --> 00:22:53,250
+infinity برضه ال limit بالساوية سفر، اذا كيف اثبت
+
+209
+00:22:53,250 --> 00:22:59,750
+او كيف اعرف ان ال limit عند ال infinity بالساوية
+
+210
+00:22:59,750 --> 00:23:04,150
+عدد او عند السالب infinity بالساوية عدد ما؟
+
+211
+00:23:04,790 --> 00:23:10,050
+التعريفات هذه ما مرت لسه علينا فنحتاج ان احنا نعرف
+
+212
+00:23:10,050 --> 00:23:18,950
+او ناخد هذه التعريفات اذا دلوقت نقصح التعريف هذا
+
+213
+00:23:18,950 --> 00:23:28,570
+ناخد
+
+214
+00:23:28,570 --> 00:23:29,070
+definition
+
+215
+00:23:48,740 --> 00:24:03,120
+فالتعريف let f be function from A to R and
+
+216
+00:24:03,120 --> 00:24:10,780
+الفترة من A إلى ماله نهاية تكون داخل المجموعة A
+
+217
+00:24:10,780 --> 00:24:13,640
+for some A ينتمي إلى R
+
+218
+00:24:20,670 --> 00:24:32,430
+فبنعرف و نقول ان ال ينتمي لار is
+
+219
+00:24:32,430 --> 00:24:36,710
+a limit of
+
+220
+00:24:36,710 --> 00:24:46,950
+ال function f as x tends to infinity and right و
+
+221
+00:24:46,950 --> 00:24:52,540
+بنكتب في الحالة هذه ان ال limitلـ f of x as x
+
+222
+00:24:52,540 --> 00:24:58,820
+tends to infinity بالساوية لعدد L إذا تحقق الشرط
+
+223
+00:24:58,820 --> 00:25:06,960
+التالي لكل إبسلون for any إبسلون أكبر من 0 نقدر
+
+224
+00:25:06,960 --> 00:25:14,660
+نلاقي capital K عدد حقيقي يعتمد على إبسلون وهذا
+
+225
+00:25:14,660 --> 00:25:23,260
+العدد أكبر من العدد A اللي هو عدد حقيقيمعين بحيث
+
+226
+00:25:23,260 --> 00:25:33,980
+أنه لكل لو كان ال X أكبر من ال K فهذا بتضمن أنه
+
+227
+00:25:33,980 --> 00:25:39,780
+absolute F of X minus L أصغر من ال given epsilon
+
+228
+00:25:39,780 --> 00:25:44,280
+تمام؟
+
+229
+00:25:44,280 --> 00:25:46,120
+بالمثل ممكن أعرف
+
+230
+00:25:52,400 --> 00:25:58,400
+العرف ما معناه ان ال limit لل function f لما x
+
+231
+00:25:58,400 --> 00:26:04,120
+تقول لسالب infinity بالساوي عدد L في الحالة هذه
+
+232
+00:26:04,120 --> 00:26:13,200
+باشترط ان المجموع المجال يحتوي على فترة زي هذه
+
+233
+00:26:13,200 --> 00:26:17,840
+بدأت
+
+234
+00:26:17,840 --> 00:26:19,300
+فترة هذه فترة زي هذه
+
+235
+00:26:22,770 --> 00:26:30,990
+هنا قلنا بدل infinity نبدلها بال-infinity وهنا بال
+
+236
+00:26:30,990 --> 00:26:37,030
+-infinity ونقول
+
+237
+00:26:37,030 --> 00:26:46,190
+إنه يوجد K المرة هذه بدل أكبر من A أصغر من A وهذه
+
+238
+00:26:46,190 --> 00:26:48,150
+تتغير لكل X
+
+239
+00:26:53,500 --> 00:26:59,960
+أصغر من K أصغر
+
+240
+00:26:59,960 --> 00:27:03,760
+من Y أصغر
+
+241
+00:27:03,760 --> 00:27:06,920
+من K أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+242
+00:27:06,920 --> 00:27:06,920
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+243
+00:27:06,920 --> 00:27:07,020
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+244
+00:27:07,020 --> 00:27:07,760
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+245
+00:27:07,760 --> 00:27:07,760
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+246
+00:27:07,760 --> 00:27:07,760
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أصغر
+
+247
+00:27:07,760 --> 00:27:12,040
+من Y أصغر من Y أصغر من Y أصغر من Y أصغر من Y أص
+
+248
+00:27:12,440 --> 00:27:18,680
+Okay طيب اذا انا لان في عندي تعريفات جديدة كمان
+
+249
+00:27:18,680 --> 00:27:25,880
+مرة كل نظريات اللي أثبتناها سابقا بس
+
+250
+00:27:25,880 --> 00:27:31,500
+بدل ما X أول ل C بيصير X أول ل infinity ال
+
+251
+00:27:31,500 --> 00:27:37,040
+sequence أول شيء إذا كانت ال limit هذه أو هذه
+
+252
+00:27:37,040 --> 00:27:42,630
+موجودة بسوء عدد Lفممكن اثباتي انها unique زيها زي
+
+253
+00:27:42,630 --> 00:27:49,210
+اي two sided limit او زيها زي اي limit اخرى كذلك
+
+254
+00:27:49,210 --> 00:27:53,370
+ممكن نثبت sequential criterion لل limits زي هدول
+
+255
+00:27:53,370 --> 00:27:59,230
+وممكن النظرية زي هذه تكون صحيحة لهذا النوع من ال
+
+256
+00:27:59,230 --> 00:28:06,130
+limits okay اذا معظم النظريات معظم النظريات اللي
+
+257
+00:28:06,130 --> 00:28:13,240
+اثبتناها تكون صحيحة لهذا النوع الجديد من الـ
+
+258
+00:28:13,240 --> 00:28:17,660
+infinite نسميها infinite limits at infinity هذه
+
+259
+00:28:17,660 --> 00:28:21,980
+limits at infinity أو سالم infinity هذه كانت
+
+260
+00:28:21,980 --> 00:28:27,520
+نسميها infinite limits فمثلا
+
+261
+00:28:27,520 --> 00:28:31,900
+على سبيل المثال وليس الحصر ممكن ان احنا نكتب
+
+262
+00:28:31,900 --> 00:28:35,060
+sequential criterion لهذا النوع من ال limits
+
+263
+00:28:42,580 --> 00:28:57,860
+هي sequential theorem sequential
+
+264
+00:28:57,860 --> 00:29:03,680
+.. sequential
+
+265
+00:29:03,680 --> 00:29:07,500
+criterion
+
+266
+00:29:07,500 --> 00:29:18,320
+.. sequential criterionfor limits for limits at
+
+267
+00:29:18,320 --> 00:29:23,580
+infinity the
+
+268
+00:29:23,580 --> 00:29:29,480
+following statements are equivalent are equivalent
+
+269
+00:29:29,480 --> 00:29:40,830
+واحد limitF of X as X tends to infinity بساوي عدد
+
+270
+00:29:40,830 --> 00:29:47,970
+M اتنين for every
+
+271
+00:29:47,970 --> 00:29:48,910
+sequence
+
+272
+00:29:51,330 --> 00:29:57,750
+x in contained in a تقابل فترة مفتوحة من a إلى م
+
+273
+00:29:57,750 --> 00:30:05,030
+للإلهية such that limit x in as n tends to
+
+274
+00:30:05,030 --> 00:30:14,410
+infinity بساوي ال infinity لازم
+
+275
+00:30:14,410 --> 00:30:19,300
+يطلع عندى limit ال imageبساوي العدد L لسيكوانس Xn
+
+276
+00:30:19,300 --> 00:30:25,380
+as N times Infinity بساوي العدد L لذا هذه
+
+277
+00:30:25,380 --> 00:30:34,580
+Sequential criterion for limits at infinity وممكن
+
+278
+00:30:34,580 --> 00:30:39,860
+نثبت النظرية هذه زي ما أثبتنا Sequential criterion
+
+279
+00:30:39,860 --> 00:30:46,300
+for finite two-sided limits أو for finite one
+
+280
+00:30:46,300 --> 00:30:51,520
+-sided limitsمثلًا لو أريد أن أثبت واحد implies
+
+281
+00:30:51,520 --> 00:30:55,660
+اتنين فبقول
+
+282
+00:30:55,660 --> 00:31:05,080
+assume أنه one holds هذا
+
+283
+00:31:05,080 --> 00:31:12,440
+معناه أن ال limit لf of x as x tends to infinity
+
+284
+00:31:12,440 --> 00:31:17,120
+بسوى عدد L طيب to prove
+
+285
+00:31:32,110 --> 00:31:38,430
+to prove two holes فابد
+
+286
+00:31:38,430 --> 00:31:46,430
+أثبت لأي sequence لأي sequence بالمواصفات هذه
+
+287
+00:31:46,430 --> 00:31:57,370
+limit صورتها بساوي L فببدأ بقول let let XM contain
+
+288
+00:31:57,370 --> 00:32:08,400
+بالـ A قاطعالفترة هذه بيجيبن
+
+289
+00:32:08,400 --> 00:32:11,560
+بحيث
+
+290
+00:32:11,560 --> 00:32:20,660
+ان ال limit لسيكوينس xn هذه بساوي
+
+291
+00:32:20,660 --> 00:32:24,660
+infinity و
+
+292
+00:32:24,660 --> 00:32:27,380
+بالثبات ان ال limit صورتها بساوي ال
+
+293
+00:32:30,930 --> 00:32:34,970
+عشان أثبت أنه two holds، بدي أثبت أنه الـ limit
+
+294
+00:32:34,970 --> 00:32:45,370
+لصورة الـ xn as n times infinity بساوي n لأن هذه
+
+295
+00:32:45,370 --> 00:32:48,930
+عبارة عن sequence، بدي أثبت limit sequence بالساوي
+
+296
+00:32:48,930 --> 00:32:54,030
+عدد، بستخدم تعريف Y capital N لل limit of a
+
+297
+00:32:54,030 --> 00:33:00,120
+sequence، صح؟إذا أنا بقول let epsilon أكبر من
+
+298
+00:33:00,120 --> 00:33:07,240
+السفر be given طيب،
+
+299
+00:33:07,240 --> 00:33:16,140
+أنا عندي فارض since ال limit ل F of X as X tends
+
+300
+00:33:16,140 --> 00:33:26,080
+to infinity بتساوي العدد Lإذا من تعريف ال limit of
+
+301
+00:33:26,080 --> 00:33:31,180
+infinity اللي زيها دي هي التعريف هي تحت تقول إنه
+
+302
+00:33:31,180 --> 00:33:41,300
+for any given epsilon يوجد عدد حقيقي K يعتمد على
+
+303
+00:33:41,300 --> 00:33:47,220
+epsilon وهذا أكبر من A بحيث
+
+304
+00:33:47,220 --> 00:33:57,390
+إنه لو كان Xأكبر من الـ K بيقدي أنه absolute f of
+
+305
+00:33:57,390 --> 00:34:03,750
+x minus ال L أصغر من إبسل أسمي ال implication هذه
+
+306
+00:34:03,750 --> 00:34:08,830
+star طيب
+
+307
+00:34:08,830 --> 00:34:13,750
+أنا برضه عندي أنا فارد أن ال sequence هذه ال given
+
+308
+00:34:13,750 --> 00:34:15,590
+sequence ال limit تبعتها
+
+309
+00:34:21,580 --> 00:34:26,380
+بساوي infinity ومن
+
+310
+00:34:26,380 --> 00:34:30,540
+تعريف ان تكون ال sequence limit تبعتها infinite
+
+311
+00:34:30,540 --> 00:34:37,180
+هذا معناه ان مقدر اخلى x in اكبر من اي عدد حقيقي
+
+312
+00:34:37,180 --> 00:34:41,440
+alpha فاخد alpha هنا بساوي k مش هذا عدد حقيقي
+
+313
+00:34:41,440 --> 00:34:48,520
+محترم فاخد هوبما عنده limit لسيكوينس Xn بالساوي
+
+314
+00:34:48,520 --> 00:34:56,920
+infinity then for any real number يوجد
+
+315
+00:34:56,920 --> 00:35:06,700
+capital N عدد طبيعي يعتمد على Kهذا أشمله عدد طبيعي
+
+316
+00:35:06,700 --> 00:35:14,800
+بحيث أنه لكل n أكبر من أو ساوي capital N هذا بتضمن
+
+317
+00:35:14,800 --> 00:35:22,980
+أن ال Xn أكبر من العدد K العدد الحقيقي K نسمي ال
+
+318
+00:35:22,980 --> 00:35:25,160
+implication هذه double star
+
+319
+00:35:39,650 --> 00:35:45,850
+تمام هيك إذا هذا ناخده من كوننا ان احنا فرضين ان
+
+320
+00:35:45,850 --> 00:35:49,350
+limit ال sequence xn بالساوية infinity ومن تعريف
+
+321
+00:35:49,350 --> 00:35:56,710
+ال infinite limit لل sequence الان الان في تحول في
+
+322
+00:35:56,710 --> 00:36:05,270
+البرهار now star and double star yield
+
+323
+00:36:09,010 --> 00:36:17,430
+بيعطوني التالي لو كانت n أكبر من أو ساوى capital N
+
+324
+00:36:17,430 --> 00:36:28,590
+فهذا بيؤدي إنه xn أكبر من k هذا موجود أخدناه من
+
+325
+00:36:28,590 --> 00:36:33,930
+double star لكل n أكبر من أو ساوى capital N بيطلع
+
+326
+00:36:33,930 --> 00:36:40,980
+xn أكبر من kطيب و من ال star و هذا بيقدي باستخدام
+
+327
+00:36:40,980 --> 00:36:47,640
+ال star ال star بيقوللي لكل x لو كانت ال x او ال
+
+328
+00:36:47,640 --> 00:36:56,860
+xn أكبر من capital K هذا بيقدي ان صورتها المسافة
+
+329
+00:36:56,860 --> 00:37:01,840
+بينها و بين ال L أصغر من إبسل صح؟
+
+330
+00:37:06,200 --> 00:37:10,360
+إذا نجي نلخص كمان مرة، إيه اللي عملناها؟ أنا إيش
+
+331
+00:37:10,360 --> 00:37:14,560
+بدأ في بتلقى limited sequence F of X N لما N تقول
+
+332
+00:37:14,560 --> 00:37:18,460
+الـinfinity بالساوي عدد L فهذه البديات بإبسلون
+
+333
+00:37:18,460 --> 00:37:22,940
+أكبر من السفر given أثبتت إن يوجد capital N عدد
+
+334
+00:37:22,940 --> 00:37:27,680
+طبيعي يعتمد على الـK والـK تعتمد على إبسلون، إذا
+
+335
+00:37:27,680 --> 00:37:33,810
+الـN هذه تعتمد على الـgiven إبسلونو هذه ال N لكل
+
+336
+00:37:33,810 --> 00:37:38,050
+small n أكبر من أو ساوي ال capital N هذه طلع عند
+
+337
+00:37:38,050 --> 00:37:41,350
+المسافة بين الحد النوني لل sequence و L أصغر من
+
+338
+00:37:41,350 --> 00:37:47,750
+Epsilon بما أن Epsilon was arbitrary since Epsilon
+
+339
+00:37:47,750 --> 00:37:54,250
+أكبر من السفر was arbitraryإذا by epsilon capital
+
+340
+00:37:54,250 --> 00:37:58,650
+N definition of limit of sequence بنكون هيك حسب
+
+341
+00:37:58,650 --> 00:38:04,570
+التعريف أثبتنا أنه limit لسيكوينس F of X N as N
+
+342
+00:38:04,570 --> 00:38:09,390
+tends to infinity بالساوي لعدد N وبالتالي هيك
+
+343
+00:38:09,390 --> 00:38:16,950
+بنكون أثبتنا إذا العبارة 2 holds وهيك بنكون أثبتنا
+
+344
+00:38:16,950 --> 00:38:23,230
+أن العبارة statement 1 implies statement 2Okay
+
+345
+00:38:23,230 --> 00:38:30,010
+تمام إذا نحن ممكن نبرهن ال sequential criterion لل
+
+346
+00:38:30,010 --> 00:38:35,730
+infinite limit و لل one sided limit و لكل أنواع ال
+
+347
+00:38:35,730 --> 00:38:47,310
+limit و هاي أثبتنا جزء برهان الجزء التاني مماثل ال
+
+348
+00:38:47,310 --> 00:38:58,010
+proof of 2 implies 1is similar to
+
+349
+00:38:58,010 --> 00:39:03,410
+original
+
+350
+00:39:03,410 --> 00:39:08,690
+proof or proof of
+
+351
+00:39:08,690 --> 00:39:14,810
+original original
+
+352
+00:39:14,810 --> 00:39:21,470
+sequential criterion original sequential criterion
+
+353
+00:39:21,470 --> 00:39:26,530
+مععمل التعديلات اللازمة فانا بقولكم انكم ترجعوا ل
+
+354
+00:39:26,530 --> 00:39:30,910
+sequential criterion الأساسية تقرأوا البرهان تبعها
+
+355
+00:39:30,910 --> 00:39:36,050
+كيف انا برهان اتنين ده او احدو تعملوا التعديلات ..
+
+356
+00:39:36,050 --> 00:39:41,070
+ازاي نبران واحد بدي لاتنين .. okay تمام .. اذا ان
+
+357
+00:39:41,070 --> 00:39:44,910
+هذه تعتبر sequential criterion لل limits at
+
+358
+00:39:44,910 --> 00:39:50,710
+infinity بالمثل ممكن ان احنا نحصل على sequential
+
+359
+00:39:50,710 --> 00:39:57,810
+criterion لل limits at negative infinity يعني ال
+
+360
+00:39:57,810 --> 00:40:04,930
+..يعني هذه ممكن تبدلها ب negative infinity وهذه
+
+361
+00:40:04,930 --> 00:40:10,570
+ممكن تبدلها ب negative infinity وهذه ممكن تبدلها ب
+
+362
+00:40:10,570 --> 00:40:15,670
+سالب infinity إلى a وهذه ممكن تبدلها ب negative
+
+363
+00:40:15,670 --> 00:40:19,790
+infinity وهذه
+
+364
+00:40:19,790 --> 00:40:23,550
+ممكن تبدلها بسالب
+
+365
+00:40:23,550 --> 00:40:24,030
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+366
+00:40:24,030 --> 00:40:24,130
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+367
+00:40:24,130 --> 00:40:24,750
+تبدلها بسالب infinity إلى a وهذه ممكن تبدلها بسالب
+
+368
+00:40:24,750 --> 00:40:24,830
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+369
+00:40:24,830 --> 00:40:24,830
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+370
+00:40:24,830 --> 00:40:25,250
+تبدلها بسالب infinity إلى a وهذه ممكن تبدلها بسالب
+
+371
+00:40:25,250 --> 00:40:26,690
+infinity إلى a وهذه ممكن تبدلها بسالب infinity إلى
+
+372
+00:40:26,690 --> 00:40:27,830
+a وهذه ممكن تبدلها بسالب infinity إلى a وهذه ممكن
+
+373
+00:40:27,830 --> 00:40:38,590
+تبدلها بسوالبرنامج طبعا مشابه للنظرية السابقة ناخد
+
+374
+00:40:38,590 --> 00:40:44,150
+أمثلة طبعا
+
+375
+00:40:44,150 --> 00:40:52,230
+في نظريات كتيرة صحيحة لنوع هذا من ال limits فلو
+
+376
+00:40:52,230 --> 00:40:58,940
+احتجنا زي ال squeeze theorem زي ال comparisontest
+
+377
+00:40:58,940 --> 00:41:04,240
+او الاخر فمثلا
+
+378
+00:41:04,240 --> 00:41:23,600
+ناخد بعض الأمثلة مثلا
+
+379
+00:41:23,600 --> 00:41:25,080
+ناخد examples
+
+380
+00:41:41,280 --> 00:41:50,920
+F of X يساوي واحد على X و X لا يساوي ساقر Show
+
+381
+00:41:50,920 --> 00:41:54,720
+that limit
+
+382
+00:41:54,720 --> 00:42:03,660
+F of X as X tends to infinity يساوي ساقر و كذلك
+
+383
+00:42:03,660 --> 00:42:11,410
+limitلف of x as x tends to negative infinity بساوة
+
+384
+00:42:11,410 --> 00:42:16,950
+ستة المظبوط
+
+385
+00:42:16,950 --> 00:42:24,370
+ال function واحد على x ثانية فلما
+
+386
+00:42:24,370 --> 00:42:29,070
+x تقول infinity واحد على x بتقولها ستة لما x
+
+387
+00:42:29,070 --> 00:42:32,690
+تقولها سالب infinity برضه ال function واحد على x
+
+388
+00:42:32,690 --> 00:42:39,550
+تقول إلى ستةفلو بدي اثبات الجزء الأول فممكن استخدم
+
+389
+00:42:39,550 --> 00:42:45,590
+التعريف او استخدم الـ sequential criterion فمثلا
+
+390
+00:42:45,590 --> 00:42:56,850
+to use a definition لو بدي استخدم التعريف مثلا
+
+391
+00:42:58,750 --> 00:43:02,270
+Limit f of x من x سواء و لا انا كنت بتساوي سفر
+
+392
+00:43:02,270 --> 00:43:10,950
+فبابدأ حسب التعريف بابدأ بإبسلون أكبر من السفر لت
+
+393
+00:43:10,950 --> 00:43:18,510
+إبسلون أكبر من السفر بكلمة و بعدين بدأ أثبت أنه في
+
+394
+00:43:18,510 --> 00:43:26,420
+ك يعتمد على إبسلون ف choose كارتة الكعلى انه واحد
+
+395
+00:43:26,420 --> 00:43:30,840
+على ابسلان فهذا تطلع عدد موجب ويعتمد على ابسلان
+
+396
+00:43:30,840 --> 00:43:36,780
+وال ا طبعا هنا في السؤال هذا هي السفر يعني هنا ال
+
+397
+00:43:36,780 --> 00:43:40,680
+domain تبعد لكل العداد الحقيقية مع ده السفرفممكن
+
+398
+00:43:40,680 --> 00:43:46,440
+اخد السفر الفقرة هذه contained in ال domain تبع
+
+399
+00:43:46,440 --> 00:43:51,100
+الـ a اللي هو كل العداد الحقيقية من عدد السفر هنا
+
+400
+00:43:51,100 --> 00:43:52,620
+لأي أكبر من أكبر من أكبر من أكبر من أكبر من أكبر
+
+401
+00:43:52,620 --> 00:43:53,060
+من أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+402
+00:43:53,060 --> 00:43:53,980
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+403
+00:43:53,980 --> 00:43:57,560
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+404
+00:43:57,560 --> 00:44:03,280
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+405
+00:44:03,280 --> 00:44:06,340
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+406
+00:44:06,340 --> 00:44:06,340
+أكبر من أكبر من أكبر من أكبر من أكبر من أكبر من
+
+407
+00:44:06,340 --> 00:44:08,320
+أكبر من أكبر من أكبر
+
+408
+00:44:09,640 --> 00:44:19,540
+إن واحد على اكس أصغر من واحد على كم و هذا بدوره
+
+409
+00:44:19,540 --> 00:44:25,560
+بقدر ان absolute of f of x minus الصفر ال ال هنا
+
+410
+00:44:25,560 --> 00:44:31,480
+هو الصفر فهذا بيطلع بساوي absolute واحد على اكس
+
+411
+00:44:31,480 --> 00:44:34,420
+فهذا عبارة عن واحد على اكس
+
+412
+00:44:45,380 --> 00:44:48,660
+و هذا أقل من واحد على كي و واحد على كي أقل من
+
+413
+00:44:48,660 --> 00:44:54,400
+إبسلو و هذا أصغر طبعا من واحد على كي طبعا عندي ال
+
+414
+00:44:54,400 --> 00:44:59,520
+X هنا أكبر من كي لحظة X أكبر من كي و ال K موجة بقى
+
+415
+00:45:03,660 --> 00:45:08,560
+القيمة المطلقة ل 1 على X هي 1 على X إطرح سفر
+
+416
+00:45:08,560 --> 00:45:14,760
+مابتعملش حاجة هذا أصغر من 1 على K ومن هنا 1 على K
+
+417
+00:45:14,760 --> 00:45:15,580
+بساوي Y
+
+418
+00:45:18,310 --> 00:45:22,630
+إذن هاني أثبتت لي أي epsilon أكبر من سفر يوجد k
+
+419
+00:45:22,630 --> 00:45:27,590
+عدد حقيقي يعتمد على epsilon بحيث لكل x أكبر من k
+
+420
+00:45:27,590 --> 00:45:33,190
+طلع absolute f of x minus L أصغر من epsilon okay
+
+421
+00:45:33,190 --> 00:45:39,430
+إذن هذا معناه إن ال limit حسب التعريف limit واحد
+
+422
+00:45:39,430 --> 00:45:46,030
+على x لما x تقول إلى infinity بساوي سفر
+
+423
+00:45:49,020 --> 00:45:53,960
+بالمثل ممكن نثبت الجزء التاني نفس البرهان مع
+
+424
+00:45:53,960 --> 00:46:02,520
+التعديل في تعريف limit at سالب infinity ممكن
+
+425
+00:46:02,520 --> 00:46:10,180
+برضه نستخدم sequential criterion لو بدأت
+
+426
+00:46:10,180 --> 00:46:15,900
+استخدم sequential criterion لإثبات
+
+427
+00:46:15,900 --> 00:46:26,470
+limitبأخد بقول ان هنا let xn be sequence contained
+
+428
+00:46:26,470 --> 00:46:35,250
+in 0 و infinity بحيث انه limit xn تساوي infinity
+
+429
+00:46:35,250 --> 00:46:39,570
+اذا
+
+430
+00:46:39,570 --> 00:46:48,340
+limit f of xn has n times infinityطبعا هذا بيقدي
+
+431
+00:46:48,340 --> 00:46:51,500
+هذا
+
+432
+00:46:51,500 --> 00:46:58,320
+بيقدي انه limit واحد على xn بساوي سفر exercise
+
+433
+00:46:58,320 --> 00:47:02,480
+أخدناها أخدنا انه limit sequence xn بساوي infinity
+
+434
+00:47:02,480 --> 00:47:06,990
+if and only if limit مقلوب السيكوانس بساوي سفرالان
+
+435
+00:47:06,990 --> 00:47:13,670
+limit f of xn بساوي limit واحد على xn as n tends
+
+436
+00:47:13,670 --> 00:47:21,150
+to infinity وهذا بيساوي ستة لأي
+
+437
+00:47:21,150 --> 00:47:26,510
+sequence نهايتها infinity نهاية سورتها بيساوي
+
+438
+00:47:26,510 --> 00:47:27,330
+العدد L
+
+439
+00:47:35,720 --> 00:47:42,000
+بنطلع ال limit ل ال function f of x as x tends to
+
+440
+00:47:42,000 --> 00:47:47,200
+infinity بساوية 0 إذا هذا برهان تاني using
+
+441
+00:47:47,200 --> 00:47:55,620
+sequential criterion okay واضح مفهوم مثال
+
+442
+00:47:55,620 --> 00:47:56,160
+تاني
+
+443
+00:48:05,300 --> 00:48:10,520
+بناخد g of x بساوي
+
+444
+00:48:10,520 --> 00:48:15,820
+واحد على extra g of x لا يساوي صغير فبدنا نثبت ان
+
+445
+00:48:15,820 --> 00:48:21,660
+ال limit ل g of x لما x تقول ل infinity و لما x
+
+446
+00:48:21,660 --> 00:48:27,280
+تقول ل سالب infinity بساوي صغير برضه ممكن نستخدم
+
+447
+00:48:27,280 --> 00:48:32,440
+sequential criterion لثبات الجزء الأول أو التاني
+
+448
+00:48:37,280 --> 00:48:42,020
+هذا كان limit xn بالساوية infinity فlimit 1 على xn
+
+449
+00:48:42,020 --> 00:48:46,300
+بالساوية infinity بساوية سفر وبالتالي limit 1 على
+
+450
+00:48:46,300 --> 00:48:57,500
+xn تربية يعني هذا بيقدر وهذا
+
+451
+00:48:57,500 --> 00:49:05,370
+بيقدر ان limit1 على xn ترجية لما انتقل ل infinity
+
+452
+00:49:05,370 --> 00:49:16,510
+بساوي limit 1 على xn ضرب limit 1 على xn وهذا بساوي
+
+453
+00:49:16,510 --> 00:49:27,310
+0 ضرب 0 بساوي 0 و limit g ل xn as n tends to
+
+454
+00:49:27,310 --> 00:49:33,810
+infinity بساوي limit1 على x in third يعني بالساعة
+
+455
+00:49:33,810 --> 00:49:40,990
+سفر صح إذا by sequential criterion
+
+456
+00:49:40,990 --> 00:49:44,030
+أثبتت
+
+457
+00:49:44,030 --> 00:49:49,130
+أنه لأي sequence x in حدودها موجب أو نهايتها
+
+458
+00:49:49,130 --> 00:49:57,970
+infinity ف limit صورتها بالساعة سفر هذا معناه أن
+
+459
+00:49:57,970 --> 00:50:06,540
+ال limitلـ function g of x لما x تقول انفينيتي
+
+460
+00:50:06,540 --> 00:50:10,980
+بساوي ستة هنا نريد أن نكون أخرجنا الجزء الأول
+
+461
+00:50:10,980 --> 00:50:14,800
+باستخدام sequential criterion بالمثل و كنا نستخدم
+
+462
+00:50:14,800 --> 00:50:18,280
+sequential criterion اللي اتبعت الجزء التالي بس
+
+463
+00:50:18,280 --> 00:50:25,580
+هنا هناخد x الموجودة في الفترة هذه و هكذاو نفس
+
+464
+00:50:25,580 --> 00:50:29,860
+النظرية اللى خدناها في القصة السابقة بالكون صحيحة
+
+465
+00:50:29,860 --> 00:50:33,760
+هذا بقى قدر المقلوب ال sequence إذا كانت limit ال
+
+466
+00:50:33,760 --> 00:50:37,320
+sequence infinity فlimit المقلوب سفر وبالتالي
+
+467
+00:50:37,320 --> 00:50:43,420
+limit المقلوب المربع بساوة سفر ممكن كمان نستخدم
+
+468
+00:50:43,420 --> 00:50:48,440
+squeeze theorem ممكن نستخدم squeeze theoremفمثلا
+
+469
+00:50:48,440 --> 00:50:56,100
+لو بدنا نبره كمان واحد بطريقة تانية فممكن ان احنا
+
+470
+00:50:56,100 --> 00:51:05,360
+note that for x أكبر من واحد لما خدت x أكبر من
+
+471
+00:51:05,360 --> 00:51:11,700
+واحد بطلع عندي دايما x تربيه أكبر من أو ساوي x
+
+472
+00:51:13,190 --> 00:51:18,610
+وبالتالي هذا بيقدي ان واحد على اكس تربيه اكبر من
+
+473
+00:51:18,610 --> 00:51:23,930
+او ساوي واحد على اكس وطبعا اكبر من السفر لكل اكس
+
+474
+00:51:23,930 --> 00:51:29,310
+اكبر من واحد طب احنا لسه مثل في الجد شوية ان limit
+
+475
+00:51:29,310 --> 00:51:33,810
+ال function واحد على اكس لما اكس تقول الى infinity
+
+476
+00:51:33,810 --> 00:51:40,930
+بساوي سفر صح؟لسه 130 في المثال الأول هذا المثال
+
+477
+00:51:40,930 --> 00:51:44,630
+الثاني في المثال السابق قصدنا ان limit ال function
+
+478
+00:51:44,630 --> 00:51:48,790
+واحد على x لما x تقول infinity تساوي سفر و limit
+
+479
+00:51:48,790 --> 00:51:54,090
+الدالة ثابت سفر لما x تقول infinity تبقى سفر اذا
+
+480
+00:51:54,090 --> 00:52:03,790
+by squeeze theorem by
+
+481
+00:52:03,790 --> 00:52:09,590
+squeeze theorem for limits at infinityLimited دالة
+
+482
+00:52:09,590 --> 00:52:19,130
+المحصورة اللي هي واحد على اكس تربيه as X tends to
+
+483
+00:52:19,130 --> 00:52:24,310
+infinity بساوي تمام
+
+484
+00:52:24,310 --> 00:52:32,490
+okay واضحو بالمثل ممكن نعطي براهين زي هذا او زي
+
+485
+00:52:32,490 --> 00:52:36,290
+هذا اما باستخدام squeeze theorem او sequential
+
+486
+00:52:36,290 --> 00:52:43,790
+criterion او حتى definition okay واضح في اي سؤال
+
+487
+00:52:43,790 --> 00:52:49,050
+في اي استفسار okay هنوقف اذا هنا و المرة الجاية في
+
+488
+00:52:49,050 --> 00:52:53,190
+بعض انواع ال infinite limits هنتكلم عنهم يعني
+
+489
+00:52:53,190 --> 00:52:59,640
+باختصارو بعدين نحاول نجمل section اربعة تلاتة
+
+490
+00:52:59,640 --> 00:53:04,100
+وبالتالي ننهي ال chapter اللي هو chapter اربعة و
+
+491
+00:53:04,100 --> 00:53:08,620
+بعدين نبدأ في chapter خمسة اللي هو اخر chapter في
+
+492
+00:53:08,620 --> 00:53:15,240
+المخرج هو اهم chapter طبعا في حد عنده اي سؤال او
+
+493
+00:53:15,240 --> 00:53:19,640
+استفسار؟ شكرا لاصغاكم و نشوف ان شاء الله المرة
+
+494
+00:53:19,640 --> 00:53:20,040
+القادمة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/bwcuptIkF-o.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/bwcuptIkF-o.srt
new file mode 100644
index 0000000000000000000000000000000000000000..75641b2c20a862beacae33e65c65c39503fdf964
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/bwcuptIkF-o.srt
@@ -0,0 +1,1647 @@
+1
+00:00:20,940 --> 00:00:27,720
+بسم الله الرحمن الرحيم إن شاء الله اليوم هناخد في
+
+2
+00:00:27,720 --> 00:00:34,940
+اللقاء الأول مناقشة، والمناقشة هذه هتكون على ال
+
+3
+00:00:34,940 --> 00:00:42,360
+Section أربعة و Section خمسة من ال Chapter تلاتة
+
+4
+00:00:42,360 --> 00:00:47,930
+في الكتاب. إحنا لقينا قبل كده Section تلاتة واحد و
+
+5
+00:00:47,930 --> 00:00:52,070
+تلاتة اتنين تلاتة تلاتة مظبوط؟ تلاتة تلاتة تلاتة
+
+6
+00:00:52,070 --> 00:00:56,010
+تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة
+
+7
+00:00:56,010 --> 00:00:56,190
+تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة
+
+8
+00:00:56,190 --> 00:01:01,490
+تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة
+
+9
+00:01:01,490 --> 00:01:02,530
+تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة تلاتة
+
+10
+00:01:02,530 --> 00:01:07,770
+تلاتة تلاتة تلاتة
+
+11
+00:01:14,970 --> 00:01:20,490
+فمين عندها أي سؤال في ال Section تلاتة تلاتة تلاتة
+
+12
+00:01:20,490 --> 00:01:28,290
+أربعة أو تلاتة خمسة Section
+
+13
+00:01:28,290 --> 00:01:29,590
+تلاتة تلاتة
+
+14
+00:01:47,340 --> 00:01:54,900
+أي سؤال؟ ستة ستة؟ طب هذا مش مقرر، لأ أنا .. عفواً
+
+15
+00:01:54,900 --> 00:02:02,500
+أنا بتطلع في مكان تاني، تلاتة تلاتة، سؤال ستة،
+
+16
+00:02:02,500 --> 00:02:06,400
+مطلوب هذا السؤال؟ ما كتبتليش أنت الأسئلة المطلوبة
+
+17
+00:02:06,400 --> 00:02:11,580
+اللي بتتكتب؟ موجودة في ال Syllabus .. موجودة في ال
+
+18
+00:02:11,580 --> 00:02:15,760
+Syllabus فهذا السؤال مش من ضمن الأسئلة اللي ..
+
+19
+00:02:15,760 --> 00:02:17,920
+اللي إحنا .. يعني إنتوا بتطلعوش على ال Syllabus 
+
+20
+00:02:17,920 --> 00:02:21,440
+فيه على الصفحة بتاعت ال Syllabus وفيه المسائل
+
+21
+00:02:21,440 --> 00:02:25,800
+المطلوبة، فبإمكانكم تعرفوا المسائل المطلوبة لكل
+
+22
+00:02:25,800 --> 00:02:30,640
+Section من هنا لآخر ال Course، فهذا السؤال السادس
+
+23
+00:02:30,640 --> 00:02:33,580
+بالذات مش مطلوب، لكن فيه أسئلة تانية ممكن أحللك
+
+24
+00:02:33,580 --> 00:02:42,160
+أربعة أو تلاتة زيه، فإيه رأيك؟ تلاتة 
+
+25
+00:02:42,160 --> 00:02:45,580
+أو أربعة يكونوا نفس الفكرة أه، تلاتة أو أربعة شو
+
+26
+00:02:45,580 --> 00:02:52,720
+بدك تلاتة ولا أربعة؟ الحل
+
+27
+00:02:52,720 --> 00:03:00,880
+السؤال أربعة مثلاً؟ هي الحل السؤال الرابع Section
+
+28
+00:03:00,880 --> 00:03:11,920
+تلاتة تلاتة في السؤال هذا Let x واحد بساوي واحد
+
+29
+00:03:11,920 --> 00:03:17,920
+and xn
+
+30
+00:03:17,920 --> 00:03:24,660
+plus one بساوي Square
+
+31
+00:03:24,660 --> 00:03:28,420
+Root لاتنين plus xn
+
+32
+00:03:31,530 --> 00:03:39,530
+for n belong to N show
+
+33
+00:03:39,530 --> 00:03:52,370
+أن ال Sequence xn converges and find its limit
+
+34
+00:03:55,460 --> 00:04:00,220
+بنثبت أولاً أن ال Sequence هذي Converges ونجيب
+
+35
+00:04:00,220 --> 00:04:10,220
+ال Limit بتاعتها، فمثلاً
+
+36
+00:04:10,220 --> 00:04:15,500
+زي هذه بنطبق عليها ال Monotone Convergence
+
+37
+00:04:15,500 --> 00:04:23,340
+Theorem، ال Monotone Convergence Theorem، فلازم نثبت
+
+38
+00:04:23,340 --> 00:04:29,560
+حاجتين، إن ال Sequence هذه Convergent لازم نثبت
+
+39
+00:04:29,560 --> 00:04:33,900
+إنها Convergent، لازم نثبت إنها Monotone يعني
+
+40
+00:04:33,900 --> 00:04:40,100
+Increasing أو Decreasing، و Bounded، فلحظوا
+
+41
+00:04:40,100 --> 00:04:47,270
+إنتوا لو بدي أحسب أول يعني الحل .. لحظة x واحد
+
+42
+00:04:47,270 --> 00:04:52,510
+بساوي واحد، طب x اتنين .. خد n بساوي واحد بيطلع جذر
+
+43
+00:04:52,510 --> 00:05:00,130
+اتنين زائد .. جذر اتنين زائد x واحد اللي هو جذر
+
+44
+00:05:00,130 --> 00:05:10,400
+التلاتة .. x تلاتة بساوي جذر اتنين زائد x اتنين، و
+
+45
+00:05:10,400 --> 00:05:16,740
+يساوي جذر اتنين زائد جذر
+
+46
+00:05:16,740 --> 00:05:24,220
+التلاتة وهكذا
+
+47
+00:05:24,220 --> 00:05:29,640
+اللي
+
+48
+00:05:29,640 --> 00:05:38,120
+بعده x أربعة بساوي جذر اتنين زائد x تلاتة
+
+49
+00:05:56,800 --> 00:06:04,520
+و هذا بيساوي جذر اتنين زائد اللي
+
+50
+00:06:04,520 --> 00:06:10,500
+هو جذر التربيع إيه اللي اتنين زائد جذر التلاتة
+
+51
+00:06:27,010 --> 00:06:33,310
+فال .. العدد يعني هذا دايماً بيكون أقل من أو يساوي
+
+52
+00:06:33,310 --> 00:06:42,010
+اتنين، أقل من أو يساوي اتنين صح؟ آه، و هذا أصغر من أو
+
+53
+00:06:42,010 --> 00:06:45,410
+يساوي اتنين، يعني جذر التلاتة أصغر من اتنين صح؟ آه
+
+54
+00:06:45,410 --> 00:06:50,750
+فهذا أكيد أصغر من أو يساوي اتنين، و هذا
+
+55
+00:06:53,300 --> 00:06:58,180
+أصغر من أو يساوي اتنين، وبالتالي هذا أصغر من أو يساوي
+
+56
+00:06:58,180 --> 00:07:06,560
+جذر الأربعة اللي هو أصغر من أو يساوي اتنين. هذا و
+
+57
+00:07:06,560 --> 00:07:11,160
+طبعاً وكل واحد من هدول أكبر من أو يساوي الواحد، إذاً
+
+58
+00:07:11,160 --> 00:07:18,040
+ممكن إحنا نثبت الآن By Induction Claim إن xn أكبر
+
+59
+00:07:18,040 --> 00:07:24,320
+من أو يساوي الواحد، أصغر من أو يساوي اتنين For
+
+60
+00:07:24,320 --> 00:07:30,590
+every n belong to N، آه وهذا ممكن تبرهنه By
+
+61
+00:07:30,590 --> 00:07:34,650
+Induction زي ما شوفنا في الأمثلة صح؟ طبعاً طبعاً
+
+62
+00:07:34,650 --> 00:07:43,730
+طبعاً طبعاً طبعاً
+
+63
+00:07:43,730 --> 00:07:43,970
+طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً
+
+64
+00:07:43,970 --> 00:07:44,150
+طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً
+
+65
+00:07:44,150 --> 00:07:48,410
+طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً طبعاً
+
+66
+00:07:48,410 --> 00:07:55,130
+طبعاً طبعاً طبعاً طبعاً
+
+67
+00:07:55,130 --> 00:07:57,910
+طب
+
+68
+00:08:01,490 --> 00:08:11,110
+الآن من أعلى الاختصارات الموجودة في
+
+69
+00:08:11,110 --> 00:08:17,050
+الوثيقة الاختصار
+
+70
+00:08:17,050 --> 00:08:20,290
+Xn مرتبط
+
+71
+00:08:23,380 --> 00:08:27,360
+بعدين بنلاحظ إنه قيمة ال Sequence بتكبر كل ما آه
+
+72
+00:08:27,360 --> 00:08:34,580
+فممكن إثبات إنه Claim تاني إذا ال Sequence Bounded
+
+73
+00:08:34,580 --> 00:08:39,240
+Claim x
+
+74
+00:08:39,240 --> 00:08:45,740
+in أصغر
+
+75
+00:08:45,740 --> 00:08:50,460
+منها ويساوي x in زائد واحد For every n
+
+76
+00:08:54,650 --> 00:09:02,290
+|xN is increasing وهذا ممكن إثباته برضه By
+
+77
+00:09:02,290 --> 00:09:11,410
+Induction By Induction هنا هنا Proof Use Induction
+
+78
+00:09:11,410 --> 00:09:14,630
+Use
+
+79
+00:09:14,630 --> 00:09:20,610
+Induction on M فالحالة
+
+80
+00:09:21,890 --> 00:09:30,890
+الحالة n بيساوي واحد أنا عندي x واحد بيساوي واحد و
+
+81
+00:09:30,890 --> 00:09:34,810
+x واحد
+
+82
+00:09:34,810 --> 00:09:40,510
+زائد واحد اللي هو x اتنين بيساوي جذر التلاتة، جذر
+
+83
+00:09:40,510 --> 00:09:47,810
+التلاتة، وهذا بالتأكيد أكبر من واحد اللي هو x واحد
+
+84
+00:09:50,170 --> 00:09:57,350
+إذاً هيطلع عندي x2 أكبر من أو يساوي x1 إذاً
+
+85
+00:09:57,350 --> 00:10:04,170
+العبارة هذه صحيحة عند m بساوي واحد، Assume
+
+86
+00:10:04,170 --> 00:10:09,970
+Induction Hypothesis الفرض بتاع ال Induction، Assume
+
+87
+00:10:09,970 --> 00:10:13,910
+أنه
+
+88
+00:10:22,570 --> 00:10:24,510
+أصغر من أو يساوي
+
+89
+00:10:35,580 --> 00:10:40,400
+بنثبت صحة الحق برضه عند n بساوي k زائد واحد، إذاً
+
+90
+00:10:40,400 --> 00:10:46,060
+Show xk زائد واحد أصلاً لو يساوي xk زائد واحد زائد
+
+91
+00:10:46,060 --> 00:10:55,700
+واحد عن x زائد اتنين We Have لدينا لاحظوا
+
+92
+00:10:55,700 --> 00:11:01,180
+xk زائد اتنين من ال Definition، من ال Definition
+
+93
+00:11:01,180 --> 00:11:03,360
+بتاع ال Sequence
+
+94
+00:11:06,860 --> 00:11:12,440
+من ال Inductive Formula أو ال Recursive Formula xk
+
+95
+00:11:12,440 --> 00:11:19,840
+زائد اتنين بساوي جذر التربيعي لاتنين زائد x k زائد
+
+96
+00:11:19,840 --> 00:11:28,000
+واحد صح، و By Induction Hypothesis من الفرض بتاع ال
+
+97
+00:11:28,000 --> 00:11:35,020
+Induction هذا بساوي جذر اتنين و xk زائد واحد هذا
+
+98
+00:11:35,020 --> 00:11:44,080
+إيه؟ هيطلع أصغر من xk زائد واحد أكبر من أو يساوي xk
+
+99
+00:11:44,080 --> 00:11:50,380
+أنا xk أنا نحط شريط أكبر xk زائد اتنين
+
+100
+00:11:55,180 --> 00:12:03,320
+فهذا أكبر من أو يساوي x .. هبدل xk زيادة واحد بأكبر
+
+101
+00:12:03,320 --> 00:12:07,660
+من أو يساوي xk من ال Induction Hypothesis من الفرض
+
+102
+00:12:07,660 --> 00:12:12,150
+بتاع ال Induction وهذا By ال Definition باستخدام الـ
+
+103
+00:12:12,150 --> 00:12:18,310
+Definition بتاع ال Sequence، الجذر هذا بساوي xk زي
+
+104
+00:12:18,310 --> 00:12:24,190
+واحدة، إذاً هنأ إثبتنا إن x sub k plus two bigger
+
+105
+00:12:24,190 --> 00:12:30,710
+than or equal to x sub k plus one، إذاً This
+
+106
+00:12:30,710 --> 00:12:37,250
+Completes This
+
+107
+00:12:37,250 --> 00:12:38,610
+Completes The Induction
+
+108
+00:12:43,320 --> 00:12:48,000
+وبالتالي إذا هيك بنكون أثبتنا ال Claim بتاعي إن
+
+109
+00:12:48,000 --> 00:12:51,780
+أنا في عندي Two Claims، أول شيء Sequence Bounded
+
+110
+00:12:51,780 --> 00:12:57,200
+والتاني بيقول إن ال Sequence Increasing، وبالتالي
+
+111
+00:12:57,200 --> 00:13:05,200
+Therefore إذاً
+
+112
+00:13:05,200 --> 00:13:09,580
+لو سمينا هذا Claim One وهذا Claim Two
+
+113
+00:13:17,790 --> 00:13:27,210
+فإذا Now It Claims
+
+114
+00:13:27,210 --> 00:13:32,930
+One And Two And ال Monotone Convergence Theorem
+
+115
+00:13:32,930 --> 00:13:41,530
+بيقدّوا إن Sequence Xn Convergence، Say
+
+116
+00:13:41,530 --> 00:13:52,080
+دعنا نسمي ال Limit بتاعتها Limit Xn بساوي x، طبعاً
+
+117
+00:13:52,080 --> 00:13:56,820
+هذا بيطلع عدد حقيقي الآن
+
+118
+00:13:56,820 --> 00:14:07,420
+بنوجد قيمة Limit x هذه، فبنرجع To Find To Find x
+
+119
+00:14:07,420 --> 00:14:12,880
+لإيجاد قيمة ال x بناخد
+
+120
+00:14:12,880 --> 00:14:21,340
+ال Limit Take Limit Of
+
+121
+00:14:21,340 --> 00:14:28,220
+Both Sides Of
+
+122
+00:14:35,540 --> 00:14:41,820
+بناخد ال Limit لطرفين Of Both Sides Of المعادلة xn
+
+123
+00:14:41,820 --> 00:14:50,280
+زي الواحد بساوي Square Root ل Two Plus Xn To
+
+124
+00:14:50,280 --> 00:14:50,800
+Get
+
+125
+00:14:53,550 --> 00:14:59,490
+Limit Xn Plus One As N Tends To Infinity بساوي
+
+126
+00:14:59,490 --> 00:15:02,890
+Limit الجذر التربيعي ممكن تدخل ال Limit تحت الجذر
+
+127
+00:15:02,890 --> 00:15:08,210
+التربيعي وباستخدام قوانين النهايات Limit الاتنين
+
+128
+00:15:08,210 --> 00:15:13,810
+بيطلع اتنين زي Limit Xn As N Tends To Infinity، طيب
+
+129
+00:15:13,810 --> 00:15:19,170
+أنا عندي Limit Xn زي واحد بساوي x
+
+130
+00:15:26,140 --> 00:15:36,560
+هذه المعادلة بيصير
+
+131
+00:15:36,560 --> 00:15:41,100
+x تربيع Negative x Negative اتنين Negative اتنين Negative
+
+132
+00:15:41,100 --> 00:15:45,160
+اتنين Negative اتنين Negative اتنين Negative اتنين Negative
+
+133
+00:15:45,160 --> 00:15:58,780
+اتنين Negative اتنين Negative اتنين Negative اتنين
+
+134
+00:16:01,690 --> 00:16:08,750
+ال Xn أكبر من أو يساوي واحد، أصغر من أو يساوي اتنين
+
+135
+00:16:08,750 --> 00:16:18,250
+لكل n، هذا من Claim One بيقدي إن ال Limit حسب
+
+136
+00:16:18,250 --> 00:16:25,270
+نظرية سابقة، هذا بيقدي إن ال Limit ل Xn أصغر من أو
+
+137
+00:16:25,270 --> 00:16:31,750
+يساوي اتنين، أكبر من أو يساوي الواحد، وطبعاً ال Limit
+
+138
+00:16:31,750 --> 00:16:37,110
+قلنا ل Xn بساوي x، إذاً واحد أصغر من أو يساوي x أصغر
+
+139
+00:16:37,110 --> 00:16:42,190
+من أو يساوي اتنين، طب أنا عندي خيارين إما x بساوي
+
+140
+00:16:42,190 --> 00:16:49,450
+اتنين أو x بساوي سالب واحد، مش ممكن ال x محصورة بين 
+
+141
+00:16:49,450 --> 00:16:54,090
+واحد و اثنين إذا هذه الإجابة غير مقبولة وبالتالي
+
+142
+00:16:54,090 --> 00:16:59,710
+الـ x بساوي اثنين وهيك بنكون أثبتنا إن الـ sequence
+
+143
+00:16:59,710 --> 00:17:03,650
+is convergent و هي وجدنا الـ limit تبعتها بالمثل
+
+144
+00:17:03,650 --> 00:17:09,130
+ممكن نحل باقي الأسئلة زي مثلا سؤال واحد و اثنين و
+
+145
+00:17:09,130 --> 00:17:16,190
+ثلاثة و خمسة و ستة و سبعة okay
+
+146
+00:17:16,190 --> 00:17:24,550
+تمام في أي أسئلة ثانية في هذا الـ section السؤال تسعة
+
+147
+00:17:37,220 --> 00:18:05,540
+سؤال تسعة هاي
+
+148
+00:18:05,540 --> 00:18:13,770
+السؤال تسعة، الـ section ثلاثة ثلاثة، السؤال هذا بيقول
+
+149
+00:18:13,770 --> 00:18:22,970
+let a be an infinite subset of R و الـ set هذه bounded
+
+150
+00:18:22,970 --> 00:18:32,090
+أو bounded above محدودة
+
+151
+00:18:34,790 --> 00:18:42,530
+من أعلى and
+
+152
+00:18:42,530 --> 00:18:50,170
+let الـ U
+
+153
+00:18:50,170 --> 00:18:53,610
+بساوي الـ supremum للـ set A طبعا الـ set A bounded
+
+154
+00:18:56,560 --> 00:19:00,740
+above وبالتالي by الـ supremum property الـ supremum
+
+155
+00:19:00,740 --> 00:19:07,480
+تبعها exist دعنا نسميه U المطلوب
+
+156
+00:19:07,480 --> 00:19:14,240
+show that show
+
+157
+00:19:14,240 --> 00:19:22,340
+there exist يوجد an increasing ... an increasing
+
+158
+00:19:22,340 --> 00:19:29,900
+subsequence
+
+159
+00:19:29,900 --> 00:19:33,180
+أو sequence there exists an increasing sequence xn
+
+160
+00:19:33,180 --> 00:19:40,120
+contained in A such that الـ limit لسيكوينس xn
+
+161
+00:19:40,120 --> 00:19:51,480
+بساوي U
+
+162
+00:19:51,480 --> 00:19:52,880
+بساوي U
+
+163
+00:20:03,070 --> 00:20:08,150
+I prove من
+
+164
+00:20:08,150 --> 00:20:12,630
+خواص الـ supremum
+
+165
+00:20:12,630 --> 00:20:19,850
+احنا
+
+166
+00:20:19,850 --> 00:20:27,650
+فيها أنا كان لمبة دخلناها قبل هيك وهذه اللمبة بتقول
+
+167
+00:20:27,650 --> 00:20:28,230
+أنه
+
+168
+00:20:42,440 --> 00:20:51,320
+لما أخذناها في أول chapter بتقول أنه ... أفندم؟
+
+169
+00:20:51,320 --> 00:21:05,800
+بنقول an upper bound an upper bound U of الـ set A is
+
+170
+00:21:05,800 --> 00:21:09,540
+the supremum
+
+171
+00:21:11,820 --> 00:21:21,120
+the supremum of a, if and only if لكل
+
+172
+00:21:21,120 --> 00:21:29,260
+إبسلون أكبر من الصفر يوجد x إبسلون ينتمي إلى a
+
+173
+00:21:29,260 --> 00:21:33,680
+بحيث أنه بحيث
+
+174
+00:21:33,680 --> 00:21:37,920
+أنه u سالب إبسلون أصغر من x إبسلون
+
+175
+00:21:40,930 --> 00:21:46,950
+نظبط صح؟ نطبقها لأن أنا عندي هاي U بساوي الـ
+
+176
+00:21:46,950 --> 00:21:52,670
+supremum لـ A إذا لو أخدت for epsilon بساوي واحد
+
+177
+00:21:52,670 --> 00:22:04,190
+أكبر من الصفر يوجد X واحد ينتمي إلى A such that U
+
+178
+00:22:04,190 --> 00:22:17,500
+سالب واحد أصغر من x واحد next
+
+179
+00:22:17,500 --> 00:22:21,240
+for
+
+180
+00:22:21,240 --> 00:22:27,360
+for
+
+181
+00:22:27,360 --> 00:22:33,400
+epsilon بساوي نص لو
+
+182
+00:22:33,400 --> 00:22:37,200
+أخدت الـ epsilon هذه بساوي نص ف choose
+
+183
+00:22:40,760 --> 00:22:51,040
+choose by above لمّا X2
+
+184
+00:22:51,040 --> 00:23:01,180
+تنتمي إلى A وممكن نختار X2 أكبر من أو يساوي X1
+
+185
+00:23:01,180 --> 00:23:08,580
+الأولى هي such that U سالب نص الإبسلون أصغر من X2
+
+186
+00:23:19,440 --> 00:23:29,340
+بعدين now for إبسلون بساوي واحد على ثلاثة أكبر من
+
+187
+00:23:29,340 --> 00:23:34,220
+صفر it choose إذا في اللمّة هذه خدوا إبسلون بساوي
+
+188
+00:23:34,220 --> 00:23:39,520
+ثلث it choose X ثلاثة ينتمي إلى A
+
+189
+00:23:42,410 --> 00:23:48,590
+بحيث انه X ثلاثة هذا ممكن اختاره أكبر من أو يساوي X
+
+190
+00:23:48,590 --> 00:24:03,090
+اثنين and U اللي هو U سالب ثلث أصغر من X ثلاثة صح؟
+
+191
+00:24:03,090 --> 00:24:14,810
+continuing in this process لو استمرنا بالعملية الـ
+
+192
+00:24:14,810 --> 00:24:27,910
+continuing in this process we
+
+193
+00:24:27,910 --> 00:24:35,230
+get by induction عملية
+
+194
+00:24:35,230 --> 00:24:36,790
+استقرائية that
+
+195
+00:24:46,820 --> 00:24:52,740
+for epsilon بساوي واحد على ك أكبر من الصفر
+
+196
+00:24:56,070 --> 00:25:04,030
+there exists xk أكبر من أو يساوي xk زائد واحد such
+
+197
+00:25:04,030 --> 00:25:14,950
+that absolute u سالب واحد على k أصغر من xk وهذا
+
+198
+00:25:14,950 --> 00:25:23,530
+صحيح for every k ينتمي إلى n تمام؟
+
+199
+00:25:32,820 --> 00:25:40,120
+طيب أنا عندي ...
+
+200
+00:25:40,120 --> 00:25:50,560
+خلّيني أمسح اللمّة هذه طيب
+
+201
+00:25:50,560 --> 00:25:56,700
+إذا أنا عندي U نيجاتيف واحد على K طلع أصغر من XK
+
+202
+00:26:00,780 --> 00:26:08,580
+و الـ XK هذه أصغر من أو يساوي الـ U لأن الـ U هو الـ
+
+203
+00:26:08,580 --> 00:26:16,240
+supremum لـ A و XK عنصر في A و الـ U upper bound للـ
+
+204
+00:26:16,240 --> 00:26:22,840
+A ف ... و XK عنصر في A إذا الـ XK لازم يكون أصغر من
+
+205
+00:26:22,840 --> 00:26:31,010
+أو يساوي الـ U و الـ U أصغر من أو يساوي أو أصغر من u
+
+206
+00:26:31,010 --> 00:26:35,790
+زائد واحد على k الكلام هذا صحيح for all k belong
+
+207
+00:26:35,790 --> 00:26:44,830
+to n مظبوط صحيح so احنا أثبتنا هيك أن يوجد
+
+208
+00:26:44,830 --> 00:26:51,810
+sequence يوجد increasing sequence
+
+209
+00:26:51,810 --> 00:27:01,940
+x in او xk مهم XK contained in A such that absolute
+
+210
+00:27:01,940 --> 00:27:10,820
+XK minus U أصغر من واحد على K for all K تنتمي إلى
+
+211
+00:27:10,820 --> 00:27:19,920
+N طيب
+
+212
+00:27:19,920 --> 00:27:23,400
+ما هذا عبارة عن واحد على K في واحد
+
+213
+00:27:26,730 --> 00:27:32,150
+هذا أصغر من أو يساوي واحد في واحد على K لكل K ينتمي
+
+214
+00:27:32,150 --> 00:27:42,410
+ل N إذا hence by previous theorem اللي هي نظرية
+
+215
+00:27:42,410 --> 00:27:47,410
+فاكرينها اثنين أربعة في الـ notes نظرية اثنين أربعة
+
+216
+00:27:47,410 --> 00:27:48,810
+في الـ notes تبعتنا
+
+217
+00:27:54,490 --> 00:28:02,870
+with c بساوي واحد أكبر من صفر and a n او a k a k
+
+218
+00:28:02,870 --> 00:28:12,410
+بساوي واحد على k tends to zero we get تديني
+
+219
+00:28:22,310 --> 00:28:27,970
+نحصل على أن الـ limit ل xk as k tends to infinity
+
+220
+00:28:27,970 --> 00:28:35,470
+بساوي الـ U وهذا هو المطلوب مآم؟ واضح؟ لأن هذا هو
+
+221
+00:28:35,470 --> 00:28:40,490
+البرهان واضح البرهان؟ في أي استفسار؟ في أي شيء مش
+
+222
+00:28:40,490 --> 00:28:49,730
+واضح؟ طيب ماشي الحال خلينا نشوف هاي سؤال ثلاثة
+
+223
+00:29:01,490 --> 00:29:09,710
+سؤال اثنين section ثلاثة ثلاثة فهنا عندي x واحد
+
+224
+00:29:09,710 --> 00:29:17,070
+أنا عندي x واحد عدد أكبر من واحد و x n plus one
+
+225
+00:29:17,070 --> 00:29:22,290
+بساوي بنعرفه
+
+226
+00:29:22,290 --> 00:29:33,410
+على أنه اثنين سالب واحد على x n لكل n عدد طبيعي
+
+227
+00:29:33,410 --> 00:29:45,270
+show اثبتي أن الـ sequence x in is
+
+228
+00:29:45,270 --> 00:29:50,150
+bounded and
+
+229
+00:29:50,150 --> 00:29:56,810
+monotone يعني إما increasing أو
+
+230
+00:29:56,810 --> 00:29:57,330
+decreasing
+
+231
+00:30:01,980 --> 00:30:09,140
+بعدين find الـ limit find its
+
+232
+00:30:09,140 --> 00:30:17,980
+limit إذا إزاي نعملها في سؤال السؤال
+
+233
+00:30:17,980 --> 00:30:24,220
+الرابع إزاي السؤال الرابع okay هنا
+
+234
+00:30:24,220 --> 00:30:25,980
+في بس يعني الـ trick
+
+235
+00:30:32,350 --> 00:30:37,550
+لو كتبنا أول ثلاثة أربع خمس حدود نقدر نشوف وين يعني
+
+236
+00:30:37,550 --> 00:30:41,870
+الـ sequence محصورة بين أي أعداد ايه هو الـ upper
+
+237
+00:30:41,870 --> 00:30:46,950
+bound و الـ lower bound للـ sequence فمثلا لو بدي
+
+238
+00:30:46,950 --> 00:30:52,390
+احسب الحد رقم أي حد واحد أكبر من واحد طب الحد
+
+239
+00:30:52,390 --> 00:31:00,470
+الثاني بساوي اثنين سالب واحد على x واحد لأن أنا
+
+240
+00:31:00,470 --> 00:31:06,090
+عندي هنا باخد n بالساعة واحد تعطيني x اثنين طب أنا
+
+241
+00:31:06,090 --> 00:31:12,210
+عندي x واحد أكبر من واحد هذا بيؤدي إلى أن مخلوق x
+
+242
+00:31:12,210 --> 00:31:25,390
+واحد أصغر من واحد وبالتالي
+
+243
+00:31:33,910 --> 00:31:44,010
+إذا وهذا طبعا عدد موجب أكيد و أصغر من واحد إذا
+
+244
+00:31:44,010 --> 00:31:48,590
+أنا بطرح ... بطرح من الـ ... من الـ ... من الاثنين عدد
+
+245
+00:31:48,590 --> 00:31:54,730
+موجب و أصغر من واحد فهذا هيكون يعني أكيد أكبر من
+
+246
+00:31:54,730 --> 00:32:01,540
+واحد آه لأ هذا بدي يكون أصغر من اثنين هذا بالتأكيد
+
+247
+00:32:01,540 --> 00:32:05,980
+أصغر من اثنين لأن هذا عدد موجب هذا عدد موجب في
+
+248
+00:32:05,980 --> 00:32:09,480
+النهاية بغض النظر أكبر من واحد ولا أصغر من واحد
+
+249
+00:32:09,480 --> 00:32:15,440
+لما أطرح أنا عدد موجب من عدد العدد بيصغر صح فإذا x
+
+250
+00:32:15,440 --> 00:32:23,900
+اثنين أصغر من اثنين طب و x ثلاثة بساوي اثنين سالب
+
+251
+00:32:23,900 --> 00:32:25,540
+واحد على x اثنين
+
+252
+00:32:30,440 --> 00:32:35,840
+برضه هذا عدد موجب فهيكون هذا أصغر من ايه من اثنين
+
+253
+00:32:35,840 --> 00:32:44,960
+و هكذا إذا و طبعا x واحد وهذا
+
+254
+00:32:44,960 --> 00:32:51,740
+العدد أصغر من واحد فهذا أكيد هيطلع أكبر من الواحد
+
+255
+00:32:51,740 --> 00:32:56,460
+إذا هذا هيكون أكبر من واحد و هذا أكبر من واحد إذا
+
+256
+00:32:56,460 --> 00:33:00,880
+واضح أن حدود الـ sequence هتكون محصورة بين واحد و
+
+257
+00:33:00,880 --> 00:33:09,880
+اثنين إذا بقدر أنا أعمل ادعي بقدر ادعي أقول claim
+
+258
+00:33:09,880 --> 00:33:13,720
+واحد
+
+259
+00:33:13,720 --> 00:33:16,200
+أن الـ ...
+
+260
+00:33:21,340 --> 00:33:26,800
+إن Xn أكبر من أو يساوي الواحد أصغر من أو يساوي اثنين
+
+261
+00:33:26,800 --> 00:33:31,840
+لكل N طبعا
+
+262
+00:33:31,840 --> 00:33:41,140
+هذا يعني ممكن برهانه by induction طيب
+
+263
+00:33:41,140 --> 00:33:46,280
+الـ ... إذا هنا prove it
+
+264
+00:33:49,440 --> 00:34:00,540
+prove it by induction الكلام
+
+265
+00:34:00,540 --> 00:34:05,860
+الثاني أن
+
+266
+00:34:05,860 --> 00:34:14,020
+الـ sequence تبعتي بتطلع decreasing x
+
+267
+00:34:14,020 --> 00:34:15,820
+in is decreasing
+
+268
+00:34:21,770 --> 00:34:27,350
+يعني xn أكبر من أو يساوي xn زائد one for every n
+
+269
+00:34:27,350 --> 00:34:33,590
+belonging to n to
+
+270
+00:34:33,590 --> 00:34:41,110
+see this لبرهان ذلك to see this لبرهان ذلك note
+
+271
+00:34:41,110 --> 00:34:47,550
+that note
+
+272
+00:34:47,550 --> 00:34:48,130
+first
+
+273
+00:34:51,100 --> 00:35:05,320
+لاحظي أولاً أن x n ناقص واحد لكل تربيع لو أخدت حد
+
+274
+00:35:05,320 --> 00:35:11,120
+رقم n و طرحت منه واحد و ربعته هذا مربع كامل فهذا
+
+275
+00:35:11,120 --> 00:35:17,560
+أكيد أكبر من أو يساوي صفر أي مربع كامل أي مربع
+
+276
+00:35:17,560 --> 00:35:23,290
+كامل لأي عدد حقيقي بيطلع غير سالب طبعا هذا لما
+
+277
+00:35:23,290 --> 00:35:30,610
+نربعه بيطلع x x n squared ناقص اثنين x n زائد واحد
+
+278
+00:35:30,610 --> 00:35:40,470
+وهذا مجموع من
+
+279
+00:35:40,470 --> 00:35:44,690
+المتباينة هذه بنستنتج هذا بيؤدي
+
+280
+00:35:55,680 --> 00:36:10,260
+هذا بيؤدي أن اثنين X N زائد واحد أو 
+
+281
+00:36:10,260 --> 00:36:18,840
+هذا بيقدي أنه Xn+1 أكبر من أو يساوي Xn
+
+282
+00:36:18,840 --> 00:36:24,340
+تربية ودي هذا عن ناحية التانية لاحظي هذا كله أكبر 
+
+283
+00:36:24,340 --> 00:36:27,820
+من أو يساوي صفر ودي هذا عن ناحية التانية بطلع Xn
+
+284
+00:36:27,820 --> 00:36:33,720
+تربية أكبر من أو يساوي 2Xn - 1 وهذا
+
+285
+00:36:33,720 --> 00:36:40,660
+الكلام صحيح لكل N وهذا
+
+286
+00:36:40,660 --> 00:36:55,140
+بقدربدوره انه Xn+1 - Xn 
+
+287
+00:36:55,140 --> 00:37:06,760
+أكبر من أو يساوي 2(-1)
+
+288
+00:37:06,760 --> 00:37:07,820
+على Xn
+
+289
+00:37:18,300 --> 00:37:23,720
+إذا أنا جسمي إضرب في 1 على Xn الـ Xn هنا عدد ال
+
+290
+00:37:23,720 --> 00:37:30,200
+.. الـ Xn موجة بقى الـ Xn كلها عدد موجة بقى لاحظي 
+
+291
+00:37:30,200 --> 00:37:40,640
+انت هنا .. إن احنا ضربنا في Xn عدد موجب أو لأ جسمنا
+
+292
+00:37:40,640 --> 00:37:48,360
+أو ضربنا في 1 على Xn هذا عدد موجب فبطلع
+
+293
+00:37:48,360 --> 00:37:52,100
+عندي Xn وشريط المتتابعة طبق زي ما هي و 2
+
+294
+00:37:52,100 --> 00:37:56,500
+-1 على Xn الآن هذا by definition of the
+
+295
+00:37:56,500 --> 00:38:00,860
+sequence من الـ recursive formula هي الـ recursive
+
+296
+00:38:00,860 --> 00:38:04,720
+formula أو الـ inductive formula بتقول إن هذا الفرق
+
+297
+00:38:04,720 --> 00:38:09,900
+بتطلع Xn+1 إذا الكلام هذا صحيح لكل n
+
+298
+00:38:09,900 --> 00:38:15,900
+وبالتالي إذا هيطلع عندي Xn أكبر من أو يساوي Xn+1
+
+299
+00:38:15,900 --> 00:38:24,140
+لكل n إذا الـ sequence Xn is decreasing متناقصة
+
+300
+00:38:26,080 --> 00:38:35,420
+الآن من الـ claims 1 و 2 تطلع now X
+
+301
+00:38:35,420 --> 00:38:40,700
+n is
+
+302
+00:38:40,700 --> 00:38:44,460
+decreasing and bounded
+
+303
+00:38:48,230 --> 00:38:52,870
+حسب claim 1 و claim 2 so by monotone
+
+304
+00:38:52,870 --> 00:39:03,610
+convergence theorem Xn converges say limit Xn
+
+305
+00:39:03,610 --> 00:39:10,670
+بالساوي X for some X ينتمي إلى R الآن بنجيب قيمة
+
+306
+00:39:10,670 --> 00:39:22,760
+الـ limit اللي هي الـ X فنرجع للـ inductive formula to
+
+307
+00:39:22,760 --> 00:39:34,360
+find X we have من الـ inductive formula أنا عندي الـ
+
+308
+00:39:34,360 --> 00:39:44,140
+limit لـ Xn+1 equals 2 بالساوي 2
+
+309
+00:39:44,140 --> 00:39:50,820
+-1 على limit Xn هذا لما آخذ الـ limit
+
+310
+00:39:50,820 --> 00:39:57,140
+للطرفين في الـ inductive formula طب هذا طرف الشمال
+
+311
+00:39:57,140 --> 00:40:03,530
+بالساوي X والطرف اليمين 1 على X الآن حل المعادلة
+
+312
+00:40:03,530 --> 00:40:08,510
+هاد في X اضرب في X بطلع عندي X تربيع - 2X
+
+313
+00:40:08,510 --> 00:40:14,770
++1 بالساوي 0 وانت بتحلل الى X - 2
+
+314
+00:40:14,770 --> 00:40:25,550
+X - 1 الكل تربيع بالساوي 0 فبطلع X بالساوي
+
+315
+00:40:25,550 --> 00:40:31,630
+1 وهذا صحيح لأن الـ Xn لاحظوا في claim 1 Xn
+
+316
+00:40:31,630 --> 00:40:35,150
+معصورة بين 1 و 2 إذا الـ limit تبقى بتطلع 
+
+317
+00:40:35,150 --> 00:40:40,530
+معصورة بين 1 و 2 فالجواب 1 مقبول إذا هذا
+
+318
+00:40:40,530 --> 00:40:49,890
+هو هي الـ limit طلعت بالساوي 1 تمام في
+
+319
+00:40:49,890 --> 00:40:51,010
+أي أسئلة تانية
+
+320
+00:41:02,130 --> 00:41:06,830
+في معناه وجهة إنحال كمان سؤال إذا بتحبه سيكشن 
+
+321
+00:41:06,830 --> 00:41:08,210
+البعيدة 3 4
+
+322
+00:41:26,730 --> 00:41:33,870
+سيكشن 3 4 فش ولا سؤال عندكم
+
+323
+00:41:33,870 --> 00:41:44,030
+فحللكم 
+
+324
+00:41:44,030 --> 00:41:49,390
+سؤال يحداش لانه واضح إن انتوا مش دارسين فنحل
+
+325
+00:41:49,390 --> 00:41:52,370
+السؤال هيك أنا هحلكم يعني من هندي
+
+326
+00:41:57,800 --> 00:42:07,280
+إذن سؤال 11 سيكشن 3 4 suppose
+
+327
+00:42:07,280 --> 00:42:10,860
+افترضي
+
+328
+00:42:10,860 --> 00:42:19,320
+إن Xn أكبر من أو يساوي 0 for every natural number
+
+329
+00:42:19,320 --> 00:42:30,510
+n and الـ limit للـ sequence (-1)^n Xn
+
+330
+00:42:30,510 --> 00:42:34,150
+exists
+
+331
+00:42:34,150 --> 00:42:39,110
+show
+
+332
+00:42:39,110 --> 00:42:48,490
+برهنة إن الـ sequence Xn convergence
+
+333
+00:43:42,520 --> 00:43:51,900
+Okay خلينا نشوف الـ .. طيب احنا نشوف solution
+
+334
+00:43:55,820 --> 00:44:00,460
+say احنا فرضين إن الـ limit للـ sequence هذه exist
+
+335
+00:44:00,460 --> 00:44:07,520
+فافترضي إن الـ limit للـ sequence (-1)^n as n في
+
+336
+00:44:07,520 --> 00:44:14,720
+Xn as n tends to infinity الـ limit للـ sequence هذه
+
+337
+00:44:14,720 --> 00:44:19,620
+بيساوي X for some X ينتمي الـ R
+
+338
+00:44:27,140 --> 00:44:39,900
+الآن then the subsequences الـ subsequences اللي هي 
+
+339
+00:44:39,900 --> 00:44:49,780
+لو (-1)
+
+340
+00:44:49,780 --> 00:44:57,030
+^2n و (-1)^(2n-1) in X2n و X2n-1 in هذه الـ converge لـ
+
+341
+00:44:57,030 --> 00:45:02,030
+X هذه الـ subsequence من الـ sequence هذه بس حدودها 
+
+342
+00:45:02,030 --> 00:45:09,210
+الزوجية and كمان الـ subsequence اللي حدودها
+
+343
+00:45:09,210 --> 00:45:10,050
+الفردية
+
+344
+00:45:18,430 --> 00:45:22,850
+برضه converge لـ X لأن في عندي نظرية بتقول إذا كانت
+
+345
+00:45:22,850 --> 00:45:27,070
+الـ sequence convergent لـ X فأي subsequence منها
+
+346
+00:45:27,070 --> 00:45:31,270
+بتكون convergent لنفس الـ X هذه subsequence من الـ
+
+347
+00:45:31,270 --> 00:45:37,890
+sequence هذه أخدت الحدود الزوجية وهذه subsequence
+
+348
+00:45:37,890 --> 00:45:41,470
+من الـ sequence هذه اللي هي مدة تالية الحدود 
+
+349
+00:45:41,470 --> 00:45:49,920
+الفردية طيب من هنا لاحظوا (-1)^2n
+
+350
+00:45:49,920 --> 00:45:54,820
+تطلع 1 وبالتالي الـ sequence هذه هي نفس الـ
+
+351
+00:45:54,820 --> 00:46:04,500
+sequence X2n عفواً X2n و (-1) لما
+
+352
+00:46:04,500 --> 00:46:08,220
+يكون الأس تبعها فرد تطلع -1 إذا هذه -
+
+353
+00:46:08,220 --> 00:46:14,040
+يعني - الـ sequence X2n - 1
+
+354
+00:46:16,730 --> 00:46:35,630
+تمام؟ طيب أنا عندي من الفرض by
+
+355
+00:46:35,630 --> 00:46:40,990
+hypothesis من الفرض أنا عندي Xn أكبر من أو يساوي
+
+356
+00:46:40,990 --> 00:46:54,340
+0 لكل n في N فهذا بيدي إنه X2N و أيضا X2N-1 أكبر
+
+357
+00:46:54,340 --> 00:47:01,940
+من أو يساوي 0 لكل N في الـ natural numbers وهذا
+
+358
+00:47:01,940 --> 00:47:11,040
+بيقدي بدوره إلى إنه الـ X اللي هي من هنا X بالساوي
+
+359
+00:47:11,040 --> 00:47:14,500
+ليه بالساوي limit X2N
+
+360
+00:47:16,870 --> 00:47:22,690
+هي عندي هذا
+
+361
+00:47:22,690 --> 00:47:32,230
+هو هذا من هنا بطلع عندي limit X2N بالساوي X ومن هنا 
+
+362
+00:47:32,230 --> 00:47:41,810
+بطلع عندي limit - X2N - 1 بالساوي X أو 
+
+363
+00:47:45,510 --> 00:47:48,610
+الـ -1 بيطلع برة الـ limit فبطلع عندي limit 
+
+364
+00:47:48,610 --> 00:47:56,770
+X2N - 1 بالساوي - X مظبوط؟
+
+365
+00:47:56,770 --> 00:48:02,170
+إذا هي عندي أنا limit X2N - 1 هذا بيطلع 
+
+366
+00:48:02,170 --> 00:48:10,450
+بالساوي - X أو لأ في الأول الـ X بالساوي limit X2N
+
+367
+00:48:10,450 --> 00:48:15,280
+وهذا بيطلع أكبر من أو يساوي 0 لأن أنا عندي X2N
+
+368
+00:48:15,280 --> 00:48:19,900
+أكبر من أو يساوي 0 لكل n فلما تكون الـ sequence 
+
+369
+00:48:19,900 --> 00:48:23,800
+حدودها غير سالبة فالـ limit تبعتها إذا كانت 
+
+370
+00:48:23,800 --> 00:48:27,660
+convergent الـ limit تبعتها تطلع غير سالبة وكذلك
+
+371
+00:48:27,660 --> 00:48:36,980
+- X اللي هي بالساوي limit X2N - 1 برضه أنا 
+
+372
+00:48:36,980 --> 00:48:42,850
+عندي X2N - 1 كلهم أعداد غير سالبة والـ 
+
+373
+00:48:42,850 --> 00:48:46,550
+sequence هذه convergent إذا الـ limit تبعتها بتطلع 
+
+374
+00:48:46,550 --> 00:48:50,450
+غير سالبة إذا
+
+375
+00:48:50,450 --> 00:48:55,950
+أنا من هنا بطلع عندي حاجتين X أكبر من أو يساوي 0
+
+376
+00:48:55,950 --> 00:49:04,280
+and - X and - X أكبر من أو يساوي 0
+
+377
+00:49:04,280 --> 00:49:10,060
+هذا بيؤدي إن X أكبر من أو يساوي 0 and اضرب في
+
+378
+00:49:10,060 --> 00:49:15,520
+-1 بيطلع X أصغر من أو يساوي 0 الآن أنا في 
+
+379
+00:49:15,520 --> 00:49:19,740
+عندي عدد حقيقي أكبر من أو يساوي 0 and أصغر من أو
+
+380
+00:49:19,740 --> 00:49:30,300
+يساوي 0 بيؤدي إن X بالساوي 0 تمام إذا طلع عندي 
+
+381
+00:49:30,300 --> 00:49:31,020
+هنا الـ
+
+382
+00:49:34,250 --> 00:49:40,070
+الـ limit هذه X بالساوي 0 الآن
+
+383
+00:49:40,070 --> 00:49:51,850
+تعالوا نثبت now given |Xn| > 0 it
+
+384
+00:49:51,850 --> 00:49:59,030
+choose how they exist capital 
+
+385
+00:49:59,030 --> 00:50:01,550
+N عدد طبيعي
+
+386
+00:50:04,240 --> 00:50:11,660
+بحيث إن لكل n أكبر من أو يساوي N هذا بيقدي
+
+387
+00:50:11,660 --> 00:50:23,140
+إن |Xn - 0| بالساوي |(-1)^n Xn - 0| لأن 
+
+388
+00:50:23,140 --> 00:50:30,960
+|Xn - 0| لأن 
+
+389
+00:50:30,960 --> 00:50:33,340
+هنا المفروض اكتب هنا since
+
+390
+00:50:36,490 --> 00:50:44,610
+مش احنا فرضين إنه الـ since limit (-1)^n قص ان 
+
+391
+00:50:44,610 --> 00:50:54,750
+في Xn بالساوي X بالساوي 0 احنا 
+
+392
+00:50:54,750 --> 00:50:58,910
+من الفرض أنا عندي إن الـ limit هذه موجودة تمام
+
+393
+00:50:58,910 --> 00:51:04,270
+وفرضناها X واثبتنا إن الـ limit تبعتها طلعت X 
+
+394
+00:51:04,270 --> 00:51:10,220
+بالساوي 0 الآن بما إنه limit الـ sequence هذه بالساوي
+
+395
+00:51:10,220 --> 00:51:14,220
+0، إذا من تعريف الـ limit for any given epsilon
+
+396
+00:51:14,220 --> 00:51:18,540
+يوجد N يعتمد على epsilon بحيث لكل N أكبر 
+
+397
+00:51:18,540 --> 00:51:22,960
+من أو يساوي N بطلع المسافة بين الحد العام 
+
+398
+00:51:22,960 --> 00:51:30,200
+والـ limit X اللي هي 0 أصغر من epsilon فهيك أنا 
+
+399
+00:51:30,200 --> 00:51:36,540
+بطلع عندي |Xn - 0| أصغر من epsilon إذا 
+
+400
+00:51:36,540 --> 00:51:40,120
+هنا أثبتت إنه لأي epsilon أكبر من الـ 0 يوجد
+
+401
+00:51:40,120 --> 00:51:44,100
+N يعتمد على epsilon بحيث لكل N أكبر منه
+
+402
+00:51:44,100 --> 00:51:48,140
+يساوي N المسافة بين Xn و 0 أصغر من epsilon
+
+403
+00:51:48,140 --> 00:51:53,040
+فهذا بيقدي إنه هذا معناه حسب تعريف epsilon N
+
+404
+00:51:53,040 --> 00:51:58,260
+إن الـ limit للـ sequence Xn موجودة وبالساوي
+
+405
+00:51:58,260 --> 00:52:01,890
+0 وبالتالي هيك أثبتنا إن الـ sequence Xn
+
+406
+00:52:01,890 --> 00:52:06,390
+convergence ومش هيك وبس ونهايتها كمان بالساوي
+
+407
+00:52:06,390 --> 00:52:15,170
+0 okay تمام إذا هذا حل السؤال 11 ونكتفي بهذا
+
+408
+00:52:15,170 --> 00:52:21,690
+القدر من حل المسائل ونكمل إن شاء الله حل
+
+409
+00:52:21,690 --> 00:52:26,290
+المسائل في المناقشة القادمة يوم السبت الجاي أو يوم
+
+410
+00:52:26,290 --> 00:52:32,830
+الخميس مع الشعبات التانية فنوقف هنا ونواصل إن شاء
+
+411
+00:52:32,830 --> 00:52:40,790
+الله يوم السبت الجاي تكمل المناقشة السكاشن اللي هي
+
+412
+00:52:40,790 --> 00:52:43,550
+3 4 و 3 5 و 3 6 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/bwcuptIkF-o_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/bwcuptIkF-o_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..d4fddae3c1e9af7827736a74bebc69ab5013427d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/bwcuptIkF-o_raw.json
@@ -0,0 +1 @@
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"end": 221.61, "word": " ان", "probability": 0.484375}, {"start": 221.61, "end": 221.77, "word": " ال", "probability": 0.82958984375}, {"start": 221.77, "end": 222.25, "word": " sequence", "probability": 0.845703125}, {"start": 222.25, "end": 223.07, "word": " xn", "probability": 0.629638671875}, {"start": 223.07, "end": 225.47, "word": " converges", "probability": 0.82763671875}, {"start": 225.47, "end": 228.01, "word": " and", "probability": 0.88427734375}, {"start": 228.01, "end": 228.89, "word": " find", "probability": 0.79052734375}, {"start": 228.89, "end": 231.87, "word": " its", "probability": 0.830078125}, {"start": 231.87, "end": 232.37, "word": " limit", "probability": 0.958984375}], "temperature": 1.0}, {"id": 9, "seek": 25970, "start": 235.46, "end": 259.7, "text": "بنثبت أولا أن الـ sequence هذي convergence ونجيب ال limit تبعتها فمثلة زي هذه بنطبق عليها الـ monotone convergence theorem الـ monotone convergence theorem", "tokens": [3555, 1863, 12984, 3555, 2655, 5551, 12610, 995, 14739, 2423, 39184, 8310, 8032, 8848, 1829, 32181, 4032, 1863, 7435, 1829, 3555, 2423, 4948, 6055, 3555, 34268, 11296, 6156, 2304, 12984, 37977, 30767, 1829, 29538, 44945, 9566, 3555, 4587, 25894, 11296, 2423, 39184, 1108, 310, 546, 32181, 20904, 2423, 39184, 1108, 310, 546, 32181, 20904], "avg_logprob": -0.39857954653826627, "compression_ratio": 1.5071428571428571, "no_speech_prob": 0.0, "words": [{"start": 235.46, "end": 236.46, "word": "بنثبت", "probability": 0.6174560546875}, {"start": 236.46, "end": 236.86, "word": " أولا", "probability": 0.7041015625}, {"start": 236.86, "end": 237.06, "word": " أن", "probability": 0.407470703125}, {"start": 237.06, "end": 237.2, "word": " الـ", "probability": 0.53564453125}, {"start": 237.2, "end": 237.5, "word": " sequence", "probability": 0.25830078125}, {"start": 237.5, "end": 237.94, "word": " هذي", "probability": 0.4167887369791667}, {"start": 237.94, "end": 238.46, "word": " convergence", "probability": 0.366943359375}, {"start": 238.46, "end": 240.22, "word": " ونجيب", "probability": 0.84931640625}, {"start": 240.22, "end": 240.48, "word": " ال", "probability": 0.4345703125}, {"start": 240.48, "end": 240.9, "word": " limit", "probability": 0.66064453125}, {"start": 240.9, "end": 242.6, "word": " تبعتها", "probability": 0.840087890625}, {"start": 242.6, "end": 250.22, "word": " فمثلة", "probability": 0.6878662109375}, {"start": 250.22, "end": 250.5, "word": " زي", "probability": 0.74267578125}, {"start": 250.5, "end": 250.96, "word": " هذه", "probability": 0.8955078125}, {"start": 250.96, "end": 253.48, "word": " بنطبق", "probability": 0.86328125}, {"start": 253.48, "end": 253.98, "word": " عليها", "probability": 0.99267578125}, {"start": 253.98, "end": 254.3, "word": " الـ", "probability": 0.689208984375}, {"start": 254.3, "end": 254.8, "word": " monotone", "probability": 0.8152669270833334}, {"start": 254.8, "end": 255.5, "word": " convergence", "probability": 0.9482421875}, {"start": 255.5, "end": 256.2, "word": " theorem", "probability": 0.8857421875}, {"start": 256.2, "end": 257.3, "word": " الـ", "probability": 0.66796875}, {"start": 257.3, "end": 257.84, "word": " monotone", "probability": 0.9580078125}, {"start": 257.84, "end": 259.22, "word": " convergence", "probability": 0.95654296875}, {"start": 259.22, "end": 259.7, "word": " theorem", "probability": 0.90576171875}], "temperature": 1.0}, {"id": 10, "seek": 28206, "start": 262.1, "end": 282.06, "text": "فلازم نثبت حاجتين ان ال sequence هذه convergent لازم نثبت انها convergent لازم نثبت انها monotone يعني increasing او decreasing و bounded فلحظوا انتوا لو بدى احسب اول يعني", "tokens": [5172, 1211, 31377, 2304, 8717, 12984, 3555, 2655, 11331, 26108, 2655, 9957, 16472, 2423, 8310, 29538, 9652, 6930, 5296, 31377, 2304, 8717, 12984, 3555, 2655, 16472, 11296, 416, 331, 6930, 5296, 31377, 2304, 8717, 12984, 3555, 2655, 16472, 11296, 1108, 310, 546, 37495, 22653, 5662, 1975, 2407, 23223, 4032, 37498, 6156, 1211, 5016, 19913, 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" نثبت", "probability": 0.99267578125}, {"start": 269.56, "end": 270.38, "word": " انها", "probability": 0.911865234375}, {"start": 270.38, "end": 270.86, "word": " convergent", "probability": 0.670166015625}, {"start": 270.86, "end": 271.2, "word": " لازم", "probability": 0.9319661458333334}, {"start": 271.2, "end": 271.58, "word": " نثبت", "probability": 0.9967041015625}, {"start": 271.58, "end": 271.96, "word": " انها", "probability": 0.94140625}, {"start": 271.96, "end": 273.6, "word": " monotone", "probability": 0.7742513020833334}, {"start": 273.6, "end": 273.9, "word": " يعني", "probability": 0.835693359375}, {"start": 273.9, "end": 274.4, "word": " increasing", "probability": 0.91796875}, {"start": 274.4, "end": 274.72, "word": " او", "probability": 0.851318359375}, {"start": 274.72, "end": 275.22, "word": " decreasing", "probability": 0.974609375}, {"start": 275.22, "end": 275.62, "word": " و", "probability": 0.339111328125}, {"start": 275.62, "end": 275.98, "word": " bounded", "probability": 0.8974609375}, {"start": 275.98, "end": 280.1, "word": " فلحظوا", "probability": 0.9037109375}, {"start": 280.1, "end": 280.42, "word": " انتوا", "probability": 0.7921549479166666}, {"start": 280.42, "end": 280.56, "word": " لو", "probability": 0.9697265625}, {"start": 280.56, "end": 280.84, "word": " بدى", "probability": 0.5989990234375}, {"start": 280.84, "end": 281.2, "word": " احسب", "probability": 0.9275716145833334}, {"start": 281.2, "end": 281.64, "word": " اول", "probability": 0.895263671875}, {"start": 281.64, "end": 282.06, "word": " يعني", "probability": 0.887451171875}], "temperature": 1.0}, {"id": 11, "seek": 30873, "start": 284.39, "end": 308.73, "text": "الحل .. لحظة x واحد بساوي واحد طب x اتنين .. خد in بساوي واحد بطلع جدر اتنين زائد .. جدر اتنين زائد x واحد اللي هو جدر التلاتة .. x تلاتة بساوي جدر اتنين زائد x اتنين", "tokens": [6027, 5016, 1211, 4386, 5296, 5016, 19913, 3660, 2031, 36764, 24401, 4724, 3794, 995, 45865, 36764, 24401, 23032, 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او ساوي XK انا XK انا نحط شريط اكبر XK زياد اتنين", "tokens": [2407, 1783, 42, 30767, 1829, 18513, 36764, 24401, 23758, 1975, 1829, 3224, 22807, 39896, 9566, 1211, 3615, 1975, 9381, 17082, 2288, 9154, 1783, 42, 30767, 1829, 18513, 36764, 24401, 1975, 4117, 26890, 9154, 1975, 2407, 8608, 995, 45865, 1783, 42, 1975, 8315, 1783, 42, 1975, 8315, 8717, 5016, 9566, 13412, 16572, 9566, 1975, 4117, 26890, 1783, 42, 30767, 1829, 18513, 1975, 2655, 1863, 9957], "avg_logprob": -0.23221153846153847, "compression_ratio": 1.6759259259259258, "no_speech_prob": 0.0, "words": [{"start": 692.94, "end": 693.24, "word": "و", "probability": 0.91650390625}, {"start": 693.24, "end": 693.82, "word": " XK", "probability": 0.685302734375}, {"start": 693.82, "end": 694.26, "word": " زياد", "probability": 0.67919921875}, {"start": 694.26, "end": 694.72, "word": " واحد", "probability": 0.814697265625}, {"start": 694.72, "end": 695.02, "word": " هذا", "probability": 0.640625}, {"start": 695.02, "end": 696.42, "word": " ايه؟", "probability": 0.77935791015625}, {"start": 696.42, "end": 697.0, "word": " هيطلع", "probability": 0.8773193359375}, {"start": 697.0, "end": 697.58, "word": " اصغر", "probability": 0.9010009765625}, {"start": 697.58, "end": 697.88, "word": " من", "probability": 0.99560546875}, {"start": 697.88, "end": 700.78, "word": " XK", "probability": 0.5908203125}, {"start": 700.78, "end": 701.16, "word": " زياد", "probability": 0.86474609375}, {"start": 701.16, "end": 701.7, "word": " واحد", "probability": 0.985107421875}, {"start": 701.7, "end": 702.32, "word": " اكبر", "probability": 0.9236653645833334}, {"start": 702.32, "end": 702.62, "word": " من", "probability": 0.98779296875}, {"start": 702.62, "end": 702.84, "word": " او", "probability": 0.830810546875}, {"start": 702.84, "end": 703.46, "word": " ساوي", "probability": 0.8484700520833334}, {"start": 703.46, "end": 704.08, "word": " XK", "probability": 0.9619140625}, {"start": 704.08, "end": 706.5, "word": " انا", 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"probability": 0.799072265625}, {"start": 721.18, "end": 721.66, "word": " زيادة", "probability": 0.72869873046875}, {"start": 721.66, "end": 722.12, "word": " واحد", "probability": 0.962646484375}, {"start": 722.12, "end": 723.32, "word": " بأكبر", "probability": 0.8829345703125}, {"start": 723.32, "end": 723.48, "word": " من", "probability": 0.97998046875}, {"start": 723.48, "end": 723.66, "word": " أو", "probability": 0.94677734375}, {"start": 723.66, "end": 724.1, "word": " ساوي", "probability": 0.958984375}, {"start": 724.1, "end": 724.68, "word": " XK", "probability": 0.96435546875}, {"start": 724.68, "end": 724.88, "word": " من", "probability": 0.95947265625}, {"start": 724.88, "end": 725.02, "word": " ال", "probability": 0.52099609375}, {"start": 725.02, "end": 725.4, "word": " induction", "probability": 0.98291015625}, {"start": 725.4, "end": 726.08, "word": " hypothesis", "probability": 0.93115234375}, {"start": 726.08, "end": 727.22, "word": " من", "probability": 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" إذاً", "probability": 0.5673014322916666}, {"start": 750.21, "end": 750.71, "word": " this", "probability": 0.7666015625}, {"start": 750.71, "end": 753.59, "word": " completes", "probability": 0.9609375}, {"start": 753.59, "end": 757.25, "word": " this", "probability": 0.63623046875}, {"start": 757.25, "end": 757.89, "word": " completes", "probability": 0.98046875}, {"start": 757.89, "end": 758.17, "word": " the", "probability": 0.73046875}, {"start": 758.17, "end": 758.61, "word": " induction", "probability": 0.966796875}], "temperature": 1.0}, {"id": 29, "seek": 78958, "start": 763.32, "end": 789.58, "text": "وبالتالي إذا هيك بنكون أثبتنا ال claim بتاعي إن أنا في عندي two claims أول شي sequence bounded والتاني بيقول إن ال sequence increasing وبالتالي therefore إذا لو سمينا هذا claim one وهذا claim two", "tokens": [37746, 6027, 2655, 6027, 1829, 11933, 15730, 39896, 4117, 44945, 30544, 5551, 12984, 3555, 2655, 8315, 2423, 3932, 39894, 45761, 1829, 36145, 41850, 8978, 18871, 16254, 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802.23, "word": " it", "probability": 0.385009765625}, {"start": 802.23, "end": 807.21, "word": " claims", "probability": 0.87744140625}, {"start": 807.21, "end": 807.77, "word": " one", "probability": 0.67822265625}, {"start": 807.77, "end": 808.51, "word": " and", "probability": 0.96484375}, {"start": 808.51, "end": 809.35, "word": " two", "probability": 0.9521484375}, {"start": 809.35, "end": 810.47, "word": " and", "probability": 0.9111328125}, {"start": 810.47, "end": 810.79, "word": " الـ", "probability": 0.625732421875}, {"start": 810.79, "end": 811.39, "word": " monotone", "probability": 0.8624674479166666}, {"start": 811.39, "end": 812.27, "word": " convergence", "probability": 0.9619140625}, {"start": 812.27, "end": 812.93, "word": " theorem", "probability": 0.912109375}, {"start": 812.93, "end": 814.03, "word": " بيقدّوا", "probability": 0.6564453125}, {"start": 814.03, "end": 814.23, "word": " إن", "probability": 0.406494140625}, {"start": 814.23, "end": 814.71, "word": " sequence", "probability": 0.414794921875}, {"start": 814.71, "end": 815.25, "word": " x", "probability": 0.454833984375}, {"start": 815.25, "end": 815.65, "word": " in", "probability": 0.418701171875}, {"start": 815.65, "end": 818.35, "word": " convergence", "probability": 0.61376953125}, {"start": 818.35, "end": 821.53, "word": " say", "probability": 0.75048828125}, {"start": 821.53, "end": 824.29, "word": " دعنا", "probability": 0.9607747395833334}, {"start": 824.29, "end": 825.07, "word": " نسمي", "probability": 0.9755859375}, {"start": 825.07, "end": 825.19, "word": " ال", "probability": 0.84033203125}, {"start": 825.19, "end": 825.45, "word": " limit", "probability": 0.9443359375}, {"start": 825.45, "end": 826.13, "word": " تبعتها", "probability": 0.91455078125}], "temperature": 1.0}, {"id": 31, "seek": 85332, "start": 827.62, "end": 853.32, "text": "limit x in بساوي x طبعا هذا بيطلع عدد حقيقي الان بنوجد قيمة limit x هذه فبنرجع to find to find x لإيجاد قيمة ال x بناخد ال limit", 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find", "probability": 0.9501953125}, {"start": 843.64, "end": 846.62, "word": " to", "probability": 0.71435546875}, {"start": 846.62, "end": 847.04, "word": " find", "probability": 0.9501953125}, {"start": 847.04, "end": 847.42, "word": " x", "probability": 0.962890625}, {"start": 847.42, "end": 848.06, "word": " لإيجاد", "probability": 0.95}, {"start": 848.06, "end": 848.58, "word": " قيمة", "probability": 0.99560546875}, {"start": 848.58, "end": 848.74, "word": " ال", "probability": 0.91162109375}, {"start": 848.74, "end": 849.06, "word": " x", "probability": 0.41748046875}, {"start": 849.06, "end": 852.88, "word": " بناخد", "probability": 0.93408203125}, {"start": 852.88, "end": 853.02, "word": " ال", "probability": 0.7470703125}, {"start": 853.02, "end": 853.32, "word": " limit", "probability": 0.9599609375}], "temperature": 1.0}, {"id": 32, "seek": 86822, "start": 855.58, "end": 868.22, "text": "take limit of both sides of", "tokens": [27612, 4948, 295, 1293, 4881, 295], "avg_logprob": -0.35881695577076506, "compression_ratio": 0.8181818181818182, "no_speech_prob": 0.0, "words": [{"start": 855.58, "end": 856.2, "word": "take", "probability": 0.271728515625}, {"start": 856.2, "end": 857.92, "word": " limit", "probability": 0.970703125}, {"start": 857.92, "end": 861.34, "word": " of", "probability": 0.96435546875}, {"start": 861.34, "end": 861.72, "word": " both", "probability": 0.91796875}, {"start": 861.72, "end": 862.32, "word": " sides", "probability": 0.919921875}, {"start": 862.32, "end": 868.22, "word": " of", "probability": 0.9091796875}], "temperature": 1.0}, {"id": 33, "seek": 89080, "start": 875.54, "end": 890.8, "text": "بناخد ال limit لطرفين of both sides of المعادلة xn زي الواحد بساوي square root ل two plus xn to get", "tokens": [3555, 1863, 47283, 3215, 2423, 4948, 5296, 9566, 28480, 9957, 295, 1293, 4881, 295, 9673, 3615, 18513, 37977, 2031, 77, 30767, 1829, 2423, 14407, 24401, 4724, 3794, 995, 45865, 3732, 5593, 5296, 732, 1804, 2031, 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"probability": 0.74853515625}, {"start": 915.37, "end": 915.61, "word": " limit", "probability": 0.9873046875}, {"start": 915.61, "end": 916.21, "word": " xn", "probability": 0.98388671875}, {"start": 916.21, "end": 916.53, "word": " زي", "probability": 0.928955078125}, {"start": 916.53, "end": 917.21, "word": " واحد", "probability": 0.919677734375}, {"start": 917.21, "end": 918.79, "word": " بساوي", "probability": 0.9385986328125}, {"start": 918.79, "end": 919.17, "word": " x", "probability": 0.7412109375}], "temperature": 1.0}, {"id": 35, "seek": 93862, "start": 926.14, "end": 938.62, "text": "هذه المعادلة بيصير x تربية negative x", "tokens": [3224, 24192, 9673, 3615, 18513, 37977, 4724, 1829, 9381, 13546, 2031, 6055, 2288, 21292, 3660, 3671, 2031], "avg_logprob": -0.6831597122881148, "compression_ratio": 0.8923076923076924, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 926.14, "end": 927.34, "word": "هذه", "probability": 0.174652099609375}, {"start": 927.34, "end": 927.88, "word": " المعادلة", "probability": 0.83001708984375}, {"start": 927.88, "end": 936.56, "word": " بيصير", "probability": 0.551513671875}, {"start": 936.56, "end": 936.88, "word": " x", "probability": 0.57763671875}, {"start": 936.88, "end": 937.48, "word": " تربية", "probability": 0.5003814697265625}, {"start": 937.48, "end": 937.92, "word": " negative", "probability": 0.66552734375}, {"start": 937.92, "end": 938.62, "word": " x", "probability": 0.974609375}], "temperature": 1.0}, {"id": 36, "seek": 95878, "start": 939.46, "end": 958.78, "text": "نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين نجاتف اتنين", "tokens": [1863, 7435, 9307, 5172, 1975, 2655, 1863, 9957, 8717, 7435, 9307, 5172, 1975, 2655, 1863, 9957, 8717, 7435, 9307, 5172, 1975, 2655, 1863, 9957, 8717, 7435, 9307, 5172, 1975, 2655, 1863, 9957, 8717, 7435, 9307, 5172, 1975, 2655, 1863, 9957, 8717, 7435, 9307, 5172, 1975, 2655, 1863, 9957, 8717, 7435, 9307, 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"probability": 0.869384765625}, {"start": 942.38, "end": 942.72, "word": " اتنين", "probability": 0.9755859375}, {"start": 942.72, "end": 943.06, "word": " نجاتف", "probability": 0.8880615234375}, {"start": 943.06, "end": 943.64, "word": " اتنين", "probability": 0.9832763671875}, {"start": 943.64, "end": 945.16, "word": " نجاتف", "probability": 0.896728515625}, {"start": 945.16, "end": 945.84, "word": " اتنين", "probability": 0.9871826171875}, {"start": 945.84, "end": 946.5, "word": " نجاتف", "probability": 0.907470703125}, {"start": 946.5, "end": 947.16, "word": " اتنين", "probability": 0.9888916015625}, {"start": 947.16, "end": 949.82, "word": " نجاتف", "probability": 0.9178466796875}, {"start": 949.82, "end": 950.86, "word": " اتنين", "probability": 0.989501953125}, {"start": 950.86, "end": 953.54, "word": " نجاتف", "probability": 0.9259033203125}, {"start": 953.54, "end": 958.78, "word": " اتنين", "probability": 0.98974609375}], "temperature": 1.0}, {"id": 37, "seek": 98815, "start": 961.69, "end": 988.15, "text": "الـ xn أكبر من أو ساوي واحد أصغر من أو ساوي اتنين لكل n هذا من claim one بيقدي ان ال limit حسب نظرية سابقة هذا بيقدي ان ال limit ل xn أصغر من أو ساوي اتنين أكبر من أو ساوي الواحد", "tokens": [6027, 39184, 2031, 77, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 36764, 24401, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 1975, 2655, 1863, 9957, 5296, 28820, 297, 23758, 9154, 3932, 472, 4724, 1829, 4587, 16254, 16472, 2423, 4948, 11331, 35457, 8717, 19913, 2288, 10632, 8608, 16758, 28671, 23758, 4724, 1829, 4587, 16254, 16472, 2423, 4948, 5296, 2031, 77, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 1975, 2655, 1863, 9957, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2423, 14407, 24401], "avg_logprob": -0.12000363623333532, "compression_ratio": 1.972972972972973, "no_speech_prob": 0.0, "words": [{"start": 961.69, "end": 962.13, "word": "الـ", "probability": 0.676513671875}, {"start": 962.13, "end": 962.87, "word": " xn", "probability": 0.49072265625}, {"start": 962.87, "end": 963.69, "word": " أكبر", "probability": 0.9269205729166666}, {"start": 963.69, "end": 963.93, "word": " من", "probability": 0.9228515625}, {"start": 963.93, "end": 964.17, "word": " أو", "probability": 0.97119140625}, {"start": 964.17, "end": 964.73, "word": " ساوي", "probability": 0.8427734375}, {"start": 964.73, "end": 965.35, "word": " واحد", "probability": 0.92431640625}, {"start": 965.35, "end": 967.13, "word": " أصغر", "probability": 0.950439453125}, {"start": 967.13, "end": 967.37, "word": " من", "probability": 0.99658203125}, {"start": 967.37, "end": 967.59, "word": " أو", "probability": 0.9794921875}, {"start": 967.59, "end": 968.23, "word": " ساوي", "probability": 0.9601236979166666}, {"start": 968.23, "end": 968.75, "word": " اتنين", "probability": 0.95263671875}, {"start": 968.75, "end": 969.35, "word": " لكل", "probability": 0.969970703125}, {"start": 969.35, "end": 969.75, "word": " n", "probability": 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ممكن نحل باقي الأسئلة زي مثلا سؤال واحد و اتنين و تلاتة و خمسة و ستة و سابعة okay تمام في أي أسئلة تانية في هذا section السؤال تسعة", "tokens": [2407, 39896, 4117, 44945, 30544, 5551, 12984, 3555, 2655, 8315, 36145, 2423, 8310, 307, 9652, 6930, 4032, 39896, 49610, 3215, 8315, 2423, 4948, 6055, 3555, 34268, 11296, 20666, 2304, 12984, 1211, 3714, 43020, 8717, 5016, 1211, 4724, 995, 38436, 16247, 3794, 19986, 37977, 30767, 1829, 50113, 15040, 8608, 33604, 6027, 36764, 24401, 4032, 1975, 2655, 1863, 9957, 4032, 6055, 1211, 9307, 3660, 4032, 16490, 2304, 3794, 3660, 4032, 8608, 2655, 3660, 4032, 8608, 16758, 27884, 1392, 46811, 10943, 8978, 36632, 5551, 3794, 19986, 37977, 6055, 7649, 10632, 8978, 23758, 3541, 21136, 33604, 6027, 6055, 3794, 27884], "avg_logprob": -0.1481153350515464, "compression_ratio": 1.6093023255813954, "no_speech_prob": 0.0, "words": [{"start": 1017.51, "end": 1017.73, "word": "و", "probability": 0.84912109375}, {"start": 1017.73, "end": 1017.95, "word": " هيك", 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0.779296875}, {"start": 1039.15, "end": 1039.81, "word": " أسئلة", "probability": 0.982666015625}, {"start": 1039.81, "end": 1040.23, "word": " تانية", "probability": 0.9939778645833334}, {"start": 1040.23, "end": 1040.43, "word": " في", "probability": 0.95263671875}, {"start": 1040.43, "end": 1040.65, "word": " هذا", "probability": 0.82080078125}, {"start": 1040.65, "end": 1041.21, "word": " section", "probability": 0.56787109375}, {"start": 1041.21, "end": 1044.01, "word": " السؤال", "probability": 0.85498046875}, {"start": 1044.01, "end": 1044.55, "word": " تسعة", "probability": 0.7825520833333334}], "temperature": 1.0}, {"id": 40, "seek": 108659, "start": 1057.22, "end": 1086.6, "text": "سؤال تسعة هاي السؤال تسعة", "tokens": [3794, 33604, 6027, 6055, 3794, 27884, 8032, 47302, 21136, 33604, 6027, 6055, 3794, 27884], "avg_logprob": -0.17851562897364298, "compression_ratio": 1.2105263157894737, "no_speech_prob": 0.0, "words": [{"start": 1057.22, "end": 1057.92, "word": "سؤال", "probability": 0.9353841145833334}, {"start": 1057.92, "end": 1058.46, "word": " تسعة", "probability": 0.9454752604166666}, {"start": 1058.46, "end": 1085.54, "word": " هاي", "probability": 0.4742431640625}, {"start": 1085.54, "end": 1086.08, "word": " السؤال", "probability": 0.873046875}, {"start": 1086.08, "end": 1086.6, "word": " تسعة", "probability": 0.9884440104166666}], "temperature": 1.0}, {"id": 41, "seek": 111208, "start": 1087.85, "end": 1112.09, "text": "السيقشن تلاتة تلاتة السؤال هذا بيقول let a be infinite infinite subset of R و ال set هذه bounded", "tokens": [6027, 3794, 1829, 4587, 8592, 1863, 6055, 1211, 9307, 3660, 6055, 1211, 9307, 3660, 21136, 33604, 6027, 23758, 4724, 1829, 39648, 718, 257, 312, 13785, 13785, 25993, 295, 497, 4032, 2423, 992, 29538, 37498], "avg_logprob": -0.28950894049235754, "compression_ratio": 1.303921568627451, "no_speech_prob": 0.0, "words": [{"start": 1087.85, "end": 1088.75, "word": "السيقشن", "probability": 0.6556803385416666}, {"start": 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1106.75, "end": 1107.37, "word": " R", "probability": 0.66748046875}, {"start": 1107.37, "end": 1109.47, "word": " و", "probability": 0.425048828125}, {"start": 1109.47, "end": 1109.87, "word": " ال", "probability": 0.8203125}, {"start": 1109.87, "end": 1110.27, "word": " set", "probability": 0.9541015625}, {"start": 1110.27, "end": 1110.93, "word": " هذه", "probability": 0.448486328125}, {"start": 1110.93, "end": 1112.09, "word": " bounded", "probability": 0.9130859375}], "temperature": 1.0}, {"id": 42, "seek": 113360, "start": 1114.79, "end": 1133.61, "text": "أو bounded above محدودة من أعلى and let ال U", "tokens": [10721, 2407, 37498, 3673, 3714, 24401, 23328, 3660, 9154, 5551, 3615, 23942, 293, 718, 2423, 624], "avg_logprob": -0.26769301470588236, "compression_ratio": 0.8695652173913043, "no_speech_prob": 0.0, "words": [{"start": 1114.79, "end": 1115.15, "word": "أو", "probability": 0.6826171875}, {"start": 1115.15, "end": 1115.53, "word": " bounded", "probability": 0.55078125}, 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contained in A such that ال limit لسيكوينس xn بساوي U", "tokens": [1829, 29245, 3215, 364, 5662, 4386, 282, 5662, 13924, 655, 1975, 2407, 8310, 456, 8198, 364, 5662, 8310, 2031, 77, 16212, 294, 316, 1270, 300, 2423, 4948, 5296, 3794, 1829, 4117, 2407, 9957, 3794, 2031, 77, 4724, 3794, 995, 45865, 624], "avg_logprob": -0.31733631236212595, "compression_ratio": 1.3333333333333333, "no_speech_prob": 0.0, "words": [{"start": 1165.26, "end": 1166.66, "word": "يوجد", "probability": 0.9783528645833334}, {"start": 1166.66, "end": 1167.28, "word": " an", "probability": 0.405029296875}, {"start": 1167.28, "end": 1167.94, "word": " increasing", "probability": 0.873046875}, {"start": 1167.94, "end": 1169.0, "word": " ..an", "probability": 0.3450927734375}, {"start": 1169.0, "end": 1169.9, "word": " increasing", "probability": 0.95947265625}, {"start": 1169.9, "end": 1173.18, "word": " subsequence", "probability": 0.93505859375}, {"start": 1173.18, "end": 1174.04, "word": " او", "probability": 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"probability": 0.687744140625}, {"start": 1412.9, "end": 1413.16, "word": " هذه", "probability": 0.6943359375}, {"start": 1413.16, "end": 1413.48, "word": " خدوا", "probability": 0.9676106770833334}, {"start": 1413.48, "end": 1413.84, "word": " إبسلون", "probability": 0.98193359375}, {"start": 1413.84, "end": 1414.22, "word": " بساوي", "probability": 0.9580078125}, {"start": 1414.22, "end": 1414.64, "word": " تلت", "probability": 0.9622395833333334}, {"start": 1414.64, "end": 1415.38, "word": " it", "probability": 0.861328125}, {"start": 1415.38, "end": 1416.0, "word": " choose", "probability": 0.9111328125}, {"start": 1416.0, "end": 1417.42, "word": " X", "probability": 0.47900390625}, {"start": 1417.42, "end": 1418.12, "word": " تلاتة", "probability": 0.942138671875}, {"start": 1418.12, "end": 1418.84, "word": " ينتمي", "probability": 0.949951171875}, {"start": 1418.84, "end": 1419.12, "word": " إلى", "probability": 0.9365234375}, {"start": 1419.12, "end": 1419.52, "word": " A", 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{"start": 1425.35, "end": 1425.83, "word": " تلاتة", "probability": 0.8157958984375}, {"start": 1425.83, "end": 1426.07, "word": " هذا", "probability": 0.59375}, {"start": 1426.07, "end": 1426.45, "word": " ممكن", "probability": 0.983154296875}, {"start": 1426.45, "end": 1427.07, "word": " اختاره", "probability": 0.924072265625}, {"start": 1427.07, "end": 1427.51, "word": " اكبر", "probability": 0.8605143229166666}, {"start": 1427.51, "end": 1427.69, "word": " من", "probability": 0.98486328125}, {"start": 1427.69, "end": 1427.85, "word": " او", "probability": 0.804443359375}, {"start": 1427.85, "end": 1428.33, "word": " ساوي", "probability": 0.9261067708333334}, {"start": 1428.33, "end": 1428.59, "word": " X", "probability": 0.72900390625}, {"start": 1428.59, "end": 1429.15, "word": " اتنين", "probability": 0.8758544921875}, {"start": 1429.15, "end": 1430.77, "word": " and", "probability": 0.806640625}, {"start": 1430.77, "end": 1433.63, "word": " U", "probability": 0.79296875}, {"start": 1433.63, "end": 1434.11, "word": " اللي", "probability": 0.949951171875}, {"start": 1434.11, "end": 1434.57, "word": " هو", "probability": 0.9873046875}, {"start": 1434.57, "end": 1436.05, "word": " U", "probability": 0.90625}, {"start": 1436.05, "end": 1436.83, "word": " سالب", "probability": 0.94287109375}, {"start": 1436.83, "end": 1438.31, "word": " تلت", "probability": 0.9724934895833334}, {"start": 1438.31, "end": 1439.33, "word": " اصغر", "probability": 0.947998046875}, {"start": 1439.33, "end": 1439.53, "word": " من", "probability": 0.99609375}, {"start": 1439.53, "end": 1439.85, "word": " X", "probability": 0.994140625}, {"start": 1439.85, "end": 1440.89, "word": " تلاتة", "probability": 0.9661865234375}, {"start": 1440.89, "end": 1443.09, "word": " صح؟", "probability": 0.695556640625}, {"start": 1443.09, "end": 1445.43, "word": " continuing", "probability": 0.33984375}, {"start": 1445.43, "end": 1447.81, "word": " in", "probability": 0.35986328125}, {"start": 1447.81, "end": 1450.49, "word": " this", "probability": 0.96875}, {"start": 1450.49, "end": 1451.13, "word": " process", "probability": 0.97607421875}], "temperature": 1.0}, {"id": 53, "seek": 147679, "start": 1452.83, "end": 1476.79, "text": "لو استمرنا بالعملية الـ continuing in this process we get by induction عملية استقرائية that", "tokens": [1211, 2407, 44713, 29973, 8315, 20666, 25957, 1211, 10632, 2423, 39184, 9289, 294, 341, 1399, 321, 483, 538, 33371, 6225, 42213, 10632, 44713, 4587, 2288, 16373, 10632, 300], "avg_logprob": -0.2124191872004805, "compression_ratio": 1.1261261261261262, "no_speech_prob": 0.0, "words": [{"start": 1452.83, "end": 1453.21, "word": "لو", "probability": 0.961181640625}, {"start": 1453.21, "end": 1453.77, "word": " استمرنا", "probability": 0.93017578125}, {"start": 1453.77, "end": 1454.65, "word": " بالعملية", "probability": 0.90087890625}, {"start": 1454.65, "end": 1454.81, "word": " الـ", "probability": 0.227935791015625}, {"start": 1454.81, "end": 1455.45, "word": " continuing", "probability": 0.517578125}, {"start": 1455.45, "end": 1455.83, "word": " in", "probability": 0.88525390625}, {"start": 1455.83, "end": 1456.07, "word": " this", "probability": 0.966796875}, {"start": 1456.07, "end": 1456.69, "word": " process", "probability": 0.9853515625}, {"start": 1456.69, "end": 1467.91, "word": " we", "probability": 0.56005859375}, {"start": 1467.91, "end": 1468.45, "word": " get", "probability": 0.958984375}, {"start": 1468.45, "end": 1470.55, "word": " by", "probability": 0.95947265625}, {"start": 1470.55, "end": 1471.39, "word": " induction", "probability": 0.97705078125}, {"start": 1471.39, "end": 1475.23, "word": " عملية", "probability": 0.9830729166666666}, {"start": 1475.23, "end": 1476.09, "word": " استقرائية", "probability": 0.89296875}, {"start": 1476.09, "end": 1476.79, "word": " that", "probability": 0.8701171875}], "temperature": 1.0}, {"id": 54, "seek": 149274, "start": 1486.82, "end": 1492.74, "text": "for epsilon بساوي واحد على ك أكبر من السفر", "tokens": [2994, 17889, 4724, 3794, 995, 45865, 36764, 24401, 15844, 9122, 5551, 4117, 26890, 9154, 21136, 5172, 2288], "avg_logprob": -0.363715272810724, "compression_ratio": 0.8918918918918919, "no_speech_prob": 0.0, "words": [{"start": 1486.82, "end": 1487.34, "word": "for", "probability": 0.3291015625}, {"start": 1487.34, "end": 1487.86, "word": " epsilon", "probability": 0.48583984375}, {"start": 1487.86, "end": 1488.66, "word": " بساوي", "probability": 0.70556640625}, {"start": 1488.66, "end": 1489.22, "word": " واحد", "probability": 0.93408203125}, {"start": 1489.22, "end": 1489.42, "word": " على", "probability": 0.59228515625}, {"start": 1489.42, "end": 1489.78, "word": " ك", "probability": 0.6376953125}, {"start": 1489.78, "end": 1491.48, "word": " أكبر", "probability": 0.7938639322916666}, {"start": 1491.48, "end": 1491.66, "word": " من", "probability": 0.99169921875}, {"start": 1491.66, "end": 1492.74, "word": " السفر", "probability": 0.8361002604166666}], "temperature": 1.0}, {"id": 55, "seek": 152353, "start": 1496.07, "end": 1523.53, "text": "there exists xk أكبر من او ساوي xk زايد واحد such that absolute u سالب واحد على k أصغر من xk وهذا صحيح for every k ينتمي إلى n تمام؟", "tokens": [15456, 8198, 2031, 74, 5551, 4117, 26890, 9154, 1975, 2407, 8608, 995, 45865, 2031, 74, 30767, 995, 25708, 36764, 24401, 1270, 300, 8236, 344, 8608, 6027, 3555, 36764, 24401, 15844, 350, 5551, 9381, 17082, 2288, 9154, 2031, 74, 37037, 15730, 20328, 5016, 1829, 5016, 337, 633, 350, 7251, 29399, 2304, 1829, 30731, 297, 46811, 10943, 22807], "avg_logprob": -0.17269736842105263, "compression_ratio": 1.2751677852348993, "no_speech_prob": 0.0, "words": [{"start": 1496.07, "end": 1496.45, "word": "there", "probability": 0.355224609375}, {"start": 1496.45, "end": 1496.97, "word": " exists", "probability": 0.6552734375}, {"start": 1496.97, "end": 1498.03, "word": " xk", "probability": 0.77099609375}, {"start": 1498.03, "end": 1499.41, "word": " أكبر", "probability": 0.8792317708333334}, {"start": 1499.41, "end": 1499.63, "word": " من", "probability": 0.94189453125}, {"start": 1499.63, "end": 1501.37, "word": " او", "probability": 0.550048828125}, {"start": 1501.37, "end": 1501.65, "word": " ساوي", "probability": 0.7521158854166666}, {"start": 1501.65, "end": 1502.05, "word": " xk", "probability": 0.847412109375}, {"start": 1502.05, "end": 1502.49, "word": " زايد", "probability": 0.7390950520833334}, {"start": 1502.49, "end": 1502.99, "word": " واحد", "probability": 0.9521484375}, {"start": 1502.99, "end": 1504.03, "word": " such", "probability": 0.9326171875}, {"start": 1504.03, "end": 1505.07, "word": " that", "probability": 0.95849609375}, {"start": 1505.07, "end": 1506.35, "word": " absolute", "probability": 0.712890625}, {"start": 1506.35, "end": 1508.35, "word": " u", "probability": 0.67138671875}, {"start": 1508.35, "end": 1508.97, "word": " سالب", "probability": 0.9034830729166666}, {"start": 1508.97, "end": 1509.47, "word": " واحد", "probability": 0.982666015625}, {"start": 1509.47, "end": 1509.67, "word": " على", "probability": 0.8271484375}, {"start": 1509.67, "end": 1510.21, "word": " k", "probability": 0.69287109375}, {"start": 1510.21, "end": 1511.63, "word": " أصغر", "probability": 0.96826171875}, {"start": 1511.63, "end": 1511.99, "word": " من", "probability": 0.99560546875}, {"start": 1511.99, "end": 1513.67, "word": " xk", "probability": 0.966796875}, {"start": 1513.67, "end": 1514.95, "word": " وهذا", "probability": 0.819580078125}, {"start": 1514.95, "end": 1515.57, "word": " صحيح", "probability": 0.9969482421875}, {"start": 1515.57, "end": 1515.95, "word": " for", "probability": 0.966796875}, {"start": 1515.95, "end": 1516.49, "word": " every", "probability": 0.82421875}, {"start": 1516.49, "end": 1517.17, "word": " k", "probability": 0.95458984375}, {"start": 1517.17, "end": 1518.61, "word": " ينتمي", "probability": 0.963134765625}, {"start": 1518.61, "end": 1518.85, "word": " إلى", "probability": 0.896484375}, {"start": 1518.85, "end": 1519.17, "word": " n", "probability": 0.417724609375}, {"start": 1519.17, "end": 1523.53, "word": " تمام؟", "probability": 0.91796875}], "temperature": 1.0}, {"id": 56, "seek": 155670, "start": 1532.82, "end": 1556.7, "text": "طيب أنا عندي .. خلّيني أمسح اللمّة هذه طيب إذا أنا عندي U نيجاتيب واحد على K طلع أصغر من XK", "tokens": [9566, 1829, 3555, 41850, 18871, 16254, 4386, 16490, 1211, 11703, 9957, 1829, 5551, 2304, 3794, 5016, 13672, 2304, 11703, 3660, 29538, 23032, 1829, 3555, 11933, 15730, 41850, 18871, 16254, 624, 8717, 1829, 7435, 9307, 1829, 3555, 36764, 24401, 15844, 591, 23032, 1211, 3615, 5551, 9381, 17082, 2288, 9154, 1783, 42], "avg_logprob": -0.26639092202280085, "compression_ratio": 1.3220338983050848, "no_speech_prob": 0.0, "words": [{"start": 1532.82, "end": 1533.26, "word": "طيب", "probability": 0.8347981770833334}, {"start": 1533.26, "end": 1533.5, "word": " أنا", "probability": 0.5849609375}, {"start": 1533.5, "end": 1534.2, "word": " عندي", "probability": 0.8818359375}, {"start": 1534.2, "end": 1540.12, "word": " ..", "probability": 0.466064453125}, {"start": 1540.12, "end": 1542.6, "word": " خلّيني", "probability": 0.71220703125}, {"start": 1542.6, "end": 1543.08, "word": " أمسح", "probability": 0.943115234375}, {"start": 1543.08, "end": 1543.6, "word": " اللمّة", "probability": 0.76824951171875}, {"start": 1543.6, "end": 1544.4, "word": " هذه", "probability": 0.8505859375}, {"start": 1544.4, "end": 1550.56, "word": " طيب", "probability": 0.8932291666666666}, {"start": 1550.56, "end": 1550.92, "word": " إذا", "probability": 0.673583984375}, {"start": 1550.92, "end": 1551.36, "word": " أنا", "probability": 0.8681640625}, {"start": 1551.36, "end": 1551.88, "word": " عندي", "probability": 0.937744140625}, {"start": 1551.88, "end": 1553.1, "word": " U", "probability": 0.51513671875}, {"start": 1553.1, "end": 1553.74, "word": " نيجاتيب", "probability": 0.6347249348958334}, {"start": 1553.74, "end": 1554.2, "word": " واحد", "probability": 0.960693359375}, {"start": 1554.2, "end": 1554.42, "word": " على", "probability": 0.55712890625}, {"start": 1554.42, "end": 1554.76, "word": " K", "probability": 0.78759765625}, {"start": 1554.76, "end": 1555.16, "word": " طلع", "probability": 0.7442220052083334}, {"start": 1555.16, "end": 1555.76, "word": " أصغر", "probability": 0.98046875}, {"start": 1555.76, "end": 1556.0, "word": " من", "probability": 0.99267578125}, {"start": 1556.0, "end": 1556.7, "word": " XK", "probability": 0.760498046875}], "temperature": 1.0}, {"id": 57, "seek": 158894, "start": 1560.78, "end": 1588.94, "text": "و ال XK هذه أصغر من أو ساوي ال U لأن ال U هو ال supremum ل A و XK عنصر في A و ال U upper bound لل 6A ف .. و XK عنصر في A إذا ال XK لازم يكون أصغر من أو ساوي ال U و ال U أصغر من أو ساوي أو أصغر من", "tokens": [2407, 2423, 1783, 42, 29538, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 624, 5296, 33456, 2423, 624, 31439, 2423, 23710, 449, 5296, 316, 4032, 1783, 42, 18871, 9381, 2288, 8978, 316, 4032, 2423, 624, 6597, 5472, 24976, 1386, 32, 6156, 4386, 4032, 1783, 42, 18871, 9381, 2288, 8978, 316, 11933, 15730, 2423, 1783, 42, 5296, 31377, 2304, 7251, 30544, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 2423, 624, 4032, 2423, 624, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 34051, 5551, 9381, 17082, 2288, 9154], "avg_logprob": -0.11931046543885833, "compression_ratio": 1.9536423841059603, "no_speech_prob": 0.0, "words": [{"start": 1560.78, "end": 1561.24, "word": "و", "probability": 0.96630859375}, {"start": 1561.24, "end": 1561.56, "word": " ال", "probability": 0.77197265625}, {"start": 1561.56, "end": 1562.16, "word": " XK", "probability": 0.5606689453125}, {"start": 1562.16, "end": 1562.78, "word": " هذه", "probability": 0.4404296875}, {"start": 1562.78, "end": 1564.16, "word": " أصغر", "probability": 0.9293212890625}, {"start": 1564.16, "end": 1564.38, "word": " من", "probability": 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{"start": 1596.13, "end": 1596.47, "word": " n", "probability": 0.775390625}, {"start": 1596.47, "end": 1598.67, "word": " مظبوط", "probability": 0.7635986328125}, {"start": 1598.67, "end": 1599.25, "word": " صحيح", "probability": 0.873046875}, {"start": 1599.25, "end": 1601.81, "word": " so", "probability": 0.181640625}, {"start": 1601.81, "end": 1603.33, "word": " احنا", "probability": 0.8968098958333334}, {"start": 1603.33, "end": 1603.85, "word": " اثبتنا", "probability": 0.90634765625}, {"start": 1603.85, "end": 1604.13, "word": " هيك", "probability": 0.68505859375}, {"start": 1604.13, "end": 1604.29, "word": " ان", "probability": 0.73828125}, {"start": 1604.29, "end": 1604.83, "word": " يوجد", "probability": 0.9777018229166666}, {"start": 1604.83, "end": 1605.63, "word": " sequence", "probability": 0.9033203125}, {"start": 1605.63, "end": 1606.57, "word": " يوجد", "probability": 0.9659830729166666}, {"start": 1606.57, "end": 1608.41, "word": " increasing", "probability": 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"end": 1857.51, "text": "لو كتبنا اول تلت اربع خمس حدود نقدر نشوف وين يعني ال sequence محصورة بين اي اعداد ايه هو ال upper bound و ال lower bound لل sequence فمثلا لو بدي احسب الحد رقم اي حد واحد اكبر من واحد طب الحد التاني بساوي اتنين سالب واحد على اكس واحد", "tokens": [1211, 2407, 9122, 2655, 3555, 8315, 1975, 12610, 6055, 1211, 2655, 1975, 25513, 3615, 16490, 2304, 3794, 11331, 3215, 23328, 8717, 28543, 2288, 8717, 8592, 38688, 4032, 9957, 37495, 22653, 2423, 8310, 3714, 5016, 9381, 13063, 3660, 49374, 1975, 1829, 1975, 22488, 18513, 1975, 1829, 3224, 31439, 2423, 6597, 5472, 4032, 2423, 3126, 5472, 24976, 8310, 6156, 2304, 12984, 15040, 45164, 4724, 16254, 1975, 5016, 35457, 21542, 3215, 12602, 4587, 2304, 1975, 1829, 11331, 3215, 36764, 24401, 1975, 4117, 26890, 9154, 36764, 24401, 23032, 3555, 21542, 3215, 16712, 7649, 1829, 4724, 3794, 995, 45865, 1975, 2655, 1863, 9957, 8608, 6027, 3555, 36764, 24401, 15844, 1975, 4117, 3794, 36764, 24401], "avg_logprob": -0.15340908549048685, 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"seek": 254634, "start": 2517.8, "end": 2546.34, "text": "إذن سؤال 11 سيكشن تلاتة أربعة suppose افترضي ان xn أكبر من أو ساوي سفر for every natural number n and ال limit لل sequence", "tokens": [28814, 8848, 1863, 8608, 33604, 6027, 2975, 8608, 1829, 4117, 8592, 1863, 6055, 1211, 9307, 3660, 5551, 25513, 27884, 7297, 1975, 5172, 2655, 43042, 1829, 16472, 2031, 77, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 8608, 5172, 2288, 337, 633, 3303, 1230, 297, 293, 2423, 4948, 24976, 8310], "avg_logprob": -0.3324999874830246, "compression_ratio": 1.1712328767123288, "no_speech_prob": 0.0, "words": [{"start": 2517.8, "end": 2518.38, "word": "إذن", "probability": 0.5275472005208334}, {"start": 2518.38, "end": 2520.02, "word": " سؤال", "probability": 0.84765625}, {"start": 2520.02, "end": 2520.32, "word": " 11", "probability": 0.442626953125}, {"start": 2520.32, "end": 2521.04, "word": " سيكشن", "probability": 0.6455078125}, {"start": 2521.04, "end": 2521.62, "word": " تلاتة", "probability": 0.79742431640625}, {"start": 2521.62, "end": 2522.24, "word": " أربعة", "probability": 0.8678385416666666}, {"start": 2522.24, "end": 2527.28, "word": " suppose", "probability": 0.388916015625}, {"start": 2527.28, "end": 2530.86, "word": " افترضي", "probability": 0.87724609375}, {"start": 2530.86, "end": 2531.24, "word": " ان", "probability": 0.439697265625}, {"start": 2531.24, "end": 2532.82, "word": " xn", "probability": 0.337646484375}, {"start": 2532.82, "end": 2534.46, "word": " أكبر", "probability": 0.89404296875}, {"start": 2534.46, "end": 2534.66, "word": " من", "probability": 0.7294921875}, {"start": 2534.66, "end": 2534.8, "word": " أو", "probability": 0.958984375}, {"start": 2534.8, "end": 2535.2, "word": " ساوي", "probability": 0.87744140625}, {"start": 2535.2, "end": 2535.74, "word": " سفر", "probability": 0.82275390625}, {"start": 2535.74, "end": 2537.04, "word": " for", "probability": 0.9423828125}, {"start": 2537.04, "end": 2537.68, "word": " every", "probability": 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2564.33, "word": " sequence", "probability": 0.90087890625}, {"start": 2564.33, "end": 2565.29, "word": " xn", "probability": 0.963134765625}, {"start": 2565.29, "end": 2568.49, "word": " convergence", "probability": 0.76123046875}], "temperature": 1.0}, {"id": 93, "seek": 263190, "start": 2622.52, "end": 2631.9, "text": "Okay خلّينا نشوف ال .. طيب احنا نشوف solution", "tokens": [8297, 16490, 1211, 11703, 9957, 995, 8717, 8592, 38688, 2423, 4386, 23032, 1829, 3555, 1975, 5016, 8315, 8717, 8592, 38688, 3827], "avg_logprob": -0.26136363500898535, "compression_ratio": 1.0625, "no_speech_prob": 0.0, "words": [{"start": 2622.52, "end": 2623.44, "word": "Okay", "probability": 0.705078125}, {"start": 2623.44, "end": 2624.36, "word": " خلّينا", "probability": 0.69140625}, {"start": 2624.36, "end": 2624.82, "word": " نشوف", "probability": 0.9532877604166666}, {"start": 2624.82, "end": 2625.08, "word": " ال", "probability": 0.412353515625}, {"start": 2625.08, "end": 2625.34, "word": " ..", 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من الفرض by hypothesis من الفرض أنا عندي xn أكبر من أو ساوى سفر لكل n في n فهذا بيدّي", "tokens": [39237, 10943, 22807, 23032, 1829, 3555, 41850, 18871, 16254, 9154, 27188, 43042, 538, 17291, 9154, 27188, 43042, 41850, 18871, 16254, 2031, 77, 5551, 4117, 26890, 9154, 34051, 8608, 995, 2407, 7578, 8608, 5172, 2288, 5296, 28820, 297, 8978, 297, 6156, 3224, 15730, 4724, 25708, 11703, 1829], "avg_logprob": -0.1999667559532409, "compression_ratio": 1.3412698412698412, "no_speech_prob": 0.0, "words": [{"start": 2776.73, "end": 2777.81, "word": "تمام؟", "probability": 0.6678873697916666}, {"start": 2777.81, "end": 2778.89, "word": " طيب", "probability": 0.9392903645833334}, {"start": 2778.89, "end": 2779.81, "word": " أنا", "probability": 0.44970703125}, {"start": 2779.81, "end": 2780.15, "word": " عندي", "probability": 0.94677734375}, {"start": 2780.15, "end": 2780.31, "word": " من", "probability": 0.98828125}, {"start": 2780.31, "end": 2780.83, "word": " الفرض", "probability": 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0.8274739583333334}, {"start": 2801.41, "end": 2801.99, "word": " لكل", "probability": 0.983154296875}, {"start": 2801.99, "end": 2802.43, "word": " n", "probability": 0.79833984375}, {"start": 2802.43, "end": 2802.71, "word": " في", "probability": 0.9423828125}, {"start": 2802.71, "end": 2803.15, "word": " n", "probability": 0.75146484375}, {"start": 2803.15, "end": 2805.05, "word": " فهذا", "probability": 0.9508463541666666}, {"start": 2805.05, "end": 2805.69, "word": " بيدّي", "probability": 0.69775390625}], "temperature": 1.0}, {"id": 100, "seek": 283450, "start": 2807.56, "end": 2834.5, "text": "إنه X2N و أيضا X2N-1 أكبر من أو يساوي سفر لكل N في ال natural numbers وهذا بيقدي بدوره إلى إنه ال X اللي هي من هنا X بيساوي ليه بيساوي limit X2N", "tokens": [28814, 1863, 3224, 1783, 17, 45, 4032, 36632, 11242, 995, 1783, 17, 45, 12, 16, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 8608, 5172, 2288, 5296, 28820, 426, 8978, 2423, 3303, 3547, 37037, 15730, 4724, 1829, 4587, 16254, 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2814.86, "word": " أو", "probability": 0.95751953125}, {"start": 2814.86, "end": 2815.26, "word": " يساوي", "probability": 0.8587646484375}, {"start": 2815.26, "end": 2815.78, "word": " سفر", "probability": 0.8282877604166666}, {"start": 2815.78, "end": 2817.0, "word": " لكل", "probability": 0.9794921875}, {"start": 2817.0, "end": 2817.4, "word": " N", "probability": 0.841796875}, {"start": 2817.4, "end": 2817.88, "word": " في", "probability": 0.92578125}, {"start": 2817.88, "end": 2818.46, "word": " ال", "probability": 0.95654296875}, {"start": 2818.46, "end": 2818.76, "word": " natural", "probability": 0.3330078125}, {"start": 2818.76, "end": 2819.52, "word": " numbers", "probability": 0.70947265625}, {"start": 2819.52, "end": 2821.94, "word": " وهذا", "probability": 0.8583984375}, {"start": 2821.94, "end": 2822.58, "word": " بيقدي", "probability": 0.71112060546875}, {"start": 2822.58, "end": 2823.62, "word": " بدوره", "probability": 0.9588216145833334}, {"start": 2823.62, "end": 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limit", "probability": 0.8173828125}, {"start": 2832.28, "end": 2834.5, "word": " X2N", "probability": 0.9148763020833334}], "temperature": 1.0}, {"id": 101, "seek": 286181, "start": 2836.87, "end": 2861.81, "text": "هي عندي هذا هو هذا من هان بطلع عندي limit x2n بساوي x و من هان بطلع عندي limit سالب x2n سالب واحد بساوي x او", "tokens": [3224, 1829, 18871, 16254, 23758, 31439, 23758, 9154, 8032, 7649, 4724, 9566, 1211, 3615, 18871, 16254, 4948, 2031, 17, 77, 4724, 3794, 995, 45865, 2031, 4032, 9154, 8032, 7649, 4724, 9566, 1211, 3615, 18871, 16254, 4948, 8608, 6027, 3555, 2031, 17, 77, 8608, 6027, 3555, 36764, 24401, 4724, 3794, 995, 45865, 2031, 1975, 2407], "avg_logprob": -0.18409090909090908, "compression_ratio": 1.7835051546391754, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2836.87, "end": 2837.21, "word": "هي", "probability": 0.56597900390625}, {"start": 2837.21, "end": 2837.69, "word": " عندي", "probability": 0.824462890625}, {"start": 2837.69, "end": 2842.69, "word": " هذا", "probability": 0.666015625}, {"start": 2842.69, "end": 2843.01, "word": " هو", "probability": 0.974609375}, {"start": 2843.01, "end": 2843.49, "word": " هذا", "probability": 0.96484375}, {"start": 2843.49, "end": 2845.05, "word": " من", "probability": 0.9033203125}, {"start": 2845.05, "end": 2845.47, "word": " هان", "probability": 0.564453125}, {"start": 2845.47, "end": 2846.17, "word": " بطلع", "probability": 0.8546142578125}, {"start": 2846.17, "end": 2846.55, "word": " عندي", "probability": 0.874267578125}, {"start": 2846.55, "end": 2847.03, "word": " limit", "probability": 0.9658203125}, {"start": 2847.03, "end": 2849.89, "word": " x2n", "probability": 0.5244140625}, {"start": 2849.89, "end": 2850.43, "word": " بساوي", "probability": 0.80908203125}, {"start": 2850.43, "end": 2850.87, "word": " x", "probability": 0.919921875}, {"start": 2850.87, "end": 2851.73, "word": " و", "probability": 0.95849609375}, {"start": 2851.73, "end": 2851.93, "word": " من", 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289301, "start": 2865.51, "end": 2893.01, "text": "السالب واحد بيطلع برا ال limit فبطلع عندي limit x2n سالب واحد بساوي سالب x مظبوط؟ إذا هاي عندي أنا limit x2n سالب واحد هذا بيطلع بساوي سالب x أو لأ في الأول ال x بساوي limit x2n وهذا بيطلع أكبر من أو ساوي سفر", "tokens": [6027, 3794, 6027, 3555, 36764, 24401, 4724, 1829, 9566, 1211, 3615, 4724, 23557, 2423, 4948, 6156, 3555, 9566, 1211, 3615, 18871, 16254, 4948, 2031, 17, 77, 8608, 6027, 3555, 36764, 24401, 4724, 3794, 995, 45865, 8608, 6027, 3555, 2031, 3714, 19913, 3555, 2407, 9566, 22807, 11933, 15730, 8032, 47302, 18871, 16254, 41850, 4948, 2031, 17, 77, 8608, 6027, 3555, 36764, 24401, 23758, 4724, 1829, 9566, 1211, 3615, 4724, 3794, 995, 45865, 8608, 6027, 3555, 2031, 34051, 5296, 10721, 8978, 16247, 12610, 2423, 2031, 4724, 3794, 995, 45865, 4948, 2031, 17, 77, 37037, 15730, 4724, 1829, 9566, 1211, 3615, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.22372158792885868, "compression_ratio": 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limit تبعتها إذا كانت convergent ال limit تبعتها تطلع غير سالبة و كذلك سالب x اللي هي بتساوي limit x2n سالب واحد برضه أنا عندي x2n سالب واحد كلهم", "tokens": [1211, 33456, 41850, 18871, 16254, 2031, 17, 77, 5551, 4117, 26890, 9154, 34051, 3794, 3615, 11296, 1958, 5296, 28820, 297, 6156, 1211, 15042, 6055, 30544, 2423, 8310, 11331, 3215, 23328, 11296, 32771, 13546, 8608, 6027, 49401, 6156, 6027, 4948, 6055, 3555, 34268, 11296, 11933, 15730, 25961, 2655, 9652, 6930, 2423, 4948, 6055, 3555, 34268, 11296, 6055, 9566, 1211, 3615, 32771, 13546, 8608, 6027, 49401, 4032, 9122, 8848, 23275, 8608, 6027, 3555, 2031, 13672, 1829, 39896, 39894, 3794, 995, 45865, 4948, 2031, 17, 77, 8608, 6027, 3555, 36764, 24401, 4724, 43042, 3224, 41850, 18871, 16254, 2031, 17, 77, 8608, 6027, 3555, 36764, 24401, 28242, 16095], "avg_logprob": -0.18482142516544886, "compression_ratio": 1.8341708542713568, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2893.72, "end": 2894.1, "word": "لأن", 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0.352294921875}, {"start": 3078.82, "end": 3079.14, "word": " ساوي", "probability": 0.7947591145833334}, {"start": 3079.14, "end": 3079.54, "word": " capital", "probability": 0.64892578125}, {"start": 3079.54, "end": 3079.86, "word": " N", "probability": 0.97509765625}, {"start": 3079.86, "end": 3080.78, "word": " بطلع", "probability": 0.78887939453125}, {"start": 3080.78, "end": 3081.52, "word": " المسافة", "probability": 0.9869384765625}, {"start": 3081.52, "end": 3081.9, "word": " بين", "probability": 0.9765625}, {"start": 3081.9, "end": 3082.4, "word": " الحد", "probability": 0.98974609375}, {"start": 3082.4, "end": 3082.96, "word": " العام", "probability": 0.9833984375}, {"start": 3082.96, "end": 3084.02, "word": " وال", "probability": 0.60693359375}, {"start": 3084.02, "end": 3084.38, "word": " limit", "probability": 0.9794921875}, {"start": 3084.38, "end": 3084.78, "word": " x", "probability": 0.45361328125}, {"start": 3084.78, "end": 3084.98, "word": " اللي", "probability": 0.859375}, {"start": 3084.98, "end": 3085.16, "word": " هي", "probability": 0.86328125}, {"start": 3085.16, "end": 3085.62, "word": " سفر", "probability": 0.96337890625}, {"start": 3085.62, "end": 3086.3, "word": " أصغر", "probability": 0.9417724609375}, {"start": 3086.3, "end": 3086.48, "word": " من", "probability": 0.99365234375}, {"start": 3086.48, "end": 3086.92, "word": " epsilon", "probability": 0.73779296875}], "temperature": 1.0}, {"id": 110, "seek": 311860, "start": 3089.42, "end": 3118.6, "text": "فهيك أنا بطلع عندي absolute xn minus صفر أصغر من إبسلون إذا هان أثبتت أنه لأي إبسلون أكبر من الصفر يوجد capital N يعتمد على إبسلون بحيث لكل N أكبر منه ساوي capital N المسافة بين xn وصفر أصغر من إبسلون فهذا بيقدي أنه هذا معناه حسب تعريف epsilon capital N أنه ال limit لل sequence xn موجودة و بالساوية صفر", "tokens": [5172, 3224, 1829, 4117, 41850, 4724, 9566, 1211, 3615, 18871, 16254, 8236, 2031, 77, 3175, 20328, 5172, 2288, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 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3113.04, "word": " capital", "probability": 0.7265625}, {"start": 3113.04, "end": 3113.42, "word": " N", "probability": 0.896484375}, {"start": 3113.42, "end": 3114.24, "word": " أنه", "probability": 0.581298828125}, {"start": 3114.24, "end": 3114.32, "word": " ال", "probability": 0.4833984375}, {"start": 3114.32, "end": 3114.66, "word": " limit", "probability": 0.9169921875}, {"start": 3114.66, "end": 3115.78, "word": " لل", "probability": 0.78857421875}, {"start": 3115.78, "end": 3116.3, "word": " sequence", "probability": 0.94140625}, {"start": 3116.3, "end": 3116.94, "word": " xn", "probability": 0.962890625}, {"start": 3116.94, "end": 3117.7, "word": " موجودة", "probability": 0.97265625}, {"start": 3117.7, "end": 3117.8, "word": " و", "probability": 0.7998046875}, {"start": 3117.8, "end": 3118.26, "word": " بالساوية", "probability": 0.742529296875}, {"start": 3118.26, "end": 3118.6, "word": " صفر", "probability": 0.962890625}], "temperature": 1.0}, {"id": 111, "seek": 314841, "start": 3119.29, "end": 3148.41, "text": "وبالتالي هيك أثبتنا أن ال sequence x in convergence ومش هيك و بس و نهايتها كمان بالساوى سفر okay تمام إذا هذا حل السؤال 11 و نكتفي بهذا القدر من الحل المسائل و نكمل ان شاء الله حل المسائل في المناقشة القادمة يوم السبت الجاي أو يوم الأربع مع الشعبات تانية", "tokens": [37746, 6027, 2655, 6027, 1829, 39896, 4117, 5551, 12984, 3555, 2655, 8315, 14739, 2423, 8310, 2031, 294, 32181, 4032, 2304, 8592, 39896, 4117, 4032, 4724, 3794, 4032, 8717, 11296, 36081, 11296, 9122, 2304, 7649, 20666, 3794, 995, 2407, 7578, 8608, 5172, 2288, 1392, 46811, 10943, 11933, 15730, 23758, 11331, 1211, 21136, 33604, 6027, 2975, 4032, 8717, 4117, 2655, 41185, 39627, 15730, 25062, 3215, 2288, 9154, 21542, 1211, 9673, 3794, 16373, 1211, 4032, 8717, 24793, 1211, 16472, 13412, 16606, 21984, 11331, 1211, 9673, 3794, 16373, 1211, 8978, 9673, 8315, 4587, 8592, 3660, 25062, 18513, 46007, 7251, 20498, 21136, 3555, 2655, 25724, 47302, 34051, 7251, 20498, 16247, 25513, 3615, 20449, 25124, 3615, 3555, 9307, 6055, 7649, 10632], "avg_logprob": -0.19531250513833145, "compression_ratio": 1.7622950819672132, "no_speech_prob": 0.0, "words": [{"start": 3119.29, "end": 3119.87, "word": "وبالتالي", "probability": 0.90703125}, {"start": 3119.87, "end": 3120.09, "word": " هيك", "probability": 0.734619140625}, {"start": 3120.09, "end": 3120.55, "word": " أثبتنا", "probability": 0.9611328125}, {"start": 3120.55, "end": 3120.75, "word": " أن", "probability": 0.42529296875}, {"start": 3120.75, "end": 3120.91, "word": " ال", "probability": 0.591796875}, {"start": 3120.91, "end": 3121.19, "word": " sequence", "probability": 0.87255859375}, {"start": 3121.19, "end": 3121.61, "word": " x", "probability": 0.66845703125}, {"start": 3121.61, "end": 3121.89, "word": " in", "probability": 0.284912109375}, {"start": 3121.89, "end": 3122.89, "word": " convergence", "probability": 0.1922607421875}, {"start": 3122.89, "end": 3123.51, "word": " ومش", "probability": 0.9602864583333334}, {"start": 3123.51, "end": 3123.73, "word": " هيك", "probability": 0.976318359375}, {"start": 3123.73, "end": 3123.79, "word": " و", "probability": 0.318359375}, {"start": 3123.79, "end": 3124.13, "word": " بس", "probability": 0.907958984375}, {"start": 3124.13, "end": 3124.97, "word": " و", "probability": 0.884765625}, {"start": 3124.97, "end": 3125.51, "word": " نهايتها", "probability": 0.7979736328125}, {"start": 3125.51, "end": 3125.89, "word": " كمان", "probability": 0.88232421875}, {"start": 3125.89, "end": 3126.39, "word": " بالساوى", "probability": 0.745166015625}, {"start": 3126.39, "end": 3126.73, "word": " سفر", "probability": 0.8419596354166666}, {"start": 3126.73, "end": 3129.69, "word": " okay", "probability": 0.40869140625}, {"start": 3129.69, "end": 3130.25, "word": " تمام", "probability": 0.885009765625}, {"start": 3130.25, "end": 3131.05, "word": " إذا", "probability": 0.5662841796875}, {"start": 3131.05, "end": 3131.33, "word": " هذا", "probability": 0.66943359375}, {"start": 3131.33, "end": 3131.61, "word": " حل", "probability": 0.895751953125}, {"start": 3131.61, "end": 3132.03, "word": " السؤال", "probability": 0.8069661458333334}, {"start": 3132.03, "end": 3132.53, "word": " 11", "probability": 0.29541015625}, {"start": 3132.53, "end": 3133.13, "word": " و", "probability": 0.9267578125}, {"start": 3133.13, "end": 3134.71, "word": " نكتفي", "probability": 0.9239501953125}, {"start": 3134.71, "end": 3135.17, "word": " بهذا", "probability": 0.9775390625}, {"start": 3135.17, "end": 3135.73, "word": " القدر", "probability": 0.9913736979166666}, {"start": 3135.73, "end": 3136.03, "word": " من", "probability": 0.98583984375}, {"start": 3136.03, "end": 3136.81, "word": " الحل", "probability": 0.919677734375}, {"start": 3136.81, "end": 3138.01, "word": " المسائل", "probability": 0.9173583984375}, {"start": 3138.01, "end": 3138.83, "word": " و", "probability": 0.9638671875}, {"start": 3138.83, "end": 3140.85, "word": " نكمل", "probability": 0.8114420572916666}, {"start": 3140.85, "end": 3141.03, "word": " ان", "probability": 0.513671875}, {"start": 3141.03, "end": 3141.23, "word": " شاء", "probability": 0.982421875}, {"start": 3141.23, "end": 3141.31, "word": " الله", "probability": 0.95703125}, {"start": 3141.31, "end": 3141.69, "word": " حل", "probability": 0.98486328125}, {"start": 3141.69, "end": 3142.37, "word": " المسائل", "probability": 0.9713134765625}, {"start": 3142.37, "end": 3143.25, "word": " في", "probability": 0.890625}, {"start": 3143.25, "end": 3143.97, "word": " المناقشة", "probability": 0.989453125}, {"start": 3143.97, "end": 3144.51, "word": " القادمة", "probability": 0.9923502604166666}, {"start": 3144.51, "end": 3144.79, "word": " يوم", "probability": 0.989501953125}, {"start": 3144.79, "end": 3145.31, "word": " السبت", "probability": 0.8870442708333334}, {"start": 3145.31, "end": 3145.77, "word": " الجاي", "probability": 0.705078125}, {"start": 3145.77, "end": 3146.07, "word": " أو", "probability": 0.70263671875}, {"start": 3146.07, "end": 3146.29, "word": " يوم", "probability": 0.9873046875}, {"start": 3146.29, "end": 3146.85, "word": " الأربع", "probability": 0.8390299479166666}, {"start": 3146.85, "end": 3147.45, "word": " مع", "probability": 0.98583984375}, {"start": 3147.45, "end": 3148.01, "word": " الشعبات", "probability": 0.8763427734375}, {"start": 3148.01, "end": 3148.41, "word": " تانية", "probability": 0.8650716145833334}], "temperature": 1.0}, {"id": 112, "seek": 316355, "start": 3150.67, "end": 3163.55, "text": "فنوقف هنا ونواصل ان شاء الله يوم السبت الجاي تكمل المناخشة السكاشن اللي هي تلاتة أربعة و تلاتة خمسة و تلاتة ستة", "tokens": [5172, 1863, 30543, 5172, 34105, 4032, 1863, 2407, 33546, 1211, 16472, 13412, 16606, 21984, 7251, 20498, 21136, 3555, 2655, 25724, 47302, 6055, 24793, 1211, 9673, 8315, 9778, 8592, 3660, 21136, 4117, 33599, 1863, 13672, 1829, 39896, 6055, 1211, 9307, 3660, 5551, 25513, 27884, 4032, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 4032, 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"end": 3159.03, "word": " المناخشة", "probability": 0.7546875}, {"start": 3159.03, "end": 3159.79, "word": " السكاشن", "probability": 0.679534912109375}, {"start": 3159.79, "end": 3160.53, "word": " اللي", "probability": 0.823486328125}, {"start": 3160.53, "end": 3160.79, "word": " هي", "probability": 0.86328125}, {"start": 3160.79, "end": 3161.31, "word": " تلاتة", "probability": 0.9058837890625}, {"start": 3161.31, "end": 3161.71, "word": " أربعة", "probability": 0.8235677083333334}, {"start": 3161.71, "end": 3161.85, "word": " و", "probability": 0.76123046875}, {"start": 3161.85, "end": 3162.23, "word": " تلاتة", "probability": 0.9677734375}, {"start": 3162.23, "end": 3162.71, "word": " خمسة", "probability": 0.9884033203125}, {"start": 3162.71, "end": 3162.81, "word": " و", "probability": 0.92822265625}, {"start": 3162.81, "end": 3163.15, "word": " تلاتة", "probability": 0.9803466796875}, {"start": 3163.15, "end": 3163.55, "word": " ستة", "probability": 0.9791666666666666}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 3164.50825, "duration_after_vad": 2710.391249999989}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/db6AFymIrl8.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/db6AFymIrl8.srt
new file mode 100644
index 0000000000000000000000000000000000000000..95cd554aeb1d7ef65cb77bd16186b2d0495797a3
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/db6AFymIrl8.srt
@@ -0,0 +1,1227 @@
+1
+00:00:19,490 --> 00:00:23,130
+بسم الله الرحمن الرحيم في المحاضرة هذه هناخد يعني
+
+2
+00:00:23,130 --> 00:00:31,510
+مناقشة أو هنناقش بعض المسائل في section 4-1 و 4-2 
+
+3
+00:00:31,510 --> 00:00:41,650
+فإحدى الطالبات سألت في سؤال بقول برهن الجزء B من 
+
+4
+00:00:41,650 --> 00:00:52,390
+theorem 4.2.4 using the sequential criterion لأن هنا
+
+5
+00:00:52,390 --> 00:00:56,910
+use sequential
+
+6
+00:00:56,910 --> 00:01:01,370
+.. sequential
+
+7
+00:01:01,370 --> 00:01:06,410
+argument use 
+
+8
+00:01:06,410 --> 00:01:15,290
+sequential formulation on
+
+9
+00:01:15,290 --> 00:01:15,750
+limit
+
+10
+00:01:21,840 --> 00:01:27,040
+فالبرهان ذلك let .. هنستخدم الـ sequential
+
+11
+00:01:27,040 --> 00:01:30,380
+criterion
+
+12
+00:01:30,380 --> 00:01:39,860
+let x_n be a sequence contained in A such that
+
+13
+00:01:39,860 --> 00:01:50,000
+limit طبعا حدودها تباعتها مختلفة عن الـ C such
+
+14
+00:01:50,000 --> 00:01:56,610
+that lim x_n as n tends to infinity = c
+
+15
+00:01:56,610 --> 00:02:03,050
+خلّينا نختار sequence في المجال المشترك تبع
+
+16
+00:02:03,050 --> 00:02:08,290
+الدالتين F و H وحدودها مختلفة عن ال C ونهايتها بساوي
+
+17
+00:02:08,290 --> 00:02:08,610
+C
+
+18
+00:02:16,250 --> 00:02:21,290
+by the sequential criterion حسب sequential criterion
+
+19
+00:02:21,290 --> 00:02:32,430
+for limits to show ان ال limit ل F على H as X
+
+20
+00:02:32,430 --> 00:02:42,590
+tends to C = L على H it
+
+21
+00:02:42,590 --> 00:02:43,570
+suffices
+
+22
+00:02:45,740 --> 00:02:53,260
+it suffices to show يكفي اثبات ان ال limit لل
+
+23
+00:02:53,260 --> 00:03:02,220
+image لسيكوينس xn لما n تقول ل infinity = L
+
+24
+00:03:02,220 --> 00:03:07,020
+على H لو اثبتت الكلام هذا فحسب السيكوينش هي كتير
+
+25
+00:03:07,020 --> 00:03:11,380
+تانيا بطلع limit F على H = capital L على
+
+26
+00:03:11,380 --> 00:03:12,140
+capital H
+
+27
+00:03:17,270 --> 00:03:24,050
+فنشوف to this end to
+
+28
+00:03:24,050 --> 00:03:36,670
+this end و لإثبات ذلك يعني we
+
+29
+00:03:36,670 --> 00:03:40,710
+have from
+
+30
+00:03:40,710 --> 00:03:44,170
+the sequential criterion
+
+31
+00:03:47,320 --> 00:03:56,200
+that lim f(x_n) as n tends to infinity = 
+
+32
+00:03:56,200 --> 00:04:01,600
+L أنا عندي فارض ان lim f(x) من x أول ل c =
+
+33
+00:04:01,600 --> 00:04:06,720
+L إذا by the sequential criterion هذا بكافئ انه لأي
+
+34
+00:04:06,720 --> 00:04:12,680
+sequence x_n نهايتها c بطلع نهاية صورتها = L
+
+35
+00:04:12,680 --> 00:04:18,140
+و كذلك and أنا عندي lim الـ function h(x) من x
+
+36
+00:04:18,140 --> 00:04:22,320
+تقولها c = h by the sequential criterion كمان بره
+
+37
+00:04:22,320 --> 00:04:27,120
+طبخيها على ال function h بما أن x_n sequence
+
+38
+00:04:27,120 --> 00:04:34,460
+نهايتها c إذا نهايت صرتها under h يعني lim h(
+
+39
+00:04:34,460 --> 00:04:43,840
+x_n) as n tends to infinity = capital H hence
+
+40
+00:04:45,840 --> 00:04:57,980
+وبالتالي ال limit ل f على h(x) لما x
+
+41
+00:04:57,980 --> 00:05:10,220
+تقول ل xn لما n تقول ل infinity هذا بساوي lim
+
+42
+00:05:12,840 --> 00:05:22,580
+f(x_n) على h(x_n) لما n تقول infinity ويساوي
+
+43
+00:05:22,580 --> 00:05:29,560
+أنا عندي lim h(x_n) exist وبيستويش صفر
+
+44
+00:05:29,560 --> 00:05:35,080
+وبيستويش صفر فممكن استخدم قوانين النهايات لسيكوانس
+
+45
+00:05:35,080 --> 00:05:40,420
+فlim sequence على lim sequence = lim
+
+46
+00:05:40,420 --> 00:05:41,140
+البسط
+
+47
+00:05:44,990 --> 00:05:50,170
+lim ال sequence في ال بسط على lim ال sequence
+
+48
+00:05:50,170 --> 00:05:55,330
+في المقام و
+
+49
+00:05:55,330 --> 00:05:58,010
+lim ال sequence في المقام بيساوي صفر اذا انا
+
+50
+00:05:58,010 --> 00:06:01,090
+بقدر ايه اقول lim ال quotient بيساوي the
+
+51
+00:06:01,090 --> 00:06:07,330
+quotient of the limits وهذا بيطلع lim ال بسط
+
+52
+00:06:07,330 --> 00:06:14,920
+تطلع L lim المقام H وهذا البنيةإن حسب الـ
+
+53
+00:06:14,920 --> 00:06:19,020
+sequential criterion بيطلع lim F على H عندما X
+
+54
+00:06:19,020 --> 00:06:24,020
+تقوى ل C = L على H هذا هو المقصود الـ
+
+55
+00:06:24,020 --> 00:06:28,740
+sequential formulation واضح؟ طيب، مين عندها أسئلة
+
+56
+00:06:28,740 --> 00:06:35,540
+تانية؟ في section 4, 1 أو 4, 2؟
+
+57
+00:06:35,540 --> 00:06:40,080
+طيب،
+
+58
+00:06:40,080 --> 00:06:41,580
+خليني أمسح اللوحي الأول
+
+59
+00:07:20,820 --> 00:07:28,800
+هذه السؤال تمانية section أربعة واحد show that
+
+60
+00:07:28,800 --> 00:07:38,120
+أثبتي أنه ال limit of the square root of X الـ
+
+61
+00:07:38,120 --> 00:07:44,060
+square root function as X tends to C = ال
+
+62
+00:07:44,060 --> 00:07:48,340
+square root of C for any
+
+63
+00:07:51,920 --> 00:07:59,200
+C أكبر من السفر proof
+
+64
+00:08:02,940 --> 00:08:09,860
+أحنا في النهاية عايزين نثبت أنه الفرق بين f(x)
+
+65
+00:08:09,860 --> 00:08:16,700
+absolute الفرق بين f(x) و f(c) اللي هي جدر ال
+
+66
+00:08:16,700 --> 00:08:21,680
+c بدنا في النهاية هذا يكون أصغر من أي given
+
+67
+00:08:21,680 --> 00:08:22,140
+epsilon
+
+68
+00:08:25,220 --> 00:08:31,760
+حيث x المسافة بينها وبين z ال c أصغر من delta و
+
+69
+00:08:31,760 --> 00:08:36,480
+delta to be determined يعني هيتم تعيينها لاحقا
+
+70
+00:08:36,480 --> 00:08:42,240
+okay طب ما هذا بيساوي absolute
+
+71
+00:08:46,520 --> 00:08:54,060
+must go هاي جدر ال X - جدر ال C و بنضرب في
+
+72
+00:08:54,060 --> 00:09:03,660
+المرافق اللي هو جدر ال X + جدر ال C في جدر ال X
+
+73
+00:09:03,660 --> 00:09:10,540
++ جدر ال C بنضرب
+
+74
+00:09:10,540 --> 00:09:19,050
+بسط مقام في المرافقوهذا بيطلع بيساوي absolute x
+
+75
+00:09:19,050 --> 00:09:32,850
+- c على جدر x + جدر c وهذا
+
+76
+00:09:32,850 --> 00:09:41,810
+بيساوي absolute x - c على جدر x + جدر c
+
+77
+00:09:45,070 --> 00:09:50,090
+طبعا ال X هنا لازم تكون عدد موجب أكبر من أو يساوي
+
+78
+00:09:50,090 --> 00:09:58,470
+صفر إذا هنا عندي .. أنا عندي ال X أكبر من أو يساوي
+
+79
+00:09:58,470 --> 00:10:04,630
+صفر إذا جذر ال X أكبر من أو يساوي صفر وبالتالي جذر
+
+80
+00:10:04,630 --> 00:10:12,230
+ال X + جذر ال C أكبر من أو يساوي جذر ال C صح؟
+
+81
+00:10:12,230 --> 00:10:14,470
+وبالتالي
+
+82
+00:10:16,780 --> 00:10:23,460
+هذا بيقدر من 1 على جذر ال X زي جذر ال C أصغر من
+
+83
+00:10:23,460 --> 00:10:29,980
+أو يساوي 1 على جذر ال C إذاً هذا بيطلع أصغر من
+
+84
+00:10:29,980 --> 00:10:35,860
+أو يساوي 1 على جذر ال C في absolute X - C
+
+85
+00:10:35,860 --> 00:10:42,300
+تمام؟ الآن لما يكون هذا أصغر من Delta
+
+86
+00:10:44,960 --> 00:10:48,980
+لما يكون هذا أصغر من دلتا لما يكون هذا أصغر من
+
+87
+00:10:48,980 --> 00:10:56,080
+دلتا فهذا هيكون أصغر من 1 على جدر ال C في دلتا
+
+88
+00:10:56,080 --> 00:11:01,960
+صح؟ و لو أنا بدي أكون هذا خليه
+
+89
+00:11:05,310 --> 00:11:11,610
+وبدي في النهاية هذا يكون أصغر من epsilon صح؟ إذا
+
+90
+00:11:11,610 --> 00:11:17,830
+كيف بدي أخد ال delta؟ جدر ال c في epsilon اه okay
+
+91
+00:11:17,830 --> 00:11:23,810
+تمام؟ إذا هنا هذا بيقدي أن ال delta ممكن أخدها أي
+
+92
+00:11:23,810 --> 00:11:28,850
+عدد أصغر من أو يساوي طبعا عدد موجب وأصغر من أو يساوي
+
+93
+00:11:28,850 --> 00:11:34,840
+جدر ال c في epsilon عشان يطلع المقدار هذا أصغر من
+
+94
+00:11:34,840 --> 00:11:38,400
+إبسلون بالتالي المقدار هذا أصغر من إبسلون إذا
+
+95
+00:11:38,400 --> 00:11:42,340
+شوفتوا كيف نجيب الـ delta إذا نجيب نقول let
+
+96
+00:11:42,340 --> 00:11:51,200
+epsilon let epsilon > 0 be given choose
+
+97
+00:11:51,200 --> 00:12:01,410
+delta = جذر ال C هذا عدد موجب ضرب إمسلن فهذا
+
+98
+00:12:01,410 --> 00:12:06,370
+أكيد بيطلع عدد موجب ويعتمد على إمسلن و هيو بيعتمد
+
+99
+00:12:06,370 --> 00:12:10,870
+على إمسلن then
+
+100
+00:12:10,870 --> 00:12:20,750
+لكل X بحيث absolute x - c > 0 <
+
+101
+00:12:20,750 --> 00:12:27,830
+الـ delta هذه هذا بتضمن أن absolute جذر الـ x
+
+102
+00:12:27,830 --> 00:12:38,260
+- جذر الـ c قلنا هذا طلع أصغر من أو يساوي 1
+
+103
+00:12:38,260 --> 00:12:46,060
+على جذر C في absolute X - C وطبعا هذا الأن طلع
+
+104
+00:12:46,060 --> 00:12:57,160
+أصغر من 1 على جذر C في Delta وهذا أصغر من أو
+
+105
+00:12:57,160 --> 00:13:04,100
+يساوي الأبسلون وبالتالي
+
+106
+00:13:04,100 --> 00:13:09,480
+حسب تعريف Epsilon Delta بطلع عندى اللى .. اللى أنا
+
+107
+00:13:09,480 --> 00:13:22,200
+عايزه since
+
+108
+00:13:22,200 --> 00:13:29,620
+epsilon > 0 was arbitrary لأن we have
+
+109
+00:13:29,620 --> 00:13:36,510
+أثبتنا حسب التعريف إن ال limit لجدر ال X لما X تقول
+
+110
+00:13:36,510 --> 00:13:47,690
+إلى C = جدر ال C وهو المضمن OK تمام واضح الحل
+
+111
+00:13:47,690 --> 00:13:53,570
+واضح البرهان إذا لكل epsilon > 0 اختاري ال
+
+112
+00:13:53,570 --> 00:13:58,210
+delta اللي بتشتغل صح هي جدر ال C هذا عدد موجب ثابت
+
+113
+00:13:58,210 --> 00:14:06,940
+ضرب ال epsilon اللي احنا بدينا فيها تمام؟ okay طيب
+
+114
+00:14:06,940 --> 00:14:12,360
+في أسئلة تانية؟ في أي استفسار؟
+
+115
+00:14:12,360 --> 00:14:22,220
+في أي أسئلة تانية؟ section 4-1 أو 4-2 الناس اللي
+
+116
+00:14:22,220 --> 00:14:28,300
+بتدرس و اللي حاولة تتحل الأسئلة و عندها بعض
+
+117
+00:14:28,300 --> 00:14:34,200
+الصعوبات في حل الأسئلة مين عندها؟ أي استفسار؟
+
+118
+00:14:34,200 --> 00:14:36,520
+السؤال اتناش اربعة اتنين
+
+119
+00:15:19,790 --> 00:15:31,090
+السؤال 12 أربعة اتنين في
+
+120
+00:15:31,090 --> 00:15:38,850
+عندي function f from R to R such
+
+121
+00:15:38,850 --> 00:15:50,420
+that f(x + y) = f(x) + f(y) for
+
+122
+00:15:50,420 --> 00:16:00,780
+every x و y in R assume
+
+123
+00:16:00,780 --> 00:16:03,920
+ان
+
+124
+00:16:03,920 --> 00:16:14,620
+ال limit f(x) as x tends to zero = عدد L
+
+125
+00:16:14,620 --> 00:16:16,820
+exists
+
+126
+00:16:19,070 --> 00:16:28,110
+يعني أدب real number prove
+
+127
+00:16:28,110 --> 00:16:40,930
+حاجتي الواحد ال = صفر لان limit
+
+128
+00:16:40,930 --> 00:16:44,290
+f
+
+129
+00:16:44,290 --> 00:16:44,950
+(x)
+
+130
+00:16:48,250 --> 00:16:59,370
+كما يظهر X لـ C لجميع الـ C التانية لـ R لما
+
+131
+00:16:59,370 --> 00:17:05,770
+نثبت أن الـ limit لها تساوي صفر ثم الـ function F
+
+132
+00:17:05,770 --> 00:17:12,230
+لها limit للأعداد الحقيقية C والكتاب يعطيك hint
+
+133
+00:17:12,230 --> 00:17:20,490
+يعني إرشاد كيف يعني تبدأ الحل بطريقة صحية proof
+
+134
+00:17:20,490 --> 00:17:24,410
+فخلينا
+
+135
+00:17:24,410 --> 00:17:33,150
+نبرهن الجزء الأول لحظة
+
+136
+00:17:33,150 --> 00:17:37,780
+أن ال function هذه بتحقق الشرط هذابنسميه
+
+137
+00:17:37,780 --> 00:17:42,880
+additivity ال function f بتحافظ على عملية الجمع
+
+138
+00:17:42,880 --> 00:17:48,100
+بتاخد مجموعة حاجتين تعطي صورتها مجموعة صورهم
+
+139
+00:17:48,100 --> 00:17:52,360
+فبنقول f أي function بتحقق خاصية زي هذه بنسميها
+
+140
+00:17:52,360 --> 00:17:57,570
+additive يعني دالة جمعية بتحافظ على عملية الجمع تبع 
+
+141
+00:17:57,570 --> 00:18:03,170
+الأعداد الحقيقية فبناء على الخاصية هذه لو كانت ال
+
+142
+00:18:03,170 --> 00:18:07,450
+function f لها limit  و صفر فال limit هذه لازم 
+
+143
+00:18:07,450 --> 00:18:13,630
+تكون بساوي صفر  خمتها صفر فكيف ممكن نثبت الكلام هذا
+
+144
+00:18:13,630 --> 00:18:19,590
+نسمي
+
+145
+00:18:19,590 --> 00:18:22,190
+هذا الفرض star
+
+146
+00:18:25,940 --> 00:18:34,100
+by hypothesis star من الفرض star أنا عندي  h أو ال
+
+147
+00:18:34,100 --> 00:18:42,500
+function f of two x إيش بتساوي؟ بتساوي f of x زائد
+
+148
+00:18:42,500 --> 00:18:47,260
+x مظبوط و من الفرض star
+
+149
+00:18:51,230 --> 00:18:58,670
+f of x زائد x بيساوي f of x زائد f of x صح يعني
+
+150
+00:18:58,670 --> 00:19:13,710
+بيساوي اثنين في f of x تمام طيب
+
+151
+00:19:13,710 --> 00:19:20,130
+take limit of both sides take limit
+
+152
+00:19:22,620 --> 00:19:29,320
+of both sides as
+
+153
+00:19:29,320 --> 00:19:39,900
+x tends to zero we get نحصل على انه ال limit ل f ل
+
+154
+00:19:39,900 --> 00:19:50,380
+2x لما x تقول ل 0 بساوي 2 في ال limit ل f of x لما
+
+155
+00:19:50,380 --> 00:19:51,720
+x تقول ل 0
+
+156
+00:19:56,790 --> 00:20:01,490
+Limit f of x من اكسا اولا صفر بساوي من الفرض
+
+157
+00:20:01,490 --> 00:20:08,610
+موجودة و بساوي L في اثنين بطلع اثنين L و ال limit
+
+158
+00:20:08,610 --> 00:20:14,230
+هذه هي نفسها limit
+
+159
+00:20:17,140 --> 00:20:27,440
+لـ F of 2X لما 2X تقول لـ 0 صح؟ لما X تقول لـ 0،
+
+160
+00:20:27,440 --> 00:20:38,460
+2X تقول لـ 0 و هذه هي نفسها ال limit ل F of Y لما
+
+161
+00:20:38,460 --> 00:20:48,520
+Y تقول لـ 0 صح؟ خدي Y بساوي 2X فالمهي هنا limit f
+
+162
+00:20:48,520 --> 00:20:54,400
+of y لما y تقل ل 0 ال independent variable ده سميه
+
+163
+00:20:54,400 --> 00:20:59,760
+x سميه y it doesn't matter مش مهم اه لإن هذا برضه
+
+164
+00:20:59,760 --> 00:21:07,080
+ال limit هذه بساوي L لإن أنا أصبح عندي لإن أنا
+
+165
+00:21:07,080 --> 00:21:14,040
+أصبح عندي أنا معادلة L بساوي two L حل المعادلة هذه
+
+166
+00:21:15,510 --> 00:21:23,150
+في ال فهذا بيقدي ان ال L بيساوي صفر صح ان هنا اثبتت
+
+167
+00:21:23,150 --> 00:21:28,350
+ان العدد ال L اللي هو limit لل function if and صفر
+
+168
+00:21:28,350 --> 00:21:35,370
+بيطلع بساوي صفر صحيح ولا سهل ان
+
+169
+00:21:35,370 --> 00:21:41,710
+الكتاب بيحط رجلكم قدامكم على بداية الطريق استغلي
+
+170
+00:21:41,710 --> 00:21:42,050
+صح
+
+171
+00:21:45,430 --> 00:21:57,650
+واضح البرهان هنا تمام نبرهن الجزء الثاني طيب
+
+172
+00:21:57,650 --> 00:22:03,570
+نبرهن الجزء الثاني برضه
+
+173
+00:22:03,570 --> 00:22:13,250
+الجزء الثاني فيه له hint فنستخدم ال hint also
+
+174
+00:22:15,670 --> 00:22:24,930
+by hypothesis star حسب
+
+175
+00:22:24,930 --> 00:22:31,630
+الفرض star ال function هذه is additive وبالتالي f
+
+176
+00:22:31,630 --> 00:22:40,750
+of x هي نفسها f of x ناقص c زائد c صح؟
+
+177
+00:22:44,780 --> 00:22:52,200
+و هذا بيساوي F
+
+178
+00:22:52,200 --> 00:23:03,900
+of X ناقص C زائد F of C تمام طيب هذا صحيح for all
+
+179
+00:23:03,900 --> 00:23:13,100
+C ينتمي ل R و طبعا for all X و for all X ينتمي ل R
+
+180
+00:23:14,280 --> 00:23:19,900
+في مشكلة طيب الان نفس الحاجة اخد ال limit للطرفين
+
+181
+00:23:19,900 --> 00:23:30,540
+take limit of both sides لما x تقول ل c اذا limit
+
+182
+00:23:30,540 --> 00:23:40,780
+f of x لما x تقول ل c بساوي limit الطرفين اليومين
+
+183
+00:23:40,780 --> 00:23:44,240
+مجموعة limit مجموعة بيساوي مجموعة limits لأن limit
+
+184
+00:23:44,240 --> 00:23:48,660
+الحد الأول exist هنشوف ان limit الحد الأول موجودة
+
+185
+00:23:48,660 --> 00:23:52,320
+و limit الحد الثاني موجودة وبالتالي limit المجموعة
+
+186
+00:23:52,320 --> 00:23:59,000
+بيساوي مجموعة limits فlimit f of x ناقص c as x
+
+187
+00:23:59,000 --> 00:24:01,160
+tends to c زائد
+
+188
+00:24:08,360 --> 00:24:25,360
+limit f of x لمّا x تقول لـ c limit
+
+189
+00:24:25,360 --> 00:24:30,140
+f
+
+190
+00:24:30,140 --> 00:24:33,960
+of x لمّا x تقول لـ c
+
+191
+00:24:37,080 --> 00:24:44,860
+ال limit هذه لو أخدت y بساوي x ناقص c فبطلع x
+
+192
+00:24:44,860 --> 00:24:53,900
+بساوي y زائد c وبالتالي
+
+193
+00:24:53,900 --> 00:25:01,220
+لما x تقول ل c لما x تقول ل c هذا بيقدي ان y تقول
+
+194
+00:25:01,220 --> 00:25:12,090
+ل 0 صح؟ إذن هذه هي نفس limitF of Y لما Y تقول لـ 0
+
+195
+00:25:12,090 --> 00:25:15,550
+وذلك
+
+196
+00:25:15,550 --> 00:25:22,690
+ببتعوض عن X ناقص C بساوي Y وهذا عدد ثابت F of C
+
+197
+00:25:22,690 --> 00:25:29,950
+عدد ثابت نهايته نفس العدد الثابت F of C Y ساوي طيب
+
+198
+00:25:29,950 --> 00:25:34,510
+احنا لسه من الفرض فرضين ان limit F of Y لما Y تقول
+
+199
+00:25:34,510 --> 00:25:42,440
+لـ 0 بساوي L اللي هو صفر إذاً هذا بيساوي العدد L
+
+200
+00:25:42,440 --> 00:25:46,860
+اللي هو صفر زائد
+
+201
+00:25:46,860 --> 00:25:55,600
+F of C اللي هو F of C إذاً
+
+202
+00:25:55,600 --> 00:26:00,160
+هنا أثبتنا إنه limit F of X لما X تقول لـ C exist
+
+203
+00:26:00,160 --> 00:26:01,880
+و بيساوي F of C
+
+204
+00:26:05,410 --> 00:26:09,530
+تمام؟ إذا هين أثبتنا أن ال limit لل function عند
+
+205
+00:26:09,530 --> 00:26:17,170
+أي c exist و بساوي f of c و هذا طبعا الشرط اللي ..
+
+206
+00:26:17,170 --> 00:26:21,090
+لاحظوا أنتوا limit f of x عند أي c بتطلعت بالساوية
+
+207
+00:26:21,090 --> 00:26:25,970
+قيمة الدالة عن ال c هذا حسب chapter 5 هذا معناه أن
+
+208
+00:26:25,970 --> 00:26:31,510
+الدالة هذه تطلع continuous عند أي نقطة في المجال
+
+209
+00:26:31,510 --> 00:26:36,670
+تبعها Okay إذا النتيجة الخلاصة من ال exercise هذا
+
+210
+00:26:36,670 --> 00:26:43,030
+exercise كتير مهم وهو إنه اللي لو كان في عندى
+
+211
+00:26:43,030 --> 00:26:50,370
+function from R to R و ال function هذه additive و
+
+212
+00:26:50,370 --> 00:26:56,700
+نهايتها عند الصفر موجودة فالدال هذا بتطلع متصلة عن
+
+213
+00:26:56,700 --> 00:27:00,100
+كل الأعداد الحقيقية هذا النتيجة النهائية هذا اللي
+
+214
+00:27:00,100 --> 00:27:08,040
+أثبتناه في الجزء الثاني okay تمام؟ وهذه نتيجة مهمة
+
+215
+00:27:08,040 --> 00:27:15,020
+ومعروفة في كورسات ال real analysis المتقدمة تمام
+
+216
+00:27:15,020 --> 00:27:20,860
+واضح؟ في أي استفسار؟ إذا شفتوا هنا يعني كيف استغلنا
+
+217
+00:27:20,860 --> 00:27:26,080
+الفرض star و كيف استغلنا ال hint الإرشادات اللي
+
+218
+00:27:26,080 --> 00:27:29,860
+أعطينا إياها الكتاب فاحنا ممكن نجيبلكم في الامتحان
+
+219
+00:27:29,860 --> 00:27:38,320
+أسئلة و نعطيلكم hint عليها أساسا كبداية للحل أو
+
+220
+00:27:38,320 --> 00:27:43,570
+البرهان الصحي و الشاطرة اللي تستغلها صح Okay تمام
+
+221
+00:27:43,570 --> 00:27:48,130
+في أي أسئلة تانية section أربعة واحد أو أربعة اثنين
+
+222
+00:27:48,130 --> 00:27:50,030
+رقم تلات عشر أربعة اثنين
+
+223
+00:28:25,120 --> 00:28:33,640
+أي سؤال تلتاش سيكشن أربعة اثنين في عندي F function
+
+224
+00:28:33,640 --> 00:28:39,020
+from A to R و
+
+225
+00:28:39,020 --> 00:28:49,560
+C belong to R is a cluster point cluster point of
+
+226
+00:28:49,560 --> 00:28:50,220
+A
+
+227
+00:29:03,780 --> 00:29:18,360
+فالـ limit لـ f of x لما x تقول لـ c exists prove
+
+228
+00:29:18,360 --> 00:29:29,240
+أن ال limit ل absolute f of x لما x تقول ل c
+
+229
+00:29:29,240 --> 00:29:31,720
+بيساوي
+
+230
+00:29:34,790 --> 00:29:42,870
+بتساوي absolute limit f of x لما x تقوم بالاسم
+
+231
+00:29:42,870 --> 00:29:48,950
+where
+
+232
+00:29:48,950 --> 00:29:59,690
+حيث و where حيث absolute f الـ absolute value لأي
+
+233
+00:29:59,690 --> 00:30:05,550
+function تطلع function تانية تعريفها عند أي x أو
+
+234
+00:30:05,550 --> 00:30:13,430
+قيمتها عند أي x بساوي absolute f of x لكل x
+
+235
+00:30:27,830 --> 00:30:33,090
+بمعنى آخر إذا كان في عندي function و ال limit لها
+
+236
+00:30:33,090 --> 00:30:38,510
+ان c موجودة ف limit ال absolute value لها بتكون
+
+237
+00:30:38,510 --> 00:30:43,970
+موجودة و تساوي ال absolute value لل limit يعني
+
+238
+00:30:43,970 --> 00:30:48,090
+ممكن ابدل ال limit مع ال absolute value أو ادخل ال
+
+239
+00:30:48,090 --> 00:30:53,730
+limit داخل ال absolute value وهذه حقيقة صحيحة لأن
+
+240
+00:30:53,730 --> 00:30:56,550
+ال absolute value function متتصرة
+
+241
+00:31:02,530 --> 00:31:19,850
+Okay تمام ف ..
+
+242
+00:31:19,850 --> 00:31:30,030
+طيب يعني
+
+243
+00:31:30,030 --> 00:31:38,610
+say دعونا نفترض أن ال limit ل f of x لما x تقول إلى
+
+244
+00:31:38,610 --> 00:31:44,870
+c بساوي عدد L ينتمي ل R مش ال limit had existed
+
+245
+00:31:44,870 --> 00:31:48,830
+سميها L و
+
+246
+00:31:48,830 --> 00:31:52,430
+المطلوب we need to show
+
+247
+00:31:58,250 --> 00:32:08,570
+نحتاج أن نظهر أن الـ limit لـ absolute f of x as x
+
+248
+00:32:08,570 --> 00:32:11,510
+tends to c بساوي absolute L
+
+249
+00:32:35,630 --> 00:32:44,970
+أنا في نهاية
+
+250
+00:32:44,970 --> 00:32:52,570
+الأمر بدي أثبت أن الـ absolute بدي
+
+251
+00:32:52,570 --> 00:32:55,690
+في النهاية يكون هذا أصغر من أي جيبل إبسلون
+
+252
+00:32:58,740 --> 00:33:05,380
+لكل x بحيث ان absolute x ناقص c أكبر من صفر أصغر
+
+253
+00:33:05,380 --> 00:33:17,660
+من delta where delta to be determined يعني
+
+254
+00:33:17,660 --> 00:33:23,560
+هنحددها لاحقا هذا ايه عشان اثبت ان ال limit ال
+
+255
+00:33:23,560 --> 00:33:29,700
+function هذه يعني x بساوي absolute الفبدأثبت ان
+
+256
+00:33:29,700 --> 00:33:33,300
+absolute الفرخ هذا أصغر من epsilon عندما يكون
+
+257
+00:33:33,300 --> 00:33:39,240
+المسافة بين ال X و ال C أصغر من delta و X لا تساوي
+
+258
+00:33:39,240 --> 00:33:46,580
+ال C for some delta تعتمد على ال given epsilon طب
+
+259
+00:33:46,580 --> 00:33:50,860
+أنا عندي هذا
+
+260
+00:33:50,860 --> 00:33:57,000
+بيساوي absolute absolute F of X ناقص absolute
+
+261
+00:34:09,920 --> 00:34:16,520
+تمام؟ وهذا باستخدام ال triangle inequality أصغر من
+
+262
+00:34:16,520 --> 00:34:23,440
+أو ساوي absolute f of x ناقص L صح؟ في صورة من صور
+
+263
+00:34:23,440 --> 00:34:26,140
+ال triangle inequality يجب تقول إن هذا ال absolute
+
+264
+00:34:26,140 --> 00:34:30,140
+value ل absolute a ناقص absolute b أصغر من
+
+265
+00:34:30,140 --> 00:34:35,880
+absolute a ناقص b طب أنا عندي limit f of x لما x
+
+266
+00:34:35,880 --> 00:34:41,400
+اقولها c بالساوي L فممكن اخلي هذا اصغر من اي given
+
+267
+00:34:41,400 --> 00:34:47,540
+epsilon وبالتالي هذه بتصير اصغر من اي epsilon okay
+
+268
+00:34:47,540 --> 00:34:52,420
+تمام إذا هنا خلينا نشوف
+
+269
+00:35:03,620 --> 00:35:09,800
+let epsilon أكبر
+
+270
+00:35:09,800 --> 00:35:18,820
+من الصفر be given بما انه since ال limit احنا فرضين
+
+271
+00:35:18,820 --> 00:35:25,840
+انه limit ل F of X عند X بساوي C بساوي L إذا يوجد
+
+272
+00:35:25,840 --> 00:35:31,580
+Delta تعتمد على ابسلون عدد موجب بحيث انه لكل X
+
+273
+00:35:31,580 --> 00:35:37,060
+ينتمي إلى A و absolute X ناقص C اصغر من Delta
+
+274
+00:35:37,060 --> 00:35:45,550
+أكبر من صفر هذا بيقدي إنه absolute f of x ناقص L
+
+275
+00:35:45,550 --> 00:35:51,890
+أصغر من إبسلم نسمي هذا star نسمي ال implication
+
+276
+00:35:51,890 --> 00:36:03,570
+هذا star hence
+
+277
+00:36:08,200 --> 00:36:16,880
+by triangle inequality من متبينة المثلث x ينتمي by
+
+278
+00:36:16,880 --> 00:36:26,260
+triangle inequality and star we have لدينا لو كان
+
+279
+00:36:26,260 --> 00:36:32,680
+x ينتمي ل a و absolute x ناقص c أكبر من صفر أصغر
+
+280
+00:36:32,680 --> 00:36:48,240
+من delta فهذا بيقدي انه absolute .. absolute f of x 
+
+281
+00:36:48,240 --> 00:36:59,300
+minus absolute الـ L هذا بيساوي absolute ..
+
+282
+00:36:59,300 --> 00:37:06,060
+absolute f of x minus absolute L
+
+283
+00:37:12,840 --> 00:37:17,300
+وهذا by الـ triangle inequality by الـ triangle
+
+284
+00:37:17,300 --> 00:37:24,520
+inequality أصغر من أو يساوي absolute f of x minus
+
+285
+00:37:24,520 --> 00:37:32,300
+L و by star absolute f of x minus l أصغر من إبسلون
+
+286
+00:37:32,300 --> 00:37:41,020
+تمام؟ since إبسلون was
+
+287
+00:37:41,020 --> 00:37:41,740
+arbitrary
+
+288
+00:37:47,340 --> 00:37:53,990
+إذاً we have هكذا مكون أسبابنا أنه لأي epsilon أكبر 
+
+289
+00:37:53,990 --> 00:37:59,950
+من الصفر يوجد delta تعتمد على epsilon عدد موجب من
+
+290
+00:37:59,950 --> 00:38:06,350
+حيث لكل x ينتمي لـ a و المسافة بين x و c أصغر من
+
+291
+00:38:06,350 --> 00:38:11,630
+delta و x لا تساوي c فللـ absolute value لـ f of x
+
+292
+00:38:11,630 --> 00:38:15,930
+minus absolute L أصغر من epsilon، إذن هذا معناه
+
+293
+00:38:15,930 --> 00:38:21,460
+حسب epsilon delta definition of limit إن الـ limit
+
+294
+00:38:21,460 --> 00:38:29,780
+لـ absolute f of x as x tends to c بيساوي absolute لـ
+
+295
+00:38:29,780 --> 00:38:33,140
+L 
+
+296
+00:38:33,140 --> 00:38:38,420
+المطلوب وهذا اللي احنا عايزين نثبته هذا هو البرهان
+
+297
+00:38:39,940 --> 00:38:45,060
+إذا هنا لعب دور كبير في البرهان هو الـ triangle
+
+298
+00:38:45,060 --> 00:38:52,420
+inequality و الفرض إنه limit and c موجودة بالساوية 
+
+299
+00:38:52,420 --> 00:38:58,420
+L okay تمام؟ إذا أهمية الـ exercise هذا يعتبر نظرية
+
+300
+00:38:58,420 --> 00:39:02,120
+و النظرية هذه بتقول إذا كان في end function
+
+301
+00:39:02,120 --> 00:39:07,590
+نهايتها and c موجودة فنهاية الـ absolute value للـ
+
+302
+00:39:07,590 --> 00:39:11,610
+function بيساوي الـ absolute value للـ limit بمعنى
+
+303
+00:39:11,610 --> 00:39:15,870
+آخر أنا ممكن أدخل الـ limit داخل الـ absolute value
+
+304
+00:39:15,870 --> 00:39:22,710
+أو أبدل الـ limit مع الـ absolute value okay تمام
+
+305
+00:39:22,710 --> 00:39:26,630
+واضح؟ في أسئلة تانية؟
+
+306
+00:39:35,800 --> 00:39:43,260
+أي سؤال أي استفسار okay
+
+307
+00:39:43,260 --> 00:39:45,560
+إذا نكتفي بهذا القدر 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/db6AFymIrl8_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/db6AFymIrl8_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..df344fc146566e3feeba9c4f0e118efd09173ce2
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/db6AFymIrl8_postprocess.srt
@@ -0,0 +1,1228 @@
+1
+00:00:19,490 --> 00:00:23,130
+بسم الله الرحمن الرحيم في المحاضرة هذه هناخد يعني
+
+2
+00:00:23,130 --> 00:00:31,510
+مناقشة أو هنناقش بعض المسائل في section 4-1 و 4-2
+
+3
+00:00:31,510 --> 00:00:41,650
+فإحدى الطالبات سألت في سؤال بقول برهن الجزء B من
+
+4
+00:00:41,650 --> 00:00:52,390
+theorem 4.2.4using sequential criterion لان هنا
+
+5
+00:00:52,390 --> 00:00:56,910
+use sequential
+
+6
+00:00:56,910 --> 00:01:01,370
+.. sequential
+
+7
+00:01:01,370 --> 00:01:06,410
+argument use
+
+8
+00:01:06,410 --> 00:01:15,290
+sequential formulation on
+
+9
+00:01:15,290 --> 00:01:15,750
+limit
+
+10
+00:01:21,840 --> 00:01:27,040
+فالبرهان ذلك let .. هنستخدم الـ sequential
+
+11
+00:01:27,040 --> 00:01:30,380
+criterion
+
+12
+00:01:30,380 --> 00:01:39,860
+let x in بـ sequence contained in A such that
+
+13
+00:01:39,860 --> 00:01:50,000
+limit طبعا الحدود تباعتها مختلفة عن الـ C such
+
+14
+00:01:50,000 --> 00:01:56,610
+that limitx in as n tends to infinity بساوي c
+
+15
+00:01:56,610 --> 00:02:03,050
+خلّينا نختار sequence في المجال المشترك تبع
+
+16
+00:02:03,050 --> 00:02:08,290
+الدالتين FWH وحدودها مختلفة عن ال C ونهايتها بساوي
+
+17
+00:02:08,290 --> 00:02:08,610
+C
+
+18
+00:02:16,250 --> 00:02:21,290
+by sequential criterion حسب sequential criterion
+
+19
+00:02:21,290 --> 00:02:32,430
+for limits to show ان ال limit ل F على H as X
+
+20
+00:02:32,430 --> 00:02:42,590
+tends to C بساوي L على H it
+
+21
+00:02:42,590 --> 00:02:43,570
+suffices
+
+22
+00:02:45,740 --> 00:02:53,260
+it suffices to show يكفي اثبات ان ال limit لل
+
+23
+00:02:53,260 --> 00:03:02,220
+image لسيكوينس xn لما n تقول ل infinity بساوي L
+
+24
+00:03:02,220 --> 00:03:07,020
+على H لو اثبتت الكلام هذا فحسب السيكوينش هي كتير
+
+25
+00:03:07,020 --> 00:03:11,380
+تانيا بطلع limit F على H بساوي capital L على
+
+26
+00:03:11,380 --> 00:03:12,140
+capital H
+
+27
+00:03:17,270 --> 00:03:24,050
+فنشوف to this end to
+
+28
+00:03:24,050 --> 00:03:36,670
+this end و لإثبات ذلك يعني we
+
+29
+00:03:36,670 --> 00:03:40,710
+have from
+
+30
+00:03:40,710 --> 00:03:44,170
+sequential criterion
+
+31
+00:03:47,320 --> 00:03:56,200
+that limit f of x in as n tends to infinity بساوي
+
+32
+00:03:56,200 --> 00:04:01,600
+L أنا عندي فارض ان limit f of x من x أول ل c بساوي
+
+33
+00:04:01,600 --> 00:04:06,720
+L اذا by sequential criterion هذا بكافئ انه لأي
+
+34
+00:04:06,720 --> 00:04:12,680
+sequence x in نهايتها c بطلع نهاية صورتها بساوي L
+
+35
+00:04:12,680 --> 00:04:18,140
+و كذلك andأنا عندي limit الـ function h of x من x
+
+36
+00:04:18,140 --> 00:04:22,320
+تقولها c بساوي h by sequential criterion كمان بره
+
+37
+00:04:22,320 --> 00:04:27,120
+طبخيها على ال function h بما أن x in sequence
+
+38
+00:04:27,120 --> 00:04:34,460
+نهايتها c إذا نهايت صرتها under h يعني limit h of
+
+39
+00:04:34,460 --> 00:04:43,840
+x in as n tends to infinity بساوي capital H hence
+
+40
+00:04:45,840 --> 00:04:57,980
+وبالتالي ال limit ل f على h of x لما x
+
+41
+00:04:57,980 --> 00:05:10,220
+تقول ل xn لما n تقول ل infinity هذا بساوي limit
+
+42
+00:05:12,840 --> 00:05:22,580
+f of x,n على h of x,n لما n تقول infinity ويساوي
+
+43
+00:05:22,580 --> 00:05:29,560
+انا انا اندي limit h of x,n exist وبيستويش سفر
+
+44
+00:05:29,560 --> 00:05:35,080
+وبيستويش سفر فممكن استخدم قوانين النهايات لسيكوانس
+
+45
+00:05:35,080 --> 00:05:40,420
+فlimit sequence على limit sequence بساوي limit
+
+46
+00:05:40,420 --> 00:05:41,140
+البسط
+
+47
+00:05:44,990 --> 00:05:50,170
+limit ال sequence في ال bus على limit ال sequence
+
+48
+00:05:50,170 --> 00:05:55,330
+في المقام و
+
+49
+00:05:55,330 --> 00:05:58,010
+limit ال sequence في المقام بيساوي سفر اذا انا
+
+50
+00:05:58,010 --> 00:06:01,090
+بقدر ايه اقول limit ال quotient بيساوي the
+
+51
+00:06:01,090 --> 00:06:07,330
+quotient of the limits وهذا بيطلع limit ال bus
+
+52
+00:06:07,330 --> 00:06:14,920
+تطلع L limit المقام H وهذا البنيةإن حسب الـ
+
+53
+00:06:14,920 --> 00:06:19,020
+sequential criterion بيطلع limit F على H عندما X
+
+54
+00:06:19,020 --> 00:06:24,020
+تقوى ل C بيساوي L على H هذا هو المقصود الـ
+
+55
+00:06:24,020 --> 00:06:28,740
+sequential formulation واضح؟ طيب، مين عندها أسئلة
+
+56
+00:06:28,740 --> 00:06:35,540
+تانية؟ في section 4, 1 أو 4, 2؟
+
+57
+00:06:35,540 --> 00:06:40,080
+طيب،
+
+58
+00:06:40,080 --> 00:06:41,580
+خليني أمسح اللوحي الأول
+
+59
+00:07:20,820 --> 00:07:28,800
+هذه السؤال تمانية section أربعة واحد show that
+
+60
+00:07:28,800 --> 00:07:38,120
+أثبتي أنه ال limit of the square root of X الـ
+
+61
+00:07:38,120 --> 00:07:44,060
+square root function as X tends to C بساوي ال
+
+62
+00:07:44,060 --> 00:07:48,340
+square root of C for any
+
+63
+00:07:51,920 --> 00:07:59,200
+C أكبر من السفر proof
+
+64
+00:08:02,940 --> 00:08:09,860
+أحنا في النهاية عايزين نثبت أنه الفرق بين f of x
+
+65
+00:08:09,860 --> 00:08:16,700
+absolute الفرق بين f of x و f of c اللي هي جدر ال
+
+66
+00:08:16,700 --> 00:08:21,680
+c بدنا في النهاية هذا يكون أصغر من أي given
+
+67
+00:08:21,680 --> 00:08:22,140
+epsilon
+
+68
+00:08:25,220 --> 00:08:31,760
+حيث x المسافة بينها وبين z ال c أصغر من delta و
+
+69
+00:08:31,760 --> 00:08:36,480
+delta to be determined يعني هيتم تعيينها لاحقا
+
+70
+00:08:36,480 --> 00:08:42,240
+okay طب ما هذا بيساوي absolute
+
+71
+00:08:46,520 --> 00:08:54,060
+must go هاي جدر ال X سالب جدر ال C و بنضرب في
+
+72
+00:08:54,060 --> 00:09:03,660
+المرافق اللي هو جدر ال X زائد جدر ال C في جدر ال X
+
+73
+00:09:03,660 --> 00:09:10,540
+زائد جدر ال C بنضرب
+
+74
+00:09:10,540 --> 00:09:19,050
+بسط مقام في المرافقوهذا بيطلع بيساوي absolute x
+
+75
+00:09:19,050 --> 00:09:32,850
+minus c على جدر x plus جدر sc وهذا
+
+76
+00:09:32,850 --> 00:09:41,810
+بيساوي absolute x minus c على جدر x plus جدر sc
+
+77
+00:09:45,070 --> 00:09:50,090
+طبعا ال X هنا لازم تكون عدد موجب أكبر من أو ساوي
+
+78
+00:09:50,090 --> 00:09:58,470
+سفر إذا هنا عندي .. أنا عندي ال X أكبر من أو ساوي
+
+79
+00:09:58,470 --> 00:10:04,630
+سفر إذا جذر ال X أكبر من أو ساوي سفر وبالتالي جذر
+
+80
+00:10:04,630 --> 00:10:12,230
+ال X plus جذر ال C أكبر من أو ساوي جذر ال C صح؟
+
+81
+00:10:12,230 --> 00:10:14,470
+وبالتالي
+
+82
+00:10:16,780 --> 00:10:23,460
+هذا بيقدر من واحد على جذر ال X زي جذر ال C أصغر من
+
+83
+00:10:23,460 --> 00:10:29,980
+أو ساوي واحد على جذر ال C إذاً هذا بيطلع أصغر من
+
+84
+00:10:29,980 --> 00:10:35,860
+أو ساوي واحد على جذر ال C في absolute X minus C
+
+85
+00:10:35,860 --> 00:10:42,300
+تمام؟ الآن لما يكون هذا أصغر من Delta
+
+86
+00:10:44,960 --> 00:10:48,980
+لما يكون هذا أصغر من دلتا لما يكون هذا أصغر من
+
+87
+00:10:48,980 --> 00:10:56,080
+دلتا فهذا هيكون أصغر من واحد على جدر ال C في دلتا
+
+88
+00:10:56,080 --> 00:11:01,960
+صح؟ و لو أنا بدي أكون هذا خليه
+
+89
+00:11:05,310 --> 00:11:11,610
+وبدي في النهاية هذا يكون أصغر من epsilon صح؟ إذا
+
+90
+00:11:11,610 --> 00:11:17,830
+كيف بدي أخد ال delta؟ جدر ال c في epsilon اه okay
+
+91
+00:11:17,830 --> 00:11:23,810
+تمام؟ إذا هنا هذا بيقدي أن ال delta ممكن أخدها أي
+
+92
+00:11:23,810 --> 00:11:28,850
+عدد أصغر من أو ساوي طبعا عدد موجب وأصغر من أو ساوي
+
+93
+00:11:28,850 --> 00:11:34,840
+جدر ال c في epsilonعشان يطلع المقدار هذا أصغر من
+
+94
+00:11:34,840 --> 00:11:38,400
+إبسلون بالتالي المقدار هذا أصغر من إبسلون إذا
+
+95
+00:11:38,400 --> 00:11:42,340
+شوفتوا كيف نجيب الـ delta إذا نجيب نقول let
+
+96
+00:11:42,340 --> 00:11:51,200
+epsilon let epsilon أكبر من السفر be given choose
+
+97
+00:11:51,200 --> 00:12:01,410
+delta بتساويالجذر ال C هذا عدد موجب ضرب إمسلن فهذا
+
+98
+00:12:01,410 --> 00:12:06,370
+أكيد بيطلع عدد موجب ويعتمد على إمسلن و هيو بيعتمد
+
+99
+00:12:06,370 --> 00:12:10,870
+على إمسلن then
+
+100
+00:12:10,870 --> 00:12:20,750
+لكل Xبحيث absolute x minus c أكبر من سفر أصغر من
+
+101
+00:12:20,750 --> 00:12:27,830
+الـ delta هذه هذا بتضمن أن absolute جذر الـ x
+
+102
+00:12:27,830 --> 00:12:38,260
+minus جذر الـ c قلنا هذا طلع أصغر من أو يساويواحد
+
+103
+00:12:38,260 --> 00:12:46,060
+على جذر C في absolute X minus C وطبعا هذا الأن طلع
+
+104
+00:12:46,060 --> 00:12:57,160
+أصغر من واحد على جذر C في Delta وهذا أصغر من أو
+
+105
+00:12:57,160 --> 00:13:04,100
+يساوي الأبسلون وبالتالي
+
+106
+00:13:04,100 --> 00:13:09,480
+حسب تعريف Epsilon Deltaبطلع عندى اللى .. اللى أنا
+
+107
+00:13:09,480 --> 00:13:22,200
+عايزه since
+
+108
+00:13:22,200 --> 00:13:29,620
+epsilon أكبر من السفر was arbitrary لأن we have
+
+109
+00:13:29,620 --> 00:13:36,510
+أثبتنا حسب التعريفإن ال limit لجدر ال X لما X تقول
+
+110
+00:13:36,510 --> 00:13:47,690
+إلى C بساوي جدر ال C وهو المضمن OK تمام واضح الحل
+
+111
+00:13:47,690 --> 00:13:53,570
+واضح البرهام إذا لكل epsilon أكبر من 0 اختاري ال
+
+112
+00:13:53,570 --> 00:13:58,210
+delta اللي بتشتغل صح هي جدر ال C هذا عدد موجة ثابت
+
+113
+00:13:58,210 --> 00:14:06,940
+ضرب ال epsilon اللي احنا بدينا فيهاتمام؟ okay طيب
+
+114
+00:14:06,940 --> 00:14:12,360
+في أسئلة تانية؟ في أي استفسار؟
+
+115
+00:14:12,360 --> 00:14:22,220
+في أي أسئلة تانية؟ section 4-1 أو 4-2 الناس اللي
+
+116
+00:14:22,220 --> 00:14:28,300
+بتدرس و اللي حاولة تتحل الأسئلة و عندها بعض
+
+117
+00:14:28,300 --> 00:14:34,200
+الصعوبات في حل الأسئلةمين عندها؟ أي استفسار؟
+
+118
+00:14:34,200 --> 00:14:36,520
+السؤال اتناش اربعة اتنين
+
+119
+00:15:19,790 --> 00:15:31,090
+السؤال 12 اربعة اتنين في
+
+120
+00:15:31,090 --> 00:15:38,850
+عندي function f from R to R such
+
+121
+00:15:38,850 --> 00:15:50,420
+that f of x plus y بساوي f of x plus f of yfor
+
+122
+00:15:50,420 --> 00:16:00,780
+every x و y in R assume
+
+123
+00:16:00,780 --> 00:16:03,920
+ان
+
+124
+00:16:03,920 --> 00:16:14,620
+ال limit f of x as x tends to zero بسوى عدد L
+
+125
+00:16:14,620 --> 00:16:16,820
+exists
+
+126
+00:16:19,070 --> 00:16:28,110
+يعني أدب real number prove
+
+127
+00:16:28,110 --> 00:16:40,930
+حاجتي الواحد ال بساوي سفر لان limit
+
+128
+00:16:40,930 --> 00:16:44,290
+f
+
+129
+00:16:44,290 --> 00:16:44,950
+of x
+
+130
+00:16:48,250 --> 00:16:59,370
+كما يظهر X لـ C لجميع الـ C تلتانيه لـ R لما
+
+131
+00:16:59,370 --> 00:17:05,770
+نثبت أن الـ limit لها تساوي سفر ثم الـ function F
+
+132
+00:17:05,770 --> 00:17:12,230
+لها limitالأعداد الحقيقية C والكتاب يعطيك hint
+
+133
+00:17:12,230 --> 00:17:20,490
+يعني إرشاد كيف يعني تبدأ الحل بطريقة صحية proof
+
+134
+00:17:20,490 --> 00:17:24,410
+فخلينا
+
+135
+00:17:24,410 --> 00:17:33,150
+نبرهن الجزء الأول لحظة
+
+136
+00:17:33,150 --> 00:17:37,780
+أن ال function هذه بتحقق الشرط هذابنسميه
+
+137
+00:17:37,780 --> 00:17:42,880
+additivity ال function f بتحافظ على عملية الجمع
+
+138
+00:17:42,880 --> 00:17:48,100
+بتاخد مجموعة حاجتين تعطي صورتها مجموعة صورهم
+
+139
+00:17:48,100 --> 00:17:52,360
+فبنقول f أي function بتحقق خاصية زي هذه بنسميها
+
+140
+00:17:52,360 --> 00:17:57,570
+additive يعني دالة جمعيةبتحافظ على عملية الجمع تبع
+
+141
+00:17:57,570 --> 00:18:03,170
+الأعداد الحقيقية فبناء على الخاصية هذه لو كانت ال
+
+142
+00:18:03,170 --> 00:18:07,450
+function f لها limit and سفر فال limit هذه لازم
+
+143
+00:18:07,450 --> 00:18:13,630
+تكون بساوي سفر خمتها سفر فكيف ممكن نثبت الكلام هذا
+
+144
+00:18:13,630 --> 00:18:19,590
+نسمي
+
+145
+00:18:19,590 --> 00:18:22,190
+هذا الفرض star
+
+146
+00:18:25,940 --> 00:18:34,100
+by hypothesis star من الفرض star انا عندي h او ال
+
+147
+00:18:34,100 --> 00:18:42,500
+function f of two x ايش بتساوي؟ بتساوي f of x زاد
+
+148
+00:18:42,500 --> 00:18:47,260
+x مظبوط و من الفرض star
+
+149
+00:18:51,230 --> 00:18:58,670
+f of x زاد x بيساوي f of x زاد f of x صح يعني
+
+150
+00:18:58,670 --> 00:19:13,710
+بيساوي اتنين في f of x تمام طيب
+
+151
+00:19:13,710 --> 00:19:20,130
+take limit of both sides take limit
+
+152
+00:19:22,620 --> 00:19:29,320
+of both sides as
+
+153
+00:19:29,320 --> 00:19:39,900
+x tends to zero we get نحصل على انه ال limit ل f ل
+
+154
+00:19:39,900 --> 00:19:50,380
+2x لما x تقول ل 0 بساوي 2 في ال limit ل f of x لما
+
+155
+00:19:50,380 --> 00:19:51,720
+x تقول ل 0
+
+156
+00:19:56,790 --> 00:20:01,490
+Limit f of x من اكسا اولا سفر بساوي من الفرض
+
+157
+00:20:01,490 --> 00:20:08,610
+موجودة و بساوي L في اتنين بطلع اتنين L و ال limit
+
+158
+00:20:08,610 --> 00:20:14,230
+هذه هي نفسها limit
+
+159
+00:20:17,140 --> 00:20:27,440
+لـ F of 2X لما 2X تقول لـ 0 صح؟ لما X تقول لـ 0،
+
+160
+00:20:27,440 --> 00:20:38,460
+2X تقول لـ 0 و هذه هي نفسها ال limit ل F of Y لما
+
+161
+00:20:38,460 --> 00:20:48,520
+Y تقول لـ 0 صح؟ خدي Y بساوي 2Xفالمهي هنا limit f
+
+162
+00:20:48,520 --> 00:20:54,400
+of y لما y تقل ل 0 ال independent variable ده سميه
+
+163
+00:20:54,400 --> 00:20:59,760
+x سميه y it doesn't matter مش مهم اه لإن هذا برضه
+
+164
+00:20:59,760 --> 00:21:07,080
+ال limit هذه بساوي L لإن أنا أصبح عندي لإن أنا
+
+165
+00:21:07,080 --> 00:21:14,040
+أصبح عندي أنا معادلة L بساوي two L حل المعادلة هذه
+
+166
+00:21:15,510 --> 00:21:23,150
+في ال فهذا بيقدي ان ال بيساوي سفر صح ان هنا اثبتت
+
+167
+00:21:23,150 --> 00:21:28,350
+ان العدد ال اللي هو limit لل function if and السفر
+
+168
+00:21:28,350 --> 00:21:35,370
+بيطلع بساوي سفر صحيح ولا سهل ان
+
+169
+00:21:35,370 --> 00:21:41,710
+الكتاب بيحط رجلكم قدامكم على بداية الطريق استغلي
+
+170
+00:21:41,710 --> 00:21:42,050
+صح
+
+171
+00:21:45,430 --> 00:21:57,650
+واضح البرهان هنا تمام نبرهن الجزء التاني طيب
+
+172
+00:21:57,650 --> 00:22:03,570
+نبرهن الجزء التاني برضه
+
+173
+00:22:03,570 --> 00:22:13,250
+الجزء التاني فيه له hint فنستخدم ال hint also
+
+174
+00:22:15,670 --> 00:22:24,930
+by hypothesis star حسب
+
+175
+00:22:24,930 --> 00:22:31,630
+الفرض star ال function هذه is additive وبالتالي f
+
+176
+00:22:31,630 --> 00:22:40,750
+of x هي نفسها f of x ثالث c زائد c صح؟
+
+177
+00:22:44,780 --> 00:22:52,200
+و هذا بيساوي F
+
+178
+00:22:52,200 --> 00:23:03,900
+of X minus C زائد F of C تمام طيب هذا صحيح for all
+
+179
+00:23:03,900 --> 00:23:13,100
+C ينتمي ل R و طبعا for all X و for all X ينتمي ل R
+
+180
+00:23:14,280 --> 00:23:19,900
+فى مشكلة طيب الان نفس الحاجة اخد ال limit للطرفين
+
+181
+00:23:19,900 --> 00:23:30,540
+take limit of both sides لما x تقول ل c اذا limit
+
+182
+00:23:30,540 --> 00:23:40,780
+f of x لما x تقول ل c بساوي limitالطرف اليومين
+
+183
+00:23:40,780 --> 00:23:44,240
+مجموعة limit مجموعة بيساوي مجموعة limits لأن limit
+
+184
+00:23:44,240 --> 00:23:48,660
+الحد الأول exist هنشوف ان limit الحد الأول موجودة
+
+185
+00:23:48,660 --> 00:23:52,320
+و limit الحد التاني موجودة وبالتالي limit المجموعة
+
+186
+00:23:52,320 --> 00:23:59,000
+بيساوي مجموعة limits فlimit f of x minus c as x
+
+187
+00:23:59,000 --> 00:24:01,160
+tends to c زائد
+
+188
+00:24:08,360 --> 00:24:25,360
+limit f of x لمّا x تقول لـ c limit
+
+189
+00:24:25,360 --> 00:24:30,140
+f
+
+190
+00:24:30,140 --> 00:24:33,960
+of x لمّا x تقول لـ c
+
+191
+00:24:37,080 --> 00:24:44,860
+ال limit هذه لو أخدت y بساوي x minus c فبطلع x
+
+192
+00:24:44,860 --> 00:24:53,900
+بساوي y زائد c وبالتالي
+
+193
+00:24:53,900 --> 00:25:01,220
+لما x تقول ل c لما x تقول ل c هذا بيقدر ان y تقول
+
+194
+00:25:01,220 --> 00:25:12,090
+ل 0 صح؟ إذن هذه هي نفس limitF of Y لما Y تقول لـ 0
+
+195
+00:25:12,090 --> 00:25:15,550
+وذلك
+
+196
+00:25:15,550 --> 00:25:22,690
+ببتعوض عن X سالب C بساوي Y وهذا عدد ثابت F of C
+
+197
+00:25:22,690 --> 00:25:29,950
+عدد ثابت نهايته نفس العدد الثابت F of C Y ساوي طيب
+
+198
+00:25:29,950 --> 00:25:34,510
+احنا لسه من الفرض فرضين ان limit F of Y لما Y تقول
+
+199
+00:25:34,510 --> 00:25:42,440
+لـ 0 بساوي Lاللي هو سفر إذاً هذا بيساوي العدد L
+
+200
+00:25:42,440 --> 00:25:46,860
+اللي هو سفر زائد
+
+201
+00:25:46,860 --> 00:25:55,600
+F of C اللي هو F of C إذاً
+
+202
+00:25:55,600 --> 00:26:00,160
+هنا أثبتنا إنه limit F of X لما X تقول لـ C exist
+
+203
+00:26:00,160 --> 00:26:01,880
+و بيساوي F of C
+
+204
+00:26:05,410 --> 00:26:09,530
+تمام؟ إذا هين أثبتنا أن ال limit لل function عند
+
+205
+00:26:09,530 --> 00:26:17,170
+أي c exist و بساوي f of c و هذا طبعا الشرط اللي ..
+
+206
+00:26:17,170 --> 00:26:21,090
+لاحظوا أنتوا limit f of x عند أي c بتطلعت بالساوية
+
+207
+00:26:21,090 --> 00:26:25,970
+قيمة الدالة عن ال c هذا حسب chapter 5 هذا معناه أن
+
+208
+00:26:25,970 --> 00:26:31,510
+الدالة هذه تطلع continuous عند أي نقطة في المجال
+
+209
+00:26:31,510 --> 00:26:36,670
+تبعهاOkay إذا النتيجة الخلاصة من ال exercise هذا
+
+210
+00:26:36,670 --> 00:26:43,030
+exercise كتير مهم وهو إنه اللي لو كان في عندى
+
+211
+00:26:43,030 --> 00:26:50,370
+function from R to R و ال function هذه edited و
+
+212
+00:26:50,370 --> 00:26:56,700
+نهايتها عند السفر موجودةفالدال هذا بتطلع متصلة عن
+
+213
+00:26:56,700 --> 00:27:00,100
+كل الأعداد الحقيقية هذا النتيجة النهائية هذا اللي
+
+214
+00:27:00,100 --> 00:27:08,040
+أثبتناه في الجزء التاني okay تمام؟ وهذه نتيجة مهمة
+
+215
+00:27:08,040 --> 00:27:15,020
+ومعروفة في كورسات ال real analysis المتقدمة تمام
+
+216
+00:27:15,020 --> 00:27:20,860
+واضح؟ في أي استفسار؟إذا شفتوا هنا يعني كيف استغلنا
+
+217
+00:27:20,860 --> 00:27:26,080
+الفرض السار و كيف استغلنا ال hint الإرشادات اللي
+
+218
+00:27:26,080 --> 00:27:29,860
+أعطينا إياها الكتاب فاحنا ممكن نجيبلكم في الامتحان
+
+219
+00:27:29,860 --> 00:27:38,320
+أسئلة و نعطيلكم hint عليها أساسا كبداية للحل أو
+
+220
+00:27:38,320 --> 00:27:43,570
+البرهان الصحي و الشاطرة اللي تستغلها صحيOkay تمام
+
+221
+00:27:43,570 --> 00:27:48,130
+في أي أسئلة تانية section اربع واحد او اربع اتنين
+
+222
+00:27:48,130 --> 00:27:50,030
+رقم تلات عشر اربع اتنين
+
+223
+00:28:25,120 --> 00:28:33,640
+أي سؤال تلتاش سيكشن أربعة اتنين في عندي F function
+
+224
+00:28:33,640 --> 00:28:39,020
+from A to R و
+
+225
+00:28:39,020 --> 00:28:49,560
+C belong to R is a cluster point cluster point of
+
+226
+00:28:49,560 --> 00:28:50,220
+A
+
+227
+00:29:03,780 --> 00:29:18,360
+فالـ limit لـ f of x لما x تقول لـ c exists prove
+
+228
+00:29:18,360 --> 00:29:29,240
+أن ال limit ل absolute f of x لما x تقول ل c
+
+229
+00:29:29,240 --> 00:29:31,720
+بيساوي
+
+230
+00:29:34,790 --> 00:29:42,870
+بتساوي absolute limit f of x لما x تقوم بالاسم
+
+231
+00:29:42,870 --> 00:29:48,950
+where
+
+232
+00:29:48,950 --> 00:29:59,690
+حيث و where حيث absolute fالـ absolute value لأي
+
+233
+00:29:59,690 --> 00:30:05,550
+function تطلع function تانية تعريفها عند أي x أو
+
+234
+00:30:05,550 --> 00:30:13,430
+قيمتها عند أي x بساوي absolute f of x لكل x
+
+235
+00:30:27,830 --> 00:30:33,090
+بمعنى اخر اذا كان في عندي function و ال limit لها
+
+236
+00:30:33,090 --> 00:30:38,510
+ان c موجودة ف limit ال absolute value لها بتكون
+
+237
+00:30:38,510 --> 00:30:43,970
+موجودة و تساوي ال absolute value لل limit يعني
+
+238
+00:30:43,970 --> 00:30:48,090
+ممكن ابدل ال limit مع ال absolute value او ادخل ال
+
+239
+00:30:48,090 --> 00:30:53,730
+limit داخل ال absolute value وهذه حقيقة صحيحة لأن
+
+240
+00:30:53,730 --> 00:30:56,550
+ال absolute value function متتصرة
+
+241
+00:31:02,530 --> 00:31:19,850
+Okay تمام ف ..
+
+242
+00:31:19,850 --> 00:31:30,030
+طيب يعني
+
+243
+00:31:30,030 --> 00:31:38,610
+sayدعونا نفترض أن ال limit ل f of x لما x تقول إلى
+
+244
+00:31:38,610 --> 00:31:44,870
+c بساوي عدد L ينتمي ل R مش ال limit had existed
+
+245
+00:31:44,870 --> 00:31:48,830
+سميها L و
+
+246
+00:31:48,830 --> 00:31:52,430
+المطلوب we need to show
+
+247
+00:31:58,250 --> 00:32:08,570
+نحتاج أن نظهر أن الـ limit لـ absolute f of x as x
+
+248
+00:32:08,570 --> 00:32:11,510
+tends to c بساوي absolute L
+
+249
+00:32:35,630 --> 00:32:44,970
+أنا في نهاية
+
+250
+00:32:44,970 --> 00:32:52,570
+الأمر بدي أثبت أن الـ absolute بدي
+
+251
+00:32:52,570 --> 00:32:55,690
+في النهاية يكون هذا أصغر من أي جيبل إبسلون
+
+252
+00:32:58,740 --> 00:33:05,380
+لكل x بحيث ان absolute x minus c أكبر من سفر أصغر
+
+253
+00:33:05,380 --> 00:33:17,660
+من delta where delta to be determined يعني
+
+254
+00:33:17,660 --> 00:33:23,560
+هنحددها لاحقا هذا ايه عشان اثبت ان ال limit ال
+
+255
+00:33:23,560 --> 00:33:29,700
+function هذه يعني x بساوي absolute الفبدأثبت ان
+
+256
+00:33:29,700 --> 00:33:33,300
+absolute الفرخ هذا أصغر من epsilon عندما يكون
+
+257
+00:33:33,300 --> 00:33:39,240
+المسافة بين ال X و ال C أصغر من delta و X لا تساوي
+
+258
+00:33:39,240 --> 00:33:46,580
+ال C for some delta تعتمد على ال given epsilon طب
+
+259
+00:33:46,580 --> 00:33:50,860
+أنا عندي هذا
+
+260
+00:33:50,860 --> 00:33:57,000
+بيساوي absolute absolute F of X minus absolute
+
+261
+00:34:09,920 --> 00:34:16,520
+تمام؟ وهذا باستخدام ال triangle inequality أصغر من
+
+262
+00:34:16,520 --> 00:34:23,440
+أو ساوي absolute f of x minus L صح؟ في صورة من صور
+
+263
+00:34:23,440 --> 00:34:26,140
+ال triangle inequality يجب تقول إن هذا ال absolute
+
+264
+00:34:26,140 --> 00:34:30,140
+value ل absolute a minus absolute b أصغر من
+
+265
+00:34:30,140 --> 00:34:35,880
+absolute a minus bطب انا عندى limit f of x لما x
+
+266
+00:34:35,880 --> 00:34:41,400
+اقولها c بالساوي L فممكن اخلي هذا اصغر من اي given
+
+267
+00:34:41,400 --> 00:34:47,540
+epsilon وبالتالي هذه بتصير اصغر من اي epsilon okay
+
+268
+00:34:47,540 --> 00:34:52,420
+تمام اذا هنا خلينا نشوف
+
+269
+00:35:03,620 --> 00:35:09,800
+let epsilon أكبر
+
+270
+00:35:09,800 --> 00:35:18,820
+من السفر be givenبما انه since ال limit احنا فرضين
+
+271
+00:35:18,820 --> 00:35:25,840
+انه limit ل F of X عند X بساوي C بساوي L اذا يوجد
+
+272
+00:35:25,840 --> 00:35:31,580
+Delta تعتمد على ابسلون عدد موجب بحيث انه لكل X
+
+273
+00:35:31,580 --> 00:35:37,060
+ينتمي إلى A و absolute X minus C اصغر من Delta
+
+274
+00:35:37,060 --> 00:35:45,550
+اكبر من سفر هذا بيقديإنه absolute f of x minus L
+
+275
+00:35:45,550 --> 00:35:51,890
+أصغر من إبسلم نسمي هذا star نسمي ال implication
+
+276
+00:35:51,890 --> 00:36:03,570
+هذا star hence
+
+277
+00:36:08,200 --> 00:36:16,880
+by triangle inequality من متبينة المثلث x ينتمي by
+
+278
+00:36:16,880 --> 00:36:26,260
+triangle inequality and star we have لدينا لو كان
+
+279
+00:36:26,260 --> 00:36:32,680
+x ينتمي ل a و absolute x minus c أكبر من سفر أصغر
+
+280
+00:36:32,680 --> 00:36:48,240
+من deltaفهذا بيقدي انه absolute .. absolute f of x
+
+281
+00:36:48,240 --> 00:36:59,300
+minus absolute ال L هذا بيساوي absolute ..
+
+282
+00:36:59,300 --> 00:37:06,060
+absolute f of x minus absolute L
+
+283
+00:37:12,840 --> 00:37:17,300
+وهذا by ال triangle inequality by ال triangle
+
+284
+00:37:17,300 --> 00:37:24,520
+inequality أصغر من أوي ساوي absolute f of x minus
+
+285
+00:37:24,520 --> 00:37:32,300
+l و by star absolute f of x minus l أصغر من إبسل
+
+286
+00:37:32,300 --> 00:37:41,020
+تمام؟ since إبسلون was
+
+287
+00:37:41,020 --> 00:37:41,740
+arbitrary
+
+288
+00:37:47,340 --> 00:37:53,990
+إذاً we haveهك مكون أسبابنا أنه لأي epsilon أكبر
+
+289
+00:37:53,990 --> 00:37:59,950
+من الصفر يوجد delta تعتمد على epsilon عدد موجب من
+
+290
+00:37:59,950 --> 00:38:06,350
+حيث لكل x ينتمي ل a و المسافة بين x و c أصغر من
+
+291
+00:38:06,350 --> 00:38:11,630
+delta و x لا تساوي c فلل absolute value لf of x
+
+292
+00:38:11,630 --> 00:38:15,930
+minus absolute L أصغر من epsilon، إذن هذا معناه
+
+293
+00:38:15,930 --> 00:38:21,460
+حسب epsilon delta definition of limitإن الـ limit
+
+294
+00:38:21,460 --> 00:38:29,780
+ل absolute f of x as x tends to c بساوي absolute ل
+
+295
+00:38:29,780 --> 00:38:33,140
+هو
+
+296
+00:38:33,140 --> 00:38:38,420
+المطلوب وهذا اللي احنا عايزين نثبته هذا هو البرهان
+
+297
+00:38:39,940 --> 00:38:45,060
+إذا هنا اللاعب دور كبير في البرهان هو ال triangle
+
+298
+00:38:45,060 --> 00:38:52,420
+inequality و الفرض إنه limit and c موجودة بالساوية
+
+299
+00:38:52,420 --> 00:38:58,420
+L okay تمام؟ إذا أهمية ال exercise هذا يعتبر نظرية
+
+300
+00:38:58,420 --> 00:39:02,120
+و النظرية هذه بتقول إذا كان في end function
+
+301
+00:39:02,120 --> 00:39:07,590
+نهايتها and c موجودةفنهاية ال absolute value لل
+
+302
+00:39:07,590 --> 00:39:11,610
+function بالساوي ال absolute value لل limit بمعنى
+
+303
+00:39:11,610 --> 00:39:15,870
+أخر أنا ممكن أدخل ال limit داخل ال absolute value
+
+304
+00:39:15,870 --> 00:39:22,710
+أو أبدل ال limit مع ال absolute value okay تمام
+
+305
+00:39:22,710 --> 00:39:26,630
+واضح؟ في أسئلة تانية؟
+
+306
+00:39:35,800 --> 00:39:43,260
+أي سؤال أي استفسار okay
+
+307
+00:39:43,260 --> 00:39:45,560
+إذا نكتفي بهذا القدر
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/dw89EvC63CE_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/dw89EvC63CE_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5496e13ebeaf30fd22f549e4c9c149a94d92cda7
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/dw89EvC63CE_raw.srt
@@ -0,0 +1,1252 @@
+1
+00:00:20,910 --> 00:00:26,430
+بسم الله الرحمن الرحيم في المحاضرة اليوم هناخد
+
+2
+00:00:26,430 --> 00:00:32,890
+انتباه المقدمة البسيطة عن ال infinite series اللي
+
+3
+00:00:32,890 --> 00:00:37,890
+بدأناها في المحاضرة السابقةففي المحاضرة السابقة
+
+4
+00:00:37,890 --> 00:00:41,610
+عرفنا ما معنى ان infinite series of real numbers
+
+5
+00:00:41,610 --> 00:00:46,050
+converge معناه هذا بكافئ ان ال sequence of partial
+
+6
+00:00:46,050 --> 00:00:51,970
+sums S اللي سمناها Sn حيث ال inf partial sum هو
+
+7
+00:00:51,970 --> 00:00:56,170
+مجموع أول n من حدود ال series إذا كانت ال sequence
+
+8
+00:00:56,170 --> 00:01:00,550
+هذه convergentإذا ال sequence of partial sums
+
+9
+00:01:00,550 --> 00:01:03,990
+convergent بتكون ال series convergent، إذا ال
+
+10
+00:01:03,990 --> 00:01:06,770
+sequence of partial sums divergent بتكون ال series
+
+11
+00:01:06,770 --> 00:01:11,760
+divergentطيب لو كانت ال sequence of partial sums
+
+12
+00:01:11,760 --> 00:01:15,920
+convergent و ال limit تبعتها عدد S ففي الحالة هذه
+
+13
+00:01:15,920 --> 00:01:19,420
+بتقول ان ال series converged طبعا و ال sum تبع ال
+
+14
+00:01:19,420 --> 00:01:22,680
+series مجموعة ال series بساوي ال limit لل sequence
+
+15
+00:01:22,680 --> 00:01:26,800
+of partial sums اللي هو S اذا في الحالة هذه S is
+
+16
+00:01:26,800 --> 00:01:32,860
+the sum of the infinite series الان في عنديالنظرية
+
+17
+00:01:32,860 --> 00:01:38,220
+بتعطيني test for divergence بتسميه الانث term test
+
+18
+00:01:38,220 --> 00:01:43,360
+زي ما درسته في calculus B فالنظرية هذه بتقول لو
+
+19
+00:01:43,360 --> 00:01:46,120
+كان في عندي series و ال series كانت convergent
+
+20
+00:01:46,120 --> 00:01:50,440
+فضروري it is necessary that the limit of the inf
+
+21
+00:01:50,440 --> 00:01:57,250
+term بساوي سفروالبرهان سهل هاي ال Nth partial sum
+
+22
+00:01:57,250 --> 00:02:01,830
+مجموعة أول N من حدود ال series وهي ال N minus
+
+23
+00:02:01,830 --> 00:02:06,110
+first partial sum اللي هو مجموعة N سالب واحد من
+
+24
+00:02:06,110 --> 00:02:12,310
+الحدود الأولانية لما نيجي نطرح لما نيجي نطرح فكل
+
+25
+00:02:12,310 --> 00:02:19,910
+الحدود هذه بتروح مع بعضها يبقى عندى الفرق XNأنا
+
+26
+00:02:19,910 --> 00:02:23,650
+فارض أن ال series converge إذا ال limit لل
+
+27
+00:02:23,650 --> 00:02:27,270
+sequence of partial sums exist و بالساوية sum real
+
+28
+00:02:27,270 --> 00:02:32,570
+number S الآن باخد ال limit للطرفين في ال star هنا
+
+29
+00:02:33,630 --> 00:02:38,130
+باخد limit of both sides we get انه limit x in
+
+30
+00:02:38,130 --> 00:02:43,590
+بيساوي limit s in minus limit s in minus واحد ال
+
+31
+00:02:43,590 --> 00:02:46,930
+limit ال sequence of partial sums s وهذه هي نفس ال
+
+32
+00:02:46,930 --> 00:02:51,070
+sequence of partial sums بس حدفنا أو زودنا حد برضه
+
+33
+00:02:51,070 --> 00:02:55,170
+ال limit تبعتها s إذا ال limit ل x in بيطلع بيساوي
+
+34
+00:02:55,170 --> 00:03:01,710
+سفر كما هو مظلوم تمام؟ واضح البرهان؟ الآن في عندي
+
+35
+00:03:01,710 --> 00:03:02,310
+remark
+
+36
+00:03:10,040 --> 00:03:15,900
+the converse of
+
+37
+00:03:15,900 --> 00:03:23,140
+above theorem is
+
+38
+00:03:23,140 --> 00:03:27,860
+false عكس
+
+39
+00:03:27,860 --> 00:03:36,120
+النظرية السابقة ليس صحيحا بمعنى لو كانت x in
+
+40
+00:03:38,350 --> 00:03:45,490
+converge to zero هذا لا يؤدي بالضرورة .. لا يؤدي
+
+41
+00:03:45,490 --> 00:03:56,310
+مش شرط يؤدي ان ال series sigma x in converge وهي
+
+42
+00:03:56,310 --> 00:03:58,310
+مثال على ذلك ناخد مثال
+
+43
+00:04:17,840 --> 00:04:25,520
+example the harmonic series satisfies
+
+44
+00:04:39,920 --> 00:04:48,080
+إن ال limit ل xn عبارة عن ال limit ل 1 على n يسوى
+
+45
+00:04:48,080 --> 00:04:59,420
+سفر لكن however .. however
+
+46
+00:04:59,420 --> 00:05:04,560
+ال series ال harmonic series diverges
+
+47
+00:05:09,690 --> 00:05:14,990
+ال harmonic series is divergent مش convergent ليش؟
+
+48
+00:05:14,990 --> 00:05:22,770
+why؟
+
+49
+00:05:22,770 --> 00:05:29,330
+احنا برهننا الكلام هذا في chapter تلاتة بس كش في
+
+50
+00:05:29,330 --> 00:05:31,410
+section تلاتة خمسة
+
+51
+00:05:36,620 --> 00:05:44,980
+فاكرين ان ال sequence هذه كانت طيب نجاوب ليه ال
+
+52
+00:05:44,980 --> 00:05:52,720
+harmonic series diverges في عندي جوابين answer one
+
+53
+00:05:52,720 --> 00:06:00,660
+الإجابة الأولى في
+
+54
+00:06:00,660 --> 00:06:03,740
+عندي مثال أخدنا
+
+55
+00:06:08,530 --> 00:06:14,430
+by example لسه ماخدينه اللي كان هذا الجزء اللي
+
+56
+00:06:14,430 --> 00:06:18,590
+بتعلق فيه اللي هو ال sequence of partial sums لل
+
+57
+00:06:18,590 --> 00:06:25,470
+series هذه اثبتنا انها not cauchy صح؟ وكان هذا جزء
+
+58
+00:06:25,470 --> 00:06:31,610
+من ايه؟ من تكملة البرهان صح؟ اذا نشوف الرقم تبعها
+
+59
+00:06:31,610 --> 00:06:34,830
+بالظبط section
+
+60
+00:06:37,490 --> 00:06:50,870
+مثال تلاتة خمسة ستة الجزء C by this example the
+
+61
+00:06:50,870 --> 00:07:02,430
+sequence of partial sums Sn
+
+62
+00:07:02,430 --> 00:07:03,370
+where
+
+63
+00:07:05,750 --> 00:07:13,890
+SN بيساوي سيجما من K بيساوي واحد إلى N لواحد على K
+
+64
+00:07:13,890 --> 00:07:21,150
+اللي هو واحد زاد نص زاد واحد على N أثبتنا إن ال
+
+65
+00:07:21,150 --> 00:07:30,630
+sequence هذه is not Cauchy is not Cauchy and
+
+66
+00:07:30,630 --> 00:07:34,850
+therefore not convergent
+
+67
+00:07:40,330 --> 00:07:49,510
+حسب by cushy by cushy criterion cushy criterion
+
+68
+00:07:49,510 --> 00:07:52,950
+بتقول أي sequence of real numbers is convergent if
+
+69
+00:07:52,950 --> 00:07:56,190
+and only if it is cushy اللي إحنا أثبتنا إن ال
+
+70
+00:07:56,190 --> 00:08:01,180
+sequence هذه في المثال هذاماهياش كوشي و بالتالي
+
+71
+00:08:01,180 --> 00:08:04,460
+not convergent طب ما هذه هي ال sequence of partial
+
+72
+00:08:04,460 --> 00:08:07,680
+.. هذه هي ال sequence of partial sums لل harmonic
+
+73
+00:08:07,680 --> 00:08:11,020
+series اذا by above definition
+
+74
+00:08:15,530 --> 00:08:19,870
+طبعا ال definition المسعة اللى كتبناها أول واحد
+
+75
+00:08:19,870 --> 00:08:23,830
+مادام ال series of partial sums is not convergent
+
+76
+00:08:23,830 --> 00:08:30,150
+اذا ال series نفسها is divergent
+
+77
+00:08:33,300 --> 00:08:38,940
+Okay إذا هذا كان إحدى الإجابات و ليه .. ليه ال
+
+78
+00:08:38,940 --> 00:08:42,960
+series هي divergent هاي الإثبات تبعها هنا جاي من
+
+79
+00:08:42,960 --> 00:08:47,360
+المثال هذا إن ال sequence of partial sums طلعت not
+
+80
+00:08:47,360 --> 00:08:51,340
+Cauchy وبالتالي not convergent هذا السبب في كمان
+
+81
+00:08:51,340 --> 00:09:02,140
+سبب تاني أو حل تاني أو إجابة أخرى answer
+
+82
+00:09:02,140 --> 00:09:13,540
+twoإنه by .. في مثال آخر أثبتنا فيه by example
+
+83
+00:09:13,540 --> 00:09:17,900
+تلاتة
+
+84
+00:09:17,900 --> 00:09:24,800
+تلاتة تلاتة بي .. إذا في section تلاتة تلاتة
+
+85
+00:09:24,800 --> 00:09:31,540
+المثال تلاتة بي أثبتنا فيه إن ال sequence و
+
+86
+00:09:31,540 --> 00:09:32,480
+partial sums
+
+87
+00:09:34,910 --> 00:09:40,910
+the sequence of partial sums
+
+88
+00:09:40,910 --> 00:09:44,190
+Sn
+
+89
+00:09:44,190 --> 00:09:50,370
+is unbounded
+
+90
+00:09:50,370 --> 00:09:55,910
+أثبتنا بالمثال هذا أن ال sequence Sn of partial
+
+91
+00:09:55,910 --> 00:10:04,760
+sums اللي هي هذه is unbounded and thereforeإذا
+
+92
+00:10:04,760 --> 00:10:08,380
+كانت الـ sequence unbounded إذا بتطلع divergent أو
+
+93
+00:10:08,380 --> 00:10:16,060
+not convergent لأن sn is divergent لأن لو كانت
+
+94
+00:10:16,060 --> 00:10:20,800
+convergent بتكون bounded وبالتالي therefore ال
+
+95
+00:10:20,800 --> 00:10:28,090
+series أو ال harmonic seriesdivergent إذا هذا برضه
+
+96
+00:10:28,090 --> 00:10:33,350
+إجابة تانية بتخلي ال harmonic series divergent
+
+97
+00:10:33,350 --> 00:10:40,830
+تمام طيب نرجع ناخد أمثلة أخرى
+
+98
+00:11:08,590 --> 00:11:18,590
+أول شيء الـ geometric series الـ
+
+99
+00:11:18,590 --> 00:11:26,210
+geometric series اللي هي summation من k بساوي سفر
+
+100
+00:11:26,210 --> 00:11:32,090
+to infinity لR to K اللي الحد الأول تبعها واحد
+
+101
+00:11:32,090 --> 00:11:37,250
+والأساس تبعها R اشمالها واحد
+
+102
+00:11:40,350 --> 00:11:49,570
+converges if absolute r أصغر من one and
+
+103
+00:11:49,570 --> 00:12:06,550
+diverges and its sum its sum بيساوي واحد على واحد
+
+104
+00:12:06,550 --> 00:12:08,250
+minus r
+
+105
+00:12:14,230 --> 00:12:20,410
+and diverges if
+
+106
+00:12:20,410 --> 00:12:27,790
+absolute are أكبر من أوسى واحد أثبتنا
+
+107
+00:12:27,790 --> 00:12:31,250
+هذا الجزء المرة اللي فاتت في الأول ولا ما أثبتناه
+
+108
+00:12:31,250 --> 00:12:34,350
+مش؟
+
+109
+00:12:34,350 --> 00:12:38,010
+أثبتنا صحيح؟ okay إذا
+
+110
+00:12:44,740 --> 00:12:52,060
+proof part one was
+
+111
+00:12:52,060 --> 00:12:55,940
+proved last
+
+112
+00:12:55,940 --> 00:13:01,400
+time .. last time
+
+113
+00:13:01,400 --> 00:13:09,160
+to
+
+114
+00:13:09,160 --> 00:13:12,760
+prove two assume
+
+115
+00:13:15,240 --> 00:13:29,520
+absolute are أكبر من أو ساوي الواحد افترض
+
+116
+00:13:29,520 --> 00:13:36,400
+ان absolute are أكبر من أو ساوي الواحد طيب then
+
+117
+00:13:44,600 --> 00:13:53,300
+limit xn as n tends to infinity بيساوي limit r to
+
+118
+00:13:53,300 --> 00:13:59,220
+n as n tends to infinity لحظة
+
+119
+00:13:59,220 --> 00:14:06,940
+n to absolute r أكبر من واحد بيقدر ان ر أكبر من ..
+
+120
+00:14:06,940 --> 00:14:08,580
+أصغر من واحد
+
+121
+00:14:11,700 --> 00:14:22,700
+or R أكبر من سالب واحد او
+
+122
+00:14:22,700 --> 00:14:30,200
+R أكبر من واحد او R أصغر من سالب
+
+123
+00:14:30,200 --> 00:14:34,640
+واحد وبالتالي
+
+124
+00:14:34,640 --> 00:14:42,080
+limit R أُس N مش ممكن تساوي سفرlimit ل R أسن شفنا
+
+125
+00:14:42,080 --> 00:14:50,420
+أن هذه بساوية سفر فقط لما ال R يكون أصغر من واحد و
+
+126
+00:14:50,420 --> 00:14:57,940
+أكبر من أو بساوية سفر وبالتالي اذا by .. اذا by
+
+127
+00:14:57,940 --> 00:15:08,300
+nth by nth term test اذا الحد العام الحد النوني لل
+
+128
+00:15:08,300 --> 00:15:12,700
+series هذهلا يقول لا سفر ال limit تبعته ما بيسويش
+
+129
+00:15:12,700 --> 00:15:20,400
+سفر وبالتالي by the nth term test المسح اه اذا كان
+
+130
+00:15:20,400 --> 00:15:25,480
+limit الحد النوني بيسويش سفر فال test بتقول لي ان
+
+131
+00:15:25,480 --> 00:15:31,500
+ال series is
+
+132
+00:15:31,500 --> 00:15:37,480
+divergent ال series sigma ل R to N من N equal 0 to
+
+133
+00:15:37,480 --> 00:15:39,000
+infinity diverges
+
+134
+00:15:45,290 --> 00:15:48,990
+لأنه لو كانت convergent فالمفروض limit الحد اللوني
+
+135
+00:15:48,990 --> 00:15:55,090
+يساوي سفر وهذا مش موجود تمام؟ ان انا شوفت هنا كيف
+
+136
+00:15:55,090 --> 00:16:00,710
+أخدنا ال .. او استخدمنا ال in term test لإثبات ال
+
+137
+00:16:00,710 --> 00:16:05,150
+divergence مثال
+
+138
+00:16:05,150 --> 00:16:05,630
+تاني
+
+139
+00:16:14,400 --> 00:16:23,600
+show that ال series اللي متولدة من ال sequence
+
+140
+00:16:23,600 --> 00:16:29,840
+سالب واحد to end اللي
+
+141
+00:16:29,840 --> 00:16:37,100
+هي أول حد هيكون واحد بعدين سالب واحد بعدين واحد
+
+142
+00:16:37,100 --> 00:16:43,030
+بعدين سالب واحد و هكذاshow that this series
+
+143
+00:16:43,030 --> 00:16:55,610
+diverges تعالى
+
+144
+00:16:55,610 --> 00:16:59,510
+نشوف ال sequence of partial sums ناخد ال sequence
+
+145
+00:16:59,510 --> 00:17:03,070
+of partial sums والله ده طلعت ال sequence of
+
+146
+00:17:03,070 --> 00:17:06,810
+partial sums convergent بيكون ال series convergent
+
+147
+00:17:06,810 --> 00:17:10,730
+وال sum تبعها بيساوي ال limitللـ sequence of
+
+148
+00:17:10,730 --> 00:17:14,290
+partial sums مظبوط وإذا طلعت ال sequence of
+
+149
+00:17:14,290 --> 00:17:19,570
+partial sums divergent فال series اللي تابعة إلىها
+
+150
+00:17:19,570 --> 00:17:24,650
+بتكون divergent لذا هنا هنفحص هل ال sequence of
+
+151
+00:17:24,650 --> 00:17:29,410
+partial sums divergent ولا convergent طيب تعالوا
+
+152
+00:17:29,410 --> 00:17:34,750
+نفحص Sn ال N في partial sums هذا summation من K
+
+153
+00:17:34,750 --> 00:17:47,900
+equal 0 to Nلا سالب واحد plus k هذا xk فهذا هيطلع
+
+154
+00:17:47,900 --> 00:17:57,300
+الو قيمتين الو هيطلع قيمتين
+
+155
+00:17:57,300 --> 00:18:05,300
+لو
+
+156
+00:18:05,300 --> 00:18:14,590
+كان in evenأو n odd لو كانت n even يعني نفر بأن n
+
+157
+00:18:14,590 --> 00:18:22,470
+بساوي سفر إذن هيطلع واحد، n بساوي اتنين، هيطلع
+
+158
+00:18:22,470 --> 00:18:29,470
+اندي تلت حدود، مجموعهم برضه واحد، و هكذا، إذن
+
+159
+00:18:29,470 --> 00:18:35,160
+المجموع هذا هيطلع واحدإذا كانت ال in even أما لو
+
+160
+00:18:35,160 --> 00:18:40,680
+كانت ال in odd يعني لو أخدت in بالساوية واحدلو
+
+161
+00:18:40,680 --> 00:18:46,040
+أخدت in بالساوية واحد فهيطلع عندي حدين مجموعهم سفر
+
+162
+00:18:46,040 --> 00:18:50,960
+لو أخدت in بالساوية تلاتة هيطلع مجموع أول أربع
+
+163
+00:18:50,960 --> 00:18:56,220
+حدود برضه بيطلع مجموعهم سفر و هكذا اذا سفر ال in
+
+164
+00:18:56,220 --> 00:19:00,160
+partial sums بساوية سفر اذا كانت in odd واحد اذا
+
+165
+00:19:00,160 --> 00:19:05,650
+كانت in even وبالتالي thereforetherefore a
+
+166
+00:19:05,650 --> 00:19:11,110
+sequence of partial sums is an alternating series
+
+167
+00:19:11,110 --> 00:19:16,170
+يعني حدودها متدبدة بين صفر و واحد أو أول حد واحد
+
+168
+00:19:16,170 --> 00:19:23,710
+التاني صفر واحد صفر و هكذا which
+
+169
+00:19:23,710 --> 00:19:29,710
+is divergent هذه أمرها ما كانت convergent
+
+170
+00:19:33,470 --> 00:19:39,270
+الـ sequence هذه divergent عارفين ليه؟ لأن هي ال
+
+171
+00:19:39,270 --> 00:19:54,130
+proof الـ subsequence الحدود
+
+172
+00:19:54,130 --> 00:19:58,330
+مثلا الزوجية S2K
+
+173
+00:20:03,520 --> 00:20:08,760
+هي أول حد وهي التاني لإن الحدود الزوجية أصفار كلهم
+
+174
+00:20:08,760 --> 00:20:17,120
+صح؟ تكون بيرجل سفر and ال subsequence التانية اللي
+
+175
+00:20:17,120 --> 00:20:23,400
+حدودها فردية كلهم
+
+176
+00:20:23,400 --> 00:20:30,960
+واحد، ثابت واحد وهدي تكون بيرجل واحد نعم؟
+
+177
+00:20:31,600 --> 00:20:36,180
+وبالتالي إذا مادام في عندي two subsequences واحدة
+
+178
+00:20:36,180 --> 00:20:40,120
+converge لصفر وواحدة converge لواحد وصفر بيسويش
+
+179
+00:20:40,120 --> 00:20:46,380
+الواحد وصفر لا يساوي الواحد معناته by the
+
+180
+00:20:46,380 --> 00:20:50,400
+divergence criterion ال sequence هذه is divergent
+
+181
+00:20:51,890 --> 00:20:56,830
+لأن لو كانت ال sequence هذي convergent فمفروض أي
+
+182
+00:20:56,830 --> 00:21:00,250
+subsequence تكون convergent لنفس ال limit يعني
+
+183
+00:21:00,250 --> 00:21:03,630
+المفروض دول ال limits تكون متساويين و هذا مستحيل
+
+184
+00:21:03,630 --> 00:21:07,130
+إذا
+
+185
+00:21:07,130 --> 00:21:11,490
+ال sequence of partial sums هنا divergent وبالتالي
+
+186
+00:21:11,490 --> 00:21:17,130
+نكمل الحل هنا و بالتالي therefore the series
+
+187
+00:21:21,790 --> 00:21:27,290
+المتولدة من الـ sequence سالب واحد أس N اللي هي ال
+
+188
+00:21:27,290 --> 00:21:31,970
+series هذه is
+
+189
+00:21:31,970 --> 00:21:38,510
+divergent okay
+
+190
+00:21:38,510 --> 00:21:43,210
+تمام وبالتالي هيك ممكن أثبتنا هذا مثال على
+
+191
+00:21:43,210 --> 00:21:46,430
+divergence series إذن عشان أثبت ال series
+
+192
+00:21:46,430 --> 00:21:51,260
+divergent أو convergent بحاول أفحصالـ sequence of
+
+193
+00:21:51,260 --> 00:21:55,120
+partial sums و أفحص هل الـ sequence of partial
+
+194
+00:21:55,120 --> 00:22:01,880
+sums convergent ولا divergent ناخد
+
+195
+00:22:01,880 --> 00:22:06,760
+كمان مثال مختلف من نوع آخر مثال رقم 3
+
+196
+00:22:21,550 --> 00:22:29,550
+Discuss the convergence of
+
+197
+00:22:29,550 --> 00:22:33,110
+the
+
+198
+00:22:33,110 --> 00:22:43,010
+series sigma from N equals one to infinity لواحد
+
+199
+00:22:43,010 --> 00:22:45,130
+على N في N زايد واحد
+
+200
+00:22:53,320 --> 00:22:59,340
+لما يكون الحد العام لل series xn عبارة عن كسر زي
+
+201
+00:22:59,340 --> 00:23:04,260
+هذا و الكسر اللي زي هذا ممكن تجزئته إلى كسور جزئية
+
+202
+00:23:04,260 --> 00:23:10,220
+فبنعمل كسور جزئية الأول و بنجزئ الكسر هذا إلى
+
+203
+00:23:10,220 --> 00:23:16,600
+مجموع كسرين جزئيين اذا by partial fractions او use
+
+204
+00:23:16,600 --> 00:23:22,280
+partial fractions و هذا اتعلمته في تفاضل بقى
+
+205
+00:23:25,060 --> 00:23:30,740
+by partial fractions هي
+
+206
+00:23:30,740 --> 00:23:40,560
+عندي واحد على k في k زائد واحد بساوي a على k زائد
+
+207
+00:23:40,560 --> 00:23:42,920
+b over k plus one
+
+208
+00:23:48,420 --> 00:23:53,360
+الان لتحديد ثوابت A وB نتخلص من الكسور في المعادلة
+
+209
+00:23:53,360 --> 00:23:58,940
+هذه و ذلك بضرب طرفي المعادلة في المقام تبع الطرف
+
+210
+00:23:58,940 --> 00:24:08,530
+اليسار اذا بضرب في K في K زائد واحديطلع عندي واحد
+
+211
+00:24:08,530 --> 00:24:16,450
+بساوي a في k plus one زايد b في k الآن خد k بساوي
+
+212
+00:24:16,450 --> 00:24:22,610
+سفر هذا بيقدر ان a بساوي واحد وخد k بساوي سالب
+
+213
+00:24:22,610 --> 00:24:29,650
+واحد بيقدر ان السالب b بساوي واحد بيقدر ان b بساوي
+
+214
+00:24:29,650 --> 00:24:38,450
+سالب واحدإذا طلع أنا عندي واحد على ك في ك زائد
+
+215
+00:24:38,450 --> 00:24:45,170
+واحد بساوي واحد على ك سالب واحد على ك زائد واحد
+
+216
+00:24:45,170 --> 00:24:51,550
+الان now تعالوا نشوف ال inf partial sum أو نبحث ال
+
+217
+00:24:51,550 --> 00:24:55,430
+sequence of partial sums ونبحث هل هي convergent
+
+218
+00:24:55,430 --> 00:25:03,610
+ولا divergentفهي عندي sm بساوي summation من k
+
+219
+00:25:03,610 --> 00:25:13,170
+بساوي واحد إلى n ل xk اللي
+
+220
+00:25:13,170 --> 00:25:19,670
+هو summation من k بساوي واحد إلى n لواحد على k في
+
+221
+00:25:19,670 --> 00:25:27,210
+k زايد واحدوهذا الان باستخدام ال partial fractions
+
+222
+00:25:27,210 --> 00:25:33,810
+summation من k بساوي واحد الى n الى واحد على k
+
+223
+00:25:33,810 --> 00:25:42,790
+سالب واحد على k موجب واحد الان
+
+224
+00:25:42,790 --> 00:25:47,450
+هذا المجموع تعالوا نفرد تعالوا نفرفته let's expand
+
+225
+00:25:47,450 --> 00:25:49,490
+it يعني نفرفته
+
+226
+00:25:53,610 --> 00:25:58,830
+هذا مجموع منتهي في n من الحدود خد k بالساوي واحد
+
+227
+00:25:58,830 --> 00:26:06,930
+بطلع أول حد واحد سالب نص خد k بساوي اتنين بطلع
+
+228
+00:26:06,930 --> 00:26:16,010
+الحد البعد و نص سالب تلت و هكذا إلى الحد الأخير
+
+229
+00:26:16,010 --> 00:26:23,560
+هيكون واحد على n minus واحد على n زاد واحدبنلاحظ
+
+230
+00:26:23,560 --> 00:26:28,920
+أن المجموع هذا telescoping يعني في حدود فيها إزاي
+
+231
+00:26:28,920 --> 00:26:33,280
+تتلاشى مع الحدود اللي بعديها يعني سالب نص لاحظوا
+
+232
+00:26:33,280 --> 00:26:38,640
+بيروح مع موجب نص وسالب تلت هيروح مع موجب تلت في
+
+233
+00:26:38,640 --> 00:26:43,340
+الحد اللي يليه مباشرة وواحد على ان هذا بيروح مع
+
+234
+00:26:43,340 --> 00:26:50,510
+سالب واحد على ان في الحد اللي يصدقه مباشرةفيبقى كل
+
+235
+00:26:50,510 --> 00:26:55,650
+شي بيروح بيظل يبقى عندي واحد negative one over n
+
+236
+00:26:55,650 --> 00:27:01,410
+زايد واحد اذا انا طلع عيني اثبتت ان sn بساوي واحد
+
+237
+00:27:01,410 --> 00:27:06,410
+negative واحد على n زايد واحد الكلام هذا صحيح for
+
+238
+00:27:06,410 --> 00:27:10,480
+every natural number nأخذ قيمة كما نسميها limit of
+
+239
+00:27:10,480 --> 00:27:16,580
+both sides as n tends to infinity نحصل على قيمة Sn
+
+240
+00:27:16,580 --> 00:27:17,980
+كما نسميها limit of both sides as n tends to
+
+241
+00:27:17,980 --> 00:27:18,500
+infinity نحصل على قيمة Sn كما نسميها limit of both
+
+242
+00:27:18,500 --> 00:27:19,140
+sides as n tends to infinity نحصل على قيمة Sn كما
+
+243
+00:27:19,140 --> 00:27:19,140
+نسميها limit of both sides as n tends to infinity
+
+244
+00:27:19,140 --> 00:27:20,040
+نحصل على قيمة Sn كما نسميها limit of both sides as
+
+245
+00:27:20,040 --> 00:27:20,180
+n tends to infinity نحصل على قيمة Sn كما نسميها
+
+246
+00:27:20,180 --> 00:27:20,180
+limit of both sides as n tends to infinity نحصل
+
+247
+00:27:20,180 --> 00:27:20,180
+على قيمة Sn كما نسميها limit of both sides as n
+
+248
+00:27:20,180 --> 00:27:20,180
+tends to infinity نحصل على قيمة Sn كما نسميها
+
+249
+00:27:20,180 --> 00:27:22,340
+limit of both sides as n tends to infinity نحصل
+
+250
+00:27:22,340 --> 00:27:24,580
+على قيمة Sn كما نسميها limit of both sides as n
+
+251
+00:27:24,580 --> 00:27:29,920
+tends to infinity نحصل على قيمة Sn كما نسميها
+
+252
+00:27:29,920 --> 00:27:36,040
+limit of both sides as
+
+253
+00:27:36,380 --> 00:27:41,720
+is convergent و ال limit تبعتها بساوي واحد تمام
+
+254
+00:27:41,720 --> 00:27:46,480
+وبالتالي حسب التعريف تبع ال convergence لل
+
+255
+00:27:46,480 --> 00:27:49,680
+infinite series the infinite series اللي احنا
+
+256
+00:27:49,680 --> 00:27:54,320
+عايزين نفحصها اللي هي summation from n equals one
+
+257
+00:27:54,320 --> 00:28:00,200
+to infinity ل one over n في n plus one اشملها
+
+258
+00:28:00,200 --> 00:28:02,260
+converges
+
+259
+00:28:03,720 --> 00:28:10,920
+and ال sum تبعها يعني مجموعها بساوي واحد اللي هو
+
+260
+00:28:10,920 --> 00:28:17,360
+limit لل sequence of partial sums تمام؟
+
+261
+00:28:17,360 --> 00:28:20,800
+إذا هذا النوع من ال series بنسميه telescoping ..
+
+262
+00:28:20,800 --> 00:28:28,140
+telescoping .. telescoping series
+
+263
+00:28:31,710 --> 00:28:36,990
+such type of series is called telescoping series
+
+264
+00:28:36,990 --> 00:28:41,110
+okay تمام واضح
+
+265
+00:28:54,430 --> 00:29:00,810
+طيب هناخد بس نظرية سريعة و يعني مش .. البرهانة مش
+
+266
+00:29:00,810 --> 00:29:10,810
+هياخد الا دقيقة واحدة theorem Cauchy
+
+267
+00:29:10,810 --> 00:29:17,930
+.. Cauchy criterion for
+
+268
+00:29:17,930 --> 00:29:20,950
+series
+
+269
+00:29:25,660 --> 00:29:31,060
+أحنا أخدنا قبلها كوشي كريتيريا for sequences لأن
+
+270
+00:29:31,060 --> 00:29:37,560
+في كوشي كريتيريا for series فال .. الكوشي كريتيريا
+
+271
+00:29:37,560 --> 00:29:45,840
+for series بتنص على أنه ال series sigma
+
+272
+00:29:45,840 --> 00:29:53,100
+x in converges if
+
+273
+00:29:53,100 --> 00:29:54,500
+and only if
+
+274
+00:29:58,620 --> 00:30:05,760
+لكل ابسلون اكبر من صفر يوجد capital N يعتمد على
+
+275
+00:30:05,760 --> 00:30:13,500
+ابسلون عدد طبيعي بحيث انه لكل M اكبر من N اكبر من
+
+276
+00:30:13,500 --> 00:30:21,360
+او ساوي capital N بطلع absolute SM minus SN اللي
+
+277
+00:30:21,360 --> 00:30:28,830
+هو عبارة عن absolute Xm زائد واحد زائد xn زائد
+
+278
+00:30:28,830 --> 00:30:35,030
+اتنين زائد xm اصغر من اكسمن
+
+279
+00:30:49,160 --> 00:30:55,400
+كمان مرة ال series هذه converges if and only if
+
+280
+00:30:55,400 --> 00:31:00,440
+for every epsilon فيه capital N بحيث لكل M و N
+
+281
+00:31:00,440 --> 00:31:05,140
+أكبر من أو ساوي capital N و M بتكون أكبر من N ال
+
+282
+00:31:05,140 --> 00:31:10,520
+absolute value للفرق هذا أصغر من epsilon طب هذا هو
+
+283
+00:31:10,520 --> 00:31:11,440
+شرط كوشي
+
+284
+00:31:17,000 --> 00:31:24,920
+that is .. that is .. يعني هذا يعني أنه sigma x in
+
+285
+00:31:24,920 --> 00:31:30,640
+converges if and only if ال sequence of partial
+
+286
+00:31:30,640 --> 00:31:34,860
+sums is Cauchy
+
+287
+00:31:34,860 --> 00:31:39,880
+إذا كانت ال sequence of partial sums is Cauchy هذا
+
+288
+00:31:39,880 --> 00:31:45,280
+شرط Cauchy البرهان ينتج من Cauchy criterion for
+
+289
+00:31:45,280 --> 00:31:51,890
+sequencesطيب احنا أخدنا by definition من ال
+
+290
+00:31:51,890 --> 00:31:56,050
+definition تبع ال convergence لل series by
+
+291
+00:31:56,050 --> 00:32:00,910
+definition the series sigma
+
+292
+00:32:00,910 --> 00:32:07,510
+xn converges if and only if the sequence of
+
+293
+00:32:07,510 --> 00:32:13,870
+partial sums اللي
+
+294
+00:32:13,870 --> 00:32:14,950
+هي Sn
+
+295
+00:32:22,440 --> 00:32:33,300
+converges مظبوط؟ by Cauchy criterion
+
+296
+00:32:33,300 --> 00:32:41,990
+for sequences .. for sequences من المتالياتالـ
+
+297
+00:32:41,990 --> 00:32:52,750
+sequence sn converges if and only if sn is cauchy
+
+298
+00:32:56,430 --> 00:33:02,310
+هو بالتالي ال series sigma xn converges if and
+
+299
+00:33:02,310 --> 00:33:06,150
+only if ال sequence Sn converges طيب ما هذي
+
+300
+00:33:06,150 --> 00:33:09,690
+converges if and only if it is Cauchy، إذا ال
+
+301
+00:33:09,690 --> 00:33:14,870
+series sigma xn converges if and only if ال
+
+302
+00:33:14,870 --> 00:33:20,590
+sequence of partial sums is Cauchy، و هذا هو
+
+303
+00:33:20,590 --> 00:33:21,150
+المطلوب
+
+304
+00:33:24,120 --> 00:33:28,920
+طبعا هذا هو شرط كوشي لحظوا SM ال M هنا أكبر من N
+
+305
+00:33:28,920 --> 00:33:39,020
+فلو اتكتبت الحدود فتبعت SM هيكون X1 إلى XM SN ال N
+
+306
+00:33:39,020 --> 00:33:45,740
+أصغر من M فالحدود من X1 إلى XN لما أطرح بيضل عندي
+
+307
+00:33:45,740 --> 00:33:50,260
+كل الحدود المتشابهة بيضل عندي XN زياد واحد XN زياد
+
+308
+00:33:50,260 --> 00:33:51,840
+اتنين إلى XM
+
+309
+00:33:55,430 --> 00:34:02,690
+تمام؟ اذا بنوقف هنا ال .. هذا برهان كوشي criterion
+
+310
+00:34:02,690 --> 00:34:07,190
+for series بنوقف عند النظرية هذه و المرة الجاية ان
+
+311
+00:34:07,190 --> 00:34:14,310
+شاء الله بنكمل هناخد limit comparison test
+
+312
+00:34:21,040 --> 00:34:25,600
+هناك تجارب تجارب تجارب تجارب تجارب تجارب تجارب
+
+313
+00:34:25,600 --> 00:34:27,540
+تجارب تجارب تجارب
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU.srt
new file mode 100644
index 0000000000000000000000000000000000000000..f31d94fd9a969ed11988901f5b8f9888f1cc36be
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU.srt
@@ -0,0 +1,1327 @@
+1
+00:00:23,030 --> 00:00:31,610
+في exercise هنحاول إن شاء الله نحله وهذا ال
+
+2
+00:00:31,610 --> 00:00:37,190
+exercise مهم رقم عشرة section ثلاثة أربعة في ال
+
+3
+00:00:37,190 --> 00:00:41,470
+textbook أو الكتاب المقرر ال exercise هذا بيقول
+
+4
+00:00:41,470 --> 00:00:48,370
+let xn be a bounded sequence of real numbers. Let Sn
+
+5
+00:00:48,370 --> 00:00:53,550
+be the supremum لكل حدود ال sequence من N و M  طالعاً
+
+6
+00:00:53,550 --> 00:01:05,030
+يعني هذا عبارة عن ال supremum ل XN، X رقم N زيادة
+
+7
+00:01:05,030 --> 00:01:09,950
+واحد وهكذا، فمثلا
+
+8
+00:01:09,950 --> 00:01:15,390
+S1 هيساوي ال supremum لكل ال sequence لما N بيساوي 
+
+9
+00:01:15,390 --> 00:01:23,160
+واحد صح؟ S2 هيساوي ال supremum ل X2 و X3 إلى آخرها 
+
+10
+00:01:23,160 --> 00:01:30,260
+لأن هذا ال supremum ل ال one tail ل ال two tail، S3
+
+11
+00:01:30,260 --> 00:01:35,060
+هو ال supremum ل X3 و X4 إلى آخرها، لأن هذا ال
+
+12
+00:01:35,060 --> 00:01:39,600
+supremum لل three tail of the sequence، M tail
+
+13
+00:01:42,530 --> 00:01:47,570
+لاحظوا إن الـ Sn هذه كل ما N كبرت كل ما الـ Sn هذه
+
+14
+00:01:47,570 --> 00:01:54,130
+صغرت، و ال supremum صغرت، فبتطلع decreasing sequence، و
+
+15
+00:01:54,130 --> 00:02:00,050
+طبعاً bounded، إذا الإنفمام تبعها موجود بسبب أن S 
+
+16
+00:02:00,050 --> 00:02:07,790
+إذن هنا بنعرف ال S على إن الإنفمام لل sequence Sn
+
+17
+00:02:12,100 --> 00:02:16,480
+الآن بدنا نثبت أن يوجد subsequence XKN من sequence
+
+18
+00:02:16,480 --> 00:02:21,720
+XN وهذه ال subsequence convergent للعدد S اللي هو 
+
+19
+00:02:21,720 --> 00:02:27,720
+الإنفمام لكل S. لبرهان ذلك أنا عندي S بيساوي
+
+20
+00:02:27,720 --> 00:02:31,620
+الإنفمام من الفرض لـ sequence SN
+
+21
+00:02:34,540 --> 00:02:41,960
+أخذنا قبل هيك لمبة بتقول إنه w بيساوي infimum ل
+
+22
+00:02:41,960 --> 00:02:47,960
+set S if and only if لكل epsilon أكبر من الصفر
+
+23
+00:02:47,960 --> 00:02:57,560
+يوجد S epsilon ينتمي لـ S بحيث أن S
+
+24
+00:02:57,560 --> 00:02:58,360
+epsilon
+
+25
+00:03:01,030 --> 00:03:09,790
+أصغر من W زائد Epsilon، مظبوط هكذا؟ وفي كمان حقيقة
+
+26
+00:03:09,790 --> 00:03:14,890
+ثانية أو لمبة ثانية بتقول أن U بيساوي ال supremum ل
+
+27
+00:03:14,890 --> 00:03:21,050
+S، if and only if لكل Epsilon أكبر من الصفر يوجد S
+
+28
+00:03:21,050 --> 00:03:24,670
+Epsilon ينتمي لـ S بحيث أن
+
+29
+00:03:27,490 --> 00:03:35,870
+U ناقص epsilon أصغر من S epsilon. نظبط يعني فهنطبق
+
+30
+00:03:35,870 --> 00:03:40,750
+الكلام هذا إذا هي عندي S بيساوي N في  وملة ال set
+
+31
+00:03:40,750 --> 00:03:53,330
+هذه، اعتبر هذه ال set S فهذا هو ال W، فلو أخذت for
+
+32
+00:03:53,330 --> 00:04:04,110
+each N عدد طبيعي، لكل عدد طبيعي in there
+
+33
+00:04:04,110 --> 00:04:07,910
+exists Sn
+
+34
+00:04:07,910 --> 00:04:17,090
+ينتمي لل set هذه، ال set of all Sn  hyphen عدد طبيعي
+
+35
+00:04:17,090 --> 00:04:28,190
+بحيث إن هذا ال Sn أصغر من الإنفمام اللي هو S زيادة
+
+36
+00:04:28,190 --> 00:04:34,830
+واحد على N، اللي هو ال epsilon هذا عبارة عن ال epsilon
+
+37
+00:04:34,830 --> 00:04:42,940
+ممكن آخذ epsilon بيساوي واحد على N، هذا عدد موجب، فلأي
+
+38
+00:04:42,940 --> 00:04:47,760
+epsilon عدد موجب، يوجد عنصر في ال set اللي بأخذ لها 
+
+39
+00:04:47,760 --> 00:04:51,680
+ال infimum بحيث أن هذا العنصر أصغر من ال infimum
+
+40
+00:04:51,680 --> 00:05:05,200
+زائد epsilon، حسب الحقيقة الأولى، طيب كذلك since أيضاً
+
+41
+00:05:05,200 --> 00:05:12,890
+من الفرض احنا فرضنا إن Sn بيساوي supremum لل set of
+
+42
+00:05:12,890 --> 00:05:21,530
+all xk حيث k أكبر من أو يساوي ال n، بما أن هذا صحيح 
+
+43
+00:05:21,530 --> 00:05:28,210
+إذا أنا باستخدام الحقيقة الثانية then 
+
+44
+00:05:28,210 --> 00:05:33,410
+for epsilon لو أخدت epsilon برضه واحد على n هذا
+
+45
+00:05:33,410 --> 00:05:42,830
+عدد موجب، فبنقدر نلاقي يوجد x عنصر هنا في ال
+
+46
+00:05:42,830 --> 00:05:52,250
+sequence هذه أو في ال set هذه نسمي xkn ينتمي لـ set
+
+47
+00:05:52,250 --> 00:06:04,570
+of all xk حيث k أكبر من أو يساوي ال n such
+
+48
+00:06:04,570 --> 00:06:05,070
+that
+
+49
+00:06:08,240 --> 00:06:13,400
+ال supremum اللي هو SN ناقص الـ Y اللي هو واحد
+
+50
+00:06:13,400 --> 00:06:28,220
+على N أصغر من الـ S اللي هو العنصر XKN طيب
+
+51
+00:06:28,220 --> 00:06:30,200
+إذا من هنا
+
+52
+00:06:36,260 --> 00:06:45,240
+من المتباينة هذه والمتباينة هذه هينتج
+
+53
+00:06:45,240 --> 00:06:49,700
+it follows that
+
+54
+00:06:49,700 --> 00:07:00,020
+من المتباينتين هذول أنا عندي ال Sn
+
+55
+00:07:00,020 --> 00:07:07,730
+ناقص واحد على n أكبر من أو يساوي S ناقص 
+
+56
+00:07:07,730 --> 00:07:13,890
+واحد على N، لحظة
+
+57
+00:07:13,890 --> 00:07:20,330
+من هنا SN، الـ S، lower bound للست هذه
+
+58
+00:07:20,330 --> 00:07:25,270
+وبالتالي أي عنصر هنا أكبر من أو يساوي ال S لأن أنا
+
+59
+00:07:25,270 --> 00:07:32,230
+عندي SN أكبر من أو يساوي S اطرحي واحد على
+
+60
+00:07:32,230 --> 00:07:37,590
+N من الطرفين ونحصل على الكلام هذا، هذا بيقدي إن هذا
+
+61
+00:07:37,590 --> 00:07:45,330
+الكلام صحيح، طيب، و من هنا من المتباينة الأخير هذا
+
+62
+00:07:45,330 --> 00:07:49,650
+أصغر من xkn
+
+63
+00:07:49,650 --> 00:07:52,750
+و
+
+64
+00:07:52,750 --> 00:08:01,030
+xkn هذا عنصر في ال set هذه، عنصر في ال set هذه، و Sn
+
+65
+00:08:01,030 --> 00:08:05,450
+upper bound لل set، إذن هذا ال upper bound أكبر من
+
+66
+00:08:05,450 --> 00:08:11,790
+أو يساوي كل عناصر ال set، إذن هذا أصغر من أو يساوي Sn
+
+67
+00:08:11,790 --> 00:08:16,630
+ومن
+
+68
+00:08:16,630 --> 00:08:23,290
+المتباينة هذه SN أصغر من S زائد واحد على N
+
+69
+00:08:23,290 --> 00:08:26,490
+مظبوط؟
+
+70
+00:08:39,470 --> 00:08:50,830
+thus we obtain a subsequence، الكلام هذا صحيح لكل
+
+71
+00:08:50,830 --> 00:08:51,070
+N
+
+72
+00:08:54,210 --> 00:08:59,590
+لأن هذا الكلام صحيح لكل n، وهذا الكلام صحيح لكل n
+
+73
+00:08:59,590 --> 00:09:05,790
+لكل epsilon بيساوي واحد على n، ولكل .. لكل n ممكن
+
+74
+00:09:05,790 --> 00:09:10,450
+أكون epsilon بيساوي واحد على n وهي هنا لكل n for
+
+75
+00:09:10,450 --> 00:09:14,670
+each n، كلام هذا صحيح، إذا هذا الكلام المتباينة هذه
+
+76
+00:09:14,670 --> 00:09:21,130
+صحيحة لكل n وبالتالي أنا لكل n بقدر ألاقي xkn
+
+77
+00:09:21,130 --> 00:09:25,590
+موجود هنا و بيحقق المتباينة هذه
+
+78
+00:09:28,570 --> 00:09:34,390
+عناصرها xkn من n بيساوي واحد to infinity of ال
+
+79
+00:09:34,390 --> 00:09:43,430
+sequence xn such that بحيث أن xkn
+
+80
+00:09:43,430 --> 00:09:51,310
+أصغر من S زائد واحد على n، أكبر من S ناقص واحد على
+
+81
+00:09:51,310 --> 00:09:57,610
+n، الكلام هذا صحيح لكل n في N، طيب ماذا معناه أن 
+
+82
+00:09:57,610 --> 00:10:06,510
+absolute xkn ناقص S أصغر من واحد على n لكل n في
+
+83
+00:10:06,510 --> 00:10:14,690
+N؟ so
+
+84
+00:10:14,690 --> 00:10:17,930
+by theorem، فاكرين نظرية اثنين أربعة؟
+
+85
+00:10:25,000 --> 00:10:33,100
+with c بيساوي واحد أكبر من صفر، and ال sequence an 
+
+86
+00:10:33,100 --> 00:10:38,820
+بيساوى واحد على n tends to zero بتخلي
+
+87
+00:10:38,820 --> 00:10:44,820
+.. so by this theorem بطلع عندي limit ال sequence
+
+88
+00:10:44,820 --> 00:10:54,430
+xkn as n tends to infinity بيساوي ال S وهو المطلوب، إذن
+
+89
+00:10:54,430 --> 00:11:00,690
+هي اللي أثبتت إنه يوجد subsequence من ال sequence
+
+90
+00:11:00,690 --> 00:11:06,310
+XN وال subsequence هذه convergent إلى العدد S
+
+91
+00:11:22,860 --> 00:11:27,900
+بالمناسبة في تعريف هنا في الكتب لو شفتوا طلعتوا
+
+92
+00:11:27,900 --> 00:11:33,380
+على الكتب في الأول، واضح البرهان هنا؟ واضح البرهان؟
+
+93
+00:11:33,380 --> 00:11:42,560
+في أي شيء مش واضح؟ في
+
+94
+00:11:42,560 --> 00:11:46,860
+أي شيء مش واضح؟ اللي عندها سؤال تسألني أنا هنا مش
+
+95
+00:11:46,860 --> 00:11:51,040
+تتكلم مع اللي جنبها، في أي شيء في البرهان مش واضح؟
+
+96
+00:11:52,600 --> 00:12:03,620
+طيب إذا البرهان معناته واضح، في
+
+97
+00:12:03,620 --> 00:12:18,340
+definition هنا، definition تعريف the
+
+98
+00:12:18,340 --> 00:12:28,700
+number S in above example is
+
+99
+00:12:28,700 --> 00:12:36,260
+called limit
+
+100
+00:12:36,260 --> 00:12:40,620
+superior
+
+101
+00:12:40,620 --> 00:12:47,060
+of sequence xn
+
+102
+00:12:50,000 --> 00:12:57,160
+and we write limit
+
+103
+00:12:57,160 --> 00:13:06,800
+superior ل
+
+104
+00:13:06,800 --> 00:13:10,520
+Xn بيساوي S
+
+105
+00:13:13,780 --> 00:13:19,800
+إذا العدد S هذا في التمرين أو في ال exercise أو في
+
+106
+00:13:19,800 --> 00:13:25,560
+المثال هذا بنسميه
+
+107
+00:13:25,560 --> 00:13:30,780
+limit superior ل sequence xn، limit superior of the
+
+108
+00:13:30,780 --> 00:13:33,760
+sequence xn، بيسموها بالعربي النهاية العليا
+
+109
+00:13:33,760 --> 00:13:41,640
+للمتتالية، في نهاية عليا وفي نهاية سفلى وبالتالي
+
+110
+00:13:46,810 --> 00:13:52,670
+هذه عبارة عن ال limit يعني 
+
+111
+00:13:52,670 --> 00:13:55,750
+هذه بتطلع .. يعني ممكن إثبات باستخدام ال monotone
+
+112
+00:13:55,750 --> 00:14:01,290
+convergence theorem أن هذه بيساوي هي نفسها limit
+
+113
+00:14:01,290 --> 00:14:11,530
+S as N tends to infinity limit
+
+114
+00:14:11,530 --> 00:14:17,940
+ال supremum لل sequence هذه أو limit superior لأن
+
+115
+00:14:17,940 --> 00:14:22,060
+ال sequence هذه decreaseing، وبما أن XN bounded
+
+116
+00:14:22,060 --> 00:14:27,400
+إذا هذه bounded وبالتالي
+
+117
+00:14:27,400 --> 00:14:32,360
+limit S  بيساوي ال infimum لل sequence SN by
+
+118
+00:14:32,360 --> 00:14:39,480
+monotone convergence theorem اللي هو S تمام؟
+
+119
+00:14:39,480 --> 00:14:43,500
+بالمثل في حاجة اسمها limit inferior أو النهاية
+
+120
+00:14:43,500 --> 00:14:44,120
+السفلى
+
+121
+00:14:48,820 --> 00:14:59,140
+similarly أو remark ..similarly
+
+122
+00:14:59,140 --> 00:15:02,700
+أو using
+
+123
+00:15:02,700 --> 00:15:12,320
+this .. using a similar .. a similar argument
+
+124
+00:15:14,860 --> 00:15:20,100
+يعني باستخدام برهان مشابه لبرهان التمرين اللي فات
+
+125
+00:15:20,100 --> 00:15:27,760
+one can easily easily
+
+126
+00:15:27,760 --> 00:15:37,380
+show that if xn 
+
+127
+00:15:37,380 --> 00:15:39,880
+is contained in R is bounded
+
+128
+00:15:43,230 --> 00:15:52,710
+و SN بيساوي المرهد infimum ل XK حيث K أكبر من أو 
+
+129
+00:15:52,710 --> 00:16:03,670
+يساوي N، و S بيساوي Supremum بدل الإنفمام لل sequence
+
+130
+00:16:03,670 --> 00:16:10,450
+SN حيث N تنتمي إلى N
+
+131
+00:16:15,800 --> 00:16:21,340
+ثم يوجد subsequence
+
+132
+00:16:21,340 --> 00:16:24,700
+XKN
+
+133
+00:16:24,700 --> 00:16:29,100
+subsequence XN subsequence subsequence XN subsequence XN subsequence XN
+
+134
+00:16:29,100 --> 00:16:32,500
+subsequence XN subsequence XN subsequence XN subsequence XN subsequence XN subsequence
+
+135
+00:16:32,500 --> 00:16:35,800
+XN subsequence XN subsequence XN subsequence XN subsequence XN subsequence XN
+
+136
+00:16:35,800 --> 00:16:35,900
+subsequence XN subsequence XN subsequence XN subsequence XN subsequence XN subsequence
+
+137
+00:16:35,900 --> 00:16:36,120
+XN subsequence XN subsequence XN subsequence XN subsequence XN subsequence XN
+
+138
+00:16:36,120 --> 00:16:36,560
+subsequence XN subsequence XN subsequence XN subsequence XN subsequence XN subsequence
+
+139
+00:16:36,560 --> 00:16:38,000
+XN subsequence XN
+
+140
+00:16:46,190 --> 00:16:50,570
+برضه العدد S في الحالة هذه بنسميه limit inferior
+
+141
+00:16:50,570 --> 00:17:05,330
+the sequence x_n S in above remark is
+
+142
+00:17:05,330 --> 00:17:11,850
+called is called limit
+
+143
+00:17:14,350 --> 00:17:26,290
+limit inferior limit inferior of the sequence x_n in
+
+144
+00:17:26,290 --> 00:17:29,730
+and
+
+145
+00:17:29,730 --> 00:17:35,490
+we write limit
+
+146
+00:17:35,490 --> 00:17:40,910
+inferior of x_n equals s
+
+147
+00:17:45,400 --> 00:17:49,760
+وكمان مرة هذه limit inferior بتساوي .. ممكن اثبات
+
+148
+00:17:49,760 --> 00:17:54,080
+باستخدام الـ monotone convergence theorem ممكن
+
+149
+00:17:54,080 --> 00:18:00,200
+اثبات ان هذا بساوي limit sequence
+
+150
+00:18:00,200 --> 00:18:07,240
+S as N tends to infinityلأن هذه الـ sequence Sn
+
+151
+00:18:07,240 --> 00:18:12,280
+ممكن اثبات إنها increasing متزايدة و bounded
+
+152
+00:18:12,280 --> 00:18:15,960
+therefore by monotone convergence theorem it
+
+153
+00:18:15,960 --> 00:18:20,400
+converges and the limit of it will be equal to the supremum to
+
+154
+00:18:20,400 --> 00:18:27,840
+اللي هو Sتمام؟ إذا هذه بنتسميها limit inferior of
+
+155
+00:18:27,840 --> 00:18:32,840
+the sequence x_n وهذه limit superior هي النهاية
+
+156
+00:18:32,840 --> 00:18:37,880
+العليا الصفلة النهاية العليا طبعا إذا كانت ال
+
+157
+00:18:37,880 --> 00:18:42,780
+sequence bounded ف limit superior لها exist
+
+158
+00:18:42,780 --> 00:18:48,240
+وبالساوي العدد S وكذا لك limit inferior لها exist
+
+159
+00:18:48,240 --> 00:18:54,990
+وبالساوي العدد Sبس العدد S هذا غير عن العدد S دا
+
+160
+00:18:54,990 --> 00:18:59,610
+يعني هذا خلينا نسميه S star عشان نميزه عن العدد
+
+161
+00:18:59,610 --> 00:19:07,530
+اللي فات ده خلينا نسميه S star S star وS star هنا
+
+162
+00:19:07,530 --> 00:19:13,090
+غير مختلف مش شرط يكون هو نفس ال S في ال exercise
+
+163
+00:19:13,090 --> 00:19:20,430
+اللي فات إذا نحاولوا تثبتهحاولوا تثبتوا التمرين
+
+164
+00:19:20,430 --> 00:19:25,530
+هذا اللي هو الخاص ب limit inferior بنفس ال
+
+165
+00:19:25,530 --> 00:19:30,990
+argument by similar argument باستخدام استنتاج أو
+
+166
+00:19:30,990 --> 00:19:36,470
+برهان مشابه للبرهان اللي عملناها للجزء اللي فات
+
+167
+00:19:36,470 --> 00:19:41,840
+اللي هو تمرين عشرة okay تمام؟واضح؟ اذا هنا اليوم
+
+168
+00:19:41,840 --> 00:19:45,360
+اتعلمنا ان في حاجة اسمها النهاية العليا والنهاية
+
+169
+00:19:45,360 --> 00:19:49,360
+السفلة لل sequence إذا كانت ال sequence bounded
+
+170
+00:19:49,360 --> 00:19:53,720
+فأثبتنا هنا أن النهاية العليا تبقاتها موجودة
+
+171
+00:19:53,720 --> 00:19:56,960
+وبالساوي العدد S اللي هو ال infimum لل sequence
+
+172
+00:19:56,960 --> 00:20:01,050
+هذهإذا كانت برضه ال sequence bounded فالنهاية
+
+173
+00:20:01,050 --> 00:20:06,670
+السفلة لها أو limit inferior بتطلع exist و بتساوي
+
+174
+00:20:06,670 --> 00:20:10,210
+العدد s star اللي هو ال supremum لل sequence هذه
+
+175
+00:20:10,210 --> 00:20:17,810
+أو بساوي limit ال sequence s هذه تمام؟ فحاولوا
+
+176
+00:20:17,810 --> 00:20:22,310
+تبرهنوا الجزء هذا المشابه للجزء الأول يعني في أي
+
+177
+00:20:22,310 --> 00:20:23,650
+سؤال أو استفسار؟
+
+178
+00:20:26,730 --> 00:20:30,430
+طيب الآن إذا ننتقل ل section جديد هيك احنا متكون
+
+179
+00:20:30,430 --> 00:20:36,110
+خلصنا section تلاتة اربعة في الكتاب المخرج و هنبدأ
+
+180
+00:20:36,110 --> 00:20:42,730
+section جديد اللي
+
+181
+00:20:42,730 --> 00:20:51,010
+هو section تلاتة خمسة الانوان
+
+182
+00:20:51,010 --> 00:20:54,990
+تبع koshi sequences
+
+183
+00:20:57,810 --> 00:21:03,410
+متتاليات كوشي طبعا كوشي هاد عالم ألماني German
+
+184
+00:21:03,410 --> 00:21:09,050
+mathematician كان مهتم بدراسة نوع خاص من ال
+
+185
+00:21:09,050 --> 00:21:14,570
+sequences وكان بحاول يدرس التقارب و التباعت تباعهم
+
+186
+00:21:14,570 --> 00:21:18,740
+و علاقتهم بالconvergent و divergence sequences
+
+187
+00:21:18,740 --> 00:21:24,820
+العادية فهذا يعني من كان جزء من اهتماماته طبعا له
+
+188
+00:21:24,820 --> 00:21:27,320
+حاجات كتيرة في ال analysis
+
+189
+00:21:31,200 --> 00:21:35,700
+فالجماعة الرياضيات او الناس اللي جوا بعده عشان
+
+190
+00:21:35,700 --> 00:21:39,060
+يعني تقديرا للجهد اللي قامله وللدراسة اللي قاملها
+
+191
+00:21:39,060 --> 00:21:44,380
+لهذا النوع الخاص من المتتاليات سموا المتتاليات هذه
+
+192
+00:21:44,380 --> 00:21:50,780
+باسمه متتاليات كوشي او كوشي sequences فما هي
+
+193
+00:21:50,780 --> 00:21:55,800
+متتاليات كوشي؟ متى بنقول ان ال sequence كوشي؟ فهذه
+
+194
+00:21:55,800 --> 00:21:56,360
+ال definition
+
+195
+00:22:01,970 --> 00:22:08,870
+a sequence of real numbers a sequence x_n contained
+
+196
+00:22:08,870 --> 00:22:21,230
+in R is Cauchy is Cauchy if الشرط التالي بتحقق لكل
+
+197
+00:22:21,230 --> 00:22:27,970
+epsilon أكبر من الصفر يوجد capital N يعتمد على
+
+198
+00:22:27,970 --> 00:22:37,300
+epsilon عدد طبيعيبحيث أنه لو كان n و m أكبر من أو
+
+199
+00:22:37,300 --> 00:22:42,920
+ساوي capital N فهذا لازم يقدي أن المسافة بين x_n و
+
+200
+00:22:42,920 --> 00:22:52,600
+x_m أصغر من epsilon هذه هي كوشي sequence sequence of
+
+201
+00:22:52,600 --> 00:22:57,360
+real number بنسميها كوشي متتالية كوشي إذا كان لأي
+
+202
+00:22:57,360 --> 00:23:02,360
+given epsilonبقدر ألاقي عدد طبيعي يعتمد على epsilon
+
+203
+00:23:02,360 --> 00:23:08,400
+بحيث لكل N و M لكل المؤشرات اللي من capital N و N
+
+204
+00:23:08,400 --> 00:23:14,620
+طالع المسافة تبع الحدود الفرق بين حدودهم أصغر من
+
+205
+00:23:14,620 --> 00:23:19,800
+epsilon في
+
+206
+00:23:19,800 --> 00:23:23,040
+هنا أول لمة
+
+207
+00:23:33,720 --> 00:23:42,080
+لمّة واحدة عشرين اش لمّة بتقول every ..
+
+208
+00:23:42,080 --> 00:23:52,420
+every convergent .. every convergent sequence of
+
+209
+00:23:52,420 --> 00:23:58,120
+real numbers is
+
+210
+00:23:58,120 --> 00:24:08,020
+Cauchy كل متتاليةمتقاربة بتكون كشية وهي البرهان
+
+211
+00:24:08,020 --> 00:24:14,180
+prove assume
+
+212
+00:24:14,180 --> 00:24:20,160
+let x
+
+213
+00:24:20,160 --> 00:24:26,920
+in contained in R be
+
+214
+00:24:26,920 --> 00:24:32,000
+such that limit x_n equals x
+
+215
+00:24:36,590 --> 00:24:41,730
+يعني افرض ان في عندي sequence xn وconvergent ل x
+
+216
+00:24:41,730 --> 00:24:45,110
+من
+
+217
+00:24:45,110 --> 00:24:50,190
+ان اثبت ان ال sequence هذه كوشي او كوشية ف let
+
+218
+00:24:50,190 --> 00:24:54,170
+epsilon لاثبات ان ال sequence كوشي لازم نبدأ
+
+219
+00:24:54,170 --> 00:24:58,210
+بepsilon ونرد عليها بcapital N تعطين ال
+
+220
+00:24:58,210 --> 00:25:02,070
+implication هذه صح؟ مش هيك التعريب يقول ان نبدأ
+
+221
+00:25:02,070 --> 00:25:06,090
+let epsilon أكبر من الصفر be given
+
+222
+00:25:11,220 --> 00:25:15,720
+طيب من تعريف epsilon capital N للكنفرجنس ال
+
+223
+00:25:15,720 --> 00:25:20,840
+sequence xn converge لx إذا there exist لأي given
+
+224
+00:25:20,840 --> 00:25:26,020
+epsilon بما أن xn converge لx لأي epsilon يوجد
+
+225
+00:25:26,020 --> 00:25:35,540
+capital N يعتمد على epsilon عدد طبيعي بحيث أنه
+
+226
+00:25:45,980 --> 00:25:53,820
+بحيث انه لو كان n أكبر من أو ساوي capital N هذا
+
+227
+00:25:53,820 --> 00:26:00,460
+بيقدي ان absolute x_n minus x أصغر من epsilon على 2
+
+228
+00:26:00,460 --> 00:26:04,680
+نسمي ال implication هذه star
+
+229
+00:26:09,880 --> 00:26:16,500
+الان لو أخدت N و M كلا هما أعداد طبيعية أكبر من أو
+
+230
+00:26:16,500 --> 00:26:24,680
+يساوي capital N هذا بيقدي ان absolute x_N minus X
+
+231
+00:26:24,680 --> 00:26:34,020
+M هذا بيساوي absolute x_N minus x زائد x minus x_M
+
+232
+00:26:34,020 --> 00:26:39,320
+هاي المجموع هذا وهي المجموع هذا
+
+233
+00:26:49,790 --> 00:26:53,850
+إذا أنا طرحت x ورجعت x باستخدام ال triangle
+
+234
+00:26:53,850 --> 00:27:00,510
+inequality ال absolute value للمجموع أصلا أو ساوي
+
+235
+00:27:00,510 --> 00:27:04,410
+مجموع ال absolute values إذا أنا عندي absolute x_n
+
+236
+00:27:04,410 --> 00:27:14,010
+minus x زائد absolute x minus x_m طيب باستخدام ال
+
+237
+00:27:14,010 --> 00:27:20,260
+implication starأنا عندي ال N، ال N هذه أكبر من أو
+
+238
+00:27:20,260 --> 00:27:24,760
+ساوي capital N، إذن حسب ال star، absolute الفرخ
+
+239
+00:27:24,760 --> 00:27:29,740
+هذا أصغر من epsilon على اتنين كذلك أنا عندي ال M
+
+240
+00:27:29,740 --> 00:27:36,800
+أكبر من أو ساوي N، إذن by star بدل N ب M، المسافة
+
+241
+00:27:36,800 --> 00:27:42,630
+هذه تطلع أصغر من epsilon على اتنينالمجموعة epsilon
+
+242
+00:27:42,630 --> 00:27:49,350
+إذا هاني أثبتت أنه لأي epsilon أكبر من السفر في
+
+243
+00:27:49,350 --> 00:27:54,890
+capital N يوجد capital N تعتمد على epsilon بحيث
+
+244
+00:27:54,890 --> 00:27:59,910
+لكل N و M أكبر من أو ساوي capital N بطلع المسافة
+
+245
+00:27:59,910 --> 00:28:05,690
+بين الحد رقم N والحد رقم M أصغر من epsilon إذا
+
+246
+00:28:05,690 --> 00:28:08,290
+since
+
+247
+00:28:10,030 --> 00:28:17,650
+epsilon أكبر من السفر was arbitrary إن
+
+248
+00:28:17,650 --> 00:28:22,330
+الكلام هذا صحيح أثبتنا هذا الكلام صحيح لكل epsilon
+
+249
+00:28:22,330 --> 00:28:28,030
+الepsilon هذي was arbitrary عشوائية إن هيك بتكون
+
+250
+00:28:28,030 --> 00:28:34,610
+حققنا الشرط تبقى cushy sequences وبالتالي we have
+
+251
+00:28:34,610 --> 00:28:38,370
+we get that من التعريف
+
+252
+00:28:56,870 --> 00:29:04,070
+العكس تبع النظرية هذه صحيحهنشوفه بعد شوية لكن في
+
+253
+00:29:04,070 --> 00:29:11,770
+كمان نظرية او لمّة بسيطة زي هذه هنثبتها
+
+254
+00:29:11,770 --> 00:29:20,450
+الآن بتقول ان كل كوشي sequence is bounded واضح
+
+255
+00:29:20,450 --> 00:29:23,470
+البرهان؟ في اي استفسار على البرهان؟ اعتقد البرهان
+
+256
+00:29:23,470 --> 00:29:28,480
+سهل وبسيطهذا ال .. النوع من البرهان بنسميه epsilon
+
+257
+00:29:28,480 --> 00:29:37,360
+بنسميه epsilon over two argument epsilon
+
+258
+00:29:37,360 --> 00:29:44,380
+over two argument استخدامنا epsilon over two برهان
+
+259
+00:29:44,380 --> 00:29:51,140
+باستخدام epsilon over two إذن نثبت اللمة التانية
+
+260
+00:29:51,140 --> 00:29:54,320
+قبل ما نثبت عكس النظر اللي فاتت صحيح
+
+261
+00:30:00,800 --> 00:30:10,180
+lemma 22 every Cauchy sequence
+
+262
+00:30:10,180 --> 00:30:20,600
+in R is bounded والبرهان
+
+263
+00:30:20,600 --> 00:30:27,760
+تبعها شبيه إلى حد كبير ببرهان أن كل convergence
+
+264
+00:30:27,760 --> 00:30:32,110
+sequence is boundedأثبتنا قبل هيك أن كل conversion
+
+265
+00:30:32,110 --> 00:30:35,390
+sequence is bounded اليوم هأثبت أن كل Cauchy
+
+266
+00:30:35,390 --> 00:30:40,090
+sequence is bounded والبرهان مشابه للبرهان السابق
+
+267
+00:30:40,090 --> 00:30:44,950
+proof let
+
+268
+00:30:44,950 --> 00:30:54,590
+x_n contained in R be Cauchy be Cauchy sequence
+
+269
+00:30:57,010 --> 00:31:03,350
+و خلّينا ناخد epsilon then
+
+270
+00:31:03,350 --> 00:31:09,990
+من تعريف الكوشي sequence for epsilon equals one
+
+271
+00:31:09,990 --> 00:31:16,130
+أكبر من السفر يوجد capital N يعتمد على ال epsilon
+
+272
+00:31:16,130 --> 00:31:26,330
+اللي هو الواحد عدد طبيعي بحيث أنه لabsolute x_n
+
+273
+00:31:26,330 --> 00:31:34,150
+minus x_m أصغر من epsilon اللي هو واحد وهذا الكلام
+
+274
+00:31:34,150 --> 00:31:40,930
+صحيح لكل N و M أكبر من أو ساوي capital N هذا من
+
+275
+00:31:40,930 --> 00:31:49,190
+تعريف الكوشي sequence تمام؟ طيب hence
+
+276
+00:31:50,870 --> 00:31:59,410
+من المتباينة هذه بنحصل على absolute x_n minus x
+
+277
+00:31:59,410 --> 00:32:06,670
+capital N أصغر من واحد وهذا الكلام صحيحلكل N أكبر
+
+278
+00:32:06,670 --> 00:32:11,130
+منه ساوي capital N اذا انا اشعلنت هنا المتباين هي
+
+279
+00:32:11,130 --> 00:32:18,910
+دي حصلت عليها من اللي فوق و ذلك بياخذ take M equals
+
+280
+00:32:18,910 --> 00:32:24,110
+capital N انا أخدت ال M هنا بساوي capital N وهذا 
+
+281
+00:32:24,110 --> 00:32:30,430
+مسموح له فصارت هيك طب ليش أنا عملت هيك لحاجة في
+
+282
+00:32:30,430 --> 00:32:36,930
+نفسي عقوب؟ عشان فيه غرض للاستفاء البرهان أو 
+
+283
+00:32:36,930 --> 00:32:45,150
+المطلوب في البرهان فمن هنا by the triangle inequality
+
+284
+00:32:45,150 --> 00:32:52,690
+من متباينة المثلث أنا عندي لو كان n أكبر من أو يساوي
+
+285
+00:32:52,690 --> 00:32:58,150
+Capital N هذا بيقود إلى أن absolute x in بالظبط زي ما
+
+286
+00:32:58,150 --> 00:33:02,330
+عملنا في برهان كل convergence sequence is bounded
+
+287
+00:33:02,330 --> 00:33:10,010
+فهذا بنكتبه على صورة x in minus x Capital N زائد x
+
+288
+00:33:10,010 --> 00:33:16,920
+Capital N إذا أنا طرحت x Capital N ورجعتها الآن
+
+289
+00:33:16,920 --> 00:33:21,740
+هذا أصغر من أو يساوي absolute x n سالب x Capital N
+
+290
+00:33:21,740 --> 00:33:28,340
+زائد absolute x Capital N وهذا
+
+291
+00:33:28,340 --> 00:33:33,660
+هذا هو هذا المقدار لكل n أكبر من أو يساوي Capital N
+
+292
+00:33:33,660 --> 00:33:37,960
+أصغر من 1 إذا هذا أصغر من 1 زائد absolute x
+
+293
+00:33:37,960 --> 00:33:43,540
+Capital N تمام الآن
+
+294
+00:33:43,540 --> 00:33:52,020
+now let خلّينا ناخد m .. نعرف عدد m على أنه ال
+
+295
+00:33:52,020 --> 00:33:57,480
+supremum أو ال maximum للأعداد الحقيقية غير 
+
+296
+00:33:57,480 --> 00:34:05,720
+السالبة اللي هي absolute x1 absolute x2 إلى
+
+297
+00:34:05,720 --> 00:34:16,000
+absolute x رقم Capital M minus 1 وآخر حد absolute
+
+298
+00:34:16,000 --> 00:34:20,420
+واحد
+
+299
+00:34:20,420 --> 00:34:38,180
+زائد 1 زائد absolute x رقم Capital N you
+
+300
+00:34:38,180 --> 00:34:39,340
+can verify
+
+301
+00:34:42,590 --> 00:34:51,350
+بإمكانكم التحقق إنه absolute xn أصغر طبعاً العدد m
+
+302
+00:34:51,350 --> 00:34:58,210
+هذا موجب بالتأكيد هذا موجب لأن
+
+303
+00:34:58,210 --> 00:35:05,420
+الأعداد هذه كلها موجبة وبعدين absolute xn ممكن تحقق
+
+304
+00:35:05,420 --> 00:35:10,380
+أنه أخذت أي عنصر في ال sequence xn وأخذت القيمة
+
+305
+00:35:10,380 --> 00:35:17,100
+المطلقة له فهذا هيطلع أصغر من أو يساوي m لكل n في 
+
+306
+00:35:17,100 --> 00:35:17,460
+n
+
+307
+00:35:20,190 --> 00:35:25,570
+هل إيش هذا صحيح؟ لأنه لو كانت ال N هذه لو كانت ال
+
+308
+00:35:25,570 --> 00:35:31,750
+N هي index من 1 إلى Capital N لـ Capital N سالب 1
+
+309
+00:35:31,750 --> 00:35:36,130
+لو كانت ال N هذه مؤشر N ده واحد من المؤشرات 1
+
+310
+00:35:36,130 --> 00:35:40,310
+إلى Capital N سالب 1 فالقيمة المطلقة اللي هو
+
+311
+00:35:40,310 --> 00:35:45,530
+واحد من الحدود هذه وكل حد من هذه أصلاً من أو يساوي
+
+312
+00:35:45,530 --> 00:35:49,490
+ال upper bound ال M هذا upper bound لل set هذه
+
+313
+00:35:52,340 --> 00:35:57,880
+طب لو كانت ال n small n هذه بالساوي Capital N أو
+
+314
+00:35:57,880 --> 00:36:04,980
+أكبر Capital N أو أكبر هي لو كانت small n أكبر من
+
+315
+00:36:04,980 --> 00:36:10,320
+أو يساوي Capital N فبيطلع absolute xn أصغر من 1
+
+316
+00:36:10,320 --> 00:36:14,900
+زائد absolute xn إذا القيمة المطلقة هذه أصغر من 
+
+317
+00:36:14,900 --> 00:36:22,500
+1 زائد absolute xn وهذا العدد أصغر من أو يساوي ال
+
+318
+00:36:22,500 --> 00:36:26,340
+upper bound لل set اللي بتحتويه وبتحتوي عناصر
+
+319
+00:36:26,340 --> 00:36:31,240
+ثانية وبالتالي absolute xm أصغر من أو يساوي okay
+
+320
+00:36:31,240 --> 00:36:39,900
+تمام؟ إذن هذا بكمل البرهان المرة الجاية طبعاً هنشوف
+
+321
+00:36:39,900 --> 00:36:45,730
+أنه عكس ال lemma هذه صحيح ودي بنسميها Cauchy
+
+322
+00:36:45,730 --> 00:36:50,510
+criterion أو معيار Cauchy ومعيار Cauchy بيقول إن 
+
+323
+00:36:50,510 --> 00:36:54,510
+لو كانت ال sequence Cauchy بتكون أيضا convergent
+
+324
+00:36:54,510 --> 00:36:58,430
+وبالتالي بنحصل على نظرية Cauchy أو معيار Cauchy
+
+325
+00:36:58,430 --> 00:37:04,010
+Cauchy criterion وهو إن ال sequence is convergent
+
+326
+00:37:04,010 --> 00:37:09,690
+if and only if it is Cauchy إذا هذا الجزء التالي من
+
+327
+00:37:09,690 --> 00:37:14,050
+النظرية أو عكس ال lemma دي هنشوفه المرة الجاية وناخد 
+
+328
+00:37:14,050 --> 00:37:20,350
+أمثلة كيف نثبت أن ال sequence معينة is Cauchy فهذا
+
+329
+00:37:20,350 --> 00:37:22,830
+موجود في section تلاتة خمسة في الكتاب حاولوا 
+
+330
+00:37:22,830 --> 00:37:27,930
+تقرؤوا باقي ال section وتحضروه للمرة الجاية تمام؟
+
+331
+00:37:27,930 --> 00:37:30,730
+إذا انتهت هيك المحاضرة نشوفكم إن شاء الله يوم
+
+332
+00:37:30,730 --> 00:37:31,190
+اثنين 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..c3fbdee239f9044cb3883b3356558a680451508d
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/eX4Dw2M3cKU_raw.json
@@ -0,0 +1 @@
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"end": 127.79, "text": "لاحظوا إن الـ Sn هذه كل ما N كبرت كل ما ال 6 هذه زغرت و ال suprem زغرت فبتطلع decreasing sequence و طبعا bounded إذا الإنفمام تبعها موجود بسبب عدد S إذن هنا بنعرف ال S على إن الإنفمام لل sequence Sn", "tokens": [15040, 5016, 19913, 14407, 36145, 2423, 39184, 9264, 29538, 28242, 19446, 426, 9122, 26890, 2655, 28242, 19446, 2423, 1386, 29538, 30767, 17082, 43500, 4032, 2423, 23710, 30767, 17082, 43500, 6156, 3555, 2655, 9566, 1211, 3615, 23223, 8310, 4032, 23032, 3555, 3615, 995, 37498, 11933, 15730, 33688, 1863, 5172, 2304, 10943, 6055, 3555, 3615, 11296, 3714, 29245, 23328, 4724, 35457, 3555, 6225, 3215, 3215, 318, 11933, 8848, 1863, 34105, 44945, 3615, 28480, 2423, 318, 15844, 36145, 33688, 1863, 5172, 2304, 10943, 24976, 8310, 9264], "avg_logprob": -0.27492558246567134, "compression_ratio": 1.5743589743589743, "no_speech_prob": 0.0, "words": [{"start": 102.53, "end": 103.03, "word": "لاحظوا", "probability": 0.82562255859375}, {"start": 103.03, "end": 103.19, 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epsilon أكبر من الصفر يوجد S epsilon ينتمي ل S بحيث أنه S epsilon", "tokens": [10721, 9778, 3215, 8315, 10874, 16758, 1211, 39896, 4117, 32767, 49401, 39894, 39648, 14739, 3224, 261, 4724, 3794, 995, 12610, 1536, 332, 449, 5296, 992, 318, 498, 293, 787, 498, 5296, 28820, 17889, 5551, 4117, 26890, 9154, 31767, 5172, 2288, 7251, 29245, 3215, 318, 17889, 7251, 29399, 2304, 1829, 5296, 318, 4724, 5016, 1829, 12984, 14739, 3224, 318, 17889], "avg_logprob": -0.26692707687616346, "compression_ratio": 1.3092105263157894, "no_speech_prob": 0.0, "words": [{"start": 154.54, "end": 155.04, "word": "أخدنا", "probability": 0.75372314453125}, {"start": 155.04, "end": 155.36, "word": " جابل", "probability": 0.5865071614583334}, {"start": 155.36, "end": 155.62, "word": " هيك", "probability": 0.81689453125}, {"start": 155.62, "end": 156.1, "word": " لمبة", "probability": 0.4912109375}, {"start": 156.1, "end": 156.74, "word": " بتقول", "probability": 0.9384765625}, {"start": 156.74, "end": 157.44, 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{"start": 222.45, "end": 222.81, "word": " ال", "probability": 0.45166015625}, {"start": 222.81, "end": 223.09, "word": " set", "probability": 0.9677734375}, {"start": 223.09, "end": 223.69, "word": " S", "probability": 0.53271484375}, {"start": 223.69, "end": 226.05, "word": " فهذا", "probability": 0.5030924479166666}, {"start": 226.05, "end": 226.29, "word": " هو", "probability": 0.93359375}, {"start": 226.29, "end": 226.47, "word": " ال", "probability": 0.9365234375}, {"start": 226.47, "end": 226.97, "word": " W", "probability": 0.59619140625}, {"start": 226.97, "end": 228.63, "word": " فلو", "probability": 0.9827473958333334}, {"start": 228.63, "end": 229.35, "word": " أخدت", "probability": 0.9649658203125}, {"start": 229.35, "end": 233.33, "word": " for", "probability": 0.71923828125}, {"start": 233.33, "end": 234.27, "word": " each", "probability": 0.9638671875}], "temperature": 1.0}, {"id": 9, "seek": 26375, "start": 235.69, "end": 263.75, "text": "in عدد طبيعي لكل عدد طبيعي in there exist sn ينتمي لل set هذه ال set of all sn hyphen عدد طبيعي بحيث انه ال هذا ال sn", "tokens": [259, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 5296, 28820, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 294, 456, 2514, 2406, 7251, 29399, 2304, 1829, 24976, 992, 29538, 2423, 992, 295, 439, 2406, 2477, 47059, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 16472, 3224, 2423, 23758, 2423, 2406], "avg_logprob": -0.2270047203549799, "compression_ratio": 1.4782608695652173, "no_speech_prob": 0.0, "words": [{"start": 235.69, "end": 236.15, "word": "in", "probability": 0.2318115234375}, {"start": 236.15, "end": 236.69, "word": " عدد", "probability": 0.9527994791666666}, {"start": 236.69, "end": 237.47, "word": " طبيعي", "probability": 0.956298828125}, {"start": 237.47, "end": 238.85, "word": " لكل", "probability": 0.976318359375}, {"start": 238.85, "end": 239.15, "word": " عدد", "probability": 0.9889322916666666}, {"start": 239.15, "end": 239.75, "word": " طبيعي", 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epsilon برضه واحد على n هذا عدد موجب فبنقدر نلاقي يوجد", "tokens": [1211, 1211, 3854, 295, 439, 2031, 74, 11331, 1829, 12984, 350, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2423, 297, 4724, 15042, 14739, 23758, 20328, 5016, 1829, 5016, 11933, 15730, 41850, 4724, 995, 14851, 9778, 3215, 10943, 21542, 38436, 28671, 16712, 7649, 10632, 550, 337, 17889, 45164, 5551, 9778, 3215, 2655, 17889, 4724, 43042, 3224, 36764, 24401, 15844, 297, 23758, 6225, 3215, 3215, 3714, 29245, 3555, 6156, 3555, 1863, 28543, 2288, 8717, 15040, 38436, 7251, 29245, 3215], "avg_logprob": -0.19402690326111227, "compression_ratio": 1.4623655913978495, "no_speech_prob": 0.0, "words": [{"start": 312.03, "end": 312.69, "word": "للset", "probability": 0.2888997395833333}, {"start": 312.69, "end": 312.89, "word": " of", "probability": 0.95654296875}, {"start": 312.89, "end": 313.21, "word": " all", "probability": 0.95849609375}, {"start": 313.21, "end": 314.11, "word": " xk", "probability": 0.73876953125}, 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0.95703125}, {"start": 344.17, "end": 344.39, "word": " ال", "probability": 0.84326171875}, {"start": 344.39, "end": 344.65, "word": " set", "probability": 0.9677734375}, {"start": 344.65, "end": 345.17, "word": " هذه", "probability": 0.7880859375}, {"start": 345.17, "end": 347.19, "word": " نسمي", "probability": 0.9215494791666666}, {"start": 347.19, "end": 348.69, "word": " xkn", "probability": 0.4056396484375}, {"start": 348.69, "end": 350.95, "word": " ينتمي", "probability": 0.952880859375}, {"start": 350.95, "end": 352.25, "word": " لset", "probability": 0.69482421875}, {"start": 352.25, "end": 352.51, "word": " of", "probability": 0.92919921875}, {"start": 352.51, "end": 352.93, "word": " all", "probability": 0.95947265625}, {"start": 352.93, "end": 354.07, "word": " xk", "probability": 0.9296875}, {"start": 354.07, "end": 354.57, "word": " حيث", "probability": 0.984375}, {"start": 354.57, "end": 355.05, "word": " k", "probability": 0.70458984375}, {"start": 355.05, "end": 356.83, "word": " أكبر", "probability": 0.9376627604166666}, {"start": 356.83, "end": 357.11, "word": " من", "probability": 0.947265625}, {"start": 357.11, "end": 357.35, "word": " أو", "probability": 0.92919921875}, {"start": 357.35, "end": 358.15, "word": " يساوي", "probability": 0.9171142578125}, {"start": 358.15, "end": 358.47, "word": " ال", "probability": 0.904296875}, {"start": 358.47, "end": 358.85, "word": " n", "probability": 0.6220703125}, {"start": 358.85, "end": 364.57, "word": " such", "probability": 0.9169921875}, {"start": 364.57, "end": 365.07, "word": " that", "probability": 0.9580078125}], "temperature": 1.0}, {"id": 14, "seek": 39020, "start": 368.24, "end": 390.2, "text": "الـ supremum اللي هو SN minus الـ Y اللي هو واحد على N أصغر من الـ SY اللي هو العنصر X KN طيب إذا من هنا", "tokens": [6027, 39184, 23710, 449, 13672, 1829, 31439, 13955, 3175, 2423, 39184, 398, 13672, 1829, 31439, 36764, 24401, 15844, 426, 5551, 9381, 17082, 2288, 9154, 2423, 39184, 32624, 13672, 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واحد على N لحظة من هنا S N ال capital S lower bound للست هذه وبالتالي أي أنصر هنا أكبر من أو يساوي ال S لأن أنا عندي S N أكبر من أو يساوي capital S اطرحي واحد على N من الطرفين ونحصل على الكلام هذا", "tokens": [9485, 1686, 318, 3175, 36764, 24401, 15844, 426, 5296, 5016, 19913, 3660, 9154, 34105, 318, 426, 2423, 4238, 318, 3126, 5472, 24976, 14851, 29538, 46599, 6027, 2655, 6027, 1829, 36632, 14739, 9381, 2288, 34105, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 2423, 318, 5296, 33456, 41850, 18871, 16254, 318, 426, 5551, 4117, 26890, 9154, 34051, 7251, 3794, 995, 45865, 4238, 318, 1975, 9566, 2288, 5016, 1829, 36764, 24401, 15844, 426, 9154, 41950, 28480, 9957, 4032, 1863, 5016, 36520, 15844, 2423, 28820, 10943, 23758], "avg_logprob": -0.26707847797593404, "compression_ratio": 1.7010309278350515, "no_speech_prob": 0.0, "words": [{"start": 426.11, "end": 426.83, "word": "capital", "probability": 0.6727294921875}, {"start": 426.83, "end": 427.15, "word": " S", "probability": 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ال set هذه عنصر في ال set هذه و sn upper bound لل set", "tokens": [3224, 15730, 4724, 1829, 28543, 11703, 1829, 36145, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 23032, 1829, 3555, 4032, 9154, 34105, 9154, 9673, 2655, 3555, 995, 9957, 16247, 9778, 13546, 23758, 5551, 9381, 17082, 2288, 9154, 2031, 5457, 4032, 2031, 5457, 23758, 18871, 9381, 2288, 8978, 2423, 992, 29538, 18871, 9381, 2288, 8978, 2423, 992, 29538, 4032, 2406, 6597, 5472, 24976, 992], "avg_logprob": -0.19433594425208867, "compression_ratio": 1.5945945945945945, "no_speech_prob": 0.0, "words": [{"start": 455.83, "end": 456.35, "word": "هذا", "probability": 0.88623046875}, {"start": 456.35, "end": 457.15, "word": " بيقدّي", "probability": 0.639111328125}, {"start": 457.15, "end": 457.37, "word": " إن", "probability": 0.374267578125}, {"start": 457.37, "end": 457.59, "word": " هذا", "probability": 0.70751953125}, {"start": 457.59, "end": 457.93, "word": " الكلام", "probability": 0.9251302083333334}, {"start": 457.93, "end": 458.51, "word": " صحيح", "probability": 0.990478515625}, {"start": 458.51, "end": 460.01, "word": " طيب", "probability": 0.93896484375}, {"start": 460.01, "end": 460.25, "word": " و", "probability": 0.92236328125}, {"start": 460.25, "end": 460.67, "word": " من", "probability": 0.802734375}, {"start": 460.67, "end": 461.31, "word": " هنا", "probability": 0.9951171875}, {"start": 461.31, "end": 463.87, "word": " من", "probability": 0.52099609375}, {"start": 463.87, "end": 464.49, "word": " المتباين", "probability": 0.90029296875}, {"start": 464.49, "end": 464.99, "word": " الأخير", "probability": 0.8683268229166666}, {"start": 464.99, "end": 465.33, "word": " هذا", "probability": 0.4765625}, {"start": 465.33, "end": 465.89, "word": " أصغر", "probability": 0.9716796875}, {"start": 465.89, "end": 466.25, "word": " من", "probability": 0.97509765625}, {"start": 466.25, "end": 469.65, "word": " xkn", "probability": 0.4296875}, {"start": 469.65, "end": 472.75, "word": " و", "probability": 0.8388671875}, {"start": 472.75, "end": 473.57, "word": " xkn", "probability": 0.882568359375}, {"start": 473.57, "end": 473.93, "word": " هذا", "probability": 0.68115234375}, {"start": 473.93, "end": 474.47, "word": " عنصر", "probability": 0.9767252604166666}, {"start": 474.47, "end": 474.65, "word": " في", "probability": 0.9716796875}, {"start": 474.65, "end": 474.87, "word": " ال", "probability": 0.7939453125}, {"start": 474.87, "end": 475.09, "word": " set", "probability": 0.82421875}, {"start": 475.09, "end": 475.61, "word": " هذه", "probability": 0.771484375}, {"start": 475.61, "end": 477.53, "word": " عنصر", "probability": 0.951171875}, {"start": 477.53, "end": 477.69, "word": " في", "probability": 0.9912109375}, {"start": 477.69, "end": 477.87, "word": " ال", "probability": 0.9326171875}, {"start": 477.87, "end": 478.17, "word": " set", "probability": 0.99267578125}, {"start": 478.17, "end": 478.69, "word": " هذه", "probability": 0.9384765625}, {"start": 478.69, "end": 480.45, "word": " و", "probability": 0.96875}, {"start": 480.45, "end": 481.03, "word": " sn", "probability": 0.51708984375}, {"start": 481.03, "end": 481.37, "word": " upper", "probability": 0.802734375}, {"start": 481.37, "end": 481.83, "word": " bound", "probability": 0.89208984375}, {"start": 481.83, "end": 482.07, "word": " لل", "probability": 0.81298828125}, {"start": 482.07, "end": 482.45, "word": " set", "probability": 0.95166015625}], "temperature": 1.0}, {"id": 18, "seek": 50649, "start": 483.47, "end": 506.49, "text": "إذن هذا ال upper bound أكبر من أو ساوي كل عناصر ال set إذن هذا أصغر من أو ساوي Sn ومن المتباينة هذه Sn أصغر من capital S زائد واحد على N مظبوط؟", "tokens": [28814, 8848, 1863, 23758, 2423, 6597, 5472, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 28242, 18871, 33546, 2288, 2423, 992, 11933, 8848, 1863, 23758, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 9264, 4032, 27842, 9673, 2655, 3555, 995, 9957, 3660, 29538, 9264, 5551, 9381, 17082, 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"probability": 0.98046875}, {"start": 486.17, "end": 486.61, "word": " عناصر", "probability": 0.9798177083333334}, {"start": 486.61, "end": 486.77, "word": " ال", "probability": 0.95751953125}, {"start": 486.77, "end": 487.03, "word": " set", "probability": 0.724609375}, {"start": 487.03, "end": 488.85, "word": " إذن", "probability": 0.79345703125}, {"start": 488.85, "end": 489.15, "word": " هذا", "probability": 0.96533203125}, {"start": 489.15, "end": 489.71, "word": " أصغر", "probability": 0.9835205078125}, {"start": 489.71, "end": 489.95, "word": " من", "probability": 0.99462890625}, {"start": 489.95, "end": 490.17, "word": " أو", "probability": 0.98681640625}, {"start": 490.17, "end": 490.95, "word": " ساوي", "probability": 0.9651692708333334}, {"start": 490.95, "end": 491.79, "word": " Sn", "probability": 0.3125}, {"start": 491.79, "end": 496.63, "word": " ومن", "probability": 0.873291015625}, {"start": 496.63, "end": 497.51, "word": " المتباينة", "probability": 0.944580078125}, {"start": 497.51, "end": 497.99, "word": " هذه", "probability": 0.84619140625}, {"start": 497.99, "end": 498.93, "word": " Sn", "probability": 0.83203125}, {"start": 498.93, "end": 500.43, "word": " أصغر", "probability": 0.9915771484375}, {"start": 500.43, "end": 500.67, "word": " من", "probability": 0.9931640625}, {"start": 500.67, "end": 501.13, "word": " capital", "probability": 0.6728515625}, {"start": 501.13, "end": 501.65, "word": " S", "probability": 0.9541015625}, {"start": 501.65, "end": 502.23, "word": " زائد", "probability": 0.8816731770833334}, {"start": 502.23, "end": 502.71, "word": " واحد", "probability": 0.890625}, {"start": 502.71, "end": 502.91, "word": " على", "probability": 0.69287109375}, {"start": 502.91, "end": 503.29, "word": " N", "probability": 0.63134765625}, {"start": 503.29, "end": 506.49, "word": " مظبوط؟", "probability": 0.7701416015625}], "temperature": 1.0}, {"id": 19, "seek": 53107, "start": 519.47, "end": 531.07, "text": "thus we obtain a subsequence الكلام هذا صحيح لكل ان", "tokens": [392, 301, 321, 12701, 257, 13924, 655, 2423, 28820, 10943, 23758, 20328, 5016, 1829, 5016, 5296, 28820, 16472], "avg_logprob": -0.3137335557686655, "compression_ratio": 0.971830985915493, "no_speech_prob": 0.0, "words": [{"start": 519.47, "end": 520.87, "word": "thus", "probability": 0.57012939453125}, {"start": 520.87, "end": 522.27, "word": " we", "probability": 0.77392578125}, {"start": 522.27, "end": 523.17, "word": " obtain", "probability": 0.92236328125}, {"start": 523.17, "end": 525.89, "word": " a", "probability": 0.77587890625}, {"start": 525.89, "end": 526.91, "word": " subsequence", "probability": 0.963623046875}, {"start": 526.91, "end": 527.43, "word": " الكلام", "probability": 0.7784830729166666}, {"start": 527.43, "end": 527.77, "word": " هذا", "probability": 0.8408203125}, {"start": 527.77, "end": 528.71, "word": " صحيح", "probability": 0.989990234375}, {"start": 528.71, "end": 530.83, "word": " لكل", "probability": 0.7152099609375}, {"start": 530.83, "end": 531.07, "word": " ان", "probability": 0.552734375}], "temperature": 1.0}, {"id": 20, "seek": 55227, "start": 534.21, "end": 552.27, "text": "لأن هذا الكلام صحيح لكل n و هذا الكلام صحيح لكل n لكل epsilon بالساعة واحد على n و لكل .. لكل n ممكن أكون epsilon بالساعة واحد على n وهي هنا لكل n for each n كلام هذا صحيح", "tokens": [1211, 33456, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 297, 4032, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 297, 5296, 28820, 17889, 20666, 3794, 995, 27884, 36764, 24401, 15844, 297, 4032, 5296, 28820, 4386, 5296, 28820, 297, 3714, 43020, 5551, 30544, 17889, 20666, 3794, 995, 27884, 36764, 24401, 15844, 297, 37037, 1829, 34105, 5296, 28820, 297, 337, 1184, 297, 28242, 10943, 23758, 20328, 5016, 1829, 5016], "avg_logprob": -0.20644263208728947, "compression_ratio": 1.9640287769784173, "no_speech_prob": 0.0, "words": [{"start": 534.21, "end": 534.63, "word": "لأن", "probability": 0.777587890625}, {"start": 534.63, "end": 534.89, "word": " هذا", "probability": 0.8828125}, {"start": 534.89, "end": 535.31, "word": " الكلام", "probability": 0.9007161458333334}, {"start": 535.31, "end": 535.85, "word": " صحيح", "probability": 0.9932861328125}, {"start": 535.85, "end": 536.33, "word": " لكل", "probability": 0.973388671875}, {"start": 536.33, "end": 536.69, "word": " n", "probability": 0.35546875}, {"start": 536.69, "end": 537.75, "word": " و", "probability": 0.304443359375}, {"start": 537.75, "end": 537.99, "word": " هذا", "probability": 0.6171875}, {"start": 537.99, "end": 538.41, "word": " الكلام", "probability": 0.9228515625}, {"start": 538.41, "end": 538.91, "word": " صحيح", "probability": 0.986328125}, {"start": 538.91, "end": 539.29, "word": " لكل", "probability": 0.990234375}, {"start": 539.29, "end": 539.59, "word": " n", "probability": 0.68896484375}, {"start": 539.59, "end": 541.53, "word": " لكل", "probability": 0.749755859375}, {"start": 541.53, "end": 541.95, "word": " epsilon", "probability": 0.5400390625}, {"start": 541.95, "end": 542.41, "word": " بالساعة", "probability": 0.55865478515625}, {"start": 542.41, "end": 542.73, "word": " واحد", "probability": 0.97802734375}, {"start": 542.73, "end": 542.91, "word": " على", "probability": 0.74755859375}, {"start": 542.91, "end": 543.21, "word": " n", "probability": 0.77294921875}, {"start": 543.21, "end": 543.43, "word": " و", "probability": 0.74267578125}, {"start": 543.43, "end": 543.83, "word": " لكل", "probability": 0.927978515625}, {"start": 543.83, "end": 543.99, "word": " ..", "probability": 0.28125}, {"start": 543.99, "end": 544.71, "word": " لكل", "probability": 0.9775390625}, {"start": 544.71, "end": 545.05, "word": " n", "probability": 0.9130859375}, {"start": 545.05, "end": 545.79, "word": " ممكن", "probability": 0.983154296875}, {"start": 545.79, "end": 546.23, "word": " أكون", "probability": 0.601806640625}, {"start": 546.23, "end": 546.63, "word": " epsilon", "probability": 0.8994140625}, {"start": 546.63, "end": 547.03, "word": " بالساعة", "probability": 0.9852294921875}, {"start": 547.03, "end": 547.39, "word": " واحد", "probability": 0.99365234375}, {"start": 547.39, "end": 547.55, "word": " على", "probability": 0.82763671875}, {"start": 547.55, "end": 547.85, "word": " n", "probability": 0.89111328125}, {"start": 547.85, "end": 548.91, "word": " وهي", "probability": 0.6600341796875}, {"start": 548.91, "end": 549.15, "word": " هنا", "probability": 0.96484375}, {"start": 549.15, "end": 549.63, "word": " لكل", "probability": 0.98291015625}, {"start": 549.63, "end": 549.99, "word": " n", "probability": 0.9541015625}, {"start": 549.99, "end": 550.45, "word": " for", "probability": 0.68017578125}, {"start": 550.45, "end": 550.77, "word": " each", "probability": 0.95654296875}, {"start": 550.77, "end": 551.05, "word": " n", "probability": 0.927734375}, {"start": 551.05, "end": 551.43, "word": " كلام", "probability": 0.7374267578125}, {"start": 551.43, "end": 551.69, "word": " هذا", "probability": 0.9619140625}, {"start": 551.69, "end": 552.27, "word": " صحيح", "probability": 0.996826171875}], "temperature": 1.0}, {"id": 21, "seek": 56559, "start": 552.43, "end": 565.59, "text": "أذا هذا الكلام المتباين هذه صحيحة لكل n وبالتالي أنا لكل n بقدر ألاقي xkn موجود هنا و بيحقق المتباين هذه", "tokens": [10721, 15730, 23758, 2423, 28820, 10943, 9673, 2655, 3555, 995, 9957, 29538, 20328, 5016, 1829, 5016, 3660, 5296, 28820, 297, 46599, 6027, 2655, 6027, 1829, 41850, 5296, 28820, 297, 4724, 28543, 2288, 5551, 15040, 38436, 2031, 5457, 3714, 29245, 23328, 34105, 4032, 4724, 1829, 5016, 4587, 4587, 9673, 2655, 3555, 995, 9957, 29538], "avg_logprob": -0.2951388905445735, "compression_ratio": 1.5913043478260869, "no_speech_prob": 0.0, "words": [{"start": 552.43, "end": 553.03, "word": "أذا", "probability": 0.25567626953125}, {"start": 553.03, "end": 553.31, "word": " هذا", "probability": 0.46142578125}, {"start": 553.31, "end": 553.83, "word": " الكلام", "probability": 0.8639322916666666}, {"start": 553.83, "end": 554.43, "word": " المتباين", "probability": 0.893359375}, {"start": 554.43, "end": 554.67, "word": " هذه", "probability": 0.235107421875}, {"start": 554.67, "end": 555.19, "word": " صحيحة", "probability": 0.9236328125}, {"start": 555.19, "end": 555.57, "word": " لكل", "probability": 0.971923828125}, {"start": 555.57, "end": 555.85, "word": " n", "probability": 0.21337890625}, {"start": 555.85, "end": 557.75, "word": " وبالتالي", "probability": 0.898046875}, {"start": 557.75, "end": 558.27, "word": " أنا", "probability": 0.34375}, {"start": 558.27, "end": 558.97, "word": " لكل", "probability": 0.98583984375}, {"start": 558.97, "end": 559.25, "word": " n", "probability": 0.8046875}, {"start": 559.25, "end": 559.69, "word": " بقدر", "probability": 0.9737955729166666}, {"start": 559.69, "end": 560.09, "word": " ألاقي", "probability": 0.67822265625}, {"start": 560.09, "end": 561.13, "word": " xkn", "probability": 0.552001953125}, {"start": 561.13, "end": 562.41, "word": " موجود", "probability": 0.9646809895833334}, {"start": 562.41, "end": 562.87, "word": " هنا", "probability": 0.978515625}, {"start": 562.87, "end": 563.21, "word": " و", "probability": 0.736328125}, {"start": 563.21, "end": 563.97, "word": " بيحقق", "probability": 0.83232421875}, {"start": 563.97, "end": 565.23, "word": " المتباين", "probability": 0.98447265625}, {"start": 565.23, "end": 565.59, "word": " هذه", "probability": 0.572265625}], "temperature": 1.0}, {"id": 22, "seek": 59487, "start": 568.57, "end": 594.87, "text": "عناصرها xkn من n بالساوية واحد to infinity of ال sequence xn such that بحيث أنه xkn أصغر من s زايد واحد على n أكبر من s سالب واحد على n الكلام هذا صحيح لكل n في n", "tokens": [3615, 1863, 33546, 2288, 11296, 2031, 5457, 9154, 297, 20666, 3794, 995, 2407, 10632, 36764, 24401, 281, 13202, 295, 2423, 8310, 2031, 77, 1270, 300, 4724, 5016, 1829, 12984, 14739, 3224, 2031, 5457, 5551, 9381, 17082, 2288, 9154, 262, 30767, 995, 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{"start": 726.36, "end": 729.08, "word": " تعريف", "probability": 0.9562174479166666}, {"start": 729.08, "end": 738.34, "word": " the", "probability": 0.409423828125}, {"start": 738.34, "end": 739.06, "word": " number", "probability": 0.984375}, {"start": 739.06, "end": 739.6, "word": " is", "probability": 0.397705078125}], "temperature": 1.0}, {"id": 28, "seek": 76706, "start": 741.0, "end": 767.06, "text": "in above example is called limit superior of sequence x in", "tokens": [259, 3673, 1365, 307, 1219, 4948, 13028, 295, 8310, 2031, 294], "avg_logprob": -0.3378906374176343, "compression_ratio": 1.0, "no_speech_prob": 0.0, "words": [{"start": 741.0, "end": 741.48, "word": "in", "probability": 0.08599853515625}, {"start": 741.48, "end": 742.12, "word": " above", "probability": 0.93212890625}, {"start": 742.12, "end": 744.94, "word": " example", "probability": 0.94970703125}, {"start": 744.94, "end": 748.7, "word": " is", "probability": 0.91064453125}, {"start": 748.7, "end": 749.34, "word": " called", "probability": 0.9130859375}, {"start": 749.34, "end": 756.26, "word": " limit", "probability": 0.95849609375}, {"start": 756.26, "end": 760.62, "word": " superior", "probability": 0.97412109375}, {"start": 760.62, "end": 763.42, "word": " of", "probability": 0.98095703125}, {"start": 763.42, "end": 764.58, "word": " sequence", "probability": 0.68408203125}, {"start": 764.58, "end": 766.62, "word": " x", "probability": 0.84130859375}, {"start": 766.62, "end": 767.06, "word": " in", "probability": 0.45751953125}], "temperature": 1.0}, {"id": 29, "seek": 79052, "start": 770.0, "end": 790.52, "text": "and we write limit superior ل Xn بساوي S", "tokens": [474, 321, 2464, 4948, 13028, 5296, 1783, 77, 4724, 3794, 995, 45865, 318], "avg_logprob": -0.37890625638621195, "compression_ratio": 0.8363636363636363, "no_speech_prob": 0.0, "words": [{"start": 770.0, "end": 770.74, "word": "and", "probability": 0.1981201171875}, {"start": 770.74, "end": 771.98, "word": " we", "probability": 0.92041015625}, {"start": 771.98, "end": 772.64, "word": " write", "probability": 0.94970703125}, {"start": 772.64, "end": 777.16, "word": " limit", "probability": 0.849609375}, {"start": 777.16, "end": 777.94, "word": " superior", "probability": 0.97216796875}, {"start": 777.94, "end": 786.8, "word": " ل", "probability": 0.53564453125}, {"start": 786.8, "end": 788.02, "word": " Xn", "probability": 0.287353515625}, {"start": 788.02, "end": 790.06, "word": " بساوي", "probability": 0.8883056640625}, {"start": 790.06, "end": 790.52, "word": " S", "probability": 0.8544921875}], "temperature": 1.0}, {"id": 30, "seek": 82164, "start": 793.78, "end": 821.64, "text": "إذا العدد S هذا في التمرين أو في ال exercise أو في المثال هذا بنسميه limit superior ل sequence xn limit superior of the sequence xn بيسموها بالعربي النهاية العليا للمتتالية في نهاية عالية و في نهاية سفلة وبالتالي", "tokens": [28814, 15730, 18863, 3215, 3215, 318, 23758, 8978, 16712, 29973, 9957, 34051, 8978, 2423, 5380, 34051, 8978, 9673, 12984, 6027, 23758, 44945, 38251, 1829, 3224, 4948, 13028, 5296, 8310, 2031, 77, 4948, 13028, 295, 264, 8310, 2031, 77, 4724, 1829, 38251, 2407, 11296, 20666, 3615, 2288, 21292, 28239, 11296, 10632, 18863, 20292, 995, 5296, 19528, 2655, 2655, 6027, 10632, 8978, 8717, 11296, 10632, 6225, 6027, 10632, 4032, 8978, 8717, 11296, 10632, 8608, 5172, 37977, 46599, 6027, 2655, 6027, 1829], "avg_logprob": -0.17539062052965165, "compression_ratio": 1.884393063583815, "no_speech_prob": 0.0, "words": [{"start": 793.78, "end": 794.12, "word": "إذا", "probability": 0.635498046875}, {"start": 794.12, "end": 794.68, "word": " العدد", "probability": 0.8746744791666666}, {"start": 794.68, "end": 795.0, "word": " S", "probability": 0.57373046875}, {"start": 795.0, "end": 795.42, "word": " هذا", "probability": 0.662109375}, {"start": 795.42, "end": 795.96, "word": " في", "probability": 0.830078125}, {"start": 795.96, "end": 796.7, "word": " التمرين", "probability": 0.9456380208333334}, {"start": 796.7, "end": 796.88, "word": " أو", "probability": 0.892578125}, {"start": 796.88, "end": 797.04, "word": " في", "probability": 0.8662109375}, {"start": 797.04, "end": 798.74, "word": " ال", "probability": 0.90869140625}, {"start": 798.74, "end": 799.16, "word": " exercise", "probability": 0.2783203125}, {"start": 799.16, "end": 799.7, "word": " أو", "probability": 0.890625}, {"start": 799.7, "end": 799.8, "word": " في", "probability": 0.9052734375}, {"start": 799.8, "end": 800.3, "word": " المثال", "probability": 0.9192708333333334}, {"start": 800.3, "end": 800.7, "word": " هذا", "probability": 0.90966796875}, {"start": 800.7, "end": 805.56, "word": " بنسميه", "probability": 0.8690185546875}, {"start": 805.56, "end": 805.86, "word": " limit", "probability": 0.78125}, {"start": 805.86, "end": 806.58, "word": " superior", "probability": 0.9697265625}, {"start": 806.58, "end": 806.82, "word": " ل", "probability": 0.7705078125}, {"start": 806.82, "end": 807.28, "word": " sequence", "probability": 0.8662109375}, {"start": 807.28, "end": 807.92, "word": " xn", "probability": 0.453857421875}, {"start": 807.92, "end": 809.58, "word": " limit", "probability": 0.7861328125}, {"start": 809.58, "end": 810.22, "word": " superior", "probability": 0.98193359375}, {"start": 810.22, "end": 810.6, "word": " of", "probability": 0.9677734375}, {"start": 810.6, "end": 810.78, "word": " the", "probability": 0.48974609375}, {"start": 810.78, "end": 811.18, "word": " sequence", "probability": 0.98046875}, {"start": 811.18, "end": 811.62, "word": " xn", "probability": 0.982177734375}, {"start": 811.62, "end": 812.12, "word": " بيسموها", "probability": 0.85146484375}, {"start": 812.12, "end": 812.6, "word": " بالعربي", "probability": 0.9654541015625}, {"start": 812.6, "end": 813.24, "word": " النهاية", "probability": 0.8720703125}, {"start": 813.24, "end": 813.76, "word": " العليا", "probability": 0.8361002604166666}, {"start": 813.76, "end": 814.66, "word": " للمتتالية", "probability": 0.9151204427083334}, {"start": 814.66, "end": 815.4, "word": " في", "probability": 0.6005859375}, {"start": 815.4, "end": 815.9, "word": " نهاية", "probability": 0.9580078125}, {"start": 815.9, "end": 816.26, "word": " عالية", "probability": 0.8069661458333334}, {"start": 816.26, "end": 816.36, "word": " و", "probability": 0.9189453125}, {"start": 816.36, "end": 816.48, "word": " في", "probability": 0.654296875}, {"start": 816.48, "end": 816.8, "word": " نهاية", "probability": 0.98681640625}, {"start": 816.8, "end": 817.3, "word": " سفلة", "probability": 0.9132486979166666}, {"start": 817.3, "end": 821.64, "word": " وبالتالي", "probability": 0.918359375}], "temperature": 1.0}, {"id": 31, "seek": 85517, "start": 826.81, "end": 855.17, "text": "هذه عبارة عن الـ limit يعني هذه بتطلع .. يعني ممكن أثبات باستخدام الـ monotone convergence theorem أن هذه بالساوي هي نفسها limit S as N tends to infinity limit ال supremum لل sequence هذه أو limit superior", 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"temperature": 1.0}, {"id": 37, "seek": 103184, "start": 1006.19, "end": 1031.85, "text": "برضه العدد S في الحالة هذه بنسميه limit inferior لل sequence XM S in above remark is called is called limit", "tokens": [26890, 11242, 3224, 18863, 3215, 3215, 318, 8978, 21542, 6027, 3660, 29538, 44945, 38251, 1829, 3224, 4948, 24249, 24976, 8310, 1783, 44, 318, 294, 3673, 7942, 307, 1219, 307, 1219, 4948], "avg_logprob": -0.2546386672183871, "compression_ratio": 1.1065573770491803, "no_speech_prob": 0.0, "words": [{"start": 1006.19, "end": 1006.79, "word": "برضه", "probability": 0.8009440104166666}, {"start": 1006.79, "end": 1007.37, "word": " العدد", "probability": 0.8702799479166666}, {"start": 1007.37, "end": 1007.59, "word": " S", "probability": 0.525390625}, {"start": 1007.59, "end": 1007.77, "word": " في", "probability": 0.86328125}, {"start": 1007.77, "end": 1008.21, "word": " الحالة", "probability": 0.9542643229166666}, {"start": 1008.21, "end": 1008.59, "word": " هذه", "probability": 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0.93115234375}, {"start": 1025.97, "end": 1028.91, "word": " is", "probability": 0.320556640625}, {"start": 1028.91, "end": 1029.53, "word": " called", "probability": 0.91259765625}, {"start": 1029.53, "end": 1031.85, "word": " limit", "probability": 0.95751953125}], "temperature": 1.0}, {"id": 38, "seek": 106091, "start": 1034.35, "end": 1060.91, "text": "limit inferior limit inferior of ال sequence x in and we write limit inferior ل x in بساوي s", "tokens": [4197, 270, 24249, 4948, 24249, 295, 2423, 8310, 2031, 294, 293, 321, 2464, 4948, 24249, 5296, 2031, 294, 4724, 3794, 995, 45865, 262], "avg_logprob": -0.2872721267243226, "compression_ratio": 1.2987012987012987, "no_speech_prob": 0.0, "words": [{"start": 1034.35, "end": 1035.01, "word": "limit", "probability": 0.640869140625}, {"start": 1035.01, "end": 1035.83, "word": " inferior", "probability": 0.85205078125}, {"start": 1035.83, "end": 1036.63, "word": " limit", "probability": 0.386962890625}, {"start": 1036.63, "end": 1039.59, "word": " inferior", "probability": 0.876953125}, {"start": 1039.59, "end": 1042.25, "word": " of", "probability": 0.9033203125}, {"start": 1042.25, "end": 1042.67, "word": " ال", "probability": 0.468017578125}, {"start": 1042.67, "end": 1043.31, "word": " sequence", "probability": 0.724609375}, {"start": 1043.31, "end": 1045.77, "word": " x", "probability": 0.7333984375}, {"start": 1045.77, "end": 1046.29, "word": " in", "probability": 0.52978515625}, {"start": 1046.29, "end": 1049.73, "word": " and", "probability": 0.7802734375}, {"start": 1049.73, "end": 1050.17, "word": " we", "probability": 0.90771484375}, {"start": 1050.17, "end": 1050.79, "word": " write", "probability": 0.939453125}, {"start": 1050.79, "end": 1055.49, "word": " limit", "probability": 0.97216796875}, {"start": 1055.49, "end": 1058.05, "word": " inferior", "probability": 0.91552734375}, {"start": 1058.05, "end": 1058.61, "word": " ل", "probability": 0.94287109375}, {"start": 1058.61, "end": 1058.97, "word": " x", "probability": 0.72119140625}, {"start": 1058.97, "end": 1059.43, "word": " in", "probability": 0.86083984375}, {"start": 1059.43, "end": 1060.43, "word": " بساوي", "probability": 0.7960205078125}, {"start": 1060.43, "end": 1060.91, "word": " s", "probability": 0.57421875}], "temperature": 1.0}, {"id": 39, "seek": 108368, "start": 1065.4, "end": 1083.68, "text": "وكمان مرة هذه limit inferior بتساوي .. ممكن اثبات باستخدام الـ monotone convergence theorem ممكن اثبات ان هذا بساوي limit sequence S as N tends to infinity", "tokens": [2407, 24793, 7649, 3714, 25720, 29538, 4948, 24249, 39894, 3794, 995, 45865, 4386, 3714, 43020, 1975, 12984, 3555, 9307, 4724, 995, 14851, 9778, 3215, 10943, 2423, 39184, 1108, 310, 546, 32181, 20904, 3714, 43020, 1975, 12984, 3555, 9307, 16472, 23758, 4724, 3794, 995, 45865, 4948, 8310, 318, 382, 426, 12258, 281, 13202], "avg_logprob": -0.1978183906033354, "compression_ratio": 1.3439490445859872, "no_speech_prob": 0.0, "words": [{"start": 1065.4, "end": 1066.0, "word": "وكمان", "probability": 0.8640950520833334}, {"start": 1066.0, "end": 1066.46, "word": " مرة", "probability": 0.9765625}, {"start": 1066.46, "end": 1066.88, "word": " هذه", "probability": 0.59130859375}, {"start": 1066.88, "end": 1067.24, "word": " limit", "probability": 0.5283203125}, {"start": 1067.24, "end": 1068.04, "word": " inferior", "probability": 0.80078125}, {"start": 1068.04, "end": 1068.84, "word": " بتساوي", "probability": 0.8975830078125}, {"start": 1068.84, "end": 1068.94, "word": " ..", "probability": 0.37060546875}, {"start": 1068.94, "end": 1069.2, "word": " ممكن", "probability": 0.915283203125}, {"start": 1069.2, "end": 1069.76, "word": " اثبات", "probability": 0.8934326171875}, {"start": 1069.76, "end": 1071.0, "word": " باستخدام", "probability": 0.9825032552083334}, {"start": 1071.0, "end": 1071.32, "word": " الـ", "probability": 0.83203125}, {"start": 1071.32, "end": 1071.7, "word": " monotone", "probability": 0.8382161458333334}, {"start": 1071.7, "end": 1072.42, "word": " convergence", "probability": 0.97119140625}, {"start": 1072.42, "end": 1073.0, "word": " theorem", "probability": 0.919921875}, {"start": 1073.0, "end": 1074.08, "word": " ممكن", "probability": 0.888916015625}, {"start": 1074.08, "end": 1074.56, "word": " اثبات", "probability": 0.9703369140625}, {"start": 1074.56, "end": 1074.72, "word": " ان", "probability": 0.486083984375}, {"start": 1074.72, "end": 1075.0, "word": " هذا", "probability": 0.88623046875}, {"start": 1075.0, "end": 1075.76, "word": " بساوي", "probability": 0.8935546875}, {"start": 1075.76, "end": 1076.32, "word": " limit", "probability": 0.98291015625}, {"start": 1076.32, "end": 1080.2, "word": " sequence", "probability": 0.389892578125}, {"start": 1080.2, "end": 1080.8, "word": " S", "probability": 0.6123046875}, {"start": 1080.8, "end": 1082.04, "word": " as", "probability": 0.65185546875}, {"start": 1082.04, "end": 1082.66, "word": " N", "probability": 0.81396484375}, {"start": 1082.66, "end": 1083.0, "word": " tends", "probability": 0.264404296875}, {"start": 1083.0, "end": 1083.24, "word": " to", "probability": 0.85498046875}, {"start": 1083.24, "end": 1083.68, "word": " infinity", "probability": 0.84033203125}], "temperature": 1.0}, {"id": 40, "seek": 110183, "start": 1085.48, "end": 1101.84, "text": "لأن هذه الـ sequence Sn ممكن اثبات إنها increasing متزايدة و bounded therefore by monotone convergence theorem it converges و ال limit تبعتها هتساوي ال supremum إلى اللي هو S", "tokens": [1211, 33456, 29538, 2423, 39184, 8310, 9264, 3714, 43020, 1975, 12984, 3555, 9307, 36145, 11296, 5662, 44650, 11622, 995, 25708, 3660, 4032, 37498, 4412, 538, 1108, 310, 546, 32181, 20904, 309, 9652, 2880, 4032, 2423, 4948, 6055, 3555, 34268, 11296, 8032, 2655, 3794, 995, 45865, 2423, 23710, 449, 30731, 13672, 1829, 31439, 318], "avg_logprob": -0.2663483674879427, "compression_ratio": 1.2921348314606742, "no_speech_prob": 0.0, "words": [{"start": 1085.48, "end": 1085.88, "word": 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" therefore", "probability": 0.26171875}, {"start": 1093.52, "end": 1094.06, "word": " by", "probability": 0.84814453125}, {"start": 1094.06, "end": 1094.58, "word": " monotone", "probability": 0.876953125}, {"start": 1094.58, "end": 1095.18, "word": " convergence", "probability": 0.95947265625}, {"start": 1095.18, "end": 1095.64, "word": " theorem", "probability": 0.90185546875}, {"start": 1095.64, "end": 1095.96, "word": " it", "probability": 0.84423828125}, {"start": 1095.96, "end": 1096.74, "word": " converges", "probability": 0.974365234375}, {"start": 1096.74, "end": 1097.36, "word": " و", "probability": 0.6572265625}, {"start": 1097.36, "end": 1097.48, "word": " ال", "probability": 0.75}, {"start": 1097.48, "end": 1097.74, "word": " limit", "probability": 0.9111328125}, {"start": 1097.74, "end": 1098.38, "word": " تبعتها", "probability": 0.912841796875}, {"start": 1098.38, "end": 1099.2, "word": " هتساوي", "probability": 0.851611328125}, {"start": 1099.2, "end": 1099.48, "word": " ال", "probability": 0.8095703125}, {"start": 1099.48, "end": 1100.1, "word": " supremum", "probability": 0.86572265625}, {"start": 1100.1, "end": 1100.4, "word": " إلى", "probability": 0.313232421875}, {"start": 1100.4, "end": 1101.3, "word": " اللي", "probability": 0.7724609375}, {"start": 1101.3, "end": 1101.52, "word": " هو", "probability": 0.9873046875}, {"start": 1101.52, "end": 1101.84, "word": " S", "probability": 0.87939453125}], "temperature": 1.0}, {"id": 41, "seek": 113011, "start": 1103.8, "end": 1130.12, "text": "تمام؟ إذا هذه بنتسميها limit inferior of the sequence xn وهذه limit superior هي النهاية العليا الصفلة النهاية العليا طبعا إذا كانت ال sequence bounded ف limit superior لها exist وبالساوي العدد S وكذا لك limit inferior لها exist وبالساوي العدد S", "tokens": [2655, 15042, 2304, 22807, 11933, 15730, 29538, 4724, 29399, 38251, 1829, 11296, 4948, 24249, 295, 264, 8310, 2031, 77, 37037, 24192, 4948, 13028, 39896, 28239, 11296, 10632, 18863, 20292, 995, 31767, 5172, 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"probability": 0.958984375}, {"start": 1107.84, "end": 1108.04, "word": " the", "probability": 0.6484375}, {"start": 1108.04, "end": 1108.44, "word": " sequence", "probability": 0.97802734375}, {"start": 1108.44, "end": 1109.02, "word": " xn", "probability": 0.5966796875}, {"start": 1109.02, "end": 1110.5, "word": " وهذه", "probability": 0.670654296875}, {"start": 1110.5, "end": 1110.9, "word": " limit", "probability": 0.76171875}, {"start": 1110.9, "end": 1112.02, "word": " superior", "probability": 0.400634765625}, {"start": 1112.02, "end": 1112.44, "word": " هي", "probability": 0.10858154296875}, {"start": 1112.44, "end": 1112.84, "word": " النهاية", "probability": 0.88232421875}, {"start": 1112.84, "end": 1113.26, "word": " العليا", "probability": 0.7887369791666666}, {"start": 1113.26, "end": 1113.92, "word": " الصفلة", "probability": 0.51171875}, {"start": 1113.92, "end": 1114.84, "word": " النهاية", "probability": 0.8212890625}, {"start": 1114.84, "end": 1115.4, "word": " العليا", "probability": 0.89013671875}, {"start": 1115.4, "end": 1117.08, "word": " طبعا", "probability": 0.953125}, {"start": 1117.08, "end": 1117.4, "word": " إذا", "probability": 0.9482421875}, {"start": 1117.4, "end": 1117.76, "word": " كانت", "probability": 0.992919921875}, {"start": 1117.76, "end": 1117.88, "word": " ال", "probability": 0.8583984375}, {"start": 1117.88, "end": 1118.18, "word": " sequence", "probability": 0.955078125}, {"start": 1118.18, "end": 1118.9, "word": " bounded", "probability": 0.94384765625}, {"start": 1118.9, "end": 1120.12, "word": " ف", "probability": 0.97509765625}, {"start": 1120.12, "end": 1120.46, "word": " limit", "probability": 0.80859375}, {"start": 1120.46, "end": 1121.04, "word": " superior", "probability": 0.97607421875}, {"start": 1121.04, "end": 1121.52, "word": " لها", "probability": 0.6766357421875}, {"start": 1121.52, "end": 1122.78, "word": " exist", "probability": 0.8525390625}, {"start": 1122.78, "end": 1123.96, "word": " وبالساوي", "probability": 0.710205078125}, {"start": 1123.96, "end": 1124.36, "word": " العدد", "probability": 0.92626953125}, {"start": 1124.36, "end": 1124.7, "word": " S", "probability": 0.47314453125}, {"start": 1124.7, "end": 1126.1, "word": " وكذا", "probability": 0.8173828125}, {"start": 1126.1, "end": 1126.42, "word": " لك", "probability": 0.60302734375}, {"start": 1126.42, "end": 1126.6, "word": " limit", "probability": 0.96240234375}, {"start": 1126.6, "end": 1127.32, "word": " inferior", "probability": 0.962890625}, {"start": 1127.32, "end": 1127.64, "word": " لها", "probability": 0.857666015625}, {"start": 1127.64, "end": 1128.24, "word": " exist", "probability": 0.8876953125}, {"start": 1128.24, "end": 1129.12, "word": " وبالساوي", "probability": 0.9267578125}, {"start": 1129.12, "end": 1129.8, "word": " العدد", "probability": 0.9609375}, {"start": 1129.8, "end": 1130.12, "word": " S", "probability": 0.9619140625}], "temperature": 1.0}, {"id": 42, "seek": 115627, "start": 1131.23, "end": 1156.27, "text": "بس العدد S هذا غير عن العدد S دا يعني هذا خلينا نسميه S star عشان نميزه عن العدد اللي فات ده خلينا نسميه S star S star وS star هنا غير مختلف مش شرط يكون هو نفس ال S في ال exercise اللي فات إذا نحاولوا تثبته", "tokens": [3555, 3794, 18863, 3215, 3215, 318, 23758, 32771, 13546, 18871, 18863, 3215, 3215, 318, 11778, 995, 37495, 22653, 23758, 16490, 20292, 8315, 8717, 38251, 1829, 3224, 318, 3543, 6225, 8592, 7649, 8717, 2304, 1829, 11622, 3224, 18871, 18863, 3215, 3215, 13672, 1829, 6156, 9307, 11778, 3224, 16490, 20292, 8315, 8717, 38251, 1829, 3224, 318, 3543, 318, 3543, 4032, 50, 3543, 34105, 32771, 13546, 3714, 46456, 46538, 37893, 13412, 2288, 9566, 7251, 30544, 31439, 8717, 36178, 2423, 318, 8978, 2423, 5380, 13672, 1829, 6156, 9307, 11933, 15730, 8717, 5016, 995, 12610, 14407, 6055, 12984, 3555, 47395], "avg_logprob": -0.14916992203022042, "compression_ratio": 1.8555555555555556, "no_speech_prob": 0.0, "words": [{"start": 1131.23, "end": 1131.57, "word": "بس", "probability": 0.95556640625}, {"start": 1131.57, "end": 1132.03, "word": " العدد", "probability": 0.9368489583333334}, {"start": 1132.03, "end": 1132.31, "word": " S", "probability": 0.5703125}, {"start": 1132.31, "end": 1132.73, "word": " هذا", "probability": 0.6865234375}, {"start": 1132.73, "end": 1133.79, "word": " غير", "probability": 0.902099609375}, {"start": 1133.79, "end": 1133.95, "word": " عن", "probability": 0.8662109375}, {"start": 1133.95, "end": 1134.35, "word": " العدد", "probability": 0.8927408854166666}, {"start": 1134.35, "end": 1134.59, "word": " S", "probability": 0.8916015625}, {"start": 1134.59, "end": 1134.99, "word": " دا", "probability": 0.52947998046875}, {"start": 1134.99, "end": 1135.29, "word": " يعني", "probability": 0.806396484375}, {"start": 1135.29, "end": 1135.55, "word": " هذا", "probability": 0.71484375}, {"start": 1135.55, "end": 1135.83, "word": " خلينا", "probability": 0.713134765625}, {"start": 1135.83, "end": 1136.39, "word": " نسميه", 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"probability": 0.96826171875}, {"start": 1149.67, "end": 1149.95, "word": " نفس", "probability": 0.992431640625}, {"start": 1149.95, "end": 1150.11, "word": " ال", "probability": 0.96435546875}, {"start": 1150.11, "end": 1150.37, "word": " S", "probability": 0.849609375}, {"start": 1150.37, "end": 1151.39, "word": " في", "probability": 0.94775390625}, {"start": 1151.39, "end": 1151.91, "word": " ال", "probability": 0.984375}, {"start": 1151.91, "end": 1153.09, "word": " exercise", "probability": 0.89794921875}, {"start": 1153.09, "end": 1153.69, "word": " اللي", "probability": 0.99072265625}, {"start": 1153.69, "end": 1154.03, "word": " فات", "probability": 0.994140625}, {"start": 1154.03, "end": 1154.75, "word": " إذا", "probability": 0.655029296875}, {"start": 1154.75, "end": 1155.57, "word": " نحاولوا", "probability": 0.8978515625}, {"start": 1155.57, "end": 1156.27, "word": " تثبته", "probability": 0.9261474609375}], "temperature": 1.0}, {"id": 43, "seek": 117919, "start": 1157.89, "end": 1179.19, "text": "حاولوا تثبتوا التمرين هذا اللي هو الخاص ب limit inferior بنفس ال argument by similar argument باستخدام استنتاج أو برهان مشابه للبرهان اللي عملناها للجزء اللي فات اللي هو تمرين عشرة okay تمام؟", "tokens": [5016, 995, 12610, 14407, 6055, 12984, 3555, 2655, 14407, 16712, 29973, 9957, 23758, 13672, 1829, 31439, 33962, 33546, 4724, 4948, 24249, 44945, 36178, 2423, 6770, 538, 2531, 6770, 4724, 995, 14851, 9778, 3215, 10943, 44713, 29399, 26108, 34051, 4724, 2288, 3224, 7649, 37893, 16758, 3224, 24976, 26890, 3224, 7649, 13672, 1829, 6225, 42213, 8315, 11296, 24976, 7435, 11622, 38207, 13672, 1829, 6156, 9307, 13672, 1829, 31439, 6055, 29973, 9957, 6225, 8592, 25720, 1392, 46811, 10943, 22807], "avg_logprob": -0.1267248322437336, "compression_ratio": 1.6470588235294117, "no_speech_prob": 0.0, "words": [{"start": 1157.89, "end": 1158.31, "word": "حاولوا", "probability": 0.966064453125}, {"start": 1158.31, "end": 1159.05, "word": " تثبتوا", "probability": 0.91455078125}, {"start": 1159.05, "end": 1160.43, "word": " التمرين", "probability": 0.9622395833333334}, {"start": 1160.43, "end": 1160.95, "word": " هذا", "probability": 0.92724609375}, {"start": 1160.95, "end": 1162.25, "word": " اللي", "probability": 0.8798828125}, {"start": 1162.25, "end": 1162.41, "word": " هو", "probability": 0.97314453125}, {"start": 1162.41, "end": 1162.83, "word": " الخاص", "probability": 0.96142578125}, {"start": 1162.83, "end": 1162.99, "word": " ب", "probability": 0.67919921875}, {"start": 1162.99, "end": 1163.21, "word": " limit", "probability": 0.62109375}, {"start": 1163.21, "end": 1163.95, "word": " inferior", "probability": 0.81103515625}, {"start": 1163.95, "end": 1164.59, "word": " بنفس", "probability": 0.94140625}, {"start": 1164.59, "end": 1165.53, "word": " ال", "probability": 0.90966796875}, {"start": 1165.53, "end": 1166.05, "word": " argument", "probability": 0.8271484375}, {"start": 1166.05, "end": 1166.45, "word": " by", "probability": 0.4541015625}, {"start": 1166.45, "end": 1167.03, "word": " similar", "probability": 0.9228515625}, {"start": 1167.03, "end": 1167.95, "word": " argument", "probability": 0.873046875}, {"start": 1167.95, "end": 1169.97, "word": " باستخدام", "probability": 0.97314453125}, {"start": 1169.97, "end": 1170.75, "word": " استنتاج", "probability": 0.8727213541666666}, {"start": 1170.75, "end": 1170.99, "word": " أو", "probability": 0.681640625}, {"start": 1170.99, "end": 1171.55, "word": " برهان", "probability": 0.953369140625}, {"start": 1171.55, "end": 1172.55, "word": " مشابه", "probability": 0.9847005208333334}, {"start": 1172.55, "end": 1173.13, "word": " للبرهان", "probability": 0.967041015625}, {"start": 1173.13, "end": 1173.29, "word": " اللي", "probability": 0.93212890625}, {"start": 1173.29, "end": 1174.37, "word": " عملناها", "probability": 0.8873291015625}, {"start": 1174.37, "end": 1175.47, "word": " للجزء", "probability": 0.85693359375}, {"start": 1175.47, "end": 1176.11, "word": " اللي", "probability": 0.91748046875}, {"start": 1176.11, "end": 1176.47, "word": " فات", "probability": 0.989501953125}, {"start": 1176.47, "end": 1176.81, "word": " اللي", "probability": 0.954345703125}, {"start": 1176.81, "end": 1177.01, "word": " هو", "probability": 0.98828125}, {"start": 1177.01, "end": 1177.47, "word": " تمرين", "probability": 0.7637532552083334}, {"start": 1177.47, "end": 1177.89, "word": " عشرة", "probability": 0.8453776041666666}, {"start": 1177.89, "end": 1178.45, "word": " okay", "probability": 0.62255859375}, {"start": 1178.45, "end": 1179.19, "word": " تمام؟", "probability": 0.8904622395833334}], "temperature": 1.0}, {"id": 44, "seek": 119733, "start": 1179.58, "end": 1197.34, "text": "واضح؟ اذا هنا اليوم اتعلمنا ان في حاجة اسمها النهاية العليا والنهاية السفلة لل sequence إذا كانت ال sequence bounded فأثبتنا هنا أن النهاية العليا تبقاتها موجودة وبالساوي العدد S اللي هو ال infimum لل sequence هذه", "tokens": [2407, 46958, 5016, 22807, 1975, 15730, 34105, 45595, 20498, 1975, 2655, 3615, 19528, 8315, 16472, 8978, 11331, 26108, 3660, 24525, 2304, 11296, 28239, 11296, 10632, 18863, 20292, 995, 16070, 1863, 11296, 10632, 21136, 5172, 37977, 24976, 8310, 11933, 15730, 25961, 2655, 2423, 8310, 37498, 6156, 10721, 12984, 3555, 2655, 8315, 34105, 14739, 28239, 11296, 10632, 18863, 20292, 995, 6055, 3555, 4587, 9307, 11296, 3714, 29245, 23328, 3660, 46599, 6027, 3794, 995, 45865, 18863, 3215, 3215, 318, 13672, 1829, 31439, 2423, 1536, 332, 449, 24976, 8310, 29538], "avg_logprob": -0.23257902709917092, "compression_ratio": 1.809278350515464, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1179.58, "end": 1180.44, "word": "واضح؟", "probability": 0.9111328125}, {"start": 1180.44, "end": 1181.32, "word": " اذا", "probability": 0.566650390625}, {"start": 1181.32, "end": 1181.58, "word": " هنا", "probability": 0.328125}, {"start": 1181.58, "end": 1181.84, "word": " اليوم", "probability": 0.880615234375}, {"start": 1181.84, "end": 1182.32, "word": " اتعلمنا", "probability": 0.9453125}, {"start": 1182.32, "end": 1182.44, "word": " ان", "probability": 0.83251953125}, {"start": 1182.44, "end": 1182.6, "word": " في", "probability": 0.560546875}, {"start": 1182.6, "end": 1182.9, "word": " حاجة", "probability": 0.9803059895833334}, {"start": 1182.9, "end": 1183.26, "word": " اسمها", "probability": 0.9827473958333334}, {"start": 1183.26, "end": 1183.68, "word": " النهاية", "probability": 0.900390625}, {"start": 1183.68, "end": 1184.2, "word": " العليا", "probability": 0.6695963541666666}, {"start": 1184.2, "end": 1185.36, "word": " والنهاية", "probability": 0.8612060546875}, {"start": 1185.36, "end": 1185.92, "word": " السفلة", "probability": 0.8150227864583334}, {"start": 1185.92, "end": 1186.28, "word": " لل", "probability": 0.341796875}, {"start": 1186.28, "end": 1186.98, "word": " sequence", "probability": 0.86572265625}, {"start": 1186.98, "end": 1188.08, "word": " إذا", "probability": 0.6900634765625}, {"start": 1188.08, "end": 1188.42, "word": " كانت", "probability": 0.989501953125}, {"start": 1188.42, "end": 1188.54, "word": " ال", "probability": 0.951171875}, {"start": 1188.54, "end": 1188.86, "word": " sequence", "probability": 0.98876953125}, {"start": 1188.86, "end": 1189.36, "word": " bounded", "probability": 0.8984375}, {"start": 1189.36, "end": 1190.02, "word": " فأثبتنا", "probability": 0.9331868489583334}, {"start": 1190.02, "end": 1190.28, "word": " هنا", "probability": 0.92431640625}, {"start": 1190.28, "end": 1190.54, "word": " أن", "probability": 0.51611328125}, {"start": 1190.54, "end": 1191.68, "word": " النهاية", "probability": 0.89697265625}, {"start": 1191.68, "end": 1192.02, "word": " العليا", "probability": 0.9396158854166666}, {"start": 1192.02, "end": 1192.46, "word": " تبقاتها", "probability": 0.70029296875}, {"start": 1192.46, "end": 1193.72, "word": " موجودة", "probability": 0.9796142578125}, {"start": 1193.72, "end": 1194.4, "word": " وبالساوي", "probability": 0.614404296875}, {"start": 1194.4, "end": 1194.84, "word": " العدد", "probability": 0.92529296875}, {"start": 1194.84, "end": 1195.16, "word": " S", "probability": 0.5693359375}, {"start": 1195.16, "end": 1195.4, "word": " اللي", "probability": 0.865478515625}, {"start": 1195.4, "end": 1195.6, "word": " هو", "probability": 0.98828125}, {"start": 1195.6, "end": 1195.78, "word": " ال", "probability": 0.8779296875}, {"start": 1195.78, "end": 1196.36, "word": " infimum", "probability": 0.7831217447916666}, {"start": 1196.36, "end": 1196.54, "word": " لل", "probability": 0.49853515625}, {"start": 1196.54, "end": 1196.96, "word": " sequence", "probability": 0.95849609375}, {"start": 1196.96, "end": 1197.34, "word": " هذه", "probability": 0.6640625}], "temperature": 1.0}, {"id": 45, "seek": 122365, "start": 1197.99, "end": 1223.65, "text": "إذا كانت برضه ال sequence bounded فالنهاية السفلة لها أو limit inferior بتطلع exist و بتساوي العدد s star اللي هو ال supremum لل sequence هذه أو بساوي limit ال sequence s هذه تمام؟ فحاولوا تبرهنوا الجزء هذا المشابه للجزء الأول يعني في أي سؤال أو استفسار؟", "tokens": [28814, 15730, 25961, 2655, 4724, 43042, 3224, 2423, 8310, 37498, 6156, 6027, 1863, 11296, 10632, 21136, 5172, 37977, 5296, 11296, 34051, 4948, 24249, 39894, 9566, 1211, 3615, 2514, 4032, 39894, 3794, 995, 45865, 18863, 3215, 3215, 262, 3543, 13672, 1829, 31439, 2423, 23710, 449, 24976, 8310, 29538, 34051, 4724, 3794, 995, 45865, 4948, 2423, 8310, 262, 29538, 46811, 10943, 22807, 6156, 5016, 995, 12610, 14407, 6055, 26890, 3224, 1863, 14407, 25724, 11622, 38207, 23758, 9673, 8592, 16758, 3224, 24976, 7435, 11622, 38207, 16247, 12610, 37495, 22653, 8978, 36632, 8608, 33604, 6027, 34051, 44713, 36178, 9640, 22807], "avg_logprob": -0.25209408199664246, "compression_ratio": 1.628099173553719, "no_speech_prob": 0.0, "words": [{"start": 1197.99, "end": 1198.29, "word": "إذا", "probability": 0.7308349609375}, {"start": 1198.29, "end": 1198.71, "word": " كانت", "probability": 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"probability": 0.92578125}, {"start": 1204.19, "end": 1205.45, "word": " و", "probability": 0.58349609375}, {"start": 1205.45, "end": 1206.67, "word": " بتساوي", "probability": 0.9261474609375}, {"start": 1206.67, "end": 1207.15, "word": " العدد", "probability": 0.9586588541666666}, {"start": 1207.15, "end": 1207.41, "word": " s", "probability": 0.387939453125}, {"start": 1207.41, "end": 1207.85, "word": " star", "probability": 0.071044921875}, {"start": 1207.85, "end": 1208.03, "word": " اللي", "probability": 0.910888671875}, {"start": 1208.03, "end": 1208.21, "word": " هو", "probability": 0.994140625}, {"start": 1208.21, "end": 1208.39, "word": " ال", "probability": 0.45166015625}, {"start": 1208.39, "end": 1209.03, "word": " supremum", "probability": 0.4779052734375}, {"start": 1209.03, "end": 1209.25, "word": " لل", "probability": 0.473388671875}, {"start": 1209.25, "end": 1209.75, "word": " sequence", "probability": 0.78955078125}, {"start": 1209.75, "end": 1210.21, "word": " هذه", "probability": 0.68701171875}, {"start": 1210.21, "end": 1211.77, "word": " أو", "probability": 0.7744140625}, {"start": 1211.77, "end": 1212.29, "word": " بساوي", "probability": 0.892822265625}, {"start": 1212.29, "end": 1212.69, "word": " limit", "probability": 0.96826171875}, {"start": 1212.69, "end": 1212.93, "word": " ال", "probability": 0.73779296875}, {"start": 1212.93, "end": 1213.35, "word": " sequence", "probability": 0.9931640625}, {"start": 1213.35, "end": 1213.71, "word": " s", "probability": 0.59912109375}, {"start": 1213.71, "end": 1214.21, "word": " هذه", "probability": 0.354736328125}, {"start": 1214.21, "end": 1216.69, "word": " تمام؟", "probability": 0.5982259114583334}, {"start": 1216.69, "end": 1217.81, "word": " فحاولوا", "probability": 0.98935546875}, {"start": 1217.81, "end": 1218.45, "word": " تبرهنوا", "probability": 0.764794921875}, {"start": 1218.45, "end": 1218.89, "word": " الجزء", "probability": 0.9560546875}, {"start": 1218.89, "end": 1219.17, "word": " هذا", "probability": 0.951171875}, {"start": 1219.17, "end": 1219.79, "word": " المشابه", "probability": 0.897216796875}, {"start": 1219.79, "end": 1220.35, "word": " للجزء", "probability": 0.9810791015625}, {"start": 1220.35, "end": 1220.97, "word": " الأول", "probability": 0.958740234375}, {"start": 1220.97, "end": 1221.63, "word": " يعني", "probability": 0.836181640625}, {"start": 1221.63, "end": 1222.17, "word": " في", "probability": 0.58203125}, {"start": 1222.17, "end": 1222.31, "word": " أي", "probability": 0.54443359375}, {"start": 1222.31, "end": 1222.69, "word": " سؤال", "probability": 0.9879557291666666}, {"start": 1222.69, "end": 1222.87, "word": " أو", "probability": 0.86328125}, {"start": 1222.87, "end": 1223.65, "word": " استفسار؟", "probability": 0.779937744140625}], "temperature": 1.0}, {"id": 46, "seek": 125499, "start": 1226.73, "end": 1254.99, "text": "طيب الآن إذا ننتقل ل section جديد هيك احنا متكون خلصنا section تلاتة اربعة في الكتاب المخرج و هنبدأ section جديد اللي هو section تلاتة خمسة الانوان تبع koshi sequences", "tokens": [9566, 1829, 3555, 6024, 48506, 11933, 15730, 8717, 29399, 4587, 1211, 5296, 3541, 10874, 16254, 3215, 39896, 4117, 1975, 5016, 8315, 44650, 30544, 16490, 1211, 9381, 8315, 3541, 6055, 1211, 9307, 3660, 1975, 25513, 27884, 8978, 33251, 2655, 16758, 9673, 34740, 7435, 4032, 8032, 1863, 44510, 10721, 3541, 10874, 16254, 3215, 13672, 1829, 31439, 3541, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 2423, 7649, 2407, 7649, 6055, 3555, 3615, 350, 17392, 22978], "avg_logprob": -0.20555320442528338, "compression_ratio": 1.593939393939394, "no_speech_prob": 0.0, "words": [{"start": 1226.73, "end": 1227.15, "word": "طيب", "probability": 0.9679361979166666}, {"start": 1227.15, "end": 1227.55, "word": " الآن", "probability": 0.58685302734375}, {"start": 1227.55, "end": 1227.85, "word": " إذا", "probability": 0.761474609375}, {"start": 1227.85, "end": 1228.75, "word": " ننتقل", "probability": 0.9365234375}, {"start": 1228.75, "end": 1228.89, "word": " ل", "probability": 0.95849609375}, {"start": 1228.89, "end": 1229.19, "word": " section", "probability": 0.7001953125}, {"start": 1229.19, "end": 1229.75, "word": " جديد", "probability": 0.9314778645833334}, {"start": 1229.75, "end": 1230.01, "word": " هيك", "probability": 0.7008056640625}, {"start": 1230.01, "end": 1230.15, "word": " احنا", "probability": 0.853515625}, {"start": 1230.15, "end": 1230.43, "word": " متكون", "probability": 0.57122802734375}, {"start": 1230.43, "end": 1230.97, "word": " خلصنا", "probability": 0.9761962890625}, {"start": 1230.97, "end": 1231.39, "word": " section", "probability": 0.82763671875}, {"start": 1231.39, "end": 1231.87, "word": " تلاتة", "probability": 0.9210205078125}, {"start": 1231.87, "end": 1232.35, "word": " اربعة", "probability": 0.7904459635416666}, {"start": 1232.35, "end": 1233.33, "word": " في", "probability": 0.73583984375}, {"start": 1233.33, "end": 1233.79, "word": " الكتاب", "probability": 0.9641927083333334}, {"start": 1233.79, "end": 1234.55, "word": " المخرج", "probability": 0.7696940104166666}, {"start": 1234.55, "end": 1235.27, "word": " و", "probability": 0.83837890625}, {"start": 1235.27, "end": 1236.11, "word": " هنبدأ", "probability": 0.9449462890625}, {"start": 1236.11, "end": 1236.75, "word": " section", "probability": 0.8720703125}, {"start": 1236.75, "end": 1237.85, "word": " جديد", "probability": 0.9910481770833334}, {"start": 1237.85, "end": 1242.73, "word": " اللي", "probability": 0.923095703125}, {"start": 1242.73, "end": 1242.97, "word": " هو", "probability": 0.9814453125}, {"start": 1242.97, "end": 1243.49, "word": " section", "probability": 0.85205078125}, {"start": 1243.49, "end": 1244.11, "word": " تلاتة", "probability": 0.9813232421875}, {"start": 1244.11, "end": 1245.75, "word": " خمسة", "probability": 0.9871826171875}, {"start": 1245.75, "end": 1251.01, "word": " الانوان", "probability": 0.61065673828125}, {"start": 1251.01, "end": 1251.41, "word": " تبع", "probability": 0.9501953125}, {"start": 1251.41, "end": 1252.03, "word": " koshi", "probability": 0.380615234375}, {"start": 1252.03, "end": 1254.99, "word": " sequences", "probability": 0.8837890625}], "temperature": 1.0}, {"id": 47, "seek": 127575, "start": 1257.81, "end": 1275.75, "text": "متتاليات كوشي طبعا كوشي هاد عالم ألماني German mathematician كان مهتم بدراسة نوع خاص من ال sequences وكان بحاول يدرس التقارب و التباعت تباعهم و علاقتهم بال", "tokens": [2304, 2655, 2655, 6027, 1829, 9307, 9122, 2407, 8592, 1829, 23032, 3555, 3615, 995, 9122, 2407, 8592, 1829, 8032, 18513, 6225, 45340, 5551, 19528, 7649, 1829, 6521, 48281, 25961, 3714, 3224, 39237, 47525, 23557, 3794, 3660, 8717, 45367, 16490, 33546, 9154, 2423, 22978, 4032, 41361, 4724, 5016, 995, 12610, 7251, 3215, 2288, 3794, 16712, 4587, 9640, 3555, 4032, 16712, 3555, 995, 34268, 6055, 3555, 45761, 16095, 4032, 11203, 995, 38149, 16095, 20666], "avg_logprob": -0.10546875, "compression_ratio": 1.514792899408284, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 1257.81, "end": 1258.91, "word": "متتاليات", "probability": 0.9483235677083334}, {"start": 1258.91, "end": 1259.43, "word": " كوشي", "probability": 0.8240966796875}, {"start": 1259.43, "end": 1259.87, "word": " طبعا", "probability": 0.93798828125}, {"start": 1259.87, "end": 1260.25, "word": " كوشي", "probability": 0.9249267578125}, {"start": 1260.25, "end": 1260.47, "word": " هاد", "probability": 0.659423828125}, {"start": 1260.47, "end": 1260.81, "word": " عالم", "probability": 0.96875}, {"start": 1260.81, "end": 1261.75, "word": " ألماني", "probability": 0.9549560546875}, {"start": 1261.75, "end": 1263.41, "word": " German", "probability": 0.481689453125}, {"start": 1263.41, "end": 1264.17, "word": " mathematician", "probability": 0.92626953125}, {"start": 1264.17, "end": 1265.35, "word": " كان", "probability": 0.95703125}, {"start": 1265.35, "end": 1265.95, "word": " مهتم", "probability": 0.99072265625}, {"start": 1265.95, "end": 1266.87, "word": " بدراسة", "probability": 0.9686279296875}, {"start": 1266.87, "end": 1267.19, "word": " نوع", "probability": 0.995361328125}, {"start": 1267.19, "end": 1267.83, "word": " خاص", "probability": 0.992431640625}, {"start": 1267.83, "end": 1268.87, "word": " من", "probability": 0.9951171875}, {"start": 1268.87, "end": 1269.05, "word": " ال", "probability": 0.9619140625}, {"start": 1269.05, "end": 1269.69, "word": " sequences", "probability": 0.80517578125}, {"start": 1269.69, "end": 1271.87, "word": " وكان", "probability": 0.855224609375}, {"start": 1271.87, "end": 1272.41, "word": " بحاول", "probability": 0.93994140625}, {"start": 1272.41, "end": 1272.75, "word": " يدرس", "probability": 0.9976806640625}, {"start": 1272.75, "end": 1273.37, "word": " التقارب", "probability": 0.9967041015625}, {"start": 1273.37, "end": 1273.51, "word": " و", "probability": 0.87841796875}, {"start": 1273.51, "end": 1273.93, "word": " التباعت", "probability": 0.8338623046875}, {"start": 1273.93, "end": 1274.57, "word": " تباعهم", "probability": 0.8624267578125}, {"start": 1274.57, "end": 1274.89, "word": " و", "probability": 0.99267578125}, {"start": 1274.89, "end": 1275.51, "word": " علاقتهم", "probability": 0.9541015625}, {"start": 1275.51, "end": 1275.75, "word": " بال", "probability": 0.91259765625}], "temperature": 1.0}, {"id": 48, "seek": 128732, "start": 1276.54, "end": 1287.32, "text": "convergent و divergence sequences العادية فهذا يعني من كان جزء من اهتماماته طبعا له حاجات كتيرة في ال analysis", "tokens": [1671, 331, 6930, 4032, 47387, 22978, 18863, 18513, 10632, 6156, 3224, 15730, 37495, 22653, 9154, 25961, 10874, 11622, 38207, 9154, 1975, 3224, 39237, 10943, 9307, 3224, 23032, 3555, 3615, 995, 46740, 11331, 26108, 9307, 9122, 2655, 48923, 8978, 2423, 5215], "avg_logprob": -0.20141005516052246, "compression_ratio": 1.25, "no_speech_prob": 0.0, "words": [{"start": 1276.54, "end": 1277.32, "word": "convergent", "probability": 0.80029296875}, {"start": 1277.32, "end": 1277.46, "word": " و", "probability": 0.806640625}, {"start": 1277.46, "end": 1277.94, "word": " divergence", "probability": 0.64794921875}, {"start": 1277.94, "end": 1278.74, "word": " sequences", "probability": 0.307373046875}, {"start": 1278.74, "end": 1279.62, "word": " العادية", "probability": 0.9739583333333334}, {"start": 1279.62, "end": 1281.9, "word": " فهذا", "probability": 0.80712890625}, {"start": 1281.9, "end": 1282.08, "word": " يعني", "probability": 0.92919921875}, {"start": 1282.08, "end": 1282.28, "word": " من", "probability": 0.208984375}, {"start": 1282.28, "end": 1282.52, "word": " كان", "probability": 0.83056640625}, {"start": 1282.52, "end": 1283.42, "word": " جزء", "probability": 0.9664713541666666}, {"start": 1283.42, "end": 1283.6, "word": " من", "probability": 0.994140625}, {"start": 1283.6, "end": 1284.36, "word": " اهتماماته", "probability": 0.927978515625}, {"start": 1284.36, "end": 1284.66, "word": " طبعا", "probability": 0.8759765625}, {"start": 1284.66, "end": 1284.82, "word": " له", "probability": 0.342041015625}, {"start": 1284.82, "end": 1285.2, "word": " حاجات", "probability": 0.99658203125}, {"start": 1285.2, "end": 1285.66, "word": " كتيرة", "probability": 0.92138671875}, {"start": 1285.66, "end": 1286.2, "word": " في", "probability": 0.83251953125}, {"start": 1286.2, "end": 1286.46, "word": " ال", "probability": 0.96533203125}, {"start": 1286.46, "end": 1287.32, "word": " analysis", "probability": 0.814453125}], "temperature": 1.0}, {"id": 49, "seek": 131636, "start": 1291.2, "end": 1316.36, "text": "فالجماعة الرياضيات او الناس اللي جوا بعده عشان يعني تقديرا للجهد اللي قامله وللدراسة اللي قاملها لهذا النوع الخاص من المتتاليات سموا المتتاليات هذه باسمه متتاليات كوشي او كوشي sequences فما هي متتاليات كوشي؟ متى بنقول ان ال sequence كوشي؟ فهذه ال definition", "tokens": [5172, 6027, 7435, 15042, 27884, 2423, 16572, 46958, 1829, 9307, 1975, 2407, 2423, 8315, 3794, 13672, 1829, 10874, 14407, 39182, 3224, 6225, 8592, 7649, 37495, 22653, 6055, 28543, 13546, 995, 24976, 7435, 3224, 3215, 13672, 1829, 12174, 10943, 43761, 35525, 1211, 3215, 23557, 3794, 3660, 13672, 1829, 12174, 10943, 1211, 11296, 46740, 15730, 28239, 45367, 33962, 33546, 9154, 9673, 2655, 2655, 6027, 1829, 9307, 8608, 2304, 14407, 9673, 2655, 2655, 6027, 1829, 9307, 29538, 4724, 32277, 2304, 3224, 44650, 2655, 6027, 1829, 9307, 9122, 2407, 8592, 1829, 1975, 2407, 9122, 2407, 8592, 1829, 22978, 6156, 15042, 39896, 44650, 2655, 6027, 1829, 9307, 9122, 2407, 8592, 1829, 22807, 44650, 7578, 44945, 39648, 16472, 2423, 8310, 9122, 2407, 8592, 1829, 22807, 6156, 3224, 24192, 2423, 7123], "avg_logprob": -0.16300000715255739, "compression_ratio": 2.0319634703196345, "no_speech_prob": 0.0, "words": [{"start": 1291.2, "end": 1292.16, "word": "فالجماعة", "probability": 0.8064453125}, {"start": 1292.16, "end": 1292.92, "word": " الرياضيات", "probability": 0.9595703125}, {"start": 1292.92, "end": 1293.14, "word": " او", "probability": 0.73095703125}, {"start": 1293.14, "end": 1293.48, "word": " الناس", "probability": 0.98779296875}, {"start": 1293.48, "end": 1293.6, "word": " اللي", "probability": 0.876953125}, {"start": 1293.6, "end": 1293.8, "word": " جوا", "probability": 0.8857421875}, {"start": 1293.8, "end": 1294.36, "word": " بعده", "probability": 0.939453125}, {"start": 1294.36, "end": 1295.7, "word": " عشان", "probability": 0.8522135416666666}, {"start": 1295.7, "end": 1296.14, "word": " يعني", "probability": 0.720458984375}, {"start": 1296.14, "end": 1296.84, "word": " تقديرا", "probability": 0.9248046875}, {"start": 1296.84, "end": 1297.36, "word": " للجهد", "probability": 0.9254150390625}, {"start": 1297.36, "end": 1297.52, "word": " اللي", "probability": 0.92919921875}, {"start": 1297.52, "end": 1297.9, "word": " قامله", "probability": 0.76318359375}, {"start": 1297.9, "end": 1298.64, "word": " وللدراسة", "probability": 0.8310546875}, {"start": 1298.64, "end": 1298.76, "word": " اللي", "probability": 0.985595703125}, {"start": 1298.76, "end": 1299.06, "word": " قاملها", "probability": 0.8785400390625}, {"start": 1299.06, "end": 1299.44, "word": " لهذا", "probability": 0.8212890625}, {"start": 1299.44, "end": 1299.72, "word": " النوع", "probability": 0.9736328125}, {"start": 1299.72, "end": 1300.8, "word": " الخاص", "probability": 0.962158203125}, {"start": 1300.8, "end": 1301.38, "word": " من", "probability": 0.9716796875}, {"start": 1301.38, "end": 1302.36, "word": " المتتاليات", "probability": 0.9481608072916666}, {"start": 1302.36, "end": 1303.36, "word": " سموا", "probability": 0.8605143229166666}, {"start": 1303.36, "end": 1304.14, "word": " المتتاليات", "probability": 0.99267578125}, {"start": 1304.14, "end": 1304.38, "word": " هذه", "probability": 0.36962890625}, {"start": 1304.38, "end": 1305.2, "word": " باسمه", "probability": 0.9500732421875}, {"start": 1305.2, "end": 1306.2, "word": " متتاليات", "probability": 0.973046875}, {"start": 1306.2, "end": 1306.76, "word": " كوشي", "probability": 0.765350341796875}, {"start": 1306.76, "end": 1307.88, "word": " او", "probability": 0.767578125}, {"start": 1307.88, "end": 1308.36, "word": " كوشي", "probability": 0.777313232421875}, {"start": 1308.36, "end": 1308.96, "word": " sequences", "probability": 0.796875}, {"start": 1308.96, "end": 1310.58, "word": " فما", "probability": 0.932861328125}, {"start": 1310.58, "end": 1310.78, "word": " هي", "probability": 0.890625}, {"start": 1310.78, "end": 1311.56, "word": " متتاليات", "probability": 0.98603515625}, {"start": 1311.56, "end": 1312.04, "word": " كوشي؟", "probability": 0.890234375}, {"start": 1312.04, "end": 1312.28, "word": " متى", "probability": 0.952392578125}, {"start": 1312.28, "end": 1312.62, "word": " بنقول", "probability": 0.819580078125}, {"start": 1312.62, "end": 1312.8, "word": " ان", "probability": 0.8662109375}, {"start": 1312.8, "end": 1312.92, "word": " ال", "probability": 0.89990234375}, {"start": 1312.92, "end": 1313.4, "word": " sequence", "probability": 0.9814453125}, {"start": 1313.4, "end": 1315.1, "word": " كوشي؟", "probability": 0.90361328125}, {"start": 1315.1, "end": 1315.8, "word": " فهذه", "probability": 0.8163248697916666}, {"start": 1315.8, "end": 1315.94, "word": " ال", "probability": 0.88818359375}, {"start": 1315.94, "end": 1316.36, "word": " definition", "probability": 0.95361328125}], "temperature": 1.0}, {"id": 50, "seek": 135001, "start": 1321.97, "end": 1350.01, "text": "a sequence of real numbers a sequence xn contained in R is Cauchy is Cauchy if الشرط التالي بتحقق لكل إبسلون أكبر من السفر يوجد capital N يعتمد على إبسلون عدد طبيعي", "tokens": [64, 8310, 295, 957, 3547, 257, 8310, 2031, 77, 16212, 294, 497, 307, 7544, 625, 88, 307, 7544, 625, 88, 498, 25124, 2288, 9566, 16712, 6027, 1829, 39894, 5016, 4587, 4587, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 7251, 29245, 3215, 4238, 426, 7251, 34268, 2304, 3215, 15844, 11933, 3555, 3794, 1211, 11536, 6225, 3215, 3215, 23032, 21292, 3615, 1829], "avg_logprob": -0.17991728160311193, "compression_ratio": 1.3532934131736527, "no_speech_prob": 0.0, "words": [{"start": 1321.97, "end": 1323.03, "word": "a", "probability": 0.2255859375}, {"start": 1323.03, "end": 1323.79, "word": " sequence", "probability": 0.97802734375}, {"start": 1323.79, "end": 1324.17, "word": " of", "probability": 0.970703125}, {"start": 1324.17, "end": 1324.47, "word": " real", "probability": 0.96826171875}, {"start": 1324.47, "end": 1325.09, "word": " numbers", "probability": 0.89306640625}, {"start": 1325.09, "end": 1325.37, "word": " a", "probability": 0.5029296875}, {"start": 1325.37, "end": 1326.01, "word": " sequence", "probability": 0.986328125}, {"start": 1326.01, "end": 1328.05, "word": " xn", "probability": 0.432373046875}, {"start": 1328.05, "end": 1328.87, "word": " contained", "probability": 0.64404296875}, {"start": 1328.87, "end": 1329.19, "word": " in", "probability": 0.96630859375}, {"start": 1329.19, "end": 1329.73, "word": " R", "probability": 0.84716796875}, {"start": 1329.73, "end": 1331.49, "word": " is", "probability": 0.904296875}, {"start": 1331.49, "end": 1332.71, "word": " Cauchy", "probability": 0.8478190104166666}, {"start": 1332.71, "end": 1335.01, "word": " is", "probability": 0.3271484375}, {"start": 1335.01, "end": 1335.67, "word": " Cauchy", "probability": 0.9256184895833334}, {"start": 1335.67, "end": 1336.15, "word": " if", "probability": 0.7314453125}, {"start": 1336.15, "end": 1338.31, "word": " الشرط", "probability": 0.94775390625}, {"start": 1338.31, "end": 1338.73, "word": " التالي", "probability": 0.9869791666666666}, {"start": 1338.73, "end": 1340.47, "word": " بتحقق", "probability": 0.83966064453125}, {"start": 1340.47, "end": 1341.23, "word": " لكل", "probability": 0.974853515625}, {"start": 1341.23, "end": 1341.93, "word": " إبسلون", "probability": 0.740673828125}, {"start": 1341.93, "end": 1342.45, "word": " أكبر", "probability": 0.97802734375}, {"start": 1342.45, "end": 1342.67, "word": " من", "probability": 0.99560546875}, {"start": 1342.67, "end": 1343.23, "word": " السفر", "probability": 0.8453776041666666}, {"start": 1343.23, "end": 1344.25, "word": " يوجد", "probability": 0.9798177083333334}, {"start": 1344.25, "end": 1345.11, "word": " capital", "probability": 0.58251953125}, {"start": 1345.11, "end": 1345.49, "word": " N", "probability": 0.84716796875}, {"start": 1345.49, "end": 1346.31, "word": " يعتمد", "probability": 0.9110107421875}, {"start": 1346.31, "end": 1347.97, "word": " على", "probability": 0.8916015625}, {"start": 1347.97, "end": 1348.63, "word": " إبسلون", "probability": 0.9810546875}, {"start": 1348.63, "end": 1349.13, "word": " عدد", "probability": 0.9645182291666666}, {"start": 1349.13, "end": 1350.01, "word": " طبيعي", "probability": 0.9752197265625}], "temperature": 1.0}, {"id": 51, "seek": 137842, "start": 1351.52, "end": 1378.42, "text": "بحيث أنه لو كان n و m أكبر من أو ساوي capital N فهذا لازم يقدي أن المسافة بين xn و xm أصغر من إبسلون هذه هي كوشي sequence sequence of real number بنسميها كوشي متتالية كوشي إذا كان لأي given إبسلون", "tokens": [49628, 1829, 12984, 14739, 3224, 45164, 25961, 297, 4032, 275, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 6156, 3224, 15730, 5296, 31377, 2304, 7251, 4587, 16254, 14739, 9673, 3794, 31845, 3660, 49374, 2031, 77, 4032, 2031, 76, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536, 29538, 39896, 9122, 2407, 8592, 1829, 8310, 8310, 295, 957, 1230, 44945, 38251, 1829, 11296, 9122, 2407, 8592, 1829, 44650, 2655, 6027, 10632, 9122, 2407, 8592, 1829, 11933, 15730, 25961, 5296, 10721, 1829, 2212, 11933, 3555, 3794, 1211, 11536], "avg_logprob": -0.22552083482344945, "compression_ratio": 1.5276381909547738, "no_speech_prob": 0.0, "words": [{"start": 1351.52, "end": 1352.32, "word": "بحيث", "probability": 0.8313802083333334}, {"start": 1352.32, "end": 1353.02, "word": " أنه", "probability": 0.6138916015625}, {"start": 1353.02, "end": 1354.4, "word": " لو", "probability": 0.6474609375}, {"start": 1354.4, "end": 1354.98, "word": " كان", "probability": 0.97705078125}, {"start": 1354.98, "end": 1355.46, "word": " n", "probability": 0.24609375}, {"start": 1355.46, "end": 1355.7, "word": " و", "probability": 0.85986328125}, {"start": 1355.7, "end": 1356.12, "word": " m", "probability": 0.619140625}, {"start": 1356.12, "end": 1356.86, "word": " أكبر", "probability": 0.9611002604166666}, {"start": 1356.86, "end": 1357.08, "word": " من", "probability": 0.79736328125}, {"start": 1357.08, "end": 1357.3, "word": " أو", "probability": 0.9716796875}, {"start": 1357.3, "end": 1357.8, "word": " ساوي", "probability": 0.9259440104166666}, {"start": 1357.8, "end": 1358.26, "word": " capital", "probability": 0.343017578125}, {"start": 1358.26, "end": 1358.56, "word": " N", "probability": 0.6552734375}, {"start": 1358.56, "end": 1359.12, "word": " فهذا", "probability": 0.9415690104166666}, {"start": 1359.12, "end": 1359.62, "word": " لازم", "probability": 0.888671875}, {"start": 1359.62, "end": 1360.08, "word": " يقدي", "probability": 0.6351725260416666}, {"start": 1360.08, "end": 1360.94, "word": " أن", "probability": 0.6279296875}, {"start": 1360.94, "end": 1361.68, "word": " المسافة", "probability": 0.9493408203125}, {"start": 1361.68, "end": 1361.98, "word": " بين", "probability": 0.92431640625}, {"start": 1361.98, "end": 1362.72, "word": " xn", "probability": 0.682861328125}, {"start": 1362.72, "end": 1362.92, "word": " و", "probability": 0.974609375}, {"start": 1362.92, "end": 1364.0, "word": " xm", "probability": 0.931640625}, {"start": 1364.0, "end": 1365.62, "word": " أصغر", "probability": 0.983642578125}, {"start": 1365.62, "end": 1365.8, "word": " من", "probability": 0.9892578125}, {"start": 1365.8, "end": 1366.3, "word": " إبسلون", "probability": 0.7947265625}, {"start": 1366.3, "end": 1368.92, "word": " هذه", "probability": 0.359130859375}, {"start": 1368.92, "end": 1370.04, "word": " هي", "probability": 0.85205078125}, {"start": 1370.04, "end": 1371.18, "word": " كوشي", "probability": 0.7606201171875}, {"start": 1371.18, "end": 1371.68, "word": " sequence", "probability": 0.66357421875}, {"start": 1371.68, "end": 1372.26, "word": " sequence", "probability": 0.64990234375}, {"start": 1372.26, "end": 1372.6, "word": " of", "probability": 0.916015625}, {"start": 1372.6, "end": 1372.82, "word": " real", "probability": 0.9443359375}, {"start": 1372.82, "end": 1373.18, "word": " number", "probability": 0.64794921875}, {"start": 1373.18, "end": 1373.8, "word": " بنسميها", "probability": 0.79302978515625}, {"start": 1373.8, "end": 1374.88, "word": " كوشي", "probability": 0.9605712890625}, {"start": 1374.88, "end": 1375.86, "word": " متتالية", "probability": 0.926025390625}, {"start": 1375.86, "end": 1376.36, "word": " كوشي", "probability": 0.9632568359375}, {"start": 1376.36, "end": 1376.58, "word": " إذا", "probability": 0.936767578125}, {"start": 1376.58, "end": 1376.88, "word": " كان", "probability": 0.99072265625}, {"start": 1376.88, "end": 1377.36, "word": " لأي", "probability": 0.9523111979166666}, {"start": 1377.36, "end": 1377.74, "word": " given", "probability": 0.8994140625}, {"start": 1377.74, "end": 1378.42, "word": " إبسلون", "probability": 0.89736328125}], "temperature": 1.0}, {"id": 52, "seek": 140304, "start": 1379.48, "end": 1403.04, "text": "بقدر ألاقي عدد طبيعي يعتمد على epsilon بحيث لكل N و M لكل المؤشرات اللي من capital N و N طالع المسافة تبع الحدود الفرق بين حدودهم أصغر من epsilon في هنا أول لمة", "tokens": [3555, 28543, 2288, 5551, 15040, 38436, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 7251, 34268, 2304, 3215, 15844, 17889, 4724, 5016, 1829, 12984, 5296, 28820, 426, 4032, 376, 5296, 28820, 9673, 33604, 46309, 9307, 13672, 1829, 9154, 4238, 426, 4032, 426, 23032, 6027, 3615, 9673, 3794, 31845, 3660, 6055, 3555, 3615, 21542, 3215, 23328, 27188, 2288, 4587, 49374, 11331, 3215, 23328, 16095, 5551, 9381, 17082, 2288, 9154, 17889, 8978, 34105, 5551, 12610, 32767, 3660], "avg_logprob": -0.17989582856496175, "compression_ratio": 1.5144508670520231, "no_speech_prob": 0.0, "words": [{"start": 1379.48, "end": 1380.02, "word": "بقدر", "probability": 0.8699544270833334}, {"start": 1380.02, "end": 1380.36, "word": " ألاقي", "probability": 0.5559895833333334}, {"start": 1380.36, "end": 1380.72, "word": " عدد", "probability": 0.9749348958333334}, {"start": 1380.72, "end": 1381.2, "word": " طبيعي", "probability": 0.98193359375}, {"start": 1381.2, "end": 1381.74, "word": " يعتمد", "probability": 0.9010009765625}, {"start": 1381.74, "end": 1381.9, "word": " على", "probability": 0.8544921875}, {"start": 1381.9, "end": 1382.36, "word": " epsilon", "probability": 0.250732421875}, {"start": 1382.36, "end": 1384.14, "word": " بحيث", "probability": 0.95263671875}, {"start": 1384.14, "end": 1384.62, "word": " لكل", "probability": 0.981689453125}, {"start": 1384.62, "end": 1384.94, "word": " N", "probability": 0.30712890625}, {"start": 1384.94, "end": 1385.08, "word": " و", "probability": 0.9755859375}, {"start": 1385.08, "end": 1385.36, "word": " M", "probability": 0.76025390625}, {"start": 1385.36, "end": 1386.12, "word": " لكل", "probability": 0.77685546875}, {"start": 1386.12, "end": 1387.0, "word": " المؤشرات", "probability": 0.9793701171875}, {"start": 1387.0, "end": 1387.22, "word": " اللي", "probability": 0.8564453125}, {"start": 1387.22, "end": 1387.46, "word": " من", "probability": 0.9111328125}, {"start": 1387.46, "end": 1387.84, "word": " capital", "probability": 0.7119140625}, {"start": 1387.84, "end": 1388.12, "word": " N", "probability": 0.95263671875}, {"start": 1388.12, "end": 1388.26, "word": " و", "probability": 0.97705078125}, {"start": 1388.26, "end": 1388.4, "word": " N", "probability": 0.71337890625}, {"start": 1388.4, "end": 1389.08, "word": " طالع", "probability": 0.8395182291666666}, {"start": 1389.08, "end": 1390.68, "word": " المسافة", "probability": 0.92919921875}, {"start": 1390.68, "end": 1390.94, "word": " تبع", "probability": 0.7430013020833334}, {"start": 1390.94, "end": 1391.42, "word": " الحدود", "probability": 0.8191731770833334}, {"start": 1391.42, "end": 1392.8, "word": " الفرق", "probability": 0.8096516927083334}, {"start": 1392.8, "end": 1393.0, "word": " بين", "probability": 0.9755859375}, {"start": 1393.0, "end": 1393.72, "word": " حدودهم", "probability": 0.9871826171875}, {"start": 1393.72, "end": 1394.44, "word": " أصغر", "probability": 0.9736328125}, {"start": 1394.44, "end": 1394.62, "word": " من", "probability": 0.9921875}, {"start": 1394.62, "end": 1395.4, "word": " epsilon", "probability": 0.513671875}, {"start": 1395.4, "end": 1399.8, "word": " في", "probability": 0.88671875}, {"start": 1399.8, "end": 1400.22, "word": " هنا", "probability": 0.88330078125}, {"start": 1400.22, "end": 1402.5, "word": " أول", "probability": 0.926513671875}, {"start": 1402.5, "end": 1403.04, "word": " لمة", "probability": 0.555419921875}], "temperature": 1.0}, {"id": 53, "seek": 144180, "start": 1413.72, "end": 1441.8, "text": "لمّة واحدة عشرين اش لمّة بتقول every .. every convergent .. every convergent sequence of real numbers is Cauchy كل متتالية", "tokens": [19528, 11703, 3660, 36764, 24401, 3660, 6225, 46309, 9957, 1975, 8592, 32767, 11703, 3660, 39894, 39648, 633, 4386, 633, 9652, 6930, 4386, 633, 9652, 6930, 8310, 295, 957, 3547, 307, 7544, 625, 88, 28242, 44650, 2655, 6027, 10632], "avg_logprob": -0.2708333455599271, "compression_ratio": 1.21875, "no_speech_prob": 0.0, "words": [{"start": 1413.72, "end": 1414.46, "word": "لمّة", "probability": 0.6565755208333334}, {"start": 1414.46, "end": 1415.2, "word": " واحدة", "probability": 0.8639322916666666}, {"start": 1415.2, "end": 1415.76, "word": " عشرين", "probability": 0.927734375}, {"start": 1415.76, "end": 1417.48, "word": " اش", "probability": 0.4788818359375}, {"start": 1417.48, "end": 1417.9, "word": " لمّة", "probability": 0.6751302083333334}, {"start": 1417.9, "end": 1418.5, "word": " بتقول", "probability": 0.733154296875}, {"start": 1418.5, "end": 1419.04, "word": " every", "probability": 0.36181640625}, {"start": 1419.04, "end": 1422.08, "word": " ..", "probability": 0.37158203125}, {"start": 1422.08, "end": 1423.56, "word": " every", "probability": 0.6865234375}, {"start": 1423.56, "end": 1424.72, "word": " convergent", "probability": 0.914794921875}, {"start": 1424.72, "end": 1425.54, "word": " ..", "probability": 0.865234375}, {"start": 1425.54, "end": 1427.6, "word": " every", "probability": 0.7734375}, {"start": 1427.6, "end": 1428.6, "word": " convergent", "probability": 0.950927734375}, {"start": 1428.6, "end": 1429.5, "word": " sequence", "probability": 0.98486328125}, {"start": 1429.5, "end": 1432.42, "word": " of", "probability": 0.9287109375}, {"start": 1432.42, "end": 1432.78, "word": " real", "probability": 0.9697265625}, {"start": 1432.78, "end": 1433.46, "word": " numbers", "probability": 0.87646484375}, {"start": 1433.46, "end": 1438.12, "word": " is", "probability": 0.85791015625}, {"start": 1438.12, "end": 1438.72, "word": " Cauchy", "probability": 0.7654622395833334}, {"start": 1438.72, "end": 1440.76, "word": " كل", "probability": 0.8798828125}, {"start": 1440.76, "end": 1441.8, "word": " متتالية", "probability": 0.9642333984375}], "temperature": 1.0}, {"id": 54, "seek": 147200, "start": 1442.74, "end": 1472.0, "text": "متقاربة بتكون كشية وهي البرهان prove assume let x in contained in R be such that limit x in بساوي x", "tokens": [2304, 2655, 4587, 9640, 49401, 39894, 30544, 9122, 8592, 10632, 37037, 1829, 2423, 26890, 3224, 7649, 7081, 6552, 718, 2031, 294, 16212, 294, 497, 312, 1270, 300, 4948, 2031, 294, 4724, 3794, 995, 45865, 2031], "avg_logprob": -0.206163190305233, "compression_ratio": 1.1111111111111112, "no_speech_prob": 0.0, "words": [{"start": 1442.74, "end": 1443.7, "word": "متقاربة", "probability": 0.95771484375}, {"start": 1443.7, "end": 1444.18, "word": " بتكون", "probability": 0.83154296875}, {"start": 1444.18, "end": 1444.84, "word": " كشية", "probability": 0.84619140625}, {"start": 1444.84, "end": 1447.4, "word": " وهي", "probability": 0.76123046875}, {"start": 1447.4, "end": 1448.02, "word": " البرهان", "probability": 0.8287353515625}, {"start": 1448.02, "end": 1448.54, "word": " prove", "probability": 0.237060546875}, {"start": 1448.54, "end": 1454.18, "word": " assume", "probability": 0.73779296875}, {"start": 1454.18, "end": 1456.94, "word": " let", "probability": 0.85595703125}, {"start": 1456.94, "end": 1460.16, "word": " x", "probability": 0.7119140625}, {"start": 1460.16, "end": 1460.54, "word": " in", "probability": 0.46533203125}, {"start": 1460.54, "end": 1461.48, "word": " contained", "probability": 0.54638671875}, {"start": 1461.48, "end": 1461.82, "word": " in", "probability": 0.96533203125}, {"start": 1461.82, "end": 1462.32, "word": " R", "probability": 0.88671875}, {"start": 1462.32, "end": 1466.92, "word": " be", "probability": 0.79931640625}, {"start": 1466.92, "end": 1467.54, "word": " such", "probability": 0.96337890625}, {"start": 1467.54, "end": 1468.02, "word": " that", "probability": 0.95458984375}, {"start": 1468.02, "end": 1468.62, "word": " limit", "probability": 0.966796875}, {"start": 1468.62, "end": 1470.22, "word": " x", "probability": 0.966796875}, {"start": 1470.22, "end": 1470.64, "word": " in", "probability": 0.86328125}, {"start": 1470.64, "end": 1471.52, "word": " بساوي", "probability": 0.8880615234375}, {"start": 1471.52, "end": 1472.0, "word": " x", "probability": 0.7119140625}], "temperature": 1.0}, {"id": 55, "seek": 150609, "start": 1476.59, "end": 1506.09, "text": "يعني افرض ان في عندي sequence xn وconvergent ل x من ان اثبت ان ال sequence هذه كوشي او كوشية ف let epsilon لاثبات ان ال sequence كوشي لازم نبدأ بepsilon ونرد عليها بcapital N تعطين ال implication هذه صح؟ مش هيك التعريب يقول ان نبدأ let epsilon أكبر من الصفر be given", "tokens": 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"probability": 0.685546875}, {"start": 1477.47, "end": 1477.65, "word": " في", "probability": 0.66748046875}, {"start": 1477.65, "end": 1478.03, "word": " عندي", "probability": 0.87353515625}, {"start": 1478.03, "end": 1478.65, "word": " sequence", "probability": 0.71240234375}, {"start": 1478.65, "end": 1479.35, "word": " xn", "probability": 0.5126953125}, {"start": 1479.35, "end": 1481.03, "word": " وconvergent", "probability": 0.822509765625}, {"start": 1481.03, "end": 1481.25, "word": " ل", "probability": 0.86767578125}, {"start": 1481.25, "end": 1481.73, "word": " x", "probability": 0.376220703125}, {"start": 1481.73, "end": 1485.11, "word": " من", "probability": 0.49658203125}, {"start": 1485.11, "end": 1485.27, "word": " ان", "probability": 0.218017578125}, {"start": 1485.27, "end": 1485.61, "word": " اثبت", "probability": 0.8778076171875}, {"start": 1485.61, "end": 1485.87, "word": " ان", "probability": 0.90966796875}, {"start": 1485.87, "end": 1486.21, "word": " ال", 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"probability": 0.994140625}, {"start": 1684.09, "end": 1684.55, "word": " epsilon", "probability": 0.88232421875}, {"start": 1684.55, "end": 1685.69, "word": " إذا", "probability": 0.6304931640625}, {"start": 1685.69, "end": 1688.29, "word": " since", "probability": 0.7470703125}], "temperature": 1.0}, {"id": 62, "seek": 171837, "start": 1690.03, "end": 1718.37, "text": "إبسلون أكبر من السفر was arbitrary إن الكلام هذا صحيح أثبتنا هذا الكلام صحيح لكل إبسلون الإبسلون هذي was arbitrary عشوائية إن هيك بتكون حققنا الشرط تبقى cushy sequences وبالتالي we have we get that من التعريف", "tokens": [28814, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 390, 23211, 36145, 2423, 28820, 10943, 23758, 20328, 5016, 1829, 5016, 5551, 12984, 3555, 2655, 8315, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 33688, 3555, 3794, 1211, 11536, 8032, 8848, 1829, 390, 23211, 6225, 8592, 2407, 16373, 10632, 36145, 39896, 4117, 39894, 30544, 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{"start": 1714.61, "end": 1714.87, "word": " we", "probability": 0.61279296875}, {"start": 1714.87, "end": 1715.37, "word": " get", "probability": 0.94287109375}, {"start": 1715.37, "end": 1717.31, "word": " that", "probability": 0.896484375}, {"start": 1717.31, "end": 1717.61, "word": " من", "probability": 0.9921875}, {"start": 1717.61, "end": 1718.37, "word": " التعريف", "probability": 0.9947509765625}], "temperature": 1.0}, {"id": 63, "seek": 174101, "start": 1736.87, "end": 1741.01, "text": "العكس تبع النظرية هذه صحيح", "tokens": [6027, 3615, 4117, 3794, 6055, 3555, 3615, 28239, 19913, 2288, 10632, 29538, 20328, 5016, 1829, 5016], "avg_logprob": -0.38534007352941174, "compression_ratio": 0.8888888888888888, "no_speech_prob": 0.0, "words": [{"start": 1736.8700000000001, "end": 1738.19, "word": "العكس", "probability": 0.64263916015625}, {"start": 1738.19, "end": 1739.51, "word": " تبع", "probability": 0.683349609375}, {"start": 1739.51, "end": 1740.17, "word": " النظرية", "probability": 0.9066162109375}, {"start": 1740.17, "end": 1740.47, "word": " هذه", "probability": 0.250244140625}, {"start": 1740.47, "end": 1741.01, "word": " صحيح", "probability": 0.9595947265625}], "temperature": 1.0}, {"id": 64, "seek": 176455, "start": 1741.65, "end": 1764.55, "text": "هنشوفه بعد شوية لكن في كمان نظرية او لمّة بسيطة زي هذه هنثبتها الآن بتقول ان كل كوشي sequence is bounded واضح البرهان؟ في اي استفسار على البرهان؟ اعتقد البرهان سهل وبسيط", "tokens": [3224, 1863, 8592, 38688, 3224, 39182, 13412, 2407, 10632, 44381, 8978, 9122, 2304, 7649, 8717, 19913, 2288, 10632, 1975, 2407, 32767, 11703, 3660, 4724, 3794, 1829, 9566, 3660, 30767, 1829, 29538, 8032, 1863, 12984, 3555, 2655, 11296, 6024, 48506, 39894, 39648, 16472, 28242, 9122, 2407, 8592, 1829, 8310, 307, 37498, 4032, 46958, 5016, 2423, 26890, 3224, 7649, 22807, 8978, 1975, 1829, 44713, 36178, 9640, 15844, 2423, 26890, 3224, 7649, 22807, 1975, 34268, 28543, 2423, 26890, 3224, 7649, 8608, 3224, 1211, 46599, 3794, 1829, 9566], "avg_logprob": -0.1727941211532144, "compression_ratio": 1.6022099447513811, "no_speech_prob": 0.0, "words": [{"start": 1741.65, "end": 1742.53, "word": "هنشوفه", "probability": 0.90068359375}, {"start": 1742.53, "end": 1742.85, "word": " بعد", "probability": 0.89892578125}, {"start": 1742.85, "end": 1743.43, "word": " شوية", "probability": 0.9171549479166666}, {"start": 1743.43, "end": 1743.83, "word": " لكن", "probability": 0.5732421875}, {"start": 1743.83, "end": 1744.07, "word": " في", "probability": 0.8681640625}, {"start": 1744.07, "end": 1744.43, "word": " كمان", "probability": 0.90673828125}, {"start": 1744.43, "end": 1744.99, "word": " نظرية", "probability": 0.982177734375}, {"start": 1744.99, "end": 1745.29, "word": " او", "probability": 0.725341796875}, {"start": 1745.29, "end": 1745.77, "word": " لمّة", "probability": 0.6504720052083334}, {"start": 1745.77, "end": 1747.55, "word": " بسيطة", "probability": 0.98505859375}, {"start": 1747.55, "end": 1747.81, "word": " زي", "probability": 0.978759765625}, {"start": 1747.81, "end": 1748.31, "word": " هذه", "probability": 0.775390625}, {"start": 1748.31, "end": 1751.77, "word": " هنثبتها", "probability": 0.8720703125}, {"start": 1751.77, "end": 1752.29, "word": " الآن", "probability": 0.6968994140625}, {"start": 1752.29, "end": 1753.87, "word": " بتقول", "probability": 0.89697265625}, {"start": 1753.87, "end": 1754.23, "word": " ان", "probability": 0.75732421875}, {"start": 1754.23, "end": 1754.91, "word": " كل", "probability": 0.8974609375}, {"start": 1754.91, "end": 1756.35, "word": " كوشي", "probability": 0.750640869140625}, {"start": 1756.35, "end": 1756.85, "word": " sequence", "probability": 0.78466796875}, {"start": 1756.85, "end": 1757.09, "word": " is", "probability": 0.9482421875}, {"start": 1757.09, "end": 1757.59, "word": " bounded", "probability": 0.93994140625}, {"start": 1757.59, "end": 1760.45, "word": " واضح", "probability": 0.947265625}, {"start": 1760.45, "end": 1761.07, "word": " البرهان؟", "probability": 0.81103515625}, {"start": 1761.07, "end": 1761.17, "word": " في", "probability": 0.89599609375}, {"start": 1761.17, "end": 1761.35, "word": " اي", "probability": 0.819580078125}, {"start": 1761.35, "end": 1761.91, "word": " استفسار", "probability": 0.9772135416666666}, {"start": 1761.91, "end": 1762.05, "word": " على", "probability": 0.54638671875}, {"start": 1762.05, "end": 1762.69, "word": " البرهان؟", "probability": 0.93154296875}, {"start": 1762.69, "end": 1763.03, "word": " اعتقد", "probability": 0.9329427083333334}, {"start": 1763.03, "end": 1763.47, "word": " البرهان", "probability": 0.948974609375}, {"start": 1763.47, "end": 1763.97, "word": " سهل", "probability": 0.9664713541666666}, {"start": 1763.97, "end": 1764.55, "word": " وبسيط", "probability": 0.8739013671875}], "temperature": 1.0}, {"id": 65, "seek": 179432, "start": 1765.28, "end": 1794.32, "text": "هذا ال .. النوع من البرهان بنسميه epsilon بنسميه epsilon over two argument epsilon over two argument استخدامنا epsilon over two برهان باستخدام epsilon over two إذن نثبت اللمة التانية قبل ما نثبت عكس النظر اللي فاتت صحيح", "tokens": [3224, 15730, 2423, 4386, 28239, 45367, 9154, 2423, 26890, 3224, 7649, 44945, 38251, 1829, 3224, 17889, 44945, 38251, 1829, 3224, 17889, 670, 732, 6770, 17889, 670, 732, 6770, 44713, 9778, 3215, 10943, 8315, 17889, 670, 732, 4724, 2288, 3224, 7649, 4724, 995, 14851, 9778, 3215, 10943, 17889, 670, 732, 11933, 8848, 1863, 8717, 12984, 3555, 2655, 13672, 2304, 3660, 16712, 7649, 10632, 12174, 36150, 19446, 8717, 12984, 3555, 2655, 6225, 4117, 3794, 28239, 19913, 2288, 13672, 1829, 6156, 9307, 2655, 20328, 5016, 1829, 5016], "avg_logprob": -0.16351103151545804, "compression_ratio": 1.893491124260355, "no_speech_prob": 0.0, "words": [{"start": 1765.28, "end": 1765.82, "word": "هذا", "probability": 0.864013671875}, {"start": 1765.82, "end": 1765.96, "word": " ال", "probability": 0.11932373046875}, {"start": 1765.96, "end": 1765.96, "word": " ..", "probability": 0.5791015625}, {"start": 1765.96, "end": 1766.62, "word": " النوع", "probability": 0.9326171875}, {"start": 1766.62, "end": 1766.82, "word": " من", "probability": 0.98779296875}, {"start": 1766.82, "end": 1767.2, "word": " البرهان", "probability": 0.9287109375}, {"start": 1767.2, "end": 1768.02, "word": " بنسميه", "probability": 0.8914794921875}, {"start": 1768.02, "end": 1768.48, "word": " epsilon", "probability": 0.61083984375}, {"start": 1768.48, "end": 1770.46, "word": " بنسميه", "probability": 0.9468994140625}, {"start": 1770.46, "end": 1770.94, "word": " epsilon", "probability": 0.89990234375}, {"start": 1770.94, "end": 1771.58, "word": " over", "probability": 0.93115234375}, {"start": 1771.58, "end": 1772.04, "word": " two", "probability": 0.935546875}, {"start": 1772.04, "end": 1772.98, "word": " argument", "probability": 0.861328125}, {"start": 1772.98, "end": 1777.36, "word": " epsilon", "probability": 0.7001953125}, {"start": 1777.36, "end": 1777.82, "word": " over", "probability": 0.9423828125}, {"start": 1777.82, "end": 1778.18, "word": " two", "probability": 0.9580078125}, {"start": 1778.18, "end": 1778.92, "word": " argument", "probability": 0.92041015625}, {"start": 1778.92, "end": 1780.6, "word": " استخدامنا", "probability": 0.81865234375}, {"start": 1780.6, "end": 1780.92, "word": " epsilon", "probability": 0.93603515625}, {"start": 1780.92, "end": 1781.32, "word": " over", "probability": 0.9423828125}, {"start": 1781.32, "end": 1782.34, "word": " two", "probability": 0.9619140625}, {"start": 1782.34, "end": 1784.38, "word": " برهان", "probability": 0.944091796875}, {"start": 1784.38, "end": 1785.6, "word": " باستخدام", "probability": 0.989990234375}, {"start": 1785.6, "end": 1786.06, "word": " epsilon", "probability": 0.943359375}, {"start": 1786.06, "end": 1786.62, "word": " over", "probability": 0.94384765625}, {"start": 1786.62, "end": 1787.12, "word": " two", "probability": 0.955078125}, {"start": 1787.12, "end": 1789.38, "word": " إذن", "probability": 0.6326497395833334}, {"start": 1789.38, "end": 1790.32, "word": " نثبت", "probability": 0.97509765625}, {"start": 1790.32, "end": 1790.72, "word": " اللمة", "probability": 0.648193359375}, {"start": 1790.72, "end": 1791.14, "word": " التانية", "probability": 0.9847005208333334}, {"start": 1791.14, "end": 1791.42, "word": " قبل", "probability": 0.620361328125}, {"start": 1791.42, "end": 1791.56, "word": " ما", "probability": 0.947265625}, {"start": 1791.56, "end": 1791.94, "word": " نثبت", "probability": 0.994384765625}, {"start": 1791.94, "end": 1792.26, "word": " عكس", "probability": 0.99365234375}, {"start": 1792.26, "end": 1792.58, "word": " النظر", "probability": 0.6957194010416666}, {"start": 1792.58, "end": 1793.22, "word": " اللي", "probability": 0.91357421875}, {"start": 1793.22, "end": 1793.9, "word": " فاتت", "probability": 0.9724934895833334}, {"start": 1793.9, "end": 1794.32, "word": " صحيح", "probability": 0.932373046875}], "temperature": 1.0}, {"id": 66, "seek": 182914, "start": 1800.8, "end": 1829.14, "text": "لمّا 22 every Cauchy sequence in R is bounded والبرهان تبعها شبيه إلى حد كبير ببرهان أن كل convergence sequence is bounded", "tokens": [19528, 11703, 995, 5853, 633, 7544, 625, 88, 8310, 294, 497, 307, 37498, 16070, 26890, 3224, 7649, 6055, 3555, 3615, 11296, 13412, 21292, 3224, 30731, 11331, 3215, 9122, 3555, 13546, 4724, 26890, 3224, 7649, 14739, 28242, 32181, 8310, 307, 37498], "avg_logprob": -0.20445884727850194, "compression_ratio": 1.218045112781955, "no_speech_prob": 0.0, "words": [{"start": 1800.8, "end": 1801.24, "word": "لمّا", "probability": 0.4138590494791667}, {"start": 1801.24, "end": 1801.84, "word": " 22", "probability": 0.6279296875}, {"start": 1801.84, "end": 1804.02, "word": " every", "probability": 0.66259765625}, {"start": 1804.02, "end": 1806.82, "word": " Cauchy", "probability": 0.8896484375}, {"start": 1806.82, "end": 1810.18, "word": " sequence", "probability": 0.93896484375}, {"start": 1810.18, "end": 1810.56, "word": " in", "probability": 0.93017578125}, {"start": 1810.56, "end": 1811.04, "word": " R", "probability": 0.98681640625}, {"start": 1811.04, "end": 1813.6, "word": " is", "probability": 0.93115234375}, {"start": 1813.6, "end": 1814.14, "word": " bounded", "probability": 0.95361328125}, {"start": 1814.14, "end": 1820.6, "word": " والبرهان", "probability": 0.8106689453125}, {"start": 1820.6, "end": 1821.12, "word": " تبعها", "probability": 0.8929443359375}, {"start": 1821.12, "end": 1822.4, "word": " شبيه", "probability": 0.8230794270833334}, {"start": 1822.4, "end": 1822.84, "word": " إلى", "probability": 0.89013671875}, {"start": 1822.84, "end": 1823.24, "word": " حد", "probability": 0.992431640625}, {"start": 1823.24, "end": 1823.86, "word": " كبير", "probability": 0.9970703125}, {"start": 1823.86, "end": 1826.52, "word": " ببرهان", "probability": 0.965087890625}, {"start": 1826.52, "end": 1826.78, "word": " أن", "probability": 0.57958984375}, {"start": 1826.78, "end": 1827.16, "word": " كل", "probability": 0.81201171875}, {"start": 1827.16, "end": 1827.76, "word": " convergence", "probability": 0.86083984375}, {"start": 1827.76, "end": 1828.5, "word": " sequence", "probability": 0.97705078125}, {"start": 1828.5, "end": 1828.78, "word": " is", "probability": 0.9462890625}, {"start": 1828.78, "end": 1829.14, "word": " bounded", "probability": 0.94921875}], "temperature": 1.0}, {"id": 67, "seek": 185459, "start": 1830.11, "end": 1854.59, "text": "أثبتنا قبل هيك أن كل conversion sequence is bounded اليوم هأثبت أن كل Cauchy sequence is bounded والبرهان مشابه للبرهان السابق proof let xn contained in R be Cauchy be Cauchy sequence", "tokens": [10721, 12984, 3555, 2655, 8315, 12174, 36150, 39896, 4117, 14739, 28242, 14298, 8310, 307, 37498, 45595, 20498, 8032, 10721, 12984, 3555, 2655, 14739, 28242, 7544, 625, 88, 8310, 307, 37498, 16070, 26890, 3224, 7649, 37893, 16758, 3224, 24976, 26890, 3224, 7649, 21136, 16758, 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"probability": 0.80322265625}, {"start": 1834.03, "end": 1834.47, "word": " هأثبت", "probability": 0.76268310546875}, {"start": 1834.47, "end": 1834.59, "word": " أن", "probability": 0.74951171875}, {"start": 1834.59, "end": 1835.01, "word": " كل", "probability": 0.9541015625}, {"start": 1835.01, "end": 1835.39, "word": " Cauchy", "probability": 0.6856486002604166}, {"start": 1835.39, "end": 1835.87, "word": " sequence", "probability": 0.9580078125}, {"start": 1835.87, "end": 1836.21, "word": " is", "probability": 0.94580078125}, {"start": 1836.21, "end": 1836.67, "word": " bounded", "probability": 0.94091796875}, {"start": 1836.67, "end": 1837.85, "word": " والبرهان", "probability": 0.82513427734375}, {"start": 1837.85, "end": 1838.59, "word": " مشابه", "probability": 0.9651692708333334}, {"start": 1838.59, "end": 1839.45, "word": " للبرهان", "probability": 0.9324951171875}, {"start": 1839.45, "end": 1840.09, "word": " السابق", "probability": 0.9951171875}, {"start": 1840.09, "end": 1841.55, "word": " proof", "probability": 0.52099609375}, {"start": 1841.55, "end": 1844.95, "word": " let", "probability": 0.8837890625}, {"start": 1844.95, "end": 1845.95, "word": " xn", "probability": 0.3720703125}, {"start": 1845.95, "end": 1846.91, "word": " contained", "probability": 0.5693359375}, {"start": 1846.91, "end": 1847.21, "word": " in", "probability": 0.958984375}, {"start": 1847.21, "end": 1847.79, "word": " R", "probability": 0.943359375}, {"start": 1847.79, "end": 1849.53, "word": " be", "probability": 0.388671875}, {"start": 1849.53, "end": 1850.37, "word": " Cauchy", "probability": 0.908203125}, {"start": 1850.37, "end": 1852.97, "word": " be", "probability": 0.1815185546875}, {"start": 1852.97, "end": 1853.75, "word": " Cauchy", "probability": 0.9305013020833334}, {"start": 1853.75, "end": 1854.59, "word": " sequence", "probability": 0.9609375}], "temperature": 1.0}, {"id": 68, "seek": 188231, "start": 1857.01, "end": 1882.31, "text": "و خلّينا ناخد epsilon then من تعريف الكوشي sequence for epsilon بساوي واحد أكبر من السفر يوجد capital N يعتمد على ال epsilon اللي هو الواحد عدد طبيعي بحيث أنه ل", "tokens": [2407, 16490, 1211, 11703, 9957, 995, 8717, 47283, 3215, 17889, 550, 9154, 37279, 16572, 5172, 33251, 2407, 8592, 1829, 8310, 337, 17889, 4724, 3794, 995, 45865, 36764, 24401, 5551, 4117, 26890, 9154, 21136, 5172, 2288, 7251, 29245, 3215, 4238, 426, 7251, 34268, 2304, 3215, 15844, 2423, 17889, 13672, 1829, 31439, 2423, 14407, 24401, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 14739, 3224, 5296], "avg_logprob": -0.18187040222041748, "compression_ratio": 1.4137931034482758, "no_speech_prob": 0.0, "words": [{"start": 1857.01, "end": 1857.37, "word": "و", "probability": 0.9248046875}, {"start": 1857.37, "end": 1858.17, "word": " خلّينا", "probability": 0.77880859375}, {"start": 1858.17, "end": 1858.95, "word": " ناخد", "probability": 0.9739583333333334}, {"start": 1858.95, "end": 1860.01, "word": " epsilon", "probability": 0.308837890625}, {"start": 1860.01, "end": 1863.35, "word": " then", "probability": 0.35986328125}, {"start": 1863.35, "end": 1863.59, "word": " من", "probability": 0.98828125}, {"start": 1863.59, "end": 1864.03, "word": " تعريف", "probability": 0.9894205729166666}, {"start": 1864.03, "end": 1864.53, "word": " الكوشي", "probability": 0.78936767578125}, {"start": 1864.53, "end": 1865.09, "word": " sequence", "probability": 0.44482421875}, {"start": 1865.09, "end": 1867.85, "word": " for", "probability": 0.85400390625}, {"start": 1867.85, "end": 1868.47, "word": " epsilon", "probability": 0.86474609375}, {"start": 1868.47, "end": 1869.33, "word": " بساوي", "probability": 0.8668212890625}, {"start": 1869.33, "end": 1869.99, "word": " واحد", "probability": 0.941650390625}, {"start": 1869.99, "end": 1870.65, "word": " أكبر", "probability": 0.87939453125}, {"start": 1870.65, "end": 1870.87, "word": " من", "probability": 0.99365234375}, {"start": 1870.87, "end": 1871.41, "word": " السفر", "probability": 0.788330078125}, {"start": 1871.41, "end": 1873.71, "word": " يوجد", "probability": 0.9806315104166666}, {"start": 1873.71, "end": 1874.21, "word": " capital", "probability": 0.7392578125}, {"start": 1874.21, "end": 1874.69, "word": " N", "probability": 0.7666015625}, {"start": 1874.69, "end": 1875.43, "word": " يعتمد", "probability": 0.955078125}, {"start": 1875.43, "end": 1875.61, "word": " على", "probability": 0.8740234375}, {"start": 1875.61, "end": 1875.77, "word": " ال", "probability": 0.95849609375}, {"start": 1875.77, "end": 1876.13, "word": " epsilon", "probability": 0.6357421875}, {"start": 1876.13, "end": 1876.43, "word": " اللي", "probability": 0.9189453125}, {"start": 1876.43, "end": 1876.71, "word": " هو", "probability": 0.99560546875}, {"start": 1876.71, "end": 1877.35, "word": " الواحد", "probability": 0.97509765625}, {"start": 1877.35, "end": 1877.75, "word": " عدد", "probability": 0.93212890625}, {"start": 1877.75, "end": 1878.61, "word": " طبيعي", "probability": 0.97509765625}, {"start": 1878.61, "end": 1880.43, "word": " بحيث", "probability": 0.9652099609375}, {"start": 1880.43, "end": 1881.33, "word": " أنه", "probability": 0.7322998046875}, {"start": 1881.33, "end": 1882.31, "word": " ل", "probability": 0.4912109375}], "temperature": 1.0}, {"id": 69, "seek": 190919, "start": 1883.97, "end": 1909.19, "text": "absolute xn minus xm أصغر من إبسلون اللي هو واحد وهذا الكلام صحيح لكل N و M أكبر من أو ساوي capital N هذا من تعريف الكوشي sequence تمام؟ طيب hence", "tokens": [17243, 401, 1169, 2031, 77, 3175, 2031, 76, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536, 13672, 1829, 31439, 36764, 24401, 37037, 15730, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 5296, 28820, 426, 4032, 376, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 23758, 9154, 37279, 16572, 5172, 33251, 2407, 8592, 1829, 8310, 46811, 10943, 22807, 23032, 1829, 3555, 16678], "avg_logprob": -0.20997596153846154, 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convergence sequence is bounded فهذا بنكتب على صورة x in minus x capital N زائد x capital N إذا أنا طرحت x capital N و رجعتها", "tokens": [10721, 8315, 18871, 16254, 45164, 25961, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 23758, 4724, 1829, 4587, 16254, 16472, 8236, 2031, 294, 20666, 19913, 3555, 9566, 30767, 1829, 19446, 6225, 42213, 8315, 8978, 4724, 2288, 3224, 7649, 28242, 32181, 8310, 307, 37498, 6156, 3224, 15730, 44945, 4117, 2655, 3555, 15844, 20328, 13063, 3660, 2031, 294, 3175, 2031, 4238, 426, 30767, 16373, 3215, 2031, 4238, 426, 11933, 15730, 41850, 23032, 2288, 33753, 2031, 4238, 426, 4032, 12602, 7435, 34268, 11296], "avg_logprob": -0.18350074014493398, "compression_ratio": 1.5406698564593302, "no_speech_prob": 0.0, "words": [{"start": 1969.51, "end": 1969.85, "word": "أنا", "probability": 0.782958984375}, {"start": 1969.85, "end": 1970.19, "word": " عندي", "probability": 0.86083984375}, {"start": 1970.19, "end": 1970.43, "word": " لو", 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2024.18, "text": "الان هذا اصغر من او ساوي absolute x n سالب x capital n زاد absolute x capital n وهذا هذا هو هذا المقدار لكل n اكبر من او ساوي capital n اصغر من واحد اذا هذا اصغر من واحد زاد absolute x capital n تمام الان now", "tokens": [6027, 7649, 23758, 1975, 9381, 17082, 2288, 9154, 1975, 2407, 8608, 995, 45865, 8236, 2031, 297, 8608, 6027, 3555, 2031, 4238, 297, 30767, 18513, 8236, 2031, 4238, 297, 37037, 15730, 23758, 31439, 23758, 9673, 28543, 9640, 5296, 28820, 297, 1975, 4117, 26890, 9154, 1975, 2407, 8608, 995, 45865, 4238, 297, 1975, 9381, 17082, 2288, 9154, 36764, 24401, 1975, 15730, 23758, 1975, 9381, 17082, 2288, 9154, 36764, 24401, 30767, 18513, 8236, 2031, 4238, 297, 46811, 10943, 2423, 7649, 586], "avg_logprob": -0.17790743369090406, "compression_ratio": 2.019867549668874, "no_speech_prob": 0.0, "words": [{"start": 1996.28, "end": 1996.92, "word": "الان", "probability": 0.81005859375}, {"start": 1996.92, "end": 1997.36, "word": " هذا", "probability": 0.939453125}, 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واحد", "probability": 0.99267578125}, {"start": 2016.48, "end": 2016.86, "word": " زاد", "probability": 0.951416015625}, {"start": 2016.86, "end": 2017.34, "word": " absolute", "probability": 0.96533203125}, {"start": 2017.34, "end": 2017.96, "word": " x", "probability": 0.9931640625}, {"start": 2017.96, "end": 2019.04, "word": " capital", "probability": 0.8544921875}, {"start": 2019.04, "end": 2019.44, "word": " n", "probability": 0.88671875}, {"start": 2019.44, "end": 2020.1, "word": " تمام", "probability": 0.81787109375}, {"start": 2020.1, "end": 2023.54, "word": " الان", "probability": 0.784423828125}, {"start": 2023.54, "end": 2024.18, "word": " now", "probability": 0.76953125}], "temperature": 1.0}, {"id": 75, "seek": 205158, "start": 2026.08, "end": 2051.58, "text": "let خلّينا ناخد m .. نعرف عدد m على أنه ال supremum أو ال maximum للأعداد الحقيقية غير السالبة اللي هي absolute x1 absolute x2 إلى absolute x رقم capital M minus 1", "tokens": [2631, 16490, 1211, 11703, 9957, 995, 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"probability": 0.66650390625}, {"start": 2050.4, "end": 2051.0, "word": " minus", "probability": 0.4892578125}, {"start": 2051.0, "end": 2051.58, "word": " 1", "probability": 0.56591796875}], "temperature": 1.0}, {"id": 76, "seek": 207934, "start": 2053.4, "end": 2079.34, "text": "واخر حد absolute واحد زائد واحد زائد absolute x رقم capital N you can verify", "tokens": [2407, 47283, 2288, 11331, 3215, 8236, 36764, 24401, 30767, 16373, 3215, 36764, 24401, 30767, 16373, 3215, 8236, 2031, 12602, 4587, 2304, 4238, 426, 291, 393, 16888], "avg_logprob": -0.19401041777045638, "compression_ratio": 1.2625, "no_speech_prob": 0.0, "words": [{"start": 2053.4, "end": 2054.1, "word": "واخر", "probability": 0.7068684895833334}, {"start": 2054.1, "end": 2054.54, "word": " حد", "probability": 0.992431640625}, {"start": 2054.54, "end": 2056.0, "word": " absolute", "probability": 0.583984375}, {"start": 2056.0, "end": 2060.42, "word": " واحد", "probability": 0.8359375}, {"start": 2060.42, "end": 2061.1, "word": " زائد", "probability": 0.9295247395833334}, {"start": 2061.1, "end": 2062.54, "word": " واحد", "probability": 0.73486328125}, {"start": 2062.54, "end": 2063.28, "word": " زائد", "probability": 0.9832356770833334}, {"start": 2063.28, "end": 2064.74, "word": " absolute", "probability": 0.880859375}, {"start": 2064.74, "end": 2065.46, "word": " x", "probability": 0.6689453125}, {"start": 2065.46, "end": 2066.04, "word": " رقم", "probability": 0.89306640625}, {"start": 2066.04, "end": 2066.54, "word": " capital", "probability": 0.85009765625}, {"start": 2066.54, "end": 2066.98, "word": " N", "probability": 0.546875}, {"start": 2066.98, "end": 2078.18, "word": " you", "probability": 0.85595703125}, {"start": 2078.18, "end": 2078.76, "word": " can", "probability": 0.9697265625}, {"start": 2078.76, "end": 2079.34, "word": " verify", "probability": 0.96923828125}], "temperature": 1.0}, {"id": 77, "seek": 210059, "start": 2082.59, "end": 2100.59, "text": "بإمكانكم التحقق إنه absolute xn أصغر طبعاً العدد m هذا موجب بالتأكيد هذا موجب لأن العداد هذه كلها موجبة", "tokens": [3555, 28814, 2304, 41361, 24793, 16712, 5016, 4587, 4587, 36145, 3224, 8236, 2031, 77, 5551, 9381, 17082, 2288, 23032, 3555, 3615, 995, 14111, 18863, 3215, 3215, 275, 23758, 3714, 29245, 3555, 20666, 2655, 10721, 4117, 25708, 23758, 3714, 29245, 3555, 5296, 33456, 18863, 3215, 18513, 29538, 28242, 11296, 3714, 29245, 49401], "avg_logprob": -0.1906550472172407, "compression_ratio": 1.4274193548387097, "no_speech_prob": 0.0, "words": [{"start": 2082.59, "end": 2083.49, "word": "بإمكانكم", "probability": 0.79306640625}, {"start": 2083.49, "end": 2084.31, "word": " التحقق", "probability": 0.966552734375}, {"start": 2084.31, "end": 2085.03, "word": " إنه", "probability": 0.4490966796875}, {"start": 2085.03, "end": 2087.13, "word": " absolute", "probability": 0.75634765625}, {"start": 2087.13, "end": 2088.63, "word": " xn", "probability": 0.5589599609375}, {"start": 2088.63, "end": 2090.29, "word": " أصغر", "probability": 0.910888671875}, {"start": 2090.29, "end": 2090.63, "word": " طبعاً", "probability": 0.801025390625}, {"start": 2090.63, "end": 2091.07, "word": " العدد", "probability": 0.9558919270833334}, {"start": 2091.07, "end": 2091.35, "word": " m", "probability": 0.42431640625}, {"start": 2091.35, "end": 2091.67, "word": " هذا", "probability": 0.70263671875}, {"start": 2091.67, "end": 2092.11, "word": " موجب", "probability": 0.9903971354166666}, {"start": 2092.11, "end": 2092.77, "word": " بالتأكيد", "probability": 0.89560546875}, {"start": 2092.77, "end": 2092.97, "word": " هذا", "probability": 0.7890625}, {"start": 2092.97, "end": 2093.51, "word": " موجب", "probability": 0.9943033854166666}, {"start": 2093.51, "end": 2098.21, "word": " لأن", "probability": 0.85498046875}, {"start": 2098.21, "end": 2098.75, "word": " العداد", "probability": 0.9182942708333334}, {"start": 2098.75, "end": 2098.99, "word": " هذه", "probability": 0.7626953125}, {"start": 2098.99, "end": 2099.63, "word": " كلها", "probability": 0.979736328125}, {"start": 2099.63, "end": 2100.59, "word": " موجبة", "probability": 0.95361328125}], "temperature": 1.0}, {"id": 78, "seek": 211746, "start": 2102.2, "end": 2117.46, "text": "و بعدين absolute xn ممكن تحقق أنه أخدت أي عنصر في ال sequence xn و أخدت القيمة المطلقة إليه فهذا هيطلع أصغر من أو ساوي m لكل n في n", "tokens": [2407, 39182, 9957, 8236, 2031, 77, 3714, 43020, 6055, 5016, 4587, 4587, 14739, 3224, 5551, 9778, 3215, 2655, 36632, 18871, 9381, 2288, 8978, 2423, 8310, 2031, 77, 4032, 5551, 9778, 3215, 2655, 25062, 32640, 3660, 9673, 9566, 1211, 28671, 11933, 20292, 3224, 6156, 3224, 15730, 39896, 9566, 1211, 3615, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 275, 5296, 28820, 297, 8978, 297], "avg_logprob": -0.26490384615384616, "compression_ratio": 1.3725490196078431, "no_speech_prob": 0.0, "words": [{"start": 2102.2, "end": 2102.42, "word": "و", "probability": 0.340087890625}, {"start": 2102.42, "end": 2103.04, 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"probability": 0.86328125}, {"start": 2189.48, "end": 2189.62, "word": " أو", "probability": 0.73974609375}, {"start": 2189.62, "end": 2190.08, "word": " ساوي", "probability": 0.8328450520833334}, {"start": 2190.08, "end": 2191.24, "word": " okay", "probability": 0.429931640625}, {"start": 2191.24, "end": 2192.28, "word": " تمام؟", "probability": 0.686767578125}, {"start": 2192.28, "end": 2194.68, "word": " إذن", "probability": 0.5272623697916666}, {"start": 2194.68, "end": 2194.92, "word": " هذا", "probability": 0.841796875}, {"start": 2194.92, "end": 2195.52, "word": " بكمل", "probability": 0.8680013020833334}, {"start": 2195.52, "end": 2197.04, "word": " البرهجة", "probability": 0.745458984375}, {"start": 2197.04, "end": 2198.54, "word": " المرة", "probability": 0.93603515625}, {"start": 2198.54, "end": 2198.84, "word": " الجاية", "probability": 0.8902994791666666}, {"start": 2198.84, "end": 2199.2, "word": " طبعا", "probability": 0.9869384765625}, {"start": 2199.2, "end": 2199.9, "word": " هنشوف", "probability": 0.8807373046875}, {"start": 2199.9, "end": 2201.48, "word": " أنه", "probability": 0.574462890625}, {"start": 2201.48, "end": 2202.22, "word": " عكس", "probability": 0.99267578125}, {"start": 2202.22, "end": 2202.64, "word": " النمة", "probability": 0.577880859375}, {"start": 2202.64, "end": 2202.98, "word": " هذه", "probability": 0.78173828125}, {"start": 2202.98, "end": 2203.52, "word": " صحيح", "probability": 0.991943359375}], "temperature": 1.0}, {"id": 82, "seek": 222594, "start": 2204.19, "end": 2225.95, "text": "وها دي بنسميها Cauchy criterion أو معيار Cauchy ومعيار Cauchy بيقول إن لو كانت ال sequence Cauchy بتكون أيضا convergent وبالتالي بنحصل على نظرية Cauchy أو معيار Cauchy Cauchy criterion وهو إن ال sequence is convergent if and only if it is Cauchy", "tokens": [2407, 11296, 11778, 1829, 44945, 38251, 1829, 11296, 7544, 625, 88, 46691, 34051, 20449, 1829, 9640, 7544, 625, 88, 4032, 2304, 3615, 1829, 9640, 7544, 625, 88, 4724, 1829, 39648, 36145, 45164, 25961, 2655, 2423, 8310, 7544, 625, 88, 39894, 30544, 36632, 11242, 995, 9652, 6930, 46599, 6027, 2655, 6027, 1829, 44945, 5016, 36520, 15844, 8717, 19913, 2288, 10632, 7544, 625, 88, 34051, 20449, 1829, 9640, 7544, 625, 88, 7544, 625, 88, 46691, 37037, 2407, 36145, 2423, 8310, 307, 9652, 6930, 498, 293, 787, 498, 309, 307, 7544, 625, 88], "avg_logprob": -0.16595123397124992, "compression_ratio": 1.7593582887700534, "no_speech_prob": 0.0, "words": [{"start": 2204.19, "end": 2204.61, "word": "وها", "probability": 0.42633056640625}, {"start": 2204.61, "end": 2204.75, "word": " دي", "probability": 0.702392578125}, {"start": 2204.75, "end": 2205.29, "word": " بنسميها", "probability": 0.8704833984375}, {"start": 2205.29, "end": 2205.73, "word": " Cauchy", "probability": 0.7703450520833334}, {"start": 2205.73, "end": 2206.25, "word": " criterion", "probability": 0.8408203125}, {"start": 2206.25, "end": 2206.61, "word": " أو", "probability": 0.7373046875}, {"start": 2206.61, "end": 2207.07, "word": " معيار", "probability": 0.9396158854166666}, {"start": 2207.07, "end": 2207.73, "word": " Cauchy", "probability": 0.7793782552083334}, {"start": 2207.73, "end": 2209.65, "word": " ومعيار", "probability": 0.8931640625}, {"start": 2209.65, "end": 2209.97, "word": " Cauchy", "probability": 0.9440104166666666}, {"start": 2209.97, "end": 2210.29, "word": " بيقول", "probability": 0.9755859375}, {"start": 2210.29, "end": 2210.51, "word": " إن", "probability": 0.37646484375}, {"start": 2210.51, "end": 2210.73, "word": " لو", "probability": 0.92919921875}, {"start": 2210.73, "end": 2211.13, "word": " كانت", "probability": 0.989990234375}, {"start": 2211.13, "end": 2211.25, "word": " ال", "probability": 0.94482421875}, {"start": 2211.25, "end": 2211.57, "word": " sequence", "probability": 0.93408203125}, {"start": 2211.57, "end": 2212.29, "word": " Cauchy", "probability": 0.9278971354166666}, {"start": 2212.29, "end": 2213.33, "word": " بتكون", "probability": 0.91455078125}, {"start": 2213.33, "end": 2213.75, "word": " أيضا", "probability": 0.89501953125}, {"start": 2213.75, "end": 2214.51, "word": " convergent", "probability": 0.6566162109375}, {"start": 2214.51, "end": 2215.37, "word": " وبالتالي", "probability": 0.919140625}, {"start": 2215.37, "end": 2216.13, "word": " بنحصل", "probability": 0.8509114583333334}, {"start": 2216.13, "end": 2216.31, "word": " على", "probability": 0.95263671875}, {"start": 2216.31, "end": 2216.83, "word": " نظرية", "probability": 0.96484375}, {"start": 2216.83, "end": 2217.37, "word": " Cauchy", "probability": 0.9298502604166666}, {"start": 2217.37, "end": 2217.61, "word": " أو", "probability": 0.634765625}, {"start": 2217.61, "end": 2218.11, "word": " معيار", "probability": 0.9471028645833334}, {"start": 2218.11, "end": 2218.43, "word": " Cauchy", "probability": 0.9443359375}, {"start": 2218.43, "end": 2218.79, "word": " Cauchy", "probability": 0.7935384114583334}, {"start": 2218.79, "end": 2219.35, "word": " criterion", "probability": 0.97412109375}, {"start": 2219.35, "end": 2220.35, "word": " وهو", "probability": 0.785888671875}, {"start": 2220.35, "end": 2220.59, "word": " إن", "probability": 0.7744140625}, {"start": 2220.59, "end": 2221.09, "word": " ال", "probability": 0.8232421875}, {"start": 2221.09, "end": 2222.29, "word": " sequence", "probability": 0.98388671875}, {"start": 2222.29, "end": 2223.21, "word": " is", "probability": 0.92041015625}, {"start": 2223.21, "end": 2224.01, "word": " convergent", "probability": 0.964111328125}, {"start": 2224.01, "end": 2224.37, "word": " if", "probability": 0.96142578125}, {"start": 2224.37, "end": 2224.61, "word": " and", "probability": 0.93994140625}, {"start": 2224.61, "end": 2224.85, "word": " only", "probability": 0.91650390625}, {"start": 2224.85, "end": 2225.13, "word": " if", "probability": 0.9716796875}, {"start": 2225.13, "end": 2225.35, "word": " it", "probability": 0.9716796875}, {"start": 2225.35, "end": 2225.55, "word": " is", "probability": 0.943359375}, {"start": 2225.55, "end": 2225.95, "word": " Cauchy", "probability": 0.9475911458333334}], "temperature": 1.0}, {"id": 83, "seek": 225119, "start": 2227.63, "end": 2251.19, "text": "إذا هذا الجزء التالي من النظرية أو عكس اللمحة دي هنشوفه المرة الجاية وناخد أمثلة كيف نثبت أن ال sequence معينة is Cauchy فهذا موجود في section تلاتة خمسة في الكتاب حاولوا تقرؤوا باقى ال section و تحضروه للمرة الجاية تمام؟ إذا انتهت هيك المحاضرة نشوفكم ان شاء الله يوم اتنين", "tokens": [28814, 15730, 23758, 25724, 11622, 38207, 16712, 6027, 1829, 9154, 28239, 19913, 2288, 10632, 34051, 6225, 4117, 3794, 13672, 2304, 5016, 3660, 11778, 1829, 8032, 1863, 8592, 38688, 3224, 9673, 25720, 25724, 995, 10632, 4032, 1863, 47283, 3215, 5551, 2304, 12984, 37977, 9122, 33911, 8717, 12984, 3555, 2655, 14739, 2423, 8310, 20449, 9957, 3660, 307, 7544, 625, 88, 6156, 3224, 15730, 3714, 29245, 23328, 8978, 3541, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 8978, 33251, 2655, 16758, 11331, 995, 12610, 14407, 6055, 4587, 2288, 33604, 14407, 4724, 995, 4587, 7578, 2423, 3541, 4032, 6055, 5016, 11242, 32887, 3224, 5296, 19528, 25720, 25724, 995, 10632, 46811, 10943, 22807, 11933, 15730, 16472, 47395, 2655, 39896, 4117, 9673, 5016, 46958, 25720, 8717, 8592, 38688, 24793, 16472, 13412, 16606, 21984, 7251, 20498, 1975, 2655, 1863, 9957], "avg_logprob": -0.15753265470266342, "compression_ratio": 1.7195571955719557, "no_speech_prob": 0.0, "words": [{"start": 2227.63, "end": 2227.93, "word": "إذا", "probability": 0.6072998046875}, {"start": 2227.93, "end": 2228.19, "word": " هذا", "probability": 0.468505859375}, {"start": 2228.19, "end": 2229.13, "word": " الجزء", "probability": 0.93017578125}, {"start": 2229.13, "end": 2229.55, "word": " التالي", "probability": 0.92041015625}, {"start": 2229.55, "end": 2229.69, "word": " من", "probability": 0.97998046875}, {"start": 2229.69, "end": 2230.19, "word": " النظرية", "probability": 0.9583740234375}, {"start": 2230.19, "end": 2230.41, "word": " أو", "probability": 0.7900390625}, {"start": 2230.41, "end": 2230.79, "word": " عكس", "probability": 0.9923502604166666}, {"start": 2230.79, "end": 2231.39, "word": " اللمحة", "probability": 0.6533203125}, {"start": 2231.39, "end": 2231.89, "word": " دي", "probability": 0.6953125}, {"start": 2231.89, "end": 2232.79, "word": " هنشوفه", "probability": 0.85498046875}, {"start": 2232.79, "end": 2233.09, "word": " المرة", "probability": 0.83349609375}, {"start": 2233.09, "end": 2233.51, "word": " الجاية", "probability": 0.8968098958333334}, {"start": 2233.51, "end": 2234.05, "word": " وناخد", "probability": 0.87548828125}, {"start": 2234.05, "end": 2234.59, "word": " أمثلة", "probability": 0.9736328125}, {"start": 2234.59, "end": 2235.69, "word": " كيف", "probability": 0.9521484375}, {"start": 2235.69, "end": 2236.21, "word": " نثبت", "probability": 0.988037109375}, {"start": 2236.21, "end": 2236.37, "word": " أن", "probability": 0.5830078125}, {"start": 2236.37, "end": 2236.43, "word": " ال", "probability": 0.5888671875}, {"start": 2236.43, "end": 2236.75, "word": " sequence", "probability": 0.6962890625}, {"start": 2236.75, "end": 2237.53, "word": " معينة", "probability": 0.9832356770833334}, {"start": 2237.53, "end": 2237.79, "word": " is", "probability": 0.7060546875}, {"start": 2237.79, "end": 2238.71, "word": " Cauchy", "probability": 0.8107096354166666}, {"start": 2238.71, "end": 2240.35, "word": " فهذا", "probability": 0.9446614583333334}, {"start": 2240.35, "end": 2240.73, "word": " موجود", "probability": 0.9783528645833334}, {"start": 2240.73, "end": 2240.85, "word": " في", "probability": 0.94921875}, {"start": 2240.85, "end": 2241.11, "word": " section", "probability": 0.6513671875}, {"start": 2241.11, "end": 2241.55, "word": " تلاتة", "probability": 0.8714599609375}, {"start": 2241.55, "end": 2241.91, "word": " خمسة", "probability": 0.9029541015625}, {"start": 2241.91, "end": 2242.01, "word": " في", "probability": 0.88232421875}, {"start": 2242.01, "end": 2242.39, "word": " الكتاب", "probability": 0.9641927083333334}, {"start": 2242.39, "end": 2242.83, "word": " حاولوا", "probability": 0.9844970703125}, {"start": 2242.83, "end": 2243.69, "word": " تقرؤوا", "probability": 0.86240234375}, {"start": 2243.69, "end": 2244.41, "word": " باقى", "probability": 0.8858642578125}, {"start": 2244.41, "end": 2244.51, "word": " ال", "probability": 0.93408203125}, {"start": 2244.51, "end": 2244.89, "word": " section", "probability": 0.86328125}, {"start": 2244.89, "end": 2245.17, "word": " و", "probability": 0.642578125}, {"start": 2245.17, "end": 2245.69, "word": " تحضروه", "probability": 0.7701171875}, {"start": 2245.69, "end": 2246.07, "word": " للمرة", "probability": 0.9324544270833334}, {"start": 2246.07, "end": 2246.51, "word": " الجاية", "probability": 0.9524739583333334}, {"start": 2246.51, "end": 2247.93, "word": " تمام؟", "probability": 0.74169921875}, {"start": 2247.93, "end": 2248.17, "word": " إذا", "probability": 0.8623046875}, {"start": 2248.17, "end": 2248.69, "word": " انتهت", "probability": 0.828125}, {"start": 2248.69, "end": 2248.91, "word": " هيك", "probability": 0.96728515625}, {"start": 2248.91, "end": 2249.47, "word": " المحاضرة", "probability": 0.9910888671875}, {"start": 2249.47, "end": 2249.95, "word": " نشوفكم", "probability": 0.866455078125}, {"start": 2249.95, "end": 2250.11, "word": " ان", "probability": 0.5009765625}, {"start": 2250.11, "end": 2250.37, "word": " شاء", "probability": 0.97119140625}, {"start": 2250.37, "end": 2250.39, "word": " الله", "probability": 0.9658203125}, {"start": 2250.39, "end": 2250.73, "word": " يوم", "probability": 0.9912109375}, {"start": 2250.73, "end": 2251.19, "word": " اتنين", "probability": 0.8245849609375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2251.964, "duration_after_vad": 2028.139374999992}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0bd846eb2bd41368662d9da738b93b13084f08fc
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw.srt
@@ -0,0 +1,1763 @@
+1
+00:00:21,940 --> 00:00:28,180
+إذا في ال section هذا هن إن شاء الله هنتعلم ال
+
+2
+00:00:28,180 --> 00:00:34,860
+limited theorems و بعض النظريات الخاصة بالنهايات 
+
+3
+00:00:34,860 --> 00:00:43,890
+فعشان نشوف النظريات هذه ناخد الأول تعريف sequence
+
+4
+00:00:43,890 --> 00:00:49,150
+of real numbers Xn نسميها bounded إذا قدرنا نلاقي
+
+5
+00:00:49,150 --> 00:00:56,110
+عدد موجب بـ M بحيث إنه ال absolute value لكل عناصر
+
+6
+00:00:56,110 --> 00:01:00,510
+ال sequence أصغر من أو يساوي M هذا بكافئ
+
+7
+00:01:04,010 --> 00:01:10,550
+إن عناصر الـ sequence Xn تنتمي للفترة المغلقة
+
+8
+00:01:10,550 --> 00:01:19,030
+من سالب M إلى M لكل N عدد طبيعي إذن الـ sequence
+
+9
+00:01:19,030 --> 00:01:25,310
+بتكون bounded إذا قدرت أحصرها في closed interval
+
+10
+00:01:25,310 --> 00:01:29,730
+from سالب M إلى M حيث M عدد موجب
+
+11
+00:01:32,190 --> 00:01:37,030
+النظرية هذه بتقول إن أي convergence sequence of
+
+12
+00:01:37,030 --> 00:01:41,710
+real numbers is bounded كل convergence sequence
+
+13
+00:01:41,710 --> 00:01:47,050
+بتكون ضروري أن تكون bounded ولبرهان ذلك هي افرض 
+
+14
+00:01:47,050 --> 00:01:53,910
+أن sequence x in convergent to some x في R دعونا
+
+15
+00:01:53,910 --> 00:02:00,590
+ناخد y بساوي واحد هي هاد عدد موجب given من نظرية 2
+
+16
+00:02:00,590 --> 00:02:09,130
+.2 اللي هو تعريف epsilon capital N لـ limit بقول
+
+17
+00:02:09,130 --> 00:02:13,390
+ال sequence هذه ال limit تبعتها X إذا كان لأي
+
+18
+00:02:13,390 --> 00:02:19,010
+epsilon زي هذه بقدر ألاقي عدد طبيعي يعتمد على
+
+19
+00:02:19,010 --> 00:02:23,410
+epsilon اللي هو واحد بحيث لكل N أكبر من أو يساوي
+
+20
+00:02:23,410 --> 00:02:28,570
+capital N المسافة بين XN و X أصغر من ال epsilon
+
+21
+00:02:28,570 --> 00:02:30,570
+اللي هي مخدينها واحد
+
+22
+00:02:34,370 --> 00:02:42,670
+الآن absolute x in هذا عبارة عن ممكن ابدله هذا
+
+23
+00:02:42,670 --> 00:02:51,830
+بساوي absolute x in سالب x زائد x باخد
+
+24
+00:02:51,830 --> 00:02:58,840
+اتنين هدول مع بعض يعني اترحت x و رجعت by triangle
+
+25
+00:02:58,840 --> 00:03:03,520
+inequality less
+
+26
+00:03:03,520 --> 00:03:10,200
+than or equal absolute الحد الأول زائد absolute
+
+27
+00:03:10,200 --> 00:03:16,540
+الحد الثاني ومن هنا من هنا لكل n أكبر من أو يساوي
+
+28
+00:03:16,540 --> 00:03:20,680
+capital N المقدار هذا أو ال absolute value هذه
+
+29
+00:03:20,680 --> 00:03:26,790
+أصغر من واحد إذن absolute لكل n أكبر من أو يساوي
+
+30
+00:03:26,790 --> 00:03:36,730
+capital N أصغر من واحد زائد absolute X تمام؟
+
+31
+00:03:36,730 --> 00:03:40,930
+الآن لو أخدت عرفت capital M على أن ها 
+
+32
+00:03:46,510 --> 00:03:51,250
+لو عرفت capital M و capital N على أن ها ال supremum
+
+33
+00:03:51,250 --> 00:03:55,890
+أو الأكبر بين الأعداد اللي هي absolute x1 و
+
+34
+00:03:55,890 --> 00:04:01,530
+absolute x2 و هكذا إلى absolute x رقم capital N
+
+35
+00:04:01,530 --> 00:04:07,170
+سالب واحد والعدد الأخير absolute x زائد واحد
+
+36
+00:04:10,460 --> 00:04:16,660
+ففي الحالة هذه ممكن نثبت أن absolute xn أصغر من أو
+
+37
+00:04:16,660 --> 00:04:27,980
+يساوي عدد الموجب هذا لكل n لكل العداد الطبيعي هذا
+
+38
+00:04:27,980 --> 00:04:32,860
+بيجي من أربعة من المتباينة أربعة اللي هي هنا ومن
+
+39
+00:04:32,860 --> 00:04:37,680
+التعريف capital M يعني لاحظوا أنتم أن ال sequence
+
+40
+00:04:37,680 --> 00:04:50,680
+xn حدودها X1 X2 إلى X رقم capital N سالب واحد X رقم
+
+41
+00:04:50,680 --> 00:05:00,060
+capital N X رقم capital N زائد واحد و هكذا الآن 
+
+42
+00:05:00,060 --> 00:05:06,960
+لو ال N هذه كانت واحدة من الأعداد هدول المؤشر N
+
+43
+00:05:06,960 --> 00:05:14,380
+واحد من المؤشرات واحد اتنين إلى capital N سالب واحد
+
+44
+00:05:14,380 --> 00:05:18,140
+ف
+
+45
+00:05:18,140 --> 00:05:23,860
+absolute ال XN هذا هيكون
+
+46
+00:05:23,860 --> 00:05:26,980
+إما absolute X واحد أو absolute X اتنين أو
+
+47
+00:05:26,980 --> 00:05:32,740
+absolute XN وكل واحد من هدول كل واحد من هذول أصغر
+
+48
+00:05:32,740 --> 00:05:37,020
+من أو يساوي ال supreme لهم كلهم وبالتالي هذا أكيد
+
+49
+00:05:37,020 --> 00:05:44,500
+أصغر من أو يساوي .. أصغر من أو يساوي ال M تمام؟ لو
+
+50
+00:05:44,500 --> 00:05:50,180
+كان ال N .. لو كان ال N أكبر من أو يساوي capital N
+
+51
+00:05:52,740 --> 00:05:57,480
+يعني المؤشر هذا تبع ال Xn هذه المؤشر تبع لو كان
+
+52
+00:05:57,480 --> 00:06:01,700
+أكبر يعني بسوء capital N أو capital N زائد واحد أو
+
+53
+00:06:01,700 --> 00:06:07,320
+أي عدد أكبر من capital N فمن
+
+54
+00:06:07,320 --> 00:06:11,880
+الـ .. من ال implication أربعة هاي وإذا كان ال N
+
+55
+00:06:11,880 --> 00:06:15,560
+هذه أكبر من أو يساوي capital N فبطلع absolute Xn
+
+56
+00:06:15,560 --> 00:06:21,960
+فبطلع absolute Xn هذا أصغر من absolute X زائد واحد
+
+57
+00:06:22,790 --> 00:06:25,630
+و هذا العدد .. هذا العدد من هنا
+
+58
+00:06:28,510 --> 00:06:33,430
+أصغر من أو يساوي ال supremum له ولا باقي الأعداد
+
+59
+00:06:33,430 --> 00:06:37,870
+هذه وبالتالي أصغر من أو يساوي M إذن هذا العدد أصغر
+
+60
+00:06:37,870 --> 00:06:42,490
+من أو يساوي M إذن في كل الأحوال سواء ال N كانت من
+
+61
+00:06:42,490 --> 00:06:46,770
+واحد إلى capital N سالب واحد أو N أكبر من أو يساوي
+
+62
+00:06:46,770 --> 00:06:51,810
+capital N فطلع absolute xn أصغر من أو يساوي M حيث
+
+63
+00:06:51,810 --> 00:06:56,110
+M عدد موجب وبالتالي ال sequence xn is bounded حسب
+
+64
+00:06:56,110 --> 00:07:06,350
+التعريف تمام؟ إذا هذا يبقى هنا نظرية وهذا طيب
+
+65
+00:07:06,350 --> 00:07:09,010
+في بعض الملاحظات عن نظرية هذه
+
+66
+00:07:20,320 --> 00:07:24,280
+في بعض الملاحظات الملاحظة الأولى بتقول إن ال
+
+67
+00:07:24,280 --> 00:07:29,240
+converse of theorem 2.6 need not be true يعني
+
+68
+00:07:29,240 --> 00:07:34,060
+عكس النظرية السابقة مش شرط يكون صحيح طب ما هو عكس
+
+69
+00:07:34,060 --> 00:07:37,300
+النظرية النظرية بتقول كل convergence sequence is
+
+70
+00:07:37,300 --> 00:07:41,980
+bounded عكس ال statement هذا المفروض أن يكون كل
+
+71
+00:07:41,980 --> 00:07:46,990
+bounded sequence is convergent هذا مش صحيح هاي في
+
+72
+00:07:46,990 --> 00:07:55,350
+counter example يثبت أو يوضح أن عكس نظرية 2.6 ليس
+
+73
+00:07:55,350 --> 00:08:00,930
+صحيح السيكوانس إن الحد العام تبعها xn بساوي سالب
+
+74
+00:08:00,930 --> 00:08:06,550
+واحد يعني إن السيكوانس هذه bounded ليه bounded؟ لأن
+
+75
+00:08:06,550 --> 00:08:11,130
+هاي في عدد موجب capital M بساوي واحد هذا عدد موجب
+
+76
+00:08:11,130 --> 00:08:15,670
+إذا يوجد عدد موجب اللي هو الواحد بحيث إن absolute
+
+77
+00:08:15,670 --> 00:08:20,950
+xn هذا ال absolute value العدد هذا واحد واحد أصغر
+
+78
+00:08:20,950 --> 00:08:25,490
+من أو يساوي واحد إن ال absolute xn هي أصغر من أو
+
+79
+00:08:25,490 --> 00:08:28,590
+يساوي عدد موجب بواحد لكل n إذا ال sequence هي
+
+80
+00:08:28,590 --> 00:08:33,030
+bounded واضحة حسب التعريف الآن هذه ال sequence
+
+81
+00:08:33,030 --> 00:08:39,950
+بنثبت إنها ليست convergent إنها divergent فلبرهان
+
+82
+00:08:39,950 --> 00:08:44,430
+ذلك نعمل برهان بالتناقض أنا بدأ أثبت أن الـ
+
+83
+00:08:44,430 --> 00:08:48,070
+sequence هذه is not convergent ف assume ال
+
+84
+00:08:48,070 --> 00:08:52,530
+contrary يعني assume أنها convergent to some a
+
+85
+00:08:52,530 --> 00:08:56,490
+belongs to the real numbers افرض أنها convergent
+
+86
+00:08:56,490 --> 00:09:02,220
+وال limit تبعتها عدد a فلو أخدت epsilon بساوي واحد
+
+87
+00:09:02,220 --> 00:09:06,340
+عدد موجب إذا من تعريف ال convergence إذا يوجد
+
+88
+00:09:06,340 --> 00:09:09,220
+capital N يعتمد على ال epsilon اللي أنا مختاره
+
+89
+00:09:09,220 --> 00:09:13,500
+اللي هي الواحد بحيث لكل N أكبر من أو يساوي capital
+
+90
+00:09:13,500 --> 00:09:21,540
+N المسافة بين xn و a أصغر من epsilon الآن 
+
+91
+00:09:21,540 --> 00:09:25,460
+ال N هذه اللي هي أكبر من أو يساوي capital N ممكن
+
+92
+00:09:25,460 --> 00:09:30,650
+يكون عدد فردي أو زوجي فلو كانت ال end أكبر من أو يساوي
+
+93
+00:09:30,650 --> 00:09:36,270
+capital N odd فردي فمن المتباينة خمسة بيصير هذا
+
+94
+00:09:36,270 --> 00:09:42,550
+العدد سالب واحد لأن N فردي فبصير absolute سالب
+
+95
+00:09:42,550 --> 00:09:50,250
+واحد سالب a أصغر من واحد ولو رفعت ال absolute
+
+96
+00:09:50,250 --> 00:09:55,010
+value فبينتج منها أن a أكبر من سالب اتنين أصغر من
+
+97
+00:09:55,010 --> 00:10:00,590
+صفر ولو كانت ال N اللي هي أكبر من أو يساوي capital N
+
+98
+00:10:00,590 --> 00:10:06,430
+even فهذا العدد بيصير واحد وبيصير المتباين هذا
+
+99
+00:10:06,430 --> 00:10:12,610
+absolute واحد سالب a أصغر من واحد ولو رفعت .. فكيت
+
+100
+00:10:12,610 --> 00:10:15,990
+ال absolute value هنا أو حليت المتباينة هذه في a
+
+101
+00:10:15,990 --> 00:10:20,630
+فبطلع a أكبر من صفر أصغر من اتنين إذا الآن
+
+102
+00:10:20,630 --> 00:10:25,110
+المتباينتين ستة وسبعة بيعطوني تناقض ليه؟ لأن
+
+103
+00:10:25,110 --> 00:10:31,010
+المتباينة الستة بتقول لي إن a ينتمي للفترة المفتوحة
+
+104
+00:10:31,010 --> 00:10:35,810
+من صفر من سالب اتنين لصفر والمتباينة سبعة بتقول لي
+
+105
+00:10:35,810 --> 00:10:41,680
+إن العدد a ينتمي للفترة المفتوحة من صفر لـ اتنين طب
+
+106
+00:10:41,680 --> 00:10:47,780
+هذه الفترة تقاطعها مع هذه بساوي five يعني إيه
+
+107
+00:10:47,780 --> 00:10:51,720
+ينتمي للمجموع الخالي هذا contradiction هذا
+
+108
+00:10:51,720 --> 00:10:59,580
+تناقض okay تمام؟ إذا هذا التناقض بيقول إن فرضنا،
+
+109
+00:10:59,580 --> 00:11:04,540
+الفرض طبعًا إن ال sequence هذه convergent كان خطأ،
+
+110
+00:11:04,540 --> 00:11:08,160
+إن ال sequence هذه ليست convergent ال sequence
+
+111
+00:11:08,160 --> 00:11:13,960
+هذه ليست convergent إذا هذا مثال على sequence
+
+112
+00:11:13,960 --> 00:11:16,880
+bounded but not convergent إذا مش كل bounded
+
+113
+00:11:16,880 --> 00:11:21,240
+sequence is convergent لكن العكس كمان في النظرية 2
+
+114
+00:11:21,240 --> 00:11:25,760
+.6 العكس أثبتنا إنه صحيح كل convergence sequence
+
+115
+00:11:25,760 --> 00:11:30,500
+ضروري أن تكون bounded النظرية
+
+116
+00:11:30,500 --> 00:11:35,660
+دي 2.6 الملاحظة الثانية بتقول إن نظرية 2.6 هي دي
+
+117
+00:11:35,660 --> 00:11:40,040
+زي اختبار الدم اللي بيلف الأبوة ولا يثبتها
+
+118
+00:11:46,810 --> 00:11:52,550
+الملاحظة الثانية بتقول إن نظرية 2.6 ممكن نستخدمها
+
+119
+00:11:52,550 --> 00:11:56,910
+لتثبت إن certain sequences are divergent
+
+120
+00:12:02,100 --> 00:12:06,000
+يعني نظرية هذه ممكن نستخدمها لإثبات أن ال sequence
+
+121
+00:12:06,000 --> 00:12:11,880
+معينة not convergent زي اختبار الدم بينفي
+
+122
+00:12:11,880 --> 00:12:16,020
+الأبوة ولا يثبتها وهنا هذه نظرية بتنفي ال
+
+123
+00:12:16,020 --> 00:12:22,060
+convergence ولا تثبته تمام مثلًا
+
+124
+00:12:22,060 --> 00:12:27,040
+ال sequence in هذه ال sequence of national numbers
+
+125
+00:12:27,040 --> 00:12:32,930
+هذه ال sequence هذه bounded ولا مش bounded not
+
+126
+00:12:32,930 --> 00:12:36,210
+bounded unbounded الـ sequence هذه unbounded،
+
+127
+00:12:36,210 --> 00:12:43,030
+okay؟ إذا حسب نظرية 2.6 مدامها unbounded إذا
+
+128
+00:12:43,030 --> 00:12:47,750
+لازم تكون divergent لبرهان ذلك افرضي ال
+
+129
+00:12:47,750 --> 00:12:52,330
+contrary افرضي إنها convergent إذا حسب نظرية 2
+
+130
+00:12:52,330 --> 00:12:56,250
+.6 المفروض تطلع bounded لكنها ليست bounded
+
+131
+00:12:57,900 --> 00:13:03,860
+contradiction إذا ال sequence هذه كونها unbounded
+
+132
+00:13:03,860 --> 00:13:07,840
+حسب نظرية نين سيفتر بتطلع divergent واحنا طبعًا
+
+133
+00:13:07,840 --> 00:13:12,260
+عارفين إن ال limit اللي عارفين إنها divergent لأن
+
+134
+00:13:12,260 --> 00:13:17,380
+limit n as n tends to infinity بساوي infinity ليس
+
+135
+00:13:17,380 --> 00:13:23,820
+عدد حقيقي ال sequence هذه مالهاش limit okay تمام
+
+136
+00:13:30,200 --> 00:13:38,940
+تابع الـ sequence in ليش unbounded؟
+
+137
+00:13:38,940 --> 00:13:42,080
+ليش الـ sequence هذه unbounded؟ هي ال proof
+
+138
+00:13:42,080 --> 00:13:52,280
+assume إنها bounded in is bounded بدنا نعمل
+
+139
+00:13:52,280 --> 00:13:59,460
+برهان بالتناقل إذا يوجد عدد موجب M أكبر من صفر بحيث
+
+140
+00:13:59,460 --> 00:14:07,280
+إن absolute xn اللي هي n أصغر من M لكل n 
+
+141
+00:14:07,280 --> 00:14:16,380
+belonging to the N طيب
+
+142
+00:14:16,380 --> 00:14:18,280
+by Archimedean property
+
+143
+00:14:25,940 --> 00:14:33,220
+م أكبر من الصفر بتأدي لأي عدد موجب يوجد عدد طبيعي
+
+144
+00:14:33,220 --> 00:14:42,300
+n0 ينتمي إلى n بحيث أن n0 أكبر من m طبعا
+
+145
+00:14:42,300 --> 00:14:53,400
+هذا بساوي n لأن لو سميت هذه 1 وهذه 2 فمن
+
+146
+00:14:53,400 --> 00:15:02,440
+1 و 2 from 1 and 2 بيطلع عندي N0 أكبر
+
+147
+00:15:02,440 --> 00:15:09,760
+من M و M أكبر من N0 يعني N0 أكبر من N0 هذا تناقض
+
+148
+00:15:09,760 --> 00:15:16,060
+في عدد أكبر من نفسه ما فيش لأن
+
+149
+00:15:16,060 --> 00:15:18,860
+التناقض هذا بيقول أن ال sequence هذه مش ممكن تكون
+
+150
+00:15:18,860 --> 00:15:24,740
+bounded تمام استخدمنا هنا ال Archimedean property
+
+151
+00:15:26,820 --> 00:15:32,140
+طيب نشوف الـ Limit theorems نظريات
+
+152
+00:15:32,140 --> 00:15:37,660
+النهايات أو قوانين النهايات النظرية هذه بتلخص
+
+153
+00:15:37,660 --> 00:15:44,920
+قوانين النهايات المعروفة لدينا من calculus وهي
+
+154
+00:15:44,920 --> 00:15:48,660
+إن لو في عندي sequence x n convergent ل x و
+
+155
+00:15:48,660 --> 00:15:52,300
+sequence ثانية y n convergent ل y و c أي عدد 
+
+156
+00:15:52,300 --> 00:15:54,640
+حقيقي ف
+
+157
+00:15:56,610 --> 00:16:01,510
+الـ sequence اللي حدها العام مجموع
+
+158
+00:16:01,510 --> 00:16:05,690
+حدود ال sequence xn و ال sequence yn ال sequence
+
+159
+00:16:05,690 --> 00:16:09,590
+هذا convergent و ال limit تبعتها بساوي مجموع
+
+160
+00:16:09,590 --> 00:16:17,890
+النهايات كذلك نفس الشيء بالنسبة للفرق الآن
+
+161
+00:16:17,890 --> 00:16:20,830
+ال sequence اللي حدها العام حاصل ضرب
+
+162
+00:16:23,930 --> 00:16:27,690
+الـ sequence xn مع الـ sequence yn هذه بتطلع
+
+163
+00:16:27,690 --> 00:16:34,290
+convergent ونهايتها بسبب حاصل ضرب النهايات لو في 
+
+164
+00:16:34,290 --> 00:16:38,370
+ثابت limit ثابت في ال sequence بسبب ثابت في نهاية 
+
+165
+00:16:38,370 --> 00:16:45,310
+ال sequence لو كان ال zn sequence حدودها
+
+166
+00:16:45,310 --> 00:16:51,250
+كلها غير ... لا تساوي الصفر وconvergent لعدد z لا 
+
+167
+00:16:51,250 --> 00:16:59,260
+يساوي الصفرالسيكوينس XM over ZM تظهر كونفرجنت
+
+168
+00:16:59,260 --> 00:17:05,540
+ونهايتها بساوي نهاية البسط على نهاية المقام طبعا
+
+169
+00:17:05,540 --> 00:17:12,180
+براهين الفروع كلها مطلوبة منكمفانا هاي اللي
+
+170
+00:17:12,180 --> 00:17:17,460
+برهنلكم الفرع بيه وبالمثل ممكن تجدوا الفروع الأخرى
+
+171
+00:17:17,460 --> 00:17:21,580
+موجودة برهنها في الكتاب المقرر وإذا فيش مش برهن
+
+172
+00:17:21,580 --> 00:17:29,460
+فبتبرهنوه لوحدكم فاعتبروه تمرين homework exercise
+
+173
+00:17:29,460 --> 00:17:36,480
+إذا نشوف برهان الجزء بيه في الجزء بيه عايزين نثبت
+
+174
+00:17:36,480 --> 00:17:43,250
+أن limit يبقى ان في جزء بيه عايزين نثبت ان limit
+
+175
+00:17:43,250 --> 00:17:53,730
+xn في yn بساوي x في y فالبرهان 
+
+176
+00:17:53,730 --> 00:18:01,640
+ذلكناخد المتباينة هذه أو الأعداد هذه في نهاية
+
+177
+00:18:01,640 --> 00:18:07,180
+الأمر حسب تعريف epsilon capital N للنهايات بتثبت
+
+178
+00:18:07,180 --> 00:18:11,920
+أن ال absolute value للحد العام لل sequence minus
+
+179
+00:18:11,920 --> 00:18:16,080
+ال limit المقترحة بتثبت أن ال absolute value هذه
+
+180
+00:18:16,080 --> 00:18:21,880
+أصغر من أي given epsilon فالبرهان ده عليك ببدأ هي
+
+181
+00:18:21,880 --> 00:18:29,380
+absolute xn yn minus xy هطرح من الحد هذا xn في y و
+
+182
+00:18:29,380 --> 00:18:33,940
+أرجعه فكأني ماعملتش عاجب أخد الحدين هذول مع بعض و
+
+183
+00:18:33,940 --> 00:18:37,260
+الحدين هذول مع بعض by ال triangle inequality هذا
+
+184
+00:18:37,260 --> 00:18:41,160
+أصغر من أو يساوي absolute الحد الأول زائد absolute
+
+185
+00:18:41,160 --> 00:18:47,240
+الحد الثاني في الحد الأول هذا عامل مشترك xn بطلعه
+
+186
+00:18:47,750 --> 00:18:53,010
+فبيصير absolute xn في absolute yn سالب y وفي هنا
+
+187
+00:18:53,010 --> 00:18:59,270
+أعمل مشترك y فبطلعه فبيصير absolute الحد الثاني
+
+188
+00:18:59,270 --> 00:19:06,850
+بيسوي absolute xn minus x في absolute y الآن
+
+189
+00:19:06,850 --> 00:19:14,210
+بما أن xn converge ل x إذا حسب نظرية 2.6 الـ
+
+190
+00:19:14,210 --> 00:19:18,150
+sequence هذه بما أنها convergent فهي bounded فهي 
+
+191
+00:19:18,150 --> 00:19:23,930
+bounded وبالتالي بنقدر نلاقي عدد موجب M1 بحيث أن 
+
+192
+00:19:23,930 --> 00:19:26,810
+ال absolute value لحدود ال sequence أصغر من أو 
+
+193
+00:19:26,810 --> 00:19:32,370
+يساوي M1 لو أخدت capital M المaximum الأكبر بين
+
+194
+00:19:32,370 --> 00:19:38,630
+العدد الموجب M1 والعدد الموجبabsolute y ف m بيطلع
+
+195
+00:19:38,630 --> 00:19:44,450
+عدد موجب لأنه أكبر من أو يساوي هذا وهذاإذا
+
+196
+00:19:44,450 --> 00:19:49,790
+المتباينة هذه بتصير absolute xny m minus xy أصغر
+
+197
+00:19:49,790 --> 00:19:55,970
+من أو يساوي absolute xn هذا هي أصغر من أو يساوي m
+
+198
+00:19:55,970 --> 00:20:01,110
+1 و m 1 أصغر من أو يساوي capital M إذا بشيل
+
+199
+00:20:01,110 --> 00:20:06,990
+absolute xn بحط أصغر من أو يساوي capital M في 
+
+200
+00:20:06,990 --> 00:20:12,770
+absolute y n minus y كذلك absolute y هذه هي أصغر
+
+201
+00:20:12,770 --> 00:20:18,290
+من أو يساوي capital M وبالتالي الحد هذا بيصير أصغر
+
+202
+00:20:18,290 --> 00:20:22,750
+من أو يساوي capital M بدل absolute y في absolute x
+
+203
+00:20:22,750 --> 00:20:28,350
+n minus x الآن احنا من الفرض فرضين أن ال sequence
+
+204
+00:20:28,350 --> 00:20:33,630
+x n converge ل x و ال sequence y n converge ل yإذا
+
+205
+00:20:33,630 --> 00:20:37,450
+لو أخدنا أي إبسلون لت إبسلون أكبر من الصفر P given
+
+206
+00:20:37,450 --> 00:20:42,210
+بما أنه ال sequence XM converged ل X إذا يوجد N
+
+207
+00:20:42,210 --> 00:20:47,350
+1 عدد طبيعي يعتمد على إبسلون بحيث لكل N أكبر من
+
+208
+00:20:47,350 --> 00:20:51,710
+أو يساوي capital N 1 بيطلع المسافة بين XM X أصغر
+
+209
+00:20:51,710 --> 00:20:55,790
+من إبسلون أو إبسلون على 2M يعتبر هذا هو الإبسلون
+
+210
+00:20:55,790 --> 00:21:01,070
+في التعريف لأنها دا عدد موجب وبيعتمد على إبسلون ليش
+
+211
+00:21:01,070 --> 00:21:05,970
+حطينا 2M لحاجة في نفس يعقوب، عشان في النهاية أخلي
+
+212
+00:21:05,970 --> 00:21:10,610
+ال absolute value هذه أصغر من إبسلون، هنشوفها هنا،
+
+213
+00:21:10,610 --> 00:21:15,530
+بقطوة هذه طيب، إذا قلنا بما إن it's convergent
+
+214
+00:21:15,530 --> 00:21:19,230
+ل X، فلأي إبسلون فيه capital N 1، بحيث ال
+
+215
+00:21:19,230 --> 00:21:23,910
+implication هذه تتحقق كذلك بما أن yn converge ل y
+
+216
+00:21:23,910 --> 00:21:28,270
+إذا for the same إبسلون لنفس الإبسلون العدد الموجب
+
+217
+00:21:28,270 --> 00:21:33,090
+اللي هو معطى مسبقا يوجد عدد طبيعي N2 يعتمد على 
+
+218
+00:21:33,090 --> 00:21:37,770
+إبسلون بحيث لكل N أكبر منه أو يساوي capital N2 بيطلع
+
+219
+00:21:37,770 --> 00:21:42,750
+absolute yn minus y أصغر من إبسلون على 2M
+
+220
+00:21:45,110 --> 00:21:49,830
+الآن خلّينا ناخد capital N الأكبر من N 1 و N
+
+221
+00:21:49,830 --> 00:21:54,610
+2 هذي وهذي كلاهما يعتمد على epsilon إذا
+
+222
+00:21:54,610 --> 00:22:00,490
+capital N تعتمد على epsilon و عدد طبيعي الآن لكل 
+
+223
+00:22:00,490 --> 00:22:07,370
+عدد طبيعي أكبر من أو يساوي capital N هذا بيطلع و
+
+224
+00:22:07,370 --> 00:22:11,570
+capital N هي من تعريفه هذا أكبر من أو يساوي capital
+
+225
+00:22:11,570 --> 00:22:20,710
+N 1 وأيضا أكبر من أو يساوي capital N2 إذن بيطلع
+
+226
+00:22:20,710 --> 00:22:27,350
+عندي absolute xn yn minus xy أصغر من أو يساوي M و
+
+227
+00:22:27,350 --> 00:22:32,870
+absolute yn minus y بما أن small n هذي أكبر من أو
+
+228
+00:22:32,870 --> 00:22:38,410
+يساوي capital N 2 إذن من هنا بيطلع الفرق هذا 
+
+229
+00:22:38,410 --> 00:22:43,590
+أصغر من epsilon على 2M أصغر من epsilon على
+
+230
+00:22:43,590 --> 00:22:49,900
+2M في M وكذلk small n هذه أكبر من أو يساوي n
+
+231
+00:22:49,900 --> 00:22:55,460
+1 لما small n أكبر من أو يساوي n 1 بطلع بقدر
+
+232
+00:22:55,460 --> 00:23:00,880
+أشيل absolute xn minus x و أحط أصغر من epsilon على
+
+233
+00:23:00,880 --> 00:23:08,660
+2M الآن M بتروح مع M و M بتروح مع M بضل عندي
+
+234
+00:23:08,660 --> 00:23:11,700
+epsilon على 2 زائد epsilon على 2 بيطلع
+
+235
+00:23:11,700 --> 00:23:17,540
+epsilon تمام؟ و هذا اللي بدنا ياه فاكرين في بداية 
+
+236
+00:23:17,540 --> 00:23:21,160
+البرنامج احنا عايزين في النهاية نثبت ان ال
+
+237
+00:23:21,160 --> 00:23:25,700
+absolute value هذه أصغر من epsilon أصغر من epsilon
+
+238
+00:23:27,020 --> 00:23:30,340
+for any given epsilon وفعلا هي for any epsilon
+
+239
+00:23:30,340 --> 00:23:34,760
+أكبر من 0 أثبتنا أن يوجد capital N أعداد طبيعية
+
+240
+00:23:34,760 --> 00:23:38,580
+تعتمد على epsilon بحيث لكل N أكبر من أو يساوي
+
+241
+00:23:38,580 --> 00:23:43,000
+capital N طلع ال absolute value للفرق هذا أصغر من
+
+242
+00:23:43,000 --> 00:23:48,820
+epsilon، إذن حسب التعريف بيطلع limit xn في yn بساوي
+
+243
+00:23:48,820 --> 00:23:55,170
+x في y، تمام؟ إن الهدف برهان برهان بكمل برهان جزء
+
+244
+00:23:55,170 --> 00:23:59,730
+بيه بالمثل ممكن تبرهن الأجزاء الأخرى فمطلوب منكم
+
+245
+00:23:59,730 --> 00:24:04,130
+تبرهنوها أعتقد أنه في الكتاب المقرر في نظرية
+
+246
+00:24:04,130 --> 00:24:11,010
+3 2 3 in the textbook فيها براهين بعض
+
+247
+00:24:11,010 --> 00:24:16,110
+الأجزاء أو ربما كلهم فحاولوا تكتبوا البرهان الأول
+
+248
+00:24:16,110 --> 00:24:19,630
+لوحدكم وإذا ما عرفتو اقرأوا البرهان في الكتاب
+
+249
+00:24:19,630 --> 00:24:27,040
+حاولوا تفهموه إذا في أي شيء مش واضح اكتب البرهان في 
+
+250
+00:24:27,040 --> 00:24:34,740
+ورقة و أعطينيها عشان أصلحها okay
+
+251
+00:24:34,740 --> 00:24:42,560
+إذا هذه قوانين النهايات هذه قوانين النهايات طيب في
+
+252
+00:24:42,560 --> 00:24:47,480
+كمان قوانين أخرى أو خواص أخرى للنهايات
+
+253
+00:24:52,790 --> 00:24:57,850
+فمثلا في النظرية هذه النظرية هذه بتقول لو في عندي 
+
+254
+00:24:57,850 --> 00:25:01,810
+convergence sequence xn convergence ل x و ال
+
+255
+00:25:01,810 --> 00:25:06,750
+sequence xn حدودها غير سالبة كل حدودها غير سالبة ف
+
+256
+00:25:06,750 --> 00:25:11,630
+ال limit تبعتها أيضا لازم تكون غير سالبة و البرهان 
+
+257
+00:25:11,630 --> 00:25:16,590
+سهل by contradiction prove by contradiction assume
+
+258
+00:25:16,590 --> 00:25:21,170
+on the contrary أن ال x سالبة
+
+259
+00:25:24,860 --> 00:25:29,200
+أنا عايز أثبت x أكبر من أو يساوي 0 النافي تبعها x
+
+260
+00:25:29,200 --> 00:25:36,380
+أصغر من 0 صح؟ الآن خدي epsilon بالساوي سالب x إذا
+
+261
+00:25:36,380 --> 00:25:40,300
+ال x سالب هذا سالب x عدد موجبة، إذا هاي الآن في 
+
+262
+00:25:40,300 --> 00:25:44,180
+عندي epsilon موجبة وفي عندي من الفرض x n converge
+
+263
+00:25:44,180 --> 00:25:49,440
+ل x إذا من تعريف epsilon capital N للنهايات بما أن 
+
+264
+00:25:49,440 --> 00:25:52,680
+x n converge ل x إذا يوجد capital N يعتمد على ال
+
+265
+00:25:52,680 --> 00:25:56,780
+epsilon بحيث لكل n أكبر لو يساوي capital N المسافة
+
+266
+00:25:56,780 --> 00:26:02,540
+بين xn و x أصغر من y طيب فك ال absolute value هذه
+
+267
+00:26:02,540 --> 00:26:09,260
+شيلها بيصير المتباينة هذه xn minus x أصغر من y أكبر
+
+268
+00:26:09,260 --> 00:26:10,420
+من سالب y
+
+269
+00:26:14,450 --> 00:26:18,770
+خدي هذا جزء من المتباينة و ادى x على الناحية
+
+270
+00:26:18,770 --> 00:26:24,410
+الثانية فبيصير هذا بيقدي أن xn أصغر من أو يساوي x 
+
+271
+00:26:24,410 --> 00:26:29,290
+زائد epsilon الكلام هذا صحيح لكل n أكبر من أو يساوي
+
+272
+00:26:29,290 --> 00:26:32,410
+capital N تمام؟
+
+273
+00:26:34,150 --> 00:26:39,850
+طيب الآن خدي N خدي N هنا خدي ال N الصغيرة هذه
+
+274
+00:26:39,850 --> 00:26:44,130
+بيساوي capital N فبيصير X capital N أصغر من ما 
+
+275
+00:26:44,130 --> 00:26:48,970
+بيساوي X زائد Epsilon صح عوض عن ال Epsilon ال
+
+276
+00:26:48,970 --> 00:26:56,310
+Epsilon بيساوي سالب X إذن عوض عن X بسالب عن
+
+277
+00:26:56,310 --> 00:27:02,260
+Epsilon سالب X فهذا بيطلع صفر إذا أنا طلع عندي x
+
+278
+00:27:02,260 --> 00:27:06,960
+رقم capital N أصغر من صفر وهذا جدّيني
+
+279
+00:27:06,960 --> 00:27:13,180
+contradiction ليه؟ لأنه أنا فارض في النظرية أن كل
+
+280
+00:27:13,180 --> 00:27:19,000
+حدود ال sequence كلهم غير سالبين كلهم يعني حدود 
+
+281
+00:27:19,000 --> 00:27:24,750
+غير سالبة فكيف طلع الحد رقم N  Capital N سالب هذا
+
+282
+00:27:24,750 --> 00:27:30,910
+بتناقض مع الفرض هذا بكمل البرهان تمام واضح البرهان
+
+283
+00:27:30,910 --> 00:27:35,390
+طبعا على طريقة هذه إذا لو الـ sequence كانت
+
+284
+00:27:35,390 --> 00:27:39,630
+convergent وكل حدودها غير سالبة فنهايتها أيضا 
+
+285
+00:27:39,630 --> 00:27:40,850
+هتكون غير سالبة
+
+286
+00:27:46,250 --> 00:27:49,370
+طب لو كانت الـ sequence convergent وحدودها غير
+
+287
+00:27:49,370 --> 00:27:54,610
+موجبة كل حدودها غير موجبة فنهايتها أيضا هتكون غير 
+
+288
+00:27:54,610 --> 00:28:02,430
+موجبة والبرهان مشابه لبرهان النظرية السابقة النظرية
+
+289
+00:28:02,430 --> 00:28:08,730
+اللي بعدها لو
+
+290
+00:28:08,730 --> 00:28:13,390
+في عندي two sequences والتنتين convergent والحد
+
+291
+00:28:13,390 --> 00:28:16,130
+العام الأولى أصغر من أو يساوي الحد العام الثانية
+
+292
+00:28:16,130 --> 00:28:22,110
+فنهاية الأولى أصغر من أو يساوي نهاية
+
+293
+00:28:22,110 --> 00:28:31,520
+الثانية والبرهان تطبيق مباشر على نظرية اللي فاتت لو
+
+294
+00:28:31,520 --> 00:28:39,000
+أخذت أنا عندي احنا فرضنا إن xn أصغر من أو يساوي yn
+
+295
+00:28:39,000 --> 00:28:47,380
+لكل n هذا معناه إن yn minus xn أكبر من أو يساوي 
+
+296
+00:28:47,380 --> 00:28:49,700
+صفر لكل n
+
+297
+00:28:53,620 --> 00:28:59,620
+فلو أخذت zn  عرفت zn على أنه الفرق بين yn و xn
+
+298
+00:28:59,620 --> 00:29:03,580
+فقلنا الفرق هذا غير سالب لكل n إذن هذه الـ sequence
+
+299
+00:29:03,580 --> 00:29:09,700
+الجديدة حد العام تبعها zn وهذا الحد العام غير سالب
+
+300
+00:29:09,700 --> 00:29:15,530
+لكل n والـ sequence xn convergent وyn convergent
+
+301
+00:29:15,530 --> 00:29:20,510
+حسب قوانين
+
+302
+00:29:20,510 --> 00:29:27,790
+النهايات limit الفرق هذا موجود أو بيساوي فرق
+
+303
+00:29:27,790 --> 00:29:33,430
+النهايات إذا الـ sequence zn حدودها كلها غير سالبة 
+
+304
+00:29:33,430 --> 00:29:36,950
+وconvergent لأنها الفرق بين two convergent
+
+305
+00:29:36,950 --> 00:29:37,530
+sequences
+
+306
+00:29:40,870 --> 00:29:45,350
+وبالتالي حسب النظرية السابقة إذا الـ limit الـ
+
+307
+00:29:45,350 --> 00:29:50,890
+sequence zn هذه بتطلع غير سالبة طب limit الـ zn عشان
+
+308
+00:29:50,890 --> 00:29:55,830
+قوانين النهايات بيساوي limit yn سالب limit xn و أدى 
+
+309
+00:29:55,830 --> 00:29:59,230
+هذه عن ناحية الثانية فبيطلع limit yn أكبر من لو 
+
+310
+00:29:59,230 --> 00:30:06,090
+ساوي limit xn وهذا هو المطلوب okay تمام؟ راضي؟ دي
+
+311
+00:30:06,090 --> 00:30:06,730
+أي سؤال؟
+
+312
+00:30:12,550 --> 00:30:22,370
+النتيجة هذه بتقول
+
+313
+00:30:22,370 --> 00:30:26,430
+لو كان في عندي sequence والـ sequence هذي
+
+314
+00:30:26,430 --> 00:30:35,670
+convergent وحدودها تبعتها معصورة بين a و b
+
+315
+00:30:39,050 --> 00:30:45,870
+فلازم نهايتها أيضا.. نهايتها تطلع محصورة بين
+
+316
+00:30:45,870 --> 00:30:53,930
+العددين a وb فبرهان
+
+317
+00:30:53,930 --> 00:31:03,850
+النظرية هذه بنطبق نظرية 2.9 مرتين مرة
+
+318
+00:31:04,750 --> 00:31:10,170
+على الـ sequence الثابتة a اللي حدودها تبعتها أصغر 
+
+319
+00:31:10,170 --> 00:31:15,310
+من حدود الـ sequence xn هذه convergence نهايتها a
+
+320
+00:31:15,310 --> 00:31:24,470
+وهذه convergence نهايتها a إذا limit limit الـ 
+
+321
+00:31:24,470 --> 00:31:29,710
+sequence الثابتة a بيساوي a أصغر من أو يساوي limit 
+
+322
+00:31:29,710 --> 00:31:32,090
+الـ sequence xn
+
+323
+00:31:34,930 --> 00:31:38,930
+كذلك أنا عندي الـ sequence xn من الفرض أنا عندي 
+
+324
+00:31:38,930 --> 00:31:42,770
+الـ sequence xn أصغر من أو يساوي الـ sequence اللي
+
+325
+00:31:42,770 --> 00:31:48,330
+الحد العام تبعها ثابت بيه إذا الـ limit حسب النظرية 
+
+326
+00:31:48,330 --> 00:31:52,570
+8 إذا الـ limit xn أصغر من أو يساوي limit الـ 
+
+327
+00:31:52,570 --> 00:31:57,530
+sequence اللي الحد العام تبعها ثابت اللي هو b
+
+328
+00:31:59,560 --> 00:32:05,040
+من هنا بطلع الـ limit xn هي أكبر من أو يساوي a
+
+329
+00:32:05,040 --> 00:32:14,280
+أصغر من أو يساوي b وهو المفروض في
+
+330
+00:32:14,280 --> 00:32:22,180
+الـ squeeze theorem أو نظرية الـ sandwich في 
+
+331
+00:32:22,180 --> 00:32:25,160
+ناس يسموها squeeze theorem في ناس يسموها sandwich
+
+332
+00:32:25,160 --> 00:32:31,450
+theorem هذه في التفاضل والتكامل يسموها sandwich 
+
+333
+00:32:31,450 --> 00:32:39,070
+sandwich
+
+334
+00:32:39,070 --> 00:32:42,750
+theorem أو 
+
+335
+00:32:42,750 --> 00:32:46,990
+squeeze theorem فالنظرية هذه بتقول لو في عندي three
+
+336
+00:32:46,990 --> 00:32:54,710
+sequences XN, YN و ZN والعلاقة بينهم هكذا YN 
+
+337
+00:32:54,710 --> 00:33:02,050
+محصورة بين XN و ZN ولو كانت الـ sequences الحدود 
+
+338
+00:33:02,050 --> 00:33:06,790
+العامة تبعتها على الأطراف convergent يعني الـ 
+
+339
+00:33:06,790 --> 00:33:10,530
+sequence x in convergent والـ sequence z in
+
+340
+00:33:10,530 --> 00:33:14,150
+convergent والتنتين الـ limit تبعتهم متساوية
+
+341
+00:33:16,990 --> 00:33:22,390
+فلابد أو لازم أن الـ sequence المحصورة بينهم بتكون
+
+342
+00:33:22,390 --> 00:33:27,130
+أيضا convergent ونهايتها بتساوي القيمة المشتركة
+
+343
+00:33:27,130 --> 00:33:33,070
+لنهايات للنهايات limit yn بتكون موجودة ونهايتها 
+
+344
+00:33:33,070 --> 00:33:39,910
+بتساوي نهاية xn ونهاية zn البرهان 
+
+345
+00:33:39,910 --> 00:33:41,090
+برضه مش صعب
+
+346
+00:33:46,210 --> 00:33:50,630
+احنا فرضنا إن limit sequence x in موجودة وlimit
+
+347
+00:33:50,630 --> 00:33:57,130
+sequence z in موجودة وتنتين متساويات فدعونا نسمي
+
+348
+00:33:57,130 --> 00:34:05,230
+الـ limit هذه مشتركة a عدد a الآن ناخد let epsilon
+
+349
+00:34:05,230 --> 00:34:12,210
+أكبر من صفر بما أن الـ sequence xn converge لـ a،
+
+350
+00:34:12,210 --> 00:34:15,430
+إذا يوجد عدد طبيعي Capital N يعتمد على إبسلون،
+
+351
+00:34:15,430 --> 00:34:20,570
+بحيث لكل N أكبر من أو يساوي Capital N، المسافة بين xn
+
+352
+00:34:20,570 --> 00:34:27,930
+و a أصغر من إبسلون كذلك بما أن الـ sequence zm
+
+353
+00:34:27,930 --> 00:34:33,630
+converge لـ a إذا بقدر ألاقي لكل.. لنفس الـ epsilon
+
+354
+00:34:33,630 --> 00:34:38,650
+.. لنفس الـ epsilon بقدر ألاقي Capital N ممكن الـ
+
+355
+00:34:38,650 --> 00:34:42,970
+Capital N مختلف عن الـ Capital N الأولى يعني ففي
+
+356
+00:34:42,970 --> 00:34:46,830
+الحالة هذه باخد Capital N هذا الـ maximum للـ N
+
+357
+00:34:46,830 --> 00:34:49,490
+الأولى والثانية زي ما شوفنا في الدرس السابق
+
+358
+00:34:50,870 --> 00:34:54,170
+وبالتالي لكل n أكبر من أو يساوي Capital N بقدر أخلي 
+
+359
+00:34:54,170 --> 00:35:01,530
+المسافة هذه أصغر من epsilon الآن 
+
+360
+00:35:01,530 --> 00:35:08,190
+من الفرض ومن الـ implication 8 نحصل 
+
+361
+00:35:08,190 --> 00:35:11,030
+على الـ implication الجديدة هذه 
+
+362
+00:35:18,350 --> 00:35:25,630
+يعني من هنا أنا عندي xn minus a أصغر من ي وطبعا
+
+363
+00:35:25,630 --> 00:35:33,090
+أكبر من سالب ي ومن هنا أنا عندي zn minus a أصغر من 
+
+364
+00:35:33,090 --> 00:35:36,770
+ي أكبر من سالب ي 
+
+365
+00:35:39,750 --> 00:35:45,170
+فهي xn minus a أصغر من إبسلون لكل n أكبر من أو
+
+366
+00:35:45,170 --> 00:35:54,810
+ساوي Capital N إذا هذه هذه هي من هنا أو 
+
+367
+00:35:54,810 --> 00:36:00,210
+هذه عفوا أنا أخدت الجزء هذه سالب إبسلون
+
+368
+00:36:04,860 --> 00:36:12,860
+-Epsilon أصغر من Xn-A وأنا عندي Xn أصغر من يساوي
+
+369
+00:36:12,860 --> 00:36:20,000
+Yn إذا لو طرحت A من هنا وطرحت A من هنا فبيطلع عندي 
+
+370
+00:36:20,000 --> 00:36:27,080
+Xn-A أصغر من يساوي Yn-A ونفس الحاجة لو طرحت a من
+
+371
+00:36:27,080 --> 00:36:32,620
+هنا طب يطلع y n minus a أصغر من أو يساوي z n minus
+
+372
+00:36:32,620 --> 00:36:41,320
+a ومن هنا هاندي z n minus a من هذا الجزء z n 
+
+373
+00:36:41,320 --> 00:36:43,460
+minus a أصغر من إبسلون
+
+374
+00:36:48,860 --> 00:36:54,460
+إذاً هذه الـ implication بتقول باختصار لكل N أكبر
+
+375
+00:36:54,460 --> 00:37:00,240
+من أو يساوي Capital N أنا عندي طلع يطلع عندي YN
+
+376
+00:37:00,240 --> 00:37:08,620
+minus A أصغر من أبسلون أكبر من سالب أبسلون أو
+
+377
+00:37:08,620 --> 00:37:13,140
+absolute YN minus A أصغر من أبسلون لكل N أكبر من
+
+378
+00:37:13,140 --> 00:37:18,280
+أو يساوي Capital N إذن هين أثبتنا أنه for any
+
+379
+00:37:18,280 --> 00:37:22,860
+epsilon أكبر من صفر يوجد Capital N يعتمد على 
+
+380
+00:37:22,860 --> 00:37:27,080
+epsilon عدد طبيعي بحيث لكل N أكبر من أو يساوي 
+
+381
+00:37:27,080 --> 00:37:30,720
+Capital N الـ absolute value هذي أصغر من epsilon
+
+382
+00:37:30,720 --> 00:37:34,200
+إذا by epsilon Capital N definition of limit بطلع
+
+383
+00:37:34,200 --> 00:37:39,640
+عندي limit هذا معناه إن limit الـ sequence yn as n
+
+384
+00:37:39,640 --> 00:37:45,530
+tends to infinity بتساوي العدد A وهذا اللي بدنا
+
+385
+00:37:45,530 --> 00:37:51,790
+يعني.. هذا اللي بدنا يعني.. okay تمام؟ إذن هذا
+
+386
+00:37:51,790 --> 00:37:55,790
+هو برهان الـ sandwich أو الـ squeeze الـ theorem
+
+387
+00:37:55,790 --> 00:38:00,710
+تمام؟
+
+388
+00:38:00,710 --> 00:38:04,810
+هاي ناخد بعض الـ.. بعض الأمثلة.. بعض الأمثلة
+
+389
+00:38:20,690 --> 00:38:25,770
+عايزين نثبت إن limit الـ sequence لحد العام تبعها
+
+390
+00:38:25,770 --> 00:38:31,910
+الكسر هذا بيساوي ستة فزي 
+
+391
+00:38:31,910 --> 00:38:38,270
+ما كنا نعمل في تفاضل ألف أو باء بنجسم بسط مقام على
+
+392
+00:38:38,270 --> 00:38:46,130
+n لأكبر أس اللي هو n تربيع فبيصير عندي المقدار ده
+
+393
+00:38:46,130 --> 00:38:49,790
+بيساوي اتنين على n على واحد زائد.. على واحد.. 
+
+394
+00:38:49,790 --> 00:38:54,490
+واحد على n تربيع زائد واحد أنا عندي الـ sequence
+
+395
+00:38:54,490 --> 00:38:59,190
+واحد على n الـ limit تبعتها صفر إذا اتنين على n الـ 
+
+396
+00:38:59,190 --> 00:39:06,360
+limit تبعتها صفر المقام واحد على m تربيع نهايتها
+
+397
+00:39:06,360 --> 00:39:10,580
+صفر إذا واحد زائد واحد على m تربيع نهايتها واحد زائد
+
+398
+00:39:10,580 --> 00:39:19,860
+صفر بطلع واحد لا يساوي صفر تمام إذا الـ limit الكسر 
+
+399
+00:39:19,860 --> 00:39:27,000
+هذا بيساوي limit الكسر اللي هنا الآن الكسر هذا
+
+400
+00:39:27,000 --> 00:39:31,490
+بتكون من بسط مقام الـ بسط المقام عبارة عن الـ
+
+401
+00:39:31,490 --> 00:39:35,670
+sequence 2 على m المقام عبارة عن الـ sequence الحد
+
+402
+00:39:35,670 --> 00:39:41,130
+العام تبعه 1 زائد 1 على m تربيع الآن limit الـ
+
+403
+00:39:41,130 --> 00:39:44,950
+sequence في المقام شفنا بيساوي واحد لا تساوي صفر
+
+404
+00:39:44,950 --> 00:39:50,370
+عشان هيك قدرت أقول إن limit limit الكسر بيساوي 
+
+405
+00:39:50,370 --> 00:39:53,330
+limit البسط على limit المقام لكن لو limit المقام
+
+406
+00:39:53,330 --> 00:39:59,590
+بيساوى صفر فما بقدر استخدم القانون هذا ماشي الحل 
+
+407
+00:40:02,990 --> 00:40:06,970
+limit الكسر هذا بيساوي limit الكسر اللي هنا وهذا
+
+408
+00:40:06,970 --> 00:40:11,170
+طبعا لأن limit المقام بيساويش صفر فبيساوي limit الـ
+
+409
+00:40:11,170 --> 00:40:15,170
+بسط على limit المقام حسب قوانين النهايات limit الـ
+
+410
+00:40:15,170 --> 00:40:20,610
+بسط صفر limit المقام واحد إذا الـ limit في النهاية
+
+411
+00:40:20,610 --> 00:40:29,880
+نهايتها صفر تمام المثال الثاني هذا إنه نثبت إن الـ
+
+412
+00:40:29,880 --> 00:40:35,960
+sequence الحد العام تبعها xn بيساوي sin n على n الـ
+
+413
+00:40:35,960 --> 00:40:41,520
+limit تبعتها بيساوي 0 طيب احنا عارفين من تفاضل ألف
+
+414
+00:40:41,520 --> 00:40:46,060
+إن absolute sin x أصغر من أو يساوي واحد لكل x آخر
+
+415
+00:40:46,060 --> 00:40:54,660
+أو وبالتالي لكل عدد طبيعي n absolute أو sin n أكبر
+
+416
+00:40:54,660 --> 00:41:00,960
+من أو يساوي سالب واحد أصغر من أو يساوي واحد طيب لكل
+
+417
+00:41:00,960 --> 00:41:07,460
+عدد طبيعي n واحد على n عدد موجب فلو ضربنا
+
+418
+00:41:07,460 --> 00:41:12,800
+المتباينة هذه في العدد الموجب واحد على n فبيصير 
+
+419
+00:41:12,800 --> 00:41:13,820
+شكلها هكذا
+
+420
+00:41:18,240 --> 00:41:21,680
+فبتصير الـ sequence بعد ما نضربها أو المتباينة 
+
+421
+00:41:21,680 --> 00:41:25,120
+الأخيرة بعد ما نضربها في واحد على N بصير شكلها
+
+422
+00:41:25,120 --> 00:41:30,240
+هكذا سالب واحد على N أصلا ما يساوي سالب N على N
+
+423
+00:41:30,240 --> 00:41:35,120
+أصلا ما يساوي واحد على N طيب ال sequence هذه ال
+
+424
+00:41:35,120 --> 00:41:42,860
+limit تبعتها صفر و ال sequence هذه أيضا نهايتها
+
+425
+00:41:42,860 --> 00:41:45,960
+سالب واحد في صفر بيطلع صفر
+
+426
+00:41:49,290 --> 00:41:52,830
+إذا by squeeze theorem أو by sandwich theorem
+
+427
+00:41:52,830 --> 00:41:56,210
+limit ال sequence اللي في الوسط اللي محصورة في
+
+428
+00:41:56,210 --> 00:42:03,950
+الوسط بيطلع أيضا موجودة و بيساوي الصفر okay تمام
+
+429
+00:42:03,950 --> 00:42:09,150
+طب لو كانت ال limit هذه بيساوي صفر وهذه بيساوي
+
+430
+00:42:09,150 --> 00:42:13,670
+واحد ماقدرش أطبق ال squeeze theorem لازم ال
+
+431
+00:42:13,670 --> 00:42:17,630
+sequences اللي على الأطراف بس يكونوا convergent
+
+432
+00:42:17,630 --> 00:42:23,390
+ويلهم نفس ال limit تكون نفس القيمة okay تمام؟ إذا 
+
+433
+00:42:23,390 --> 00:42:30,690
+يعني هذا .. هذا بعض الأمثلة المرة الجاية طبعا
+
+434
+00:42:30,690 --> 00:42:37,350
+هنكمل .. إذا 
+
+435
+00:42:37,350 --> 00:42:42,330
+المرة الجاية هنكمل نشوف بعض نظريات النهايات و ..
+
+436
+00:42:43,750 --> 00:42:50,310
+هنخلص اللي هو section 2.2 مش يعني ضايل
+
+437
+00:42:50,310 --> 00:42:58,470
+كتير و هحللكم بعض التمارين و ناخد ال homework ل
+
+438
+00:42:58,470 --> 00:43:03,070
+section 3.2 okay تمام فحاولوا تحضروا هذا
+
+439
+00:43:03,070 --> 00:43:12,050
+الكلام للمحاضرة الجاية و نشوف مع بعض الحاجات هذه
+
+440
+00:43:12,050 --> 00:43:18,550
+نشرحها okay إذا انتهت المحاضرة نكمل إن شاء الله
+
+441
+00:43:18,550 --> 00:43:19,450
+المرة الجاية 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d3b3a2f372b2e3cc5302a93336097b21495d1084
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/nokw77ZubUw_raw.srt
@@ -0,0 +1,1764 @@
+1
+00:00:21,940 --> 00:00:28,180
+إذا في ال section هذا هن ان شاء الله هنتعلم ال
+
+2
+00:00:28,180 --> 00:00:34,860
+limited theorems و بعض النظريات الخاصة بالنهايات
+
+3
+00:00:34,860 --> 00:00:43,890
+فعشان نشوف النظريات هذه ناخد الأول تعريفsequence
+
+4
+00:00:43,890 --> 00:00:49,150
+of real numbers Xn نسميها bounded إذا قدرنا نلاقي
+
+5
+00:00:49,150 --> 00:00:56,110
+عدد موجة ب M بحيث إنه ال absolute value لكل عناصر
+
+6
+00:00:56,110 --> 00:01:00,510
+ال sequence أصغر من أو ساوي M هذا بكافئ
+
+7
+00:01:04,010 --> 00:01:10,550
+إن أن عناصر الـ sequence Xn تنتمي للفترة المغلقة
+
+8
+00:01:10,550 --> 00:01:19,030
+من سالب M إلى M لكل N عدد طبيعي إذن الـ sequence
+
+9
+00:01:19,030 --> 00:01:25,310
+بتكون bounded إذا قدرت أحصرها في closed interval
+
+10
+00:01:25,310 --> 00:01:29,730
+from سالب M إلى M حيث M عدد مجمع
+
+11
+00:01:32,190 --> 00:01:37,030
+النظرية هذه بتقول ان اي convergence sequence of
+
+12
+00:01:37,030 --> 00:01:41,710
+real numbers is bounded كل convergence sequence
+
+13
+00:01:41,710 --> 00:01:47,050
+بتكون ضروري انها تكون bounded ولبرهان ذلك هي افرض
+
+14
+00:01:47,050 --> 00:01:53,910
+ان sequence x in convergent to some x في R دعونا
+
+15
+00:01:53,910 --> 00:02:00,590
+ناخد y بساوي واحد هي هاد عدد موجب given من نظرية 2
+
+16
+00:02:00,590 --> 00:02:09,130
+اتنيناللي هو تعريف epsilon capital N لـ limit بقول
+
+17
+00:02:09,130 --> 00:02:13,390
+ال sequence هذه ال limit تبعتها X إذا كان لأي
+
+18
+00:02:13,390 --> 00:02:19,010
+epsilon زي هذه بقدر ألاقي عدد طبيعي يعتمد على
+
+19
+00:02:19,010 --> 00:02:23,410
+epsilon اللي هو واحد بحيث لكل N أكبر من أو ساوي
+
+20
+00:02:23,410 --> 00:02:28,570
+capital N المسافة بين XN و X أصغر من ال epsilon
+
+21
+00:02:28,570 --> 00:02:30,570
+اللي هي مخدينها واحد
+
+22
+00:02:34,370 --> 00:02:42,670
+الان absolute x in هذا عبارة عن ممكن ابدله هذا
+
+23
+00:02:42,670 --> 00:02:51,830
+بساوي absolute x in سالب x زاد x باخد
+
+24
+00:02:51,830 --> 00:02:58,840
+اتنين هدول مع بعض يعني اترحت x و رجعتby triangle
+
+25
+00:02:58,840 --> 00:03:03,520
+inequality less
+
+26
+00:03:03,520 --> 00:03:10,200
+than or equal absolute الحد الأول زاد absolute
+
+27
+00:03:10,200 --> 00:03:16,540
+الحد الثاني ومن هنا من هنا لكل n أكبر من أو ساوي
+
+28
+00:03:16,540 --> 00:03:20,680
+capital N المقدار هذا أو ال absolute value هذه
+
+29
+00:03:20,680 --> 00:03:26,790
+أصغر من واحدأذن absolute لكل n أكبر من لو ساوي
+
+30
+00:03:26,790 --> 00:03:36,730
+capital N أصغر من واحد واحد زائد absolute X تمام؟
+
+31
+00:03:36,730 --> 00:03:40,930
+الآن لو أخدت عرفت capital M على انها
+
+32
+00:03:46,510 --> 00:03:51,250
+لو عرفت capital N و capital M علي انها ال supremum
+
+33
+00:03:51,250 --> 00:03:55,890
+أو الأكبر بين الأعداد اللي هي absolute x1 و
+
+34
+00:03:55,890 --> 00:04:01,530
+absolute x2 و هكذا إلى absolute x رقم capital N
+
+35
+00:04:01,530 --> 00:04:07,170
+سالق واحد والعدد الأخير absolute x زائد واحد
+
+36
+00:04:10,460 --> 00:04:16,660
+ففي الحالة هذه ممكن نثبت أن absolute xn أصغر من أو
+
+37
+00:04:16,660 --> 00:04:27,980
+ساول عدد الموجب هذا لكل n لكل العداد الطبيعي هذا
+
+38
+00:04:27,980 --> 00:04:32,860
+بيجي من أربعة من المتبينة أربعة اللي هي هان ومن
+
+39
+00:04:32,860 --> 00:04:37,680
+التعريف capital M يعني لاحظوا أنتوا أن ال sequence
+
+40
+00:04:37,680 --> 00:04:50,680
+xnحدودها X1 X2 إلى X رقم capital N سالب واحد X رقم
+
+41
+00:04:50,680 --> 00:05:00,060
+capital N X رقم capital N زاد واحد و هكذا الان
+
+42
+00:05:00,060 --> 00:05:06,960
+لو ال N هذه كانت واحدة من الاعداد هدول المؤشر N
+
+43
+00:05:06,960 --> 00:05:14,380
+واحد من المؤشراتواحد اتنين الى capital N سالب واحد
+
+44
+00:05:14,380 --> 00:05:18,140
+ف
+
+45
+00:05:18,140 --> 00:05:23,860
+absolute ال XN هذا هيكون
+
+46
+00:05:23,860 --> 00:05:26,980
+اما absolute X واحد او absolute X اتنين او
+
+47
+00:05:26,980 --> 00:05:32,740
+absolute XN وكل واحد من هدولكل واحد من هذول أصغر
+
+48
+00:05:32,740 --> 00:05:37,020
+من أو ساوي ال supreme لهم كلهم وبالتالي هذا أكيد
+
+49
+00:05:37,020 --> 00:05:44,500
+أصغر من أو ساوي .. أصغر من أو ساوي ال N تمام؟ لو
+
+50
+00:05:44,500 --> 00:05:50,180
+كان ال N .. لو كان ال N أكبر من أو ساوي capital N
+
+51
+00:05:52,740 --> 00:05:57,480
+يعني المؤشر هذا تبع ال Xn هذي المؤشر تبع لو كان
+
+52
+00:05:57,480 --> 00:06:01,700
+أكبر يعني بسوء capital N أو capital N زائد واحد أو
+
+53
+00:06:01,700 --> 00:06:07,320
+أي عدد أكبر من capital N فمن
+
+54
+00:06:07,320 --> 00:06:11,880
+ال .. من ال implication أربع هاي و اذا كان ال N
+
+55
+00:06:11,880 --> 00:06:15,560
+هذي أكبر مما بسوء capital N فبطلع absolute Xn
+
+56
+00:06:15,560 --> 00:06:21,960
+فبطلع absolute Xn هذا أصغر من absolute X زائد واحد
+
+57
+00:06:22,790 --> 00:06:25,630
+و هذا العدد .. هذا العدد من هنا
+
+58
+00:06:28,510 --> 00:06:33,430
+أصغر من أو يساوي ال supremum له ولا باقي الأعداد
+
+59
+00:06:33,430 --> 00:06:37,870
+هذه وبالتالي أصغر من أو يساوي M إذن هذا العدد أصغر
+
+60
+00:06:37,870 --> 00:06:42,490
+من أو يساوي M إذن في كل الأحوال سواء ال N كانت من
+
+61
+00:06:42,490 --> 00:06:46,770
+واحد إلى capital N سالب واحد أو N أكبر من أو يساوي
+
+62
+00:06:46,770 --> 00:06:51,810
+capital N فطلع absolute xn أصغر من أو يساوي M حيث
+
+63
+00:06:51,810 --> 00:06:56,110
+M عدد موجب وبالتالي ال sequence xn is bounded حسب
+
+64
+00:06:56,110 --> 00:07:06,350
+التعريف تمام؟إذا هذا يبقى هنا نظرية و هذا طيب
+
+65
+00:07:06,350 --> 00:07:09,010
+في بعض الملاحظات عن نظرية هذه
+
+66
+00:07:20,320 --> 00:07:24,280
+في بعض الملاحظات الملاحظة الأولى بتقول ان ال
+
+67
+00:07:24,280 --> 00:07:29,240
+converts of theorem two six need not be true يعني
+
+68
+00:07:29,240 --> 00:07:34,060
+عكس النظرية السابقة مش شرط يكون صحيح طب ما هو عكس
+
+69
+00:07:34,060 --> 00:07:37,300
+النظرية النظرية بتقول كل convergence sequence is
+
+70
+00:07:37,300 --> 00:07:41,980
+bounded عكس ال statement هذا المفروض ان يكون كل
+
+71
+00:07:41,980 --> 00:07:46,990
+bounded sequence is convergent هذا مش صحو هاي في
+
+72
+00:07:46,990 --> 00:07:55,350
+counter example يثبت أو يوضح أن عكس نظرية 2.6 ليس
+
+73
+00:07:55,350 --> 00:08:00,930
+صحيحالسيكوانس ان الحد العام تبعها xn بساوي سالب
+
+74
+00:08:00,930 --> 00:08:06,550
+واحد قص ان السيكوانس هذي bounded ليه bounded؟ لأن
+
+75
+00:08:06,550 --> 00:08:11,130
+هاي في عدد موجب capital M بساوي واحد هذا عدد موجب
+
+76
+00:08:11,130 --> 00:08:15,670
+إذا يوجد عدد موجب اللي هو الواحد بحيث ان absolute
+
+77
+00:08:15,670 --> 00:08:20,950
+xn هذا ال absolute value العدد هذا واحد واحد أصغر
+
+78
+00:08:20,950 --> 00:08:25,490
+من اذا بساوي واحدإن ال absolute xn هي أصغر من أو
+
+79
+00:08:25,490 --> 00:08:28,590
+ساوي عدد موجة بواحد لكل n إذا ال sequence هي
+
+80
+00:08:28,590 --> 00:08:33,030
+bounded واضحة حسب التعريف الآن هذه ال sequence
+
+81
+00:08:33,030 --> 00:08:39,950
+بنثبت إنها ليست convergent إنها divergent فلبرهان
+
+82
+00:08:39,950 --> 00:08:44,430
+ذلك نعمل برهان بالتناقضأنا بدأ أثبت أن الـ
+
+83
+00:08:44,430 --> 00:08:48,070
+sequence هذه is not convergent ف assume ال
+
+84
+00:08:48,070 --> 00:08:52,530
+contrary يعني assume أنها convergent to some a
+
+85
+00:08:52,530 --> 00:08:56,490
+belongs to the real numbers افرض أنها convergent
+
+86
+00:08:56,490 --> 00:09:02,220
+وال limit تبعتها عدد aفلو أخدت epsilon بساوي واحد
+
+87
+00:09:02,220 --> 00:09:06,340
+عدد موجب إذا من تعريف ال convergence إذا يوجد
+
+88
+00:09:06,340 --> 00:09:09,220
+capital N يعتمد على ال epsilon اللي أنا مختاره
+
+89
+00:09:09,220 --> 00:09:13,500
+اللي هي الواحد بحيث لكل N أكبر من أو ساوي capital
+
+90
+00:09:13,500 --> 00:09:21,540
+N المسافة بين xn و a أصغر من epsilon الآن
+
+91
+00:09:21,540 --> 00:09:25,460
+ال N هذه اللي هي أكبر من أو ساوي capital N ممكن
+
+92
+00:09:25,460 --> 00:09:30,650
+يكون عدد فردي أو زوجيفلو كانت ال end أكبر من نفسه
+
+93
+00:09:30,650 --> 00:09:36,270
+capital N odd فردي فمن المتباينة خمسة بيصير هذا
+
+94
+00:09:36,270 --> 00:09:42,550
+العدد سالب واحد لأن N فردي فبصير absolute سالب
+
+95
+00:09:42,550 --> 00:09:50,250
+واحد سالب a أصغر من واحد و لو رفعت ال absolute
+
+96
+00:09:50,250 --> 00:09:55,010
+value فبينتج منها أن a أكبر من سالب اتنين أصغر من
+
+97
+00:09:55,010 --> 00:10:00,590
+سفرلو كرت ال N اللي هي أكبر ما يساوي capital N
+
+98
+00:10:00,590 --> 00:10:06,430
+even فهذا العدد بيصير واحد وبيصير المتباين هذا
+
+99
+00:10:06,430 --> 00:10:12,610
+absolute واحد سالب a أصغر من واحدلو رفعت .. فكيت
+
+100
+00:10:12,610 --> 00:10:15,990
+ال absolute value هنا أو حليت المتباينة هذه في a
+
+101
+00:10:15,990 --> 00:10:20,630
+فبطلع a أكبر من صفر أصغر من اتنين اذا الان
+
+102
+00:10:20,630 --> 00:10:25,110
+المتباينتين ستة و سبعة بيعطوني تناقض ليه؟ لأن
+
+103
+00:10:25,110 --> 00:10:31,010
+المتباينة الستة بتقوللي ان a ينتمي للفترة المفتوحة
+
+104
+00:10:31,010 --> 00:10:35,810
+من صفر من سالب اتنين لصفر والمتباينة سبعة بتقوللي
+
+105
+00:10:35,810 --> 00:10:41,680
+ان العدد a ينتمي للفترة المفتوحة من صفر لاتنينطب
+
+106
+00:10:41,680 --> 00:10:47,780
+هذه الفترة تقاطعها مع هذه بساوي five يعني إيه
+
+107
+00:10:47,780 --> 00:10:51,720
+ينتمي للمجموع الخالي، هذا contradiction، هذا
+
+108
+00:10:51,720 --> 00:10:59,580
+تناقض، okay تمام؟ إذا هذا التناقض بيقول إن فرضنا،
+
+109
+00:10:59,580 --> 00:11:04,540
+الفرض طبعنا إن ال sequence هذه convergent كان خطأ،
+
+110
+00:11:04,540 --> 00:11:08,160
+إن ال sequence هذه ليست convergent، ال sequence
+
+111
+00:11:08,160 --> 00:11:13,960
+هذه ليست convergentإذا هذا مثال على sequence
+
+112
+00:11:13,960 --> 00:11:16,880
+bounded but not convergent إذا مش كل bounded
+
+113
+00:11:16,880 --> 00:11:21,240
+sequence is convergent لكن العكس كمان في النظرية 2
+
+114
+00:11:21,240 --> 00:11:25,760
+-6 العكس أثبتنا أنه صحيح كل convergence sequence
+
+115
+00:11:25,760 --> 00:11:30,500
+ضروري أن تكون bounded النظرية
+
+116
+00:11:30,500 --> 00:11:35,660
+دي 2-6 الملاحظة التانية بتقول إن نظرية 2-6 هي دي
+
+117
+00:11:35,660 --> 00:11:40,040
+زي اختبار الدم اللي بيلف الأبوة ولا يثبتها
+
+118
+00:11:46,810 --> 00:11:52,550
+الملاحظة التانية بتقول إن نظرية 2.6 ممكن نستخدمها
+
+119
+00:11:52,550 --> 00:11:56,910
+تثبت إن certain sequences are divergent
+
+120
+00:12:02,100 --> 00:12:06,000
+يعني نظرية هذه ممكن نستخدمها لإثبات أن ال sequence
+
+121
+00:12:06,000 --> 00:12:11,880
+is معينة not convergent زي اختبار الدم بينفي
+
+122
+00:12:11,880 --> 00:12:16,020
+الأبوة و لا يثبتها و هنا هذه نظرية بتنفي ال
+
+123
+00:12:16,020 --> 00:12:22,060
+convergence و لا تثبته تمام مثلا
+
+124
+00:12:22,060 --> 00:12:27,040
+ال sequence in هذه ال sequence of national numbers
+
+125
+00:12:27,040 --> 00:12:32,930
+هذه ال sequence هذه bounded ولا مش boundednot
+
+126
+00:12:32,930 --> 00:12:36,210
+bounded, unbounded، الـ sequence هذه unbounded،
+
+127
+00:12:36,210 --> 00:12:43,030
+okay؟ إذا حسب نظرية 2.6، مدامها unbounded، إذا
+
+128
+00:12:43,030 --> 00:12:47,750
+لازم تكون divergent، لبرهان ذلك، افرضي ال
+
+129
+00:12:47,750 --> 00:12:52,330
+contrary، افرضي إنها convergent، إذا حسب نظرية 2
+
+130
+00:12:52,330 --> 00:12:56,250
+.6، المفروض تطلع bounded، لكنها ليست bounded
+
+131
+00:12:57,900 --> 00:13:03,860
+contradiction إذا ال sequence هذه كونها unbounded
+
+132
+00:13:03,860 --> 00:13:07,840
+حسب نظرية نين سيفتر بتطلع divergent و احنا طبعا
+
+133
+00:13:07,840 --> 00:13:12,260
+عارفين ان ال limit اللي عارفين انها divergent لأن
+
+134
+00:13:12,260 --> 00:13:17,380
+limit n as n tends to infinity بساوي infinity ليس
+
+135
+00:13:17,380 --> 00:13:23,820
+عدد حقيقي ال sequence هذه مالهاش limit okay تمام
+
+136
+00:13:30,200 --> 00:13:38,940
+تابع الـ sequence in ليش unbounded؟
+
+137
+00:13:38,940 --> 00:13:42,080
+ليش الـ sequence هذه unbounded؟ هاي ال proof
+
+138
+00:13:42,080 --> 00:13:52,280
+assume إنها bounded، in is bounded، بدنا نعمل
+
+139
+00:13:52,280 --> 00:13:59,460
+برهان بالتناقلإذا يوجد عدد مجد M أكبر من سفر بحيث
+
+140
+00:13:59,460 --> 00:14:07,280
+أنه absolute xn اللي هي M أصغر من M لكل M
+
+141
+00:14:07,280 --> 00:14:16,380
+belonging to N طيب
+
+142
+00:14:16,380 --> 00:14:18,280
+by Archimedean property
+
+143
+00:14:25,940 --> 00:14:33,220
+م أكبر من السفر بتأدي لأي عدد موجب يوجد عدد طبيعي
+
+144
+00:14:33,220 --> 00:14:42,300
+n0 ينتمي إلى n بحيث انه n0 أكبر من m طبعا
+
+145
+00:14:42,300 --> 00:14:53,400
+هذا بساوي n لأن لو سميت هذه واحد و هذه اتنين فمن
+
+146
+00:14:53,400 --> 00:15:02,440
+واحد و اتنين from واحدand اتنين بيطلع عندي N0 أكبر
+
+147
+00:15:02,440 --> 00:15:09,760
+من M و M أكبر من N0 يعني N0 أكبر من N0 هذا تناقض
+
+148
+00:15:09,760 --> 00:15:16,060
+فى عدد أكبر من نفسه مافيش لأن
+
+149
+00:15:16,060 --> 00:15:18,860
+التناقض هذا بيقول ان ال sequence هذه مش ممكن تكون
+
+150
+00:15:18,860 --> 00:15:24,740
+bounded تمام استخدمنا هنا ال Archimedean property
+
+151
+00:15:26,820 --> 00:15:32,140
+طيب نشوف الـ Limited theorems نظريات
+
+152
+00:15:32,140 --> 00:15:37,660
+النهايات أو قوانين النهايات النظرية هذه بتلخص
+
+153
+00:15:37,660 --> 00:15:44,920
+قوانين النهايات المعروفة لدينا من calculus بي وهي
+
+154
+00:15:44,920 --> 00:15:48,660
+إن لو في عندي sequence x in convergent ل x و
+
+155
+00:15:48,660 --> 00:15:52,300
+sequence ثانية y in convergent ل y و c أي عدد
+
+156
+00:15:52,300 --> 00:15:54,640
+حقيقي ف
+
+157
+00:15:56,610 --> 00:16:01,510
+الـ sequence اللي لحد الآن تبعها مجموعة الحد العام
+
+158
+00:16:01,510 --> 00:16:05,690
+تبع ال sequence xn و ال sequence yn ال sequence
+
+159
+00:16:05,690 --> 00:16:09,590
+هذا convergent و ال limit تبعتها بساوي مجموعة
+
+160
+00:16:09,590 --> 00:16:17,890
+النهايات كذلك نفس الشيء بالنسبة للفرق الآن
+
+161
+00:16:17,890 --> 00:16:20,830
+ال sequence اللي لحد الآن تبعها حاصل ضرب
+
+162
+00:16:23,930 --> 00:16:27,690
+الـ sequence xn مع الـ sequence yn هذه بتطلع
+
+163
+00:16:27,690 --> 00:16:34,290
+convergent ونهايتها بسبب حصر ضارب النهايات لو في
+
+164
+00:16:34,290 --> 00:16:38,370
+ثابت limit ثابت في ال sequence بسبب ثابت في نهاية
+
+165
+00:16:38,370 --> 00:16:45,310
+ال sequence لو كان ال zn sequence حدودها
+
+166
+00:16:45,310 --> 00:16:51,250
+كلها غير .. لا تساوي السفر وconvergent لعدد z لا
+
+167
+00:16:51,250 --> 00:16:59,260
+يساوي سفرالسيكوينس XM over ZM تظهر كونفرجنت
+
+168
+00:16:59,260 --> 00:17:05,540
+ونهايتها بساوي نهاية البسط على نهاية المقام طبعا
+
+169
+00:17:05,540 --> 00:17:12,180
+براهين الفروح كلها مطلوبة منكمفانا هاي اللي
+
+170
+00:17:12,180 --> 00:17:17,460
+برهنلكم الفرع بيه وبالمثل ممكن تجدوا الفروع الأخرى
+
+171
+00:17:17,460 --> 00:17:21,580
+موجودة برهنها في الكتاب المقرر وإذا فيش مش برهن
+
+172
+00:17:21,580 --> 00:17:29,460
+فبتبرهنوه لوحدكم فاعتبروه تمرين homework exercise
+
+173
+00:17:29,460 --> 00:17:36,480
+إذا نشوف برهان الجزء بيه في الجزء بيه عايزين نثبت
+
+174
+00:17:36,480 --> 00:17:43,250
+أنه limitيبقى ان في جزء بيه عايزين نثبت ان limit
+
+175
+00:17:43,250 --> 00:17:53,730
+xn في yn بساوي x في y فالبرهان
+
+176
+00:17:53,730 --> 00:18:01,640
+ذلكناخد المتباينة هذه أو الأعداد هذه في نهاية
+
+177
+00:18:01,640 --> 00:18:07,180
+الأمر حسب تعريف epsilon capital N للنهايات بتثبت
+
+178
+00:18:07,180 --> 00:18:11,920
+أن ال absolute value للحد العام لل sequence minus
+
+179
+00:18:11,920 --> 00:18:16,080
+ال limit المقترحة بتثبت أن ال absolute value هذه
+
+180
+00:18:16,080 --> 00:18:21,880
+أصغر من أي given epsilonفالبرهان ده عليك ببدأ هي
+
+181
+00:18:21,880 --> 00:18:29,380
+absolute xn yn minus xy هطرح من الحد هذا xn في y و
+
+182
+00:18:29,380 --> 00:18:33,940
+أرجعه فكأني ماعملتش عاجب أخد الحدين هذول مع بعض و
+
+183
+00:18:33,940 --> 00:18:37,260
+الحدين هذول مع بعض by ال triangle inequality هذا
+
+184
+00:18:37,260 --> 00:18:41,160
+أصغر من أو ساوي absolute الحد الأول زاد absolute
+
+185
+00:18:41,160 --> 00:18:47,240
+الحد التاني في الحد الأول هذا عامل مشترك xn بطلعه
+
+186
+00:18:47,750 --> 00:18:53,010
+فبصير absolute xn في absolute yn سالب y وفي هنا
+
+187
+00:18:53,010 --> 00:18:59,270
+اعمل مشترك y فبطلعه فبصير absolute الحد التاني
+
+188
+00:18:59,270 --> 00:19:06,850
+بيسوي absolute xn minus x في absolute ال y الان
+
+189
+00:19:06,850 --> 00:19:14,210
+بما ان xn converge ل xأذا حسب نظرية 2.6 الـ
+
+190
+00:19:14,210 --> 00:19:18,150
+sequence هذه بما أنها convergent فهي bounded فهي
+
+191
+00:19:18,150 --> 00:19:23,930
+bounded وبالتالي بنقدر نلاقي عدد موجب M1 بحيث أن
+
+192
+00:19:23,930 --> 00:19:26,810
+ال absolute value لحدود ال sequence أصغر من أو
+
+193
+00:19:26,810 --> 00:19:32,370
+ساوي M1 لو أخدت capital M المaximum الأكبر بين
+
+194
+00:19:32,370 --> 00:19:38,630
+العدد الموجب M1 والعدد الموجبabsolute y ف m بيطلع
+
+195
+00:19:38,630 --> 00:19:44,450
+عدد موجب لأنه أكبر من أو ساوي هذا وهذاأذا
+
+196
+00:19:44,450 --> 00:19:49,790
+المتباينة هذه بتصير absolute xny m minus xy أصغر
+
+197
+00:19:49,790 --> 00:19:55,970
+من أو ساوي absolute xn هذا هي أصغر من أو ساوي m
+
+198
+00:19:55,970 --> 00:20:01,110
+واحد و m واحد أصغر من أو ساوي capital M إذا بشيل
+
+199
+00:20:01,110 --> 00:20:06,990
+absolute xn بحط أصغر من أو ساوي capital Mفي
+
+200
+00:20:06,990 --> 00:20:12,770
+absolute y n minus y كذلك absolute y هذه هي أصغر
+
+201
+00:20:12,770 --> 00:20:18,290
+من أو يساوي capital M وبالتالي الحد هذا بيصير أصغر
+
+202
+00:20:18,290 --> 00:20:22,750
+من أو يساوي capital M بدل absolute y في absolute x
+
+203
+00:20:22,750 --> 00:20:28,350
+n minus x الآن احنا من الفرض فرضين ان ال sequence
+
+204
+00:20:28,350 --> 00:20:33,630
+x n converge ل x و ال sequence y n converge ل yإذا
+
+205
+00:20:33,630 --> 00:20:37,450
+لو أخدنا أي إبسلون لت إبسلون أكبر من السفر P given
+
+206
+00:20:37,450 --> 00:20:42,210
+بما أنه ال sequence XM converged ل X إذا يوجد N
+
+207
+00:20:42,210 --> 00:20:47,350
+واحد عدد طبيعي يعتمد على إبسلون بحيث لكل N أكبر من
+
+208
+00:20:47,350 --> 00:20:51,710
+أو ساوي capital N واحد بيطلع المسافة بين XM X أصغر
+
+209
+00:20:51,710 --> 00:20:55,790
+من إبسلون أو إبسلون على 2M يعتبر هذا هو الإبسلون
+
+210
+00:20:55,790 --> 00:21:01,070
+في التعريف لأنها دا عدد موجة وبيعتمد على إبسلونليش
+
+211
+00:21:01,070 --> 00:21:05,970
+حطينا 2M لحاجة في نفس يعقوب، عشان في النهاية أخلي
+
+212
+00:21:05,970 --> 00:21:10,610
+ال absolute value هذه أصغر من إبسلون، هنشوفها هنا،
+
+213
+00:21:10,610 --> 00:21:15,530
+بقطوة هذه طيب، إذا قلنا بما إن it's unconventional
+
+214
+00:21:15,530 --> 00:21:19,230
+ل X، فلأي إبسلون فيه capital N واحد، بحيث ال
+
+215
+00:21:19,230 --> 00:21:23,910
+implication هذه تتحققكذلك بما أن yn converge ل y
+
+216
+00:21:23,910 --> 00:21:28,270
+إذا for the same إبسلون لنفس الإبسلون العدد موجب
+
+217
+00:21:28,270 --> 00:21:33,090
+اللي هو معطى مسبقا يوجد عدد طبيعي N2 يعتمد على
+
+218
+00:21:33,090 --> 00:21:37,770
+إبسلون بحيث لكل N أكبر منه ساوي capital N2 بيطلع
+
+219
+00:21:37,770 --> 00:21:42,750
+absolute yn minus y أصغر من إبسلون على 2M
+
+220
+00:21:45,110 --> 00:21:49,830
+الان خلّينا ناخد capital N الأكبر من N واحد و N
+
+221
+00:21:49,830 --> 00:21:54,610
+اتنين هذي و هذي كلاهما يعتمد على epsilon اذا
+
+222
+00:21:54,610 --> 00:22:00,490
+capital N تعتمد على epsilon و عدد طبيعي الان لكل
+
+223
+00:22:00,490 --> 00:22:07,370
+عدد طبيعي أكبر من أو ساوي capital N هذا بيطلع و
+
+224
+00:22:07,370 --> 00:22:11,570
+capital N هي من تعريفه هذا أكبر من أو ساوي capital
+
+225
+00:22:11,570 --> 00:22:20,710
+N واحدو أيضا أكبر من أوساوي capital N2أذن بيطلع
+
+226
+00:22:20,710 --> 00:22:27,350
+عندي absolute xn yn minus xy أصغر من أو يساوي M و
+
+227
+00:22:27,350 --> 00:22:32,870
+absolute yn minus y بما أن small n هذي أكبر من أو
+
+228
+00:22:32,870 --> 00:22:38,410
+يساوي capital N اتنين أذن من هنا بيطلع الفرق هذا
+
+229
+00:22:38,410 --> 00:22:43,590
+أصغر من epsilon على اتنين M أصغر من epsilon على
+
+230
+00:22:43,590 --> 00:22:49,900
+اتنين M في Mو كذلك small n هذه أكبر من أو ساوي n
+
+231
+00:22:49,900 --> 00:22:55,460
+واحد لما small n أكبر من أو ساوي n واحد بطلع بقدر
+
+232
+00:22:55,460 --> 00:23:00,880
+أشيل absolute xn minus x و أحط أصغر من epsilon على
+
+233
+00:23:00,880 --> 00:23:08,660
+اتنين M الان M بتروح مع M و M بتروح مع M بضل عندي
+
+234
+00:23:08,660 --> 00:23:11,700
+epsilon على اتنين زاد epsilon على اتنين بطلع
+
+235
+00:23:11,700 --> 00:23:17,540
+epsilon تمام؟ و هذا اللي بدنا ي ..فاكرين في بداية
+
+236
+00:23:17,540 --> 00:23:21,160
+البرنامج احنا عايزين في النهاية نثبت ان ال
+
+237
+00:23:21,160 --> 00:23:25,700
+absolute value هذه اصغر من epsilon اصغر من epsilon
+
+238
+00:23:27,020 --> 00:23:30,340
+for any given epsilon وفعلا هي for any epsilon
+
+239
+00:23:30,340 --> 00:23:34,760
+أكبر من 0 أثبتنا أن يوجد capital N أعداد طبيعية
+
+240
+00:23:34,760 --> 00:23:38,580
+تامد على epsilon بحيث لكل N أكبر من أو ساوي
+
+241
+00:23:38,580 --> 00:23:43,000
+capital N طلع ال absolute value للفرق هذا أصغر من
+
+242
+00:23:43,000 --> 00:23:48,820
+epsilon، إذن حسب التعريف بطلع limit xn في yn بساوي
+
+243
+00:23:48,820 --> 00:23:55,170
+x في y، تمام؟إن الهدف برهن برهان بكمل برهان جزء
+
+244
+00:23:55,170 --> 00:23:59,730
+بيه بالمثل ممكن تبرهن الأجزاء الأخرى فمطلوب منكم
+
+245
+00:23:59,730 --> 00:24:04,130
+اتبرهنوها اعتقد انه في الكتاب المقرر في نظرية
+
+246
+00:24:04,130 --> 00:24:11,010
+تلاتة اتنين تلاتة in the textbook فيها براهين بعض
+
+247
+00:24:11,010 --> 00:24:16,110
+الأجزاء أو ربما كلهم فحاولوا تكتبوا البرهان الأول
+
+248
+00:24:16,110 --> 00:24:19,630
+لوحدكم وإذا ماعرفتوش اخرقوا البرهان في الكتاب
+
+249
+00:24:19,630 --> 00:24:27,040
+حاولوا تفهموهإذا في أي شيء مش واضح اكتب البرهان في
+
+250
+00:24:27,040 --> 00:24:34,740
+ورقة و أعطينيها عشان أصلح لكيها okay
+
+251
+00:24:34,740 --> 00:24:42,560
+إذا هذه قوانين النهايات هذه قوانين النهايات طيب في
+
+252
+00:24:42,560 --> 00:24:47,480
+كمان قوانين أخرى أو خواس أخرى للنهايات
+
+253
+00:24:52,790 --> 00:24:57,850
+فمثلا في النظرية هذه النظرية هذه بتقول لو في عندي
+
+254
+00:24:57,850 --> 00:25:01,810
+convergence sequence xn convergence ل x و ال
+
+255
+00:25:01,810 --> 00:25:06,750
+sequence xn حدودها غير سالبة كل حدودها غير سالبة ف
+
+256
+00:25:06,750 --> 00:25:11,630
+ال limit تبعتها ايضا لازم تكون غير سالبة و البرهان
+
+257
+00:25:11,630 --> 00:25:16,590
+سهل by contradiction prove by contradiction assume
+
+258
+00:25:16,590 --> 00:25:21,170
+on the contrary ان ال x سالبة
+
+259
+00:25:24,860 --> 00:25:29,200
+أنا عايز أثبت x أكبر من أو ساوى 0 النافي تبعها x
+
+260
+00:25:29,200 --> 00:25:36,380
+أصغر من 0 صح؟الان خدي epsilon بالساوي سالب x إذا
+
+261
+00:25:36,380 --> 00:25:40,300
+ال x سالب هذا سالب x عدد موجبة، إذا هاي الآن في
+
+262
+00:25:40,300 --> 00:25:44,180
+عندي epsilon موجبة وفي عندي من الفرض x n converge
+
+263
+00:25:44,180 --> 00:25:49,440
+ل x إذا من تعريف epsilon capital N للنهايات بما أن
+
+264
+00:25:49,440 --> 00:25:52,680
+x n converge ل x إذا يوجد capital N يعتمد على ال
+
+265
+00:25:52,680 --> 00:25:56,780
+epsilon بحيث لكل n أكبر لو ساوي capital Nالمسافة
+
+266
+00:25:56,780 --> 00:26:02,540
+بين xn و x أصغر من y طيب فك ال absolute value هذه
+
+267
+00:26:02,540 --> 00:26:09,260
+شيلها بصير المتباينة هذه xn minus x أصغر من y أكبر
+
+268
+00:26:09,260 --> 00:26:10,420
+من سالب y
+
+269
+00:26:14,450 --> 00:26:18,770
+خدي هذا جزء من المتباينة و ادى x على الناحية
+
+270
+00:26:18,770 --> 00:26:24,410
+التانية فبصير هذا بيقدي ان xn أصغر من أو ساوي x
+
+271
+00:26:24,410 --> 00:26:29,290
+زاد epsilon الكلام هذا صحيح لكل n أكبر من أو ساوي
+
+272
+00:26:29,290 --> 00:26:32,410
+capital N تمام؟
+
+273
+00:26:34,150 --> 00:26:39,850
+طيب الان خدي N خدي N هنا خدي ال N الصغيرة هذه
+
+274
+00:26:39,850 --> 00:26:44,130
+بيساوي capital N فبصير X capital N أصغر من ما
+
+275
+00:26:44,130 --> 00:26:48,970
+بيساوي X زاد Epsilon صح عوض عن ال Epsilon ال
+
+276
+00:26:48,970 --> 00:26:56,310
+Epsilon بيساوي سالب X إذن عوض عن X بسالب عن
+
+277
+00:26:56,310 --> 00:27:02,260
+Epsilon سالب X فهذا بيطلع sevenإذا أنا طلع عندي x
+
+278
+00:27:02,260 --> 00:27:06,960
+رقم capital N أصغر من صفر وهذا جدّيني
+
+279
+00:27:06,960 --> 00:27:13,180
+contradiction ليه؟ لأنه أنا فارق في النظرية أن كل
+
+280
+00:27:13,180 --> 00:27:19,000
+حدود ال sequence كلهم غير سالبين كلهم يعني حدود
+
+281
+00:27:19,000 --> 00:27:24,750
+غير سالبةفكيف طلع الحد رقم N capital N سالب هذا
+
+282
+00:27:24,750 --> 00:27:30,910
+بتناقض مع الفرض هذا بكمل البرهان تمام واضح البرهان
+
+283
+00:27:30,910 --> 00:27:35,390
+طبعا على طريقة هذه إذا لو ال sequence كانت
+
+284
+00:27:35,390 --> 00:27:39,630
+convergent وكل حدودها غير سالبة فنهايتها أيضا
+
+285
+00:27:39,630 --> 00:27:40,850
+هتكون غير سالبة
+
+286
+00:27:46,250 --> 00:27:49,370
+طب لو كانت ال sequence convergent و حدودها غير
+
+287
+00:27:49,370 --> 00:27:54,610
+موجبة كل حدودها غير موجبة فنهايتها ايضا هتكون غير
+
+288
+00:27:54,610 --> 00:28:02,430
+موجبة و البرهان مشابه لبرهان نظرية السابقة النظرية
+
+289
+00:28:02,430 --> 00:28:08,730
+اللي بعدها لو
+
+290
+00:28:08,730 --> 00:28:13,390
+في عندي two sequences و التنتين convergentوالحد
+
+291
+00:28:13,390 --> 00:28:16,130
+العاملة الأولى أصغر من أو ساوي الحد العاملة
+
+292
+00:28:16,130 --> 00:28:22,110
+التانية فنهاية الأولى أصغر من أو ساوي نهاية
+
+293
+00:28:22,110 --> 00:28:31,520
+التانية والبرهان تطبيق مباشر على نظرية اللى فاتتلو
+
+294
+00:28:31,520 --> 00:28:39,000
+اخدت انا عندي احنا فرضين ان xn أصغر من أو يساوي yn
+
+295
+00:28:39,000 --> 00:28:47,380
+لكل n هذا معناه ان yn minus xn أكبر من أو يساوي
+
+296
+00:28:47,380 --> 00:28:49,700
+سفر لكل n
+
+297
+00:28:53,620 --> 00:28:59,620
+فلو أخدت zn عرفت zn على أنه الفرق بين yn و xn
+
+298
+00:28:59,620 --> 00:29:03,580
+فقلنا الفرق هذا غير سالب لكل n إذن هذه ال sequence
+
+299
+00:29:03,580 --> 00:29:09,700
+جديدة حد العام تبعها zn وهذا الحد العام غير سالب
+
+300
+00:29:09,700 --> 00:29:15,530
+لكل nوالـ sequence xn convergent وyn convergent
+
+301
+00:29:15,530 --> 00:29:20,510
+حسب قوانين
+
+302
+00:29:20,510 --> 00:29:27,790
+النهايات limit الفرق هذا موجود أو بساوي فرق
+
+303
+00:29:27,790 --> 00:29:33,430
+النهايات إذا ال sequence zn حدودها كلها غير سالبة
+
+304
+00:29:33,430 --> 00:29:36,950
+وconvergent لأنها الفرق بين two convergent
+
+305
+00:29:36,950 --> 00:29:37,530
+sequences
+
+306
+00:29:40,870 --> 00:29:45,350
+وبالتالي حسب النظرية السابقة إذا ال limit ال
+
+307
+00:29:45,350 --> 00:29:50,890
+sequence zn هذه بتطلع غير سالبة طب limit ال zn عسب
+
+308
+00:29:50,890 --> 00:29:55,830
+قوانين النهيات بيساوي limit yn سالب limit xn و ادى
+
+309
+00:29:55,830 --> 00:29:59,230
+هذه عن ناحية التانية فبيطلع limit yn أكبر من لو
+
+310
+00:29:59,230 --> 00:30:06,090
+ساوي limit xn وهذا هو المظلومة okay تمام؟ راضى؟ دى
+
+311
+00:30:06,090 --> 00:30:06,730
+اي سؤال؟
+
+312
+00:30:12,550 --> 00:30:22,370
+النتيجة هذه بتقول
+
+313
+00:30:22,370 --> 00:30:26,430
+لو كان في عندي sequence و ال sequence هذي
+
+314
+00:30:26,430 --> 00:30:35,670
+convergent و الحدود تبعتها معصورة بين a و b
+
+315
+00:30:39,050 --> 00:30:45,870
+فلازم نهايتها ايضا .. نهايتها تطلع محصورة بين
+
+316
+00:30:45,870 --> 00:30:53,930
+العددين a وb فبرهان
+
+317
+00:30:53,930 --> 00:31:03,850
+النظرية هذه بنطبق نظرية 2.9 مرتين مرة
+
+318
+00:31:04,750 --> 00:31:10,170
+على الـ sequence الثابتة a اللي الحدود تبعتها أصغر
+
+319
+00:31:10,170 --> 00:31:15,310
+من حدود ال sequence xn هذه convergence نهايتها a
+
+320
+00:31:15,310 --> 00:31:24,470
+وهذه convergence نهايتها a إذا limit limit ال
+
+321
+00:31:24,470 --> 00:31:29,710
+sequence الثابتة a بساوي a أصغر من أو ساوي limit
+
+322
+00:31:29,710 --> 00:31:32,090
+ال sequence xn
+
+323
+00:31:34,930 --> 00:31:38,930
+كذلك أنا عندي الـ sequence xn من الفرض أنا عندي
+
+324
+00:31:38,930 --> 00:31:42,770
+الـ sequence xn أصغر من أو ساوي الـ sequence اللي
+
+325
+00:31:42,770 --> 00:31:48,330
+الحد العام تبعها ثابت بيه إذا ال limit حسب النظرية
+
+326
+00:31:48,330 --> 00:31:52,570
+8 إذا ال limit xn أصغر من أو ساوي limit ال
+
+327
+00:31:52,570 --> 00:31:57,530
+sequence اللي الحد العام تبعها ثابت اللي هو بيه
+
+328
+00:31:59,560 --> 00:32:05,040
+من هنا بطلع الـ delimit xn هي أكبر من أو ساوي a
+
+329
+00:32:05,040 --> 00:32:14,280
+أصغر من أو ساوي b و هو المفروض في
+
+330
+00:32:14,280 --> 00:32:22,180
+ال squeeze theorem أو نظرية الـ sandwich في
+
+331
+00:32:22,180 --> 00:32:25,160
+ناس يسموها squeeze theorem في ناس يسموها sandwich
+
+332
+00:32:25,160 --> 00:32:31,450
+theorem هذه في التفاضل والتكاملبسموها sandwich
+
+333
+00:32:31,450 --> 00:32:39,070
+sandwich
+
+334
+00:32:39,070 --> 00:32:42,750
+firm أو
+
+335
+00:32:42,750 --> 00:32:46,990
+squeeze firm فالنظرية هذه بتقول لو في عندي three
+
+336
+00:32:46,990 --> 00:32:54,710
+sequences XN, YN و ZN و العلاقة بينهم هكذا YN
+
+337
+00:32:54,710 --> 00:33:02,050
+محصورة بين XN و ZNو لو كانت الـ sequences الحدود
+
+338
+00:33:02,050 --> 00:33:06,790
+العامة تبعتها على الأطراف convergent يعني ال
+
+339
+00:33:06,790 --> 00:33:10,530
+sequence x in convergent و ال sequence z in
+
+340
+00:33:10,530 --> 00:33:14,150
+convergent و التنتين ال limit تبعتهم متساوية
+
+341
+00:33:16,990 --> 00:33:22,390
+فلابد أو لازم أن الـ sequence المحصورة بينهم بتكون
+
+342
+00:33:22,390 --> 00:33:27,130
+أيضًا convergent ونهايتها بتساوي القيمة المشتركة
+
+343
+00:33:27,130 --> 00:33:33,070
+لنهايات للنهايات limit yn بتكون موجودة ونهايتها
+
+344
+00:33:33,070 --> 00:33:39,910
+بتساوي نهاية xn ونهاية zn البرهان
+
+345
+00:33:39,910 --> 00:33:41,090
+برضه مش صعب
+
+346
+00:33:46,210 --> 00:33:50,630
+أحنا فرضين أن limit sequence x in موجودة و limit
+
+347
+00:33:50,630 --> 00:33:57,130
+sequence z in موجودة وتنتين متساويات فدعونا نسمي
+
+348
+00:33:57,130 --> 00:34:05,230
+ال limit هذه مشتركة a عدد a الأن ناخد let epsilon
+
+349
+00:34:05,230 --> 00:34:12,210
+أكبر من صفر bبما أن ال sequence xn converge ل a،
+
+350
+00:34:12,210 --> 00:34:15,430
+إذا يوجد عدد طبيعي capital N يعتمد على إبسلون،
+
+351
+00:34:15,430 --> 00:34:20,570
+بحيث لكل N أكبر من وسوء capital N، المسافة بين xn
+
+352
+00:34:20,570 --> 00:34:27,930
+و a أصغر من إبسلونكذلك بما أن ال sequence zm
+
+353
+00:34:27,930 --> 00:34:33,630
+converge ل a إذا بقدر ألاقي لكل .. لنفس ال epsilon
+
+354
+00:34:33,630 --> 00:34:38,650
+.. لنفس ال epsilon بقدر ألاقي capital N ممكن ال
+
+355
+00:34:38,650 --> 00:34:42,970
+capital N مختلف عن ال capital N الأولى نية ففي
+
+356
+00:34:42,970 --> 00:34:46,830
+الحالة هذه باخد capital N هذا ال maximum لل N
+
+357
+00:34:46,830 --> 00:34:49,490
+الأولى و التانية زي ما شوفنا في القرآن السابق
+
+358
+00:34:50,870 --> 00:34:54,170
+وبالتالي لكل n أكبر من مسافة capital N بقدر أخلي
+
+359
+00:34:54,170 --> 00:35:01,530
+المسافة هذه أصغر من epsilon الآن
+
+360
+00:35:01,530 --> 00:35:08,190
+من الفرض و من ال implication تمانية نحصل
+
+361
+00:35:08,190 --> 00:35:11,030
+على ال implication الجديدة هذه
+
+362
+00:35:18,350 --> 00:35:25,630
+يعني من هنا انا عندي xn minus a أصغر من ي وطبعا
+
+363
+00:35:25,630 --> 00:35:33,090
+أكبر من سالب ي ومن هنا انا عندي zn minus a أصغر من
+
+364
+00:35:33,090 --> 00:35:36,770
+ي أكبر من سالب ي
+
+365
+00:35:39,750 --> 00:35:45,170
+فهي xn minus a أصغر من إبسلون لكل n أكبر من أو
+
+366
+00:35:45,170 --> 00:35:54,810
+ساوية capital N إذا هذه هذه هي من هنا أو
+
+367
+00:35:54,810 --> 00:36:00,210
+هذه عفوا أنا أخدت الجزء هذه سالب إبسلون
+
+368
+00:36:04,860 --> 00:36:12,860
+-Epsilon أصغر من Xn-A وأنا عندي Xn أصغر من يساوي
+
+369
+00:36:12,860 --> 00:36:20,000
+Yn إذا لو طرحت A من هنا وطرحت A من هنا فبيطلع عندي
+
+370
+00:36:20,000 --> 00:36:27,080
+Xn-A أصغر من يساوي Yn-Aو نفس الحاجة لو طرحت a من
+
+371
+00:36:27,080 --> 00:36:32,620
+هنا طب يطلع y n minus a أصغر من أو ساوي z n minus
+
+372
+00:36:32,620 --> 00:36:41,320
+a و من هنا هاندي z n minus a من هذا الجزء z n
+
+373
+00:36:41,320 --> 00:36:43,460
+minus a أصغر من إبسن
+
+374
+00:36:48,860 --> 00:36:54,460
+إذاً هذه الـ implication بتقول باختصار لكل N أكبر
+
+375
+00:36:54,460 --> 00:37:00,240
+من أو ساوي capital N أنا عندي طلع يطلع عندي YN
+
+376
+00:37:00,240 --> 00:37:08,620
+minus A أصغر من أبسلون أكبر من سالب أبسلون أو
+
+377
+00:37:08,620 --> 00:37:13,140
+absolute YN minus A أصغر من أبسلون لكل N أكبر من
+
+378
+00:37:13,140 --> 00:37:18,280
+أو ساوي capital Nأذن هين أثبتنا أنه for any
+
+379
+00:37:18,280 --> 00:37:22,860
+epsilon أكبر من سفر يوجد capital N يعتمد على
+
+380
+00:37:22,860 --> 00:37:27,080
+epsilon عدد طبيعي بحيث لكل N أكبر من أو ساوي
+
+381
+00:37:27,080 --> 00:37:30,720
+capital N ال absolute value هذي أصغر من epsilon
+
+382
+00:37:30,720 --> 00:37:34,200
+اذا by epsilon capital N definition of limit بطلع
+
+383
+00:37:34,200 --> 00:37:39,640
+عندي limit هذا معناه ان limit ال sequence yn as n
+
+384
+00:37:39,640 --> 00:37:45,530
+tends to infinity بتساوي العدد Aو هذا اللي بدنا
+
+385
+00:37:45,530 --> 00:37:51,790
+يعني .. هذا اللي بدنا يعني .. okay تمام؟ إذن هذا
+
+386
+00:37:51,790 --> 00:37:55,790
+هو برهان ال sandwich أو ال squeeze ال theorem
+
+387
+00:37:55,790 --> 00:38:00,710
+تمام؟
+
+388
+00:38:00,710 --> 00:38:04,810
+هاي ناخد بعض ال .. بعض الأمثلة .. بعض الأمثلة
+
+389
+00:38:20,690 --> 00:38:25,770
+عايزين نثبت انه limit ال sequence لحد العام تبعها
+
+390
+00:38:25,770 --> 00:38:31,910
+الكاسر هذا بساوة ستة فزي
+
+391
+00:38:31,910 --> 00:38:38,270
+ما كنا نعمل في تفاضل ألف أو با بنجسم بسط مقام على
+
+392
+00:38:38,270 --> 00:38:46,130
+n لأكبر أساللي هو n تربية فبصير عندي المقدار ده
+
+393
+00:38:46,130 --> 00:38:49,790
+بيساوي اتنين على n على واحد زاد .. على واحد ..
+
+394
+00:38:49,790 --> 00:38:54,490
+واحد على n تربية زاد واحد انا عندي ال sequence
+
+395
+00:38:54,490 --> 00:38:59,190
+واحد على n ال limit تبعتها سفر اذا اتنين على n ال
+
+396
+00:38:59,190 --> 00:39:06,360
+limit تبعتها سفرالمقام واحد على M تربية نهايتها
+
+397
+00:39:06,360 --> 00:39:10,580
+سفر اذا واحد زاد واحد على M تربية نهايتها واحد زاد
+
+398
+00:39:10,580 --> 00:39:19,860
+سفر بطلع واحد لا يساوي سفر تمام اذا ال limit الكسر
+
+399
+00:39:19,860 --> 00:39:27,000
+هذا بساوي limit الكسر اللي هان الان الكسر هذا
+
+400
+00:39:27,000 --> 00:39:31,490
+بتكون من bust مقام ال bustالمقام عبارة عن الـ
+
+401
+00:39:31,490 --> 00:39:35,670
+sequence 2 على M المقام عبارة عن الـ sequence الحد
+
+402
+00:39:35,670 --> 00:39:41,130
+العام تبعه 1 زاد 1 على M تربيه الآن limit ال
+
+403
+00:39:41,130 --> 00:39:44,950
+sequence في المقام شفنا بالساوي واحد لا تساوي سفر
+
+404
+00:39:44,950 --> 00:39:50,370
+عشان هيك قدرت اقول ان limit limit الكسر بساوي
+
+405
+00:39:50,370 --> 00:39:53,330
+limit البسط على limit المقام لكن لو limit المقام
+
+406
+00:39:53,330 --> 00:39:59,590
+بالساوي سفر فما بقدر استخدم القانون هذا ماشي الحل
+
+407
+00:40:02,990 --> 00:40:06,970
+limit الكثر هذا بيساوي limit الكثر اللي هان وهذا
+
+408
+00:40:06,970 --> 00:40:11,170
+طبعا لأن limit المقام بيساويش سفر فبساوي limit ال
+
+409
+00:40:11,170 --> 00:40:15,170
+bus على limit المقام حسب قوانين النهايات limit ال
+
+410
+00:40:15,170 --> 00:40:20,610
+bus سفر limit المقام واحد اذا ال limit في النهاية
+
+411
+00:40:20,610 --> 00:40:29,880
+تمتها سفر تمام المثال التاني هذابإنه نثبت ان الـ
+
+412
+00:40:29,880 --> 00:40:35,960
+sequence الحد العام تبعها xn بساوي sin n على n ال
+
+413
+00:40:35,960 --> 00:40:41,520
+limit تبعتها بساوي 0 طيب احنا عارفين من تفاض الألف
+
+414
+00:40:41,520 --> 00:40:46,060
+ان absolute sin x أصغر من أو ساوي واحد لكل x أخر
+
+415
+00:40:46,060 --> 00:40:54,660
+او وبالتالي لكل عدد طبيعي n absolute او sin n أكبر
+
+416
+00:40:54,660 --> 00:41:00,960
+من أو ساوي سالب واحدأصغر من أو يساوي واحد طيب لكل
+
+417
+00:41:00,960 --> 00:41:07,460
+عدد طبيعي ان واحد على ان عدد موجب فلو ضربنا
+
+418
+00:41:07,460 --> 00:41:12,800
+المتباينة هذه في العدد الموجب واحد على ان فبصير
+
+419
+00:41:12,800 --> 00:41:13,820
+شكلها هكذا
+
+420
+00:41:18,240 --> 00:41:21,680
+فبتصير ال sequence بعد ما نضربها أو المتباينة
+
+421
+00:41:21,680 --> 00:41:25,120
+الأخيرة بعد ما نضربها في واحد على N بصير شكلها
+
+422
+00:41:25,120 --> 00:41:30,240
+هكذا سالب واحد على N أصلا ما يساوي sign N على N
+
+423
+00:41:30,240 --> 00:41:35,120
+أصلا ما يساوي واحد على N طيب ال sequence هذه ال
+
+424
+00:41:35,120 --> 00:41:42,860
+limit تبعتها سفر و ال sequence هذه أيضا نهايتها
+
+425
+00:41:42,860 --> 00:41:45,960
+سالب واحد في سفر بطلع سفر
+
+426
+00:41:49,290 --> 00:41:52,830
+إذا by squeeze theorem أو by sandwich theorem
+
+427
+00:41:52,830 --> 00:41:56,210
+limit ال sequence اللي في الوسط اللي محصورة في
+
+428
+00:41:56,210 --> 00:42:03,950
+النصر بيطلع أيضا موجودة و بيساوي السفر okay تمام
+
+429
+00:42:03,950 --> 00:42:09,150
+طب لو كانت ال limit هذه بيساوي سفر وهذه بيساوي
+
+430
+00:42:09,150 --> 00:42:13,670
+واحد ماقدرش أطبق ال squeeze theorem لازم ال
+
+431
+00:42:13,670 --> 00:42:17,630
+sequences اللي على أطرافمش بس يكونوا convergent
+
+432
+00:42:17,630 --> 00:42:23,390
+ويلهم نفس ال limit تكون نفس القيمة okay تمام؟ اذا
+
+433
+00:42:23,390 --> 00:42:30,690
+يعني هذا .. هذا بعض الأمثلة المرة الجاية طبعا
+
+434
+00:42:30,690 --> 00:42:37,350
+هنكمل .. اذا
+
+435
+00:42:37,350 --> 00:42:42,330
+المرة الجاية هنكمل نشوف بعض نظريات النهايات و ..
+
+436
+00:42:43,750 --> 00:42:50,310
+هنخلص اللي هو section اتنين اتنين مش يعني ضايل
+
+437
+00:42:50,310 --> 00:42:58,470
+كتير و هحللكم بعض التمرين و ناخد ال homework ل
+
+438
+00:42:58,470 --> 00:43:03,070
+section تلات اتنين okay تمام فحاولوا تحضروا هذا
+
+439
+00:43:03,070 --> 00:43:12,050
+الكلام للمحاضرة الجاية و نشوف مع بعض الحاجات هذهو
+
+440
+00:43:12,050 --> 00:43:18,550
+نشرحها okay إذا انتهت المحاضرة نكمل ان شاء الله
+
+441
+00:43:18,550 --> 00:43:19,450
+المرة الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..59b84f3a95df3d13fcfcc5c41951533028823fe2
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_postprocess.srt
@@ -0,0 +1,1368 @@
+1
+00:00:21,190 --> 00:00:26,170
+السلام عليكم في المحاضرة هذه ان شاء الله هناخد
+
+2
+00:00:26,170 --> 00:00:32,530
+section رقم اتنين شبطر اربعة اللي هو section اربعة
+
+3
+00:00:32,530 --> 00:00:39,250
+اتنين عنوانه limit theorems او قوانين النهايات لل
+
+4
+00:00:39,250 --> 00:00:45,070
+functions او الدوالة جبل ما نتعرض ل limit theorems
+
+5
+00:00:45,070 --> 00:00:53,310
+او قوانين النهايات للدوال بناخد تعريفما معنى ان
+
+6
+00:00:53,310 --> 00:00:59,410
+الـ function معرفة على subset of R A ما معنى انها
+
+7
+00:00:59,410 --> 00:01:06,230
+تكون bounded على جوار لنقطة C هي في C cluster
+
+8
+00:01:06,230 --> 00:01:10,610
+point للمجال تبع الدالة ما معنى ان الدالة تكون
+
+9
+00:01:10,610 --> 00:01:19,650
+bounded محصورة على جوارلنقطة C هذا معناه ان يوجد
+
+10
+00:01:19,650 --> 00:01:29,150
+Delta نبرهود جوار Delta لنقطة C بعمق Delta ويوجد
+
+11
+00:01:29,150 --> 00:01:34,570
+عدد موجب M بحيث ان ال absolute value لقيم الدالة
+
+12
+00:01:34,570 --> 00:01:39,950
+أصغر من أو ساوي العدد الموجب لكل X في مجال الدالة
+
+13
+00:01:39,950 --> 00:01:47,010
+واللي موجود في ال Delta نبرهود لل Cإذا قدرت ألاقي
+
+14
+00:01:47,010 --> 00:01:51,210
+delta neighborhood للنقطة C بحيث وعدد موجب بحيث
+
+15
+00:01:51,210 --> 00:01:55,490
+المتباين هذا تتحقق لكل X ال delta neighborhood
+
+16
+00:01:55,490 --> 00:02:00,710
+وطبعا هو موجود في ال A فبقول المدالة is bounded on
+
+17
+00:02:00,710 --> 00:02:03,190
+a neighborhood of C
+
+18
+00:02:05,840 --> 00:02:11,200
+النظرية هذه بتقول إنه لو أنا فيه أندي فانكشن زي
+
+19
+00:02:11,200 --> 00:02:16,740
+هذه و ال limit تبعتها عند ال cluster point C exist
+
+20
+00:02:16,740 --> 00:02:22,340
+فلابد ضروري جدا أن من الضروري أن تكون الدالة
+
+21
+00:02:22,340 --> 00:02:30,000
+bounded على جوار لنخطة C okay فتعالوا نلفم الكلام
+
+22
+00:02:30,000 --> 00:02:35,410
+هذا left epsilon بساوي واحد عدد موجببما أنه من
+
+23
+00:02:35,410 --> 00:02:40,050
+الفرض بما أنه limit f of x as x tends to z بيساوي
+
+24
+00:02:40,050 --> 00:02:44,870
+عدد L إذا حسب تعريف الـ epsilon دلتا لل limit of
+
+25
+00:02:44,870 --> 00:02:47,690
+function there exists delta that depends on
+
+26
+00:02:47,690 --> 00:02:51,410
+epsilon اللي هو واحد عدد موجب بحيث ال implication
+
+27
+00:02:51,410 --> 00:02:56,230
+هي تتحقق اللي أنا باستخدام ال triangle inequality
+
+28
+00:02:56,230 --> 00:03:01,330
+absolute f of xبساوي absolute F of X minus L
+
+29
+00:03:01,330 --> 00:03:06,850
+لانطرحت L رجعتها بعدين باخد اتنين هدول مع بعض By
+
+30
+00:03:06,850 --> 00:03:10,030
+the triangle equality هذا أصغر من F of X minus L
+
+31
+00:03:10,030 --> 00:03:14,510
+زاد absolute L وهذا من فوق أصغر من واحد زاد
+
+32
+00:03:14,510 --> 00:03:22,930
+absolute هذا الكلام صحيح لكل X لكل X أنتمي ل A لكل
+
+33
+00:03:22,930 --> 00:03:27,590
+X أنتمي ل A وأيضا
+
+34
+00:03:29,110 --> 00:03:39,530
+لكل X مختلفة عن الـC إذا الـX هنا لاتساوي C وأيضا
+
+35
+00:03:39,530 --> 00:03:45,170
+هذا من هنا X لاتساوي C ومن هنا هذا معناه absolute
+
+36
+00:03:45,170 --> 00:03:50,870
+X سالب C أصغر من Delta معناته الـX تنتمي للـDelta
+
+37
+00:03:50,870 --> 00:03:58,950
+مبروض لـCتمام ان هذا الكلام صحيح لكل x في a مختلف
+
+38
+00:03:58,950 --> 00:04:05,090
+عن ال c و لكل و في نفس الوقت موجود ال x في ال
+
+39
+00:04:05,090 --> 00:04:12,010
+delta neighborhood ل c طيب
+
+40
+00:04:12,010 --> 00:04:20,600
+احنا بما نثبت انه بما نثبت ان ال absolute valueأو
+
+41
+00:04:20,600 --> 00:04:35,240
+جدنا delta neighborhood لـC نحن
+
+42
+00:04:35,240 --> 00:04:43,120
+عايزين نثبت أن absolute F of X أصغر منها يساوي Mأو
+
+43
+00:04:43,120 --> 00:04:47,860
+there exists M أكبر من السفر بحيث أن absolute F of
+
+44
+00:04:47,860 --> 00:04:55,540
+X أصغر من أو ساوي M لكل X هنتمي إلى A تقاطع D
+
+45
+00:04:55,540 --> 00:05:03,240
+Delta of C مش هيك تعريف اللي هو المتبين هذا طيب we
+
+46
+00:05:03,240 --> 00:05:04,300
+have two cases
+
+47
+00:05:11,240 --> 00:05:17,160
+كاس واحد إذا كانت الـ C تنتمي إلى A ال cluster
+
+48
+00:05:17,160 --> 00:05:22,200
+point C هي ال cluster point ال C is a cluster
+
+49
+00:05:22,200 --> 00:05:29,680
+point لست A فممكن ال C تنتمي ل A و ممكن ما تنتميش
+
+50
+00:05:29,680 --> 00:05:36,000
+إذا في أنتي حالتين لو كانت ال C تنتمي ل A in this
+
+51
+00:05:36,000 --> 00:05:47,190
+case in this casechoose M بيساوي
+
+52
+00:05:47,190 --> 00:05:59,230
+ال maximum ل absolute F of C و 1 زاد absolute L
+
+53
+00:05:59,230 --> 00:06:03,790
+then
+
+54
+00:06:03,790 --> 00:06:08,630
+الحالة هذه absolute F of X
+
+55
+00:06:11,500 --> 00:06:16,520
+إذا كانت الـ X هذه هي الـ C ف absolute F of X
+
+56
+00:06:16,520 --> 00:06:21,800
+بساوي absolute F of C وبالتالي أصغر من أو يساوي
+
+57
+00:06:21,800 --> 00:06:28,400
+absolute F of C اللي هي أصغر من أو يساوي M وإذا
+
+58
+00:06:28,400 --> 00:06:34,660
+كانت الـ X هذه مختلفة عن الـ C
+
+59
+00:06:40,340 --> 00:06:45,700
+ففي الحالة هذه بيطلع absolute F of X أصغر من واحد
+
+60
+00:06:45,700 --> 00:06:51,980
+زاد absolute L وهذا العدد أصغر من أو يساوي M ففي
+
+61
+00:06:51,980 --> 00:06:57,560
+كل الأحوال هذا الكلام صحيح لكل X تنتمي إلى A تقاطة
+
+62
+00:06:57,560 --> 00:07:05,800
+U Delta of C وهو المطلوب طبعاً نشوف الحالة التانية
+
+63
+00:07:05,800 --> 00:07:13,770
+Case 2إن الـ X لا تنتمي إلى A in
+
+64
+00:07:13,770 --> 00:07:20,810
+this case we
+
+65
+00:07:20,810 --> 00:07:32,390
+have absolute F in this case we take M
+
+66
+00:07:32,390 --> 00:07:37,310
+بساوي واحد زاد absolute L
+
+67
+00:07:40,850 --> 00:07:45,550
+بالحالة هاريب ناخد M بساوية واحد زاد absolute L
+
+68
+00:07:45,550 --> 00:07:49,670
+باستخدام
+
+69
+00:07:49,670 --> 00:07:54,030
+المتباينة هذه الأخيرة اللي هي سميها star
+
+70
+00:08:03,260 --> 00:08:10,660
+بطلع عندي absolute f of x إذا كانت الـ
+
+71
+00:08:10,660 --> 00:08:17,860
+c لا تنتمي إلى a فبصير هذه a و إطرح منها c هي a
+
+72
+00:08:17,860 --> 00:08:23,760
+صح؟ وبالتالي الكلام هذا بصير أصغر من 1 زاد
+
+73
+00:08:23,760 --> 00:08:29,700
+absolute L اللي هو أصغر من أو ساوي M وهذا صحيح لكل
+
+74
+00:08:29,700 --> 00:08:36,000
+x ينتمي إلى a تقاطع بي دلتاوبسيط لأن الـ C مش
+
+75
+00:08:36,000 --> 00:08:42,320
+موجودة في إيه فاطرحها أو ماطرحاش مابتفرجش لأن هنا
+
+76
+00:08:42,320 --> 00:08:48,700
+استخدمنا star و هنا برضه استخدمنا star then by
+
+77
+00:08:48,700 --> 00:08:56,580
+star ال absolute value هنا طلعتم بيساوي absolute
+
+78
+00:08:56,580 --> 00:09:01,340
+of f of c إذا كانت x بيساوي c وإذا كانت x مختلفة
+
+79
+00:09:01,340 --> 00:09:06,080
+عن c فمن ال start absolute of f of x تطلع أصغر من
+
+80
+00:09:06,080 --> 00:09:10,000
+واحد زاد absolute ل وهادي أصغر من أو ساوي بدا في
+
+81
+00:09:10,000 --> 00:09:14,620
+الحالتين هين أثبتنا إن يوجد عدد موجة بم هي العدد
+
+82
+00:09:14,620 --> 00:09:18,680
+الموجة في الحالة الأولى هو maximum هذا عدد الموجة
+
+83
+00:09:20,790 --> 00:09:25,610
+أو في الحالة التانية M هو واحد زاد absolute L و
+
+84
+00:09:25,610 --> 00:09:30,790
+هذا عدد موجب و في الحالتين absolute F of X أصغر من
+
+85
+00:09:30,790 --> 00:09:41,090
+M لكل X في إيه تخاطر دي إذن F is bounded on
+
+86
+00:09:41,090 --> 00:09:47,950
+a neighborhoodon a neighborhood bounded on a
+
+87
+00:09:47,950 --> 00:09:53,630
+neighborhood of c وهو المطلوب اذا النظرية بتقول اي
+
+88
+00:09:53,630 --> 00:09:57,730
+function لها limit على نقطة محددة لازم تكون
+
+89
+00:09:57,730 --> 00:10:05,970
+bounded على جوار لهذه النقطة واضح المرهان؟ في اي
+
+90
+00:10:05,970 --> 00:10:16,380
+استخسار؟ مش واضحالان هذه النظرية ممكن نستخدمها في
+
+91
+00:10:16,380 --> 00:10:22,200
+إثبات أن ال limits ل functions معينة، لنقاط معينة
+
+92
+00:10:22,200 --> 00:10:31,220
+غير موجودة فمثلا على سبيل المثال example
+
+93
+00:10:31,220 --> 00:10:34,780
+show أن ال limit
+
+94
+00:10:37,670 --> 00:10:43,190
+هنا الـ function 1 على x لما x تقول سفر does not
+
+95
+00:10:43,190 --> 00:10:55,490
+exist in R إذن
+
+96
+00:10:55,490 --> 00:11:01,550
+هنا الـ function تبعتي f of x بتساوي 1 على x حيث x
+
+97
+00:11:01,550 --> 00:11:02,690
+طبعا لا يترزق
+
+98
+00:11:09,460 --> 00:11:13,900
+ومن هنا اثبت ان ال limit ل f of x عند السفر مش
+
+99
+00:11:13,900 --> 00:11:27,400
+موجودة فالبرهان ذلك by above the theorem it
+
+100
+00:11:27,400 --> 00:11:33,100
+suffices it suffices to show
+
+101
+00:11:36,350 --> 00:11:44,610
+إن الـ function تبعتنا الـ
+
+102
+00:11:44,610 --> 00:11:49,750
+function
+
+103
+00:11:49,750 --> 00:11:59,130
+f هذه اللي عرضناها that الـ function f is not is
+
+104
+00:11:59,130 --> 00:12:04,590
+not bounded on
+
+105
+00:12:06,370 --> 00:12:19,450
+اني on any delta neighborhood of zero
+
+106
+00:12:39,000 --> 00:12:44,700
+عشان أثبت أن الدالة ليست bounded ليس .. ماليهاش
+
+107
+00:12:44,700 --> 00:12:49,620
+limit عند السفر حسب النظرية يكفي أثبات أن الدالة
+
+108
+00:12:49,620 --> 00:12:58,800
+هذه is not bounded على أي جوار للسفر لأن
+
+109
+00:12:58,800 --> 00:13:05,760
+لو كانت bounded على جوار واحد للسفرلأ لو كانت ال
+
+110
+00:13:05,760 --> 00:13:11,480
+limit .. لأن لو كانت ال limit تبعتها موجودة فلازم
+
+111
+00:13:11,480 --> 00:13:17,820
+تكون bounded على جوار معين للصفر فعشان أستنتج أن
+
+112
+00:13:17,820 --> 00:13:23,000
+ال limit مش موجودة، بحيث أثبت أن ال function
+
+113
+00:13:23,000 --> 00:13:28,960
+ماهياش bounded على كل الجوارات للصفر أو على أي
+
+114
+00:13:28,960 --> 00:13:30,080
+جوار للصفر
+
+115
+00:13:37,200 --> 00:13:45,620
+فال .. اذا ال .. البرهان هنا now
+
+116
+00:13:45,620 --> 00:13:51,240
+is approved by contradiction
+
+117
+00:13:58,300 --> 00:14:04,240
+يعني حاسبكم أن تتكملوا البرهان بالتناقض افرضي انه
+
+118
+00:14:04,240 --> 00:14:11,660
+يوجد جوار يوجد
+
+119
+00:14:11,660 --> 00:14:18,860
+جوار او function bounded عليه يوجد جوار delta معين
+
+120
+00:14:18,860 --> 00:14:24,060
+و ال function هذه bounded عليه و حاول يقصلي إلى
+
+121
+00:14:24,060 --> 00:14:24,580
+تناقض
+
+122
+00:14:27,970 --> 00:14:33,810
+هذا بيعطي برهان تاني إذا نجحت طبعا في بيعطيه برهان
+
+123
+00:14:33,810 --> 00:14:37,130
+Contradiction فهيبقى بصير عندك برهان تاني إنه
+
+124
+00:14:37,130 --> 00:14:41,510
+limit ال function واحد على x لأن x أقل ل 0 غير
+
+125
+00:14:41,510 --> 00:14:45,650
+موجودة طبعا احنا أخدنا المرة اللي فاتت باستخدام ال
+
+126
+00:14:45,650 --> 00:14:50,650
+sequential criterion أثبتنا إنه ال limit لل
+
+127
+00:14:50,650 --> 00:14:54,540
+function هذه عند الصفر مش موجودةاللي فيه برهان
+
+128
+00:14:54,540 --> 00:14:57,640
+باستخدام الـ sequential criterion واليوم هيفيه
+
+129
+00:14:57,640 --> 00:15:04,960
+برهان تاني باستخدام النظرية اللي هيحاولوا
+
+130
+00:15:04,960 --> 00:15:09,680
+تبرهنوا هذا أو اذا ماعرفتوش احكولي او تعالولي على
+
+131
+00:15:09,680 --> 00:15:15,880
+المكتب خلال ساعات المكتب دي طيب
+
+132
+00:15:15,880 --> 00:15:17,880
+نرجع اللي هي نظريات النهايات
+
+133
+00:15:50,720 --> 00:16:00,200
+دع F وG يكونوا عاملين من A to R يكونوا عاملين
+
+134
+00:16:12,190 --> 00:16:21,810
+بـClusterPoint ودى ينتمي الـ R عدد حقيقي and limit
+
+135
+00:16:23,250 --> 00:16:32,030
+f of x as x tends to c بساوي n أنتمي إلى r و limit
+
+136
+00:16:32,030 --> 00:16:43,910
+g of x as x tends to c بساوي n عدد حقيقي أيضا then
+
+137
+00:16:43,910 --> 00:16:48,750
+النتيجة واحد
+
+138
+00:16:48,750 --> 00:17:01,570
+limit أو aالـ limit لـ f زائد g of x and x times c
+
+139
+00:17:01,570 --> 00:17:10,510
+بساوي n plus m و لو أخدت الفرق بين الدالتين فlimit
+
+140
+00:17:10,510 --> 00:17:13,350
+الفرق بساوي فرق ال limits
+
+141
+00:17:16,580 --> 00:17:27,340
+الـ limit لحاصل ضرب F في G as X سمسة C بساوية لضرب
+
+142
+00:17:27,340 --> 00:17:34,200
+M C
+
+143
+00:17:34,200 --> 00:17:38,960
+limit لثابت B في F
+
+144
+00:17:49,320 --> 00:17:51,640
+والجزء الأخير
+
+145
+00:18:06,970 --> 00:18:16,090
+لأ F over G of X as X tends to C يسال L over M
+
+146
+00:18:16,090 --> 00:18:22,470
+طبعا بشرق وراية لإن M لأ يسال
+
+147
+00:18:30,390 --> 00:18:33,910
+الان هذه قواعد او قوانين النهايات نفس قوانين
+
+148
+00:18:33,910 --> 00:18:38,810
+النهايات اللي أخدناها بخصوص ال sequences و ال
+
+149
+00:18:38,810 --> 00:18:43,790
+functions و بالمناسبة احنا قلنا ان ال sequence هي
+
+150
+00:18:43,790 --> 00:18:51,690
+ال function ال function بالنوع الخاص طيب في
+
+151
+00:18:51,690 --> 00:18:55,950
+طريقتين لبرهان النظرية هذه خوف one
+
+152
+00:19:03,810 --> 00:19:15,070
+يوزن epsilon delta definition زي
+
+153
+00:19:15,070 --> 00:19:21,750
+اللي ما شوفنا في حالة النهايات ال limits لل
+
+154
+00:19:21,750 --> 00:19:25,310
+sequences هنا استخدمنا epsilon capital N
+
+155
+00:19:25,310 --> 00:19:29,510
+definition هنا هنستخدم epsilon delta definition
+
+156
+00:19:29,510 --> 00:19:36,260
+فمثلا يعني لو بالبداية to showلو بدأت بالجزء الأول
+
+157
+00:19:36,260 --> 00:19:45,020
+مثلا بي show limit ل f plus g of x as x tends to c
+
+158
+00:19:45,020 --> 00:19:53,300
+بساوي n plus m let epsilon أكبر من السفر be given
+
+159
+00:19:53,300 --> 00:19:57,000
+since
+
+160
+00:20:01,260 --> 00:20:08,660
+لما يكون f of x as x tends to c بساوي L هنا هناك
+
+161
+00:20:08,660 --> 00:20:15,540
+delta one تعتمد على أبسلان عدد موزة لكل x ينتمي
+
+162
+00:20:15,540 --> 00:20:20,530
+إلى aو absolute x minus c أصغر من الـ delta و 1
+
+163
+00:20:20,530 --> 00:20:27,610
+أكبر من 0 هذا بتضمن أن absolute f of x minus l
+
+164
+00:20:27,610 --> 00:20:35,430
+أصغر من إبسلن أتنين هذا
+
+165
+00:20:35,430 --> 00:20:43,750
+من تعريف epsilon delta ال element also أيضا since
+
+166
+00:20:46,390 --> 00:20:52,030
+أنا عندي فرد من limit لل function g of x as x
+
+167
+00:20:52,030 --> 00:20:59,530
+tends to c exist و بيسوي عدد حقيقي M فليه for the
+
+168
+00:20:59,530 --> 00:21:04,750
+same epsilon for the same given epsilon من تعريف
+
+169
+00:21:04,750 --> 00:21:11,570
+ال limit يوجد Delta 2عدد موجة بيعتمد على epsilon
+
+170
+00:21:11,570 --> 00:21:19,290
+بحيث انه لكل x ينتمي إلى a بحيث absolute x minus c
+
+171
+00:21:19,290 --> 00:21:24,490
+أصغر من delta اتنين أكبر من سفر هذا بتضمن ان
+
+172
+00:21:24,490 --> 00:21:33,210
+absolute g of x negative m أصغر من epsilon على
+
+173
+00:21:33,210 --> 00:21:49,020
+اتنينالـ Sample Implication دابل الصين Now
+
+174
+00:21:49,020 --> 00:21:53,960
+لـ
+
+175
+00:21:53,960 --> 00:22:01,060
+Delta دلينا نقرح Delta على إنها ال minimum الأصغر
+
+176
+00:22:03,020 --> 00:22:09,000
+الأصغر بين دلتا واحد ودلتا اتنين طبعا دلتا واحد
+
+177
+00:22:09,000 --> 00:22:13,780
+ودلتا اتنين عدد موجب اذا الدلتا هذه عدد موجب بعدين
+
+178
+00:22:13,780 --> 00:22:18,080
+دلتا واحد ودلتا اتنين depend on epsilon اذا الدلتا
+
+179
+00:22:18,080 --> 00:22:22,320
+هذه depend on epsilon لان هي أثبتنا ان يوجد دلتا
+
+180
+00:22:22,320 --> 00:22:25,080
+عدد موجب بحيث انه
+
+181
+00:22:29,490 --> 00:22:38,050
+لكل x ينتمي إلى a إذا حدث ان absolute x minus c
+
+182
+00:22:38,050 --> 00:22:45,450
+أكبر من سفر يعني x تساوي c و أصغر من delta هذا
+
+183
+00:22:45,450 --> 00:22:49,570
+هيضمن ان
+
+184
+00:22:49,570 --> 00:23:02,710
+absoluteF plus G of X minus L زي م ال
+
+185
+00:23:02,710 --> 00:23:08,190
+absolute value هذي بما أنها تطلع أصغر من إبسل طيب
+
+186
+00:23:08,190 --> 00:23:18,780
+F زي G of X هذي عبارة عن F of X زي G of Xوباستخدام
+
+187
+00:23:18,780 --> 00:23:22,060
+الـ triangle inequality هذا أصغر من أوي ساوي
+
+188
+00:23:22,060 --> 00:23:32,360
+absolute f of x minus L زائد absolute g
+
+189
+00:23:32,360 --> 00:23:39,520
+of x minus M مظبوطة الآن
+
+190
+00:23:39,520 --> 00:23:41,080
+باستخدام الـ star
+
+191
+00:23:43,690 --> 00:23:49,910
+أنا عند ال X موجودة في A والبساطة بين X وC أصغر من
+
+192
+00:23:49,910 --> 00:23:55,760
+دلتاوالـ delta هذه أصغر من أو ساوى delta واحد فهي
+
+193
+00:23:55,760 --> 00:23:58,200
+delta ال minimum الأصغر من delta واحد يعني ال
+
+194
+00:23:58,200 --> 00:24:02,260
+delta هنا من تعريف ال delta delta أصغر من أو ساوى
+
+195
+00:24:02,260 --> 00:24:05,520
+delta واحد وأصغر من أو ساوى delta minimum طيب لما
+
+196
+00:24:05,520 --> 00:24:08,820
+يكون ال absolute value ل X minus C أصغر من delta
+
+197
+00:24:08,820 --> 00:24:14,320
+واحد أصغر من delta واحد إذا by star بطلع absolute
+
+198
+00:24:14,320 --> 00:24:20,290
+F of X minus L أصغر من X على 2و كذلك الـ delta
+
+199
+00:24:20,290 --> 00:24:26,730
+اللي قلنا أصغر من أو يساوي delta two من تعريفها طب
+
+200
+00:24:26,730 --> 00:24:30,290
+لما يكون absolute x minus c أكبر من سفر و أصغر من
+
+201
+00:24:30,290 --> 00:24:38,970
+delta اتنين اذا by double star by
+
+202
+00:24:38,970 --> 00:24:44,550
+double star بيطلع absolute g of x minus m أصغر من
+
+203
+00:24:44,550 --> 00:24:51,340
+epsilon ع اتنين مجموعة بيطلع epsilonانلخص ايش
+
+204
+00:24:51,340 --> 00:24:57,560
+اثبتنا هنا اللي اثبتناه هو ما يريد for any epsilon
+
+205
+00:24:57,560 --> 00:25:01,240
+for any given epsilon اكبر من سفر there exists
+
+206
+00:25:01,240 --> 00:25:06,040
+دلتا positive number ويعتمد على ابسلون لان دلتا
+
+207
+00:25:06,040 --> 00:25:10,500
+واحد ودلتا اتنين يعتمدوا على ابسلون بحيث لكل x هي
+
+208
+00:25:10,500 --> 00:25:15,460
+ايه وabsolute x minus c اكبر من سفر اصغر من الدلتا
+
+209
+00:25:15,460 --> 00:25:17,700
+هذه طلع absolute
+
+210
+00:25:21,470 --> 00:25:25,930
+ال function تبعتي اللي هي مجموعة f و g minus ال
+
+211
+00:25:25,930 --> 00:25:32,150
+limit المقترحها أصغر من epsilon طبعا؟
+
+212
+00:25:32,150 --> 00:25:39,970
+اذا حسب التعريف بما أن هذا صحيح since epsilon أكبر
+
+213
+00:25:39,970 --> 00:25:46,070
+من سفر was arbitrary معناته احنا أثبتنا أن هذا
+
+214
+00:25:46,070 --> 00:25:51,360
+الكلام لكل epsilon موجبةفي delta تعطيني ال
+
+215
+00:25:51,360 --> 00:25:56,980
+implication تبقى تعريف ال limit وبالتالي by
+
+216
+00:25:56,980 --> 00:26:02,740
+definition by definition أو epsilon delta
+
+217
+00:26:02,740 --> 00:26:06,800
+definition of limit بتطلع عندي ال limit لل
+
+218
+00:26:06,800 --> 00:26:14,700
+function f plus g as x tends to c بتطلع بالساوية
+
+219
+00:26:14,700 --> 00:26:21,270
+لعدد L زاوية يعني وهو المطلوبتمام؟ لأن هنا
+
+220
+00:26:21,270 --> 00:26:26,150
+استخدمنا تعريف epsilon دلتا لإثبات أن limit مجموعة
+
+221
+00:26:26,150 --> 00:26:30,070
+two functions بيساوي مجموعة limits ل two functions
+
+222
+00:26:30,070 --> 00:26:36,350
+the limit of a sum is the sum of limits بالمثل
+
+223
+00:26:36,350 --> 00:26:43,630
+ممكن البرهن باقي أعزائي النظرية similarly or the
+
+224
+00:26:43,630 --> 00:26:47,410
+proof of
+
+225
+00:26:47,410 --> 00:26:48,390
+the other
+
+226
+00:26:52,480 --> 00:27:02,220
+parts is similar .. is similar لبرهان
+
+227
+00:27:02,220 --> 00:27:07,060
+اللي احنا اخدناه وبالتالي هسيبكم انتوا تكتبوا
+
+228
+00:27:14,260 --> 00:27:20,000
+ارجعوا لبرهان ال limit لحاصل ضرب two sequences
+
+229
+00:27:20,000 --> 00:27:24,180
+بساوي لحاصل ضرب ال limits و حاولوا تجلدوا البرهان
+
+230
+00:27:24,180 --> 00:27:28,220
+نفس الحاجة برضه هذا .. هذا طبعا corollary على
+
+231
+00:27:28,220 --> 00:27:32,460
+الجزء اللي قبله والتاني هناك limit خارج قسم ال two
+
+232
+00:27:32,460 --> 00:27:38,840
+sequences بساوي خارج قسم ال limits فممكن برضه نعمم
+
+233
+00:27:38,840 --> 00:27:42,910
+البرهان تبع ال sequence ل ال functionsفحاولوا
+
+234
+00:27:42,910 --> 00:27:47,470
+تكتبوا البراهين هذه لأنه ممكن كل بساطة انتحار نصف
+
+235
+00:27:47,470 --> 00:27:52,830
+التاني او النهائي هقولك مثلا برهنك ان ال limit
+
+236
+00:27:52,830 --> 00:27:55,890
+حاصل ضرب two functions بسبب حاصل ضرب ال limit
+
+237
+00:27:55,890 --> 00:28:03,850
+using epsilon delta definition حددلك الطريقة يجب
+
+238
+00:28:03,850 --> 00:28:08,750
+انكم تكتبوا براهين باق الأجزاء تمام؟
+
+239
+00:28:10,180 --> 00:28:14,320
+طيب هذا برهان الأول باستخدام epsilon delta
+
+240
+00:28:14,320 --> 00:28:19,160
+definition لكن في برهان تاني باستخدام ال
+
+241
+00:28:19,160 --> 00:28:22,700
+sequential criterion اللي أخدناه المرة اللي فاتت
+
+242
+00:28:22,700 --> 00:28:28,880
+فنشوف البرهان التاني
+
+243
+00:28:28,880 --> 00:28:36,940
+اذا
+
+244
+00:28:36,940 --> 00:28:38,400
+proof اتنين
+
+245
+00:29:03,440 --> 00:29:09,520
+فمثلا to show البرهن
+
+246
+00:29:09,520 --> 00:29:16,180
+جزء طبعا باج الأعزاء برهنها بالمثلالمرة هذه مثلا
+
+247
+00:29:16,180 --> 00:29:26,880
+لتبعت مثلا يزر بي to show limit ل f ضرب g of x as
+
+248
+00:29:26,880 --> 00:29:35,200
+x tends to c بساوي L times M هنستخدم الـ
+
+249
+00:29:35,200 --> 00:29:43,200
+sequential criterion ف let x in be sequence in A
+
+250
+00:29:45,300 --> 00:29:55,540
+وحدودها مختلفة عن الـ C such that limit x in as n
+
+251
+00:29:55,540 --> 00:30:04,580
+tends to infinity is 7C We must show
+
+252
+00:30:04,580 --> 00:30:09,620
+عشان أثبت limit حاصل ضرب دلت هنا الـ C موجودة
+
+253
+00:30:10,370 --> 00:30:14,950
+بتساوي L في N حسب الـ sequential criterion لازم
+
+254
+00:30:14,950 --> 00:30:18,370
+أثبت أنه لأي sequence حدودها مختلفة عن النقطة C
+
+255
+00:30:18,370 --> 00:30:26,450
+ونهايتها C لازم نهاية صورتها لازم نثبت نهاية
+
+256
+00:30:26,450 --> 00:30:35,130
+صورتها limit ال F ضرب G لسيكوينس X N as N times
+
+257
+00:30:35,130 --> 00:30:42,140
+infinity بساوي L ضرب Nففتنة ان ال limit صورة ال
+
+258
+00:30:42,140 --> 00:30:47,260
+sequence xn under the function f ضارف g بالساوية L
+
+259
+00:30:47,260 --> 00:30:51,740
+M لان حسب ال sequential criterion تطلع الدالة تبعت
+
+260
+00:30:51,740 --> 00:30:56,260
+ال limit لعنصر موجودة و بالساوية العدد L في M
+
+261
+00:30:56,260 --> 00:31:02,580
+تمام؟ لان باقى نثبت ان ال limit لل image of this
+
+262
+00:31:02,580 --> 00:31:06,760
+sequence أبقى عن العدد LM طيب
+
+263
+00:31:09,150 --> 00:31:17,010
+لبرهان ذلك .. Now .. تعالوا نثبت الكلام هذا Now ال
+
+264
+00:31:17,010 --> 00:31:26,770
+limit لف ضارب g of xn as n tends to infinity بساوي
+
+265
+00:31:26,770 --> 00:31:29,810
+ال
+
+266
+00:31:29,810 --> 00:31:36,280
+limitحاصل ضرب two functions عند أي x أو xn هذا
+
+267
+00:31:36,280 --> 00:31:44,720
+عبارة عن f of xn ضرب g of xn لأن هذا من تعريف حاصل
+
+268
+00:31:44,720 --> 00:31:50,120
+ضرب two functions الآن هذه عبارة عن sequence وهذه
+
+269
+00:31:50,120 --> 00:31:54,900
+عبارة عن sequenceو limit الـ sequence الأولى exist
+
+270
+00:31:54,900 --> 00:31:59,340
+و limit الـ sequence التانية exist إذا by limit
+
+271
+00:31:59,340 --> 00:32:03,460
+theorems لsequences إن ال limit حاصل الضرب بساوي
+
+272
+00:32:03,460 --> 00:32:11,300
+حاصل ضرب ال limits فهذا بساوي limit f of xn ضرب
+
+273
+00:32:12,730 --> 00:32:21,230
+limit g of xn as n times infinity as n times
+
+274
+00:32:21,230 --> 00:32:29,850
+infinity طب إيش اللي ضمنلي ماذا يضمن إن ال limit ل
+
+275
+00:32:29,850 --> 00:32:36,270
+f of xn موجودة و limit ل g of xn موجودة لأنه أنا
+
+276
+00:32:36,270 --> 00:32:48,500
+فارد since limitf of x as x tends to c exist لأن
+
+277
+00:32:48,500 --> 00:32:57,200
+هاد exist بساوي m exist and equals m and ال limit
+
+278
+00:32:57,200 --> 00:33:03,700
+لل function g of x as x tends to c exist and بساوي
+
+279
+00:33:03,700 --> 00:33:08,300
+العدد m وعندي
+
+280
+00:33:08,300 --> 00:33:19,540
+xmو limit xn بساوي c ف by sequential criterion اذا
+
+281
+00:33:19,540 --> 00:33:26,330
+limit صورة ال xn under f موجودة صح؟ اهو limit سورة
+
+282
+00:33:26,330 --> 00:33:32,150
+ال sequence xn under g موجودة okay إذا هذا مضمون
+
+283
+00:33:32,150 --> 00:33:37,630
+وجود ال limit هذه وجود ال limit هذه مضمون لأن
+
+284
+00:33:37,630 --> 00:33:40,930
+limit ال f عن c موجودة وبالتالي و limit ال
+
+285
+00:33:40,930 --> 00:33:45,610
+sequence xn بالساوى c إذا هذه موجودة من ال
+
+286
+00:33:45,610 --> 00:33:48,790
+sequential criterion و كذلك هذه من ال sequential
+
+287
+00:33:48,790 --> 00:33:49,330
+criterion
+
+288
+00:33:52,710 --> 00:33:59,930
+طيب ما هدى ال limit الأخيرة هدى بالساوي L by
+
+289
+00:33:59,930 --> 00:34:05,050
+sequential criterion اذا كانت limit f of x اما x
+
+290
+00:34:05,050 --> 00:34:08,890
+او لا c موجودة وبالساوي L واندي فيه sequence
+
+291
+00:34:08,890 --> 00:34:13,850
+نهايتها C فنهاية صورة ال sequence under F بالساوي
+
+292
+00:34:13,850 --> 00:34:20,550
+L وكذلكنفس الحاجة بما ان limit g and c موجودة و
+
+293
+00:34:20,550 --> 00:34:25,250
+بيساوي m و xn نهايتها c اذا by sequential
+
+294
+00:34:25,250 --> 00:34:32,550
+criterion limit g سورة ال xn under g بيساوي m اذا
+
+295
+00:34:32,550 --> 00:34:42,410
+هذا by sequential criterion وهذا
+
+296
+00:34:42,410 --> 00:34:43,650
+اللي احنا عايزين نثبته
+
+297
+00:34:48,110 --> 00:34:53,630
+عايزين نثبت ان limit f ضارب g او ال image لسيكوينس
+
+298
+00:34:53,630 --> 00:34:57,890
+x in under ال function f ضارب g بالساوية ال M هاي
+
+299
+00:34:57,890 --> 00:35:02,050
+بدينا ب limit ال image لسيكوينس x in under f g
+
+300
+00:35:02,050 --> 00:35:09,110
+وطلعت هي بالساوية ال M اذا by sequential
+
+301
+00:35:26,030 --> 00:35:34,050
+البرهين الأجزاء الأخرى مماثلة ال proof of the
+
+302
+00:35:34,050 --> 00:35:37,510
+other parts
+
+303
+00:35:40,000 --> 00:35:50,560
+is similar مشابه exercise it
+
+304
+00:35:50,560 --> 00:35:59,140
+اتمرن عليهم اتمرن علي كتابة البرهيم اكتبوهم okay
+
+305
+00:35:59,140 --> 00:36:03,340
+تمام إذا مافي عندي نهائي من القرآن لlimits
+
+306
+00:36:05,580 --> 00:36:10,940
+ناخد لنا بس مثال او اتنين او خلينا ناخد corollary
+
+307
+00:36:10,940 --> 00:36:19,420
+خليني
+
+308
+00:36:19,420 --> 00:36:23,600
+بس يعني اخدلي corollary سريع على النظرية هذه
+
+309
+00:36:39,870 --> 00:36:45,390
+Corollary 1 F
+
+310
+00:36:48,310 --> 00:36:58,610
+f1,f2 إلى fm are functions from a to r و c a
+
+311
+00:36:58,610 --> 00:37:02,190
+cluster point
+
+312
+00:37:02,190 --> 00:37:14,130
+of a and limit fk of x as x tends to c بالساوي الك
+
+313
+00:37:14,130 --> 00:37:17,750
+then
+
+314
+00:37:19,410 --> 00:37:29,030
+وطبعا هنا for all k بيساوي واحد اتنين الى M then
+
+315
+00:37:29,030 --> 00:37:42,870
+واحد limit ل F1 زي F2 زي وهاكذا زي Fn of X بيساوي
+
+316
+00:37:42,870 --> 00:38:02,630
+لما X تقول ل C هذا بيساوي N1 زي NL2 زائد زائد
+
+317
+00:38:02,630 --> 00:38:08,790
+LL اتنين limit لحاصل
+
+318
+00:38:08,790 --> 00:38:21,970
+ضرب F1 ضرب F2 ضرب FNF of X as X tends to C بساوي
+
+319
+00:38:21,970 --> 00:38:30,190
+L1 ضرب L2 ضرب و هكذا ضرب LL تلاتة
+
+320
+00:38:30,190 --> 00:38:34,130
+F
+
+321
+00:38:34,130 --> 00:38:41,090
+limit F of X as X tends to C بساوي L
+
+322
+00:38:45,730 --> 00:38:55,510
+ل F of X الكل أُس N as X tends to C بساوي L أُس N
+
+323
+00:38:55,510 --> 00:39:08,870
+لبرهان
+
+324
+00:39:08,870 --> 00:39:15,920
+ال Corollary هذافهذا الجزء الأول هو نتيجة على
+
+325
+00:39:15,920 --> 00:39:27,940
+الجزء A من النظرية زائد induction on M إذاً
+
+326
+00:39:27,940 --> 00:39:38,580
+to prove one and two use above theorem and
+
+327
+00:39:38,580 --> 00:39:39,280
+induction
+
+328
+00:39:42,280 --> 00:39:49,140
+invection on edge هنا
+
+329
+00:39:49,140 --> 00:39:56,840
+في اثبات الجزء التالت to prove تلاتة
+
+330
+00:40:00,320 --> 00:40:17,960
+أخذ F1 بساوي F2 بساوي Fn بساوي F in part اتنين to
+
+331
+00:40:17,960 --> 00:40:22,960
+get the
+
+332
+00:40:22,960 --> 00:40:23,420
+result
+
+333
+00:40:30,710 --> 00:40:35,510
+يعني لدرهان الكلام هذا إذا كانت limit F عن C بساوي
+
+334
+00:40:35,510 --> 00:40:42,330
+L فعوضي او خدي F واحد هي F و F اتنين F و كلهم
+
+335
+00:40:42,330 --> 00:40:50,570
+بساوي F فكأن إذا هنا limit F to the power N and X
+
+336
+00:40:50,570 --> 00:40:55,630
+هيطلع بساوي L مضروبة في نفس M المرات اللي هو إذا
+
+337
+00:40:55,630 --> 00:41:00,960
+هذا الجزء التالت corollary على الجزء التانيOkay
+
+338
+00:41:00,960 --> 00:41:04,040
+تمام؟
+
+339
+00:41:04,040 --> 00:41:08,220
+Okay إذا بنوقف هنا وإن شاء الله المرة الجاية هنأكل
+
+340
+00:41:08,220 --> 00:41:14,340
+أمثلة على المظريات هذه ونحاول
+
+341
+00:41:14,340 --> 00:41:19,260
+ناخد قصارى أخرى لل limits of functions إذا نكتفي
+
+342
+00:41:19,260 --> 00:41:23,600
+بهذا القدر ونشوفكم إن شاء الله يوم الأثنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..59b84f3a95df3d13fcfcc5c41951533028823fe2
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/r7wN576DqQ0_raw.srt
@@ -0,0 +1,1368 @@
+1
+00:00:21,190 --> 00:00:26,170
+السلام عليكم في المحاضرة هذه ان شاء الله هناخد
+
+2
+00:00:26,170 --> 00:00:32,530
+section رقم اتنين شبطر اربعة اللي هو section اربعة
+
+3
+00:00:32,530 --> 00:00:39,250
+اتنين عنوانه limit theorems او قوانين النهايات لل
+
+4
+00:00:39,250 --> 00:00:45,070
+functions او الدوالة جبل ما نتعرض ل limit theorems
+
+5
+00:00:45,070 --> 00:00:53,310
+او قوانين النهايات للدوال بناخد تعريفما معنى ان
+
+6
+00:00:53,310 --> 00:00:59,410
+الـ function معرفة على subset of R A ما معنى انها
+
+7
+00:00:59,410 --> 00:01:06,230
+تكون bounded على جوار لنقطة C هي في C cluster
+
+8
+00:01:06,230 --> 00:01:10,610
+point للمجال تبع الدالة ما معنى ان الدالة تكون
+
+9
+00:01:10,610 --> 00:01:19,650
+bounded محصورة على جوارلنقطة C هذا معناه ان يوجد
+
+10
+00:01:19,650 --> 00:01:29,150
+Delta نبرهود جوار Delta لنقطة C بعمق Delta ويوجد
+
+11
+00:01:29,150 --> 00:01:34,570
+عدد موجب M بحيث ان ال absolute value لقيم الدالة
+
+12
+00:01:34,570 --> 00:01:39,950
+أصغر من أو ساوي العدد الموجب لكل X في مجال الدالة
+
+13
+00:01:39,950 --> 00:01:47,010
+واللي موجود في ال Delta نبرهود لل Cإذا قدرت ألاقي
+
+14
+00:01:47,010 --> 00:01:51,210
+delta neighborhood للنقطة C بحيث وعدد موجب بحيث
+
+15
+00:01:51,210 --> 00:01:55,490
+المتباين هذا تتحقق لكل X ال delta neighborhood
+
+16
+00:01:55,490 --> 00:02:00,710
+وطبعا هو موجود في ال A فبقول المدالة is bounded on
+
+17
+00:02:00,710 --> 00:02:03,190
+a neighborhood of C
+
+18
+00:02:05,840 --> 00:02:11,200
+النظرية هذه بتقول إنه لو أنا فيه أندي فانكشن زي
+
+19
+00:02:11,200 --> 00:02:16,740
+هذه و ال limit تبعتها عند ال cluster point C exist
+
+20
+00:02:16,740 --> 00:02:22,340
+فلابد ضروري جدا أن من الضروري أن تكون الدالة
+
+21
+00:02:22,340 --> 00:02:30,000
+bounded على جوار لنخطة C okay فتعالوا نلفم الكلام
+
+22
+00:02:30,000 --> 00:02:35,410
+هذا left epsilon بساوي واحد عدد موجببما أنه من
+
+23
+00:02:35,410 --> 00:02:40,050
+الفرض بما أنه limit f of x as x tends to z بيساوي
+
+24
+00:02:40,050 --> 00:02:44,870
+عدد L إذا حسب تعريف الـ epsilon دلتا لل limit of
+
+25
+00:02:44,870 --> 00:02:47,690
+function there exists delta that depends on
+
+26
+00:02:47,690 --> 00:02:51,410
+epsilon اللي هو واحد عدد موجب بحيث ال implication
+
+27
+00:02:51,410 --> 00:02:56,230
+هي تتحقق اللي أنا باستخدام ال triangle inequality
+
+28
+00:02:56,230 --> 00:03:01,330
+absolute f of xبساوي absolute F of X minus L
+
+29
+00:03:01,330 --> 00:03:06,850
+لانطرحت L رجعتها بعدين باخد اتنين هدول مع بعض By
+
+30
+00:03:06,850 --> 00:03:10,030
+the triangle equality هذا أصغر من F of X minus L
+
+31
+00:03:10,030 --> 00:03:14,510
+زاد absolute L وهذا من فوق أصغر من واحد زاد
+
+32
+00:03:14,510 --> 00:03:22,930
+absolute هذا الكلام صحيح لكل X لكل X أنتمي ل A لكل
+
+33
+00:03:22,930 --> 00:03:27,590
+X أنتمي ل A وأيضا
+
+34
+00:03:29,110 --> 00:03:39,530
+لكل X مختلفة عن الـC إذا الـX هنا لاتساوي C وأيضا
+
+35
+00:03:39,530 --> 00:03:45,170
+هذا من هنا X لاتساوي C ومن هنا هذا معناه absolute
+
+36
+00:03:45,170 --> 00:03:50,870
+X سالب C أصغر من Delta معناته الـX تنتمي للـDelta
+
+37
+00:03:50,870 --> 00:03:58,950
+مبروض لـCتمام ان هذا الكلام صحيح لكل x في a مختلف
+
+38
+00:03:58,950 --> 00:04:05,090
+عن ال c و لكل و في نفس الوقت موجود ال x في ال
+
+39
+00:04:05,090 --> 00:04:12,010
+delta neighborhood ل c طيب
+
+40
+00:04:12,010 --> 00:04:20,600
+احنا بما نثبت انه بما نثبت ان ال absolute valueأو
+
+41
+00:04:20,600 --> 00:04:35,240
+جدنا delta neighborhood لـC نحن
+
+42
+00:04:35,240 --> 00:04:43,120
+عايزين نثبت أن absolute F of X أصغر منها يساوي Mأو
+
+43
+00:04:43,120 --> 00:04:47,860
+there exists M أكبر من السفر بحيث أن absolute F of
+
+44
+00:04:47,860 --> 00:04:55,540
+X أصغر من أو ساوي M لكل X هنتمي إلى A تقاطع D
+
+45
+00:04:55,540 --> 00:05:03,240
+Delta of C مش هيك تعريف اللي هو المتبين هذا طيب we
+
+46
+00:05:03,240 --> 00:05:04,300
+have two cases
+
+47
+00:05:11,240 --> 00:05:17,160
+كاس واحد إذا كانت الـ C تنتمي إلى A ال cluster
+
+48
+00:05:17,160 --> 00:05:22,200
+point C هي ال cluster point ال C is a cluster
+
+49
+00:05:22,200 --> 00:05:29,680
+point لست A فممكن ال C تنتمي ل A و ممكن ما تنتميش
+
+50
+00:05:29,680 --> 00:05:36,000
+إذا في أنتي حالتين لو كانت ال C تنتمي ل A in this
+
+51
+00:05:36,000 --> 00:05:47,190
+case in this casechoose M بيساوي
+
+52
+00:05:47,190 --> 00:05:59,230
+ال maximum ل absolute F of C و 1 زاد absolute L
+
+53
+00:05:59,230 --> 00:06:03,790
+then
+
+54
+00:06:03,790 --> 00:06:08,630
+الحالة هذه absolute F of X
+
+55
+00:06:11,500 --> 00:06:16,520
+إذا كانت الـ X هذه هي الـ C ف absolute F of X
+
+56
+00:06:16,520 --> 00:06:21,800
+بساوي absolute F of C وبالتالي أصغر من أو يساوي
+
+57
+00:06:21,800 --> 00:06:28,400
+absolute F of C اللي هي أصغر من أو يساوي M وإذا
+
+58
+00:06:28,400 --> 00:06:34,660
+كانت الـ X هذه مختلفة عن الـ C
+
+59
+00:06:40,340 --> 00:06:45,700
+ففي الحالة هذه بيطلع absolute F of X أصغر من واحد
+
+60
+00:06:45,700 --> 00:06:51,980
+زاد absolute L وهذا العدد أصغر من أو يساوي M ففي
+
+61
+00:06:51,980 --> 00:06:57,560
+كل الأحوال هذا الكلام صحيح لكل X تنتمي إلى A تقاطة
+
+62
+00:06:57,560 --> 00:07:05,800
+U Delta of C وهو المطلوب طبعاً نشوف الحالة التانية
+
+63
+00:07:05,800 --> 00:07:13,770
+Case 2إن الـ X لا تنتمي إلى A in
+
+64
+00:07:13,770 --> 00:07:20,810
+this case we
+
+65
+00:07:20,810 --> 00:07:32,390
+have absolute F in this case we take M
+
+66
+00:07:32,390 --> 00:07:37,310
+بساوي واحد زاد absolute L
+
+67
+00:07:40,850 --> 00:07:45,550
+بالحالة هاريب ناخد M بساوية واحد زاد absolute L
+
+68
+00:07:45,550 --> 00:07:49,670
+باستخدام
+
+69
+00:07:49,670 --> 00:07:54,030
+المتباينة هذه الأخيرة اللي هي سميها star
+
+70
+00:08:03,260 --> 00:08:10,660
+بطلع عندي absolute f of x إذا كانت الـ
+
+71
+00:08:10,660 --> 00:08:17,860
+c لا تنتمي إلى a فبصير هذه a و إطرح منها c هي a
+
+72
+00:08:17,860 --> 00:08:23,760
+صح؟ وبالتالي الكلام هذا بصير أصغر من 1 زاد
+
+73
+00:08:23,760 --> 00:08:29,700
+absolute L اللي هو أصغر من أو ساوي M وهذا صحيح لكل
+
+74
+00:08:29,700 --> 00:08:36,000
+x ينتمي إلى a تقاطع بي دلتاوبسيط لأن الـ C مش
+
+75
+00:08:36,000 --> 00:08:42,320
+موجودة في إيه فاطرحها أو ماطرحاش مابتفرجش لأن هنا
+
+76
+00:08:42,320 --> 00:08:48,700
+استخدمنا star و هنا برضه استخدمنا star then by
+
+77
+00:08:48,700 --> 00:08:56,580
+star ال absolute value هنا طلعتم بيساوي absolute
+
+78
+00:08:56,580 --> 00:09:01,340
+of f of c إذا كانت x بيساوي c وإذا كانت x مختلفة
+
+79
+00:09:01,340 --> 00:09:06,080
+عن c فمن ال start absolute of f of x تطلع أصغر من
+
+80
+00:09:06,080 --> 00:09:10,000
+واحد زاد absolute ل وهادي أصغر من أو ساوي بدا في
+
+81
+00:09:10,000 --> 00:09:14,620
+الحالتين هين أثبتنا إن يوجد عدد موجة بم هي العدد
+
+82
+00:09:14,620 --> 00:09:18,680
+الموجة في الحالة الأولى هو maximum هذا عدد الموجة
+
+83
+00:09:20,790 --> 00:09:25,610
+أو في الحالة التانية M هو واحد زاد absolute L و
+
+84
+00:09:25,610 --> 00:09:30,790
+هذا عدد موجب و في الحالتين absolute F of X أصغر من
+
+85
+00:09:30,790 --> 00:09:41,090
+M لكل X في إيه تخاطر دي إذن F is bounded on
+
+86
+00:09:41,090 --> 00:09:47,950
+a neighborhoodon a neighborhood bounded on a
+
+87
+00:09:47,950 --> 00:09:53,630
+neighborhood of c وهو المطلوب اذا النظرية بتقول اي
+
+88
+00:09:53,630 --> 00:09:57,730
+function لها limit على نقطة محددة لازم تكون
+
+89
+00:09:57,730 --> 00:10:05,970
+bounded على جوار لهذه النقطة واضح المرهان؟ في اي
+
+90
+00:10:05,970 --> 00:10:16,380
+استخسار؟ مش واضحالان هذه النظرية ممكن نستخدمها في
+
+91
+00:10:16,380 --> 00:10:22,200
+إثبات أن ال limits ل functions معينة، لنقاط معينة
+
+92
+00:10:22,200 --> 00:10:31,220
+غير موجودة فمثلا على سبيل المثال example
+
+93
+00:10:31,220 --> 00:10:34,780
+show أن ال limit
+
+94
+00:10:37,670 --> 00:10:43,190
+هنا الـ function 1 على x لما x تقول سفر does not
+
+95
+00:10:43,190 --> 00:10:55,490
+exist in R إذن
+
+96
+00:10:55,490 --> 00:11:01,550
+هنا الـ function تبعتي f of x بتساوي 1 على x حيث x
+
+97
+00:11:01,550 --> 00:11:02,690
+طبعا لا يترزق
+
+98
+00:11:09,460 --> 00:11:13,900
+ومن هنا اثبت ان ال limit ل f of x عند السفر مش
+
+99
+00:11:13,900 --> 00:11:27,400
+موجودة فالبرهان ذلك by above the theorem it
+
+100
+00:11:27,400 --> 00:11:33,100
+suffices it suffices to show
+
+101
+00:11:36,350 --> 00:11:44,610
+إن الـ function تبعتنا الـ
+
+102
+00:11:44,610 --> 00:11:49,750
+function
+
+103
+00:11:49,750 --> 00:11:59,130
+f هذه اللي عرضناها that الـ function f is not is
+
+104
+00:11:59,130 --> 00:12:04,590
+not bounded on
+
+105
+00:12:06,370 --> 00:12:19,450
+اني on any delta neighborhood of zero
+
+106
+00:12:39,000 --> 00:12:44,700
+عشان أثبت أن الدالة ليست bounded ليس .. ماليهاش
+
+107
+00:12:44,700 --> 00:12:49,620
+limit عند السفر حسب النظرية يكفي أثبات أن الدالة
+
+108
+00:12:49,620 --> 00:12:58,800
+هذه is not bounded على أي جوار للسفر لأن
+
+109
+00:12:58,800 --> 00:13:05,760
+لو كانت bounded على جوار واحد للسفرلأ لو كانت ال
+
+110
+00:13:05,760 --> 00:13:11,480
+limit .. لأن لو كانت ال limit تبعتها موجودة فلازم
+
+111
+00:13:11,480 --> 00:13:17,820
+تكون bounded على جوار معين للصفر فعشان أستنتج أن
+
+112
+00:13:17,820 --> 00:13:23,000
+ال limit مش موجودة، بحيث أثبت أن ال function
+
+113
+00:13:23,000 --> 00:13:28,960
+ماهياش bounded على كل الجوارات للصفر أو على أي
+
+114
+00:13:28,960 --> 00:13:30,080
+جوار للصفر
+
+115
+00:13:37,200 --> 00:13:45,620
+فال .. اذا ال .. البرهان هنا now
+
+116
+00:13:45,620 --> 00:13:51,240
+is approved by contradiction
+
+117
+00:13:58,300 --> 00:14:04,240
+يعني حاسبكم أن تتكملوا البرهان بالتناقض افرضي انه
+
+118
+00:14:04,240 --> 00:14:11,660
+يوجد جوار يوجد
+
+119
+00:14:11,660 --> 00:14:18,860
+جوار او function bounded عليه يوجد جوار delta معين
+
+120
+00:14:18,860 --> 00:14:24,060
+و ال function هذه bounded عليه و حاول يقصلي إلى
+
+121
+00:14:24,060 --> 00:14:24,580
+تناقض
+
+122
+00:14:27,970 --> 00:14:33,810
+هذا بيعطي برهان تاني إذا نجحت طبعا في بيعطيه برهان
+
+123
+00:14:33,810 --> 00:14:37,130
+Contradiction فهيبقى بصير عندك برهان تاني إنه
+
+124
+00:14:37,130 --> 00:14:41,510
+limit ال function واحد على x لأن x أقل ل 0 غير
+
+125
+00:14:41,510 --> 00:14:45,650
+موجودة طبعا احنا أخدنا المرة اللي فاتت باستخدام ال
+
+126
+00:14:45,650 --> 00:14:50,650
+sequential criterion أثبتنا إنه ال limit لل
+
+127
+00:14:50,650 --> 00:14:54,540
+function هذه عند الصفر مش موجودةاللي فيه برهان
+
+128
+00:14:54,540 --> 00:14:57,640
+باستخدام الـ sequential criterion واليوم هيفيه
+
+129
+00:14:57,640 --> 00:15:04,960
+برهان تاني باستخدام النظرية اللي هيحاولوا
+
+130
+00:15:04,960 --> 00:15:09,680
+تبرهنوا هذا أو اذا ماعرفتوش احكولي او تعالولي على
+
+131
+00:15:09,680 --> 00:15:15,880
+المكتب خلال ساعات المكتب دي طيب
+
+132
+00:15:15,880 --> 00:15:17,880
+نرجع اللي هي نظريات النهايات
+
+133
+00:15:50,720 --> 00:16:00,200
+دع F وG يكونوا عاملين من A to R يكونوا عاملين
+
+134
+00:16:12,190 --> 00:16:21,810
+بـClusterPoint ودى ينتمي الـ R عدد حقيقي and limit
+
+135
+00:16:23,250 --> 00:16:32,030
+f of x as x tends to c بساوي n أنتمي إلى r و limit
+
+136
+00:16:32,030 --> 00:16:43,910
+g of x as x tends to c بساوي n عدد حقيقي أيضا then
+
+137
+00:16:43,910 --> 00:16:48,750
+النتيجة واحد
+
+138
+00:16:48,750 --> 00:17:01,570
+limit أو aالـ limit لـ f زائد g of x and x times c
+
+139
+00:17:01,570 --> 00:17:10,510
+بساوي n plus m و لو أخدت الفرق بين الدالتين فlimit
+
+140
+00:17:10,510 --> 00:17:13,350
+الفرق بساوي فرق ال limits
+
+141
+00:17:16,580 --> 00:17:27,340
+الـ limit لحاصل ضرب F في G as X سمسة C بساوية لضرب
+
+142
+00:17:27,340 --> 00:17:34,200
+M C
+
+143
+00:17:34,200 --> 00:17:38,960
+limit لثابت B في F
+
+144
+00:17:49,320 --> 00:17:51,640
+والجزء الأخير
+
+145
+00:18:06,970 --> 00:18:16,090
+لأ F over G of X as X tends to C يسال L over M
+
+146
+00:18:16,090 --> 00:18:22,470
+طبعا بشرق وراية لإن M لأ يسال
+
+147
+00:18:30,390 --> 00:18:33,910
+الان هذه قواعد او قوانين النهايات نفس قوانين
+
+148
+00:18:33,910 --> 00:18:38,810
+النهايات اللي أخدناها بخصوص ال sequences و ال
+
+149
+00:18:38,810 --> 00:18:43,790
+functions و بالمناسبة احنا قلنا ان ال sequence هي
+
+150
+00:18:43,790 --> 00:18:51,690
+ال function ال function بالنوع الخاص طيب في
+
+151
+00:18:51,690 --> 00:18:55,950
+طريقتين لبرهان النظرية هذه خوف one
+
+152
+00:19:03,810 --> 00:19:15,070
+يوزن epsilon delta definition زي
+
+153
+00:19:15,070 --> 00:19:21,750
+اللي ما شوفنا في حالة النهايات ال limits لل
+
+154
+00:19:21,750 --> 00:19:25,310
+sequences هنا استخدمنا epsilon capital N
+
+155
+00:19:25,310 --> 00:19:29,510
+definition هنا هنستخدم epsilon delta definition
+
+156
+00:19:29,510 --> 00:19:36,260
+فمثلا يعني لو بالبداية to showلو بدأت بالجزء الأول
+
+157
+00:19:36,260 --> 00:19:45,020
+مثلا بي show limit ل f plus g of x as x tends to c
+
+158
+00:19:45,020 --> 00:19:53,300
+بساوي n plus m let epsilon أكبر من السفر be given
+
+159
+00:19:53,300 --> 00:19:57,000
+since
+
+160
+00:20:01,260 --> 00:20:08,660
+لما يكون f of x as x tends to c بساوي L هنا هناك
+
+161
+00:20:08,660 --> 00:20:15,540
+delta one تعتمد على أبسلان عدد موزة لكل x ينتمي
+
+162
+00:20:15,540 --> 00:20:20,530
+إلى aو absolute x minus c أصغر من الـ delta و 1
+
+163
+00:20:20,530 --> 00:20:27,610
+أكبر من 0 هذا بتضمن أن absolute f of x minus l
+
+164
+00:20:27,610 --> 00:20:35,430
+أصغر من إبسلن أتنين هذا
+
+165
+00:20:35,430 --> 00:20:43,750
+من تعريف epsilon delta ال element also أيضا since
+
+166
+00:20:46,390 --> 00:20:52,030
+أنا عندي فرد من limit لل function g of x as x
+
+167
+00:20:52,030 --> 00:20:59,530
+tends to c exist و بيسوي عدد حقيقي M فليه for the
+
+168
+00:20:59,530 --> 00:21:04,750
+same epsilon for the same given epsilon من تعريف
+
+169
+00:21:04,750 --> 00:21:11,570
+ال limit يوجد Delta 2عدد موجة بيعتمد على epsilon
+
+170
+00:21:11,570 --> 00:21:19,290
+بحيث انه لكل x ينتمي إلى a بحيث absolute x minus c
+
+171
+00:21:19,290 --> 00:21:24,490
+أصغر من delta اتنين أكبر من سفر هذا بتضمن ان
+
+172
+00:21:24,490 --> 00:21:33,210
+absolute g of x negative m أصغر من epsilon على
+
+173
+00:21:33,210 --> 00:21:49,020
+اتنينالـ Sample Implication دابل الصين Now
+
+174
+00:21:49,020 --> 00:21:53,960
+لـ
+
+175
+00:21:53,960 --> 00:22:01,060
+Delta دلينا نقرح Delta على إنها ال minimum الأصغر
+
+176
+00:22:03,020 --> 00:22:09,000
+الأصغر بين دلتا واحد ودلتا اتنين طبعا دلتا واحد
+
+177
+00:22:09,000 --> 00:22:13,780
+ودلتا اتنين عدد موجب اذا الدلتا هذه عدد موجب بعدين
+
+178
+00:22:13,780 --> 00:22:18,080
+دلتا واحد ودلتا اتنين depend on epsilon اذا الدلتا
+
+179
+00:22:18,080 --> 00:22:22,320
+هذه depend on epsilon لان هي أثبتنا ان يوجد دلتا
+
+180
+00:22:22,320 --> 00:22:25,080
+عدد موجب بحيث انه
+
+181
+00:22:29,490 --> 00:22:38,050
+لكل x ينتمي إلى a إذا حدث ان absolute x minus c
+
+182
+00:22:38,050 --> 00:22:45,450
+أكبر من سفر يعني x تساوي c و أصغر من delta هذا
+
+183
+00:22:45,450 --> 00:22:49,570
+هيضمن ان
+
+184
+00:22:49,570 --> 00:23:02,710
+absoluteF plus G of X minus L زي م ال
+
+185
+00:23:02,710 --> 00:23:08,190
+absolute value هذي بما أنها تطلع أصغر من إبسل طيب
+
+186
+00:23:08,190 --> 00:23:18,780
+F زي G of X هذي عبارة عن F of X زي G of Xوباستخدام
+
+187
+00:23:18,780 --> 00:23:22,060
+الـ triangle inequality هذا أصغر من أوي ساوي
+
+188
+00:23:22,060 --> 00:23:32,360
+absolute f of x minus L زائد absolute g
+
+189
+00:23:32,360 --> 00:23:39,520
+of x minus M مظبوطة الآن
+
+190
+00:23:39,520 --> 00:23:41,080
+باستخدام الـ star
+
+191
+00:23:43,690 --> 00:23:49,910
+أنا عند ال X موجودة في A والبساطة بين X وC أصغر من
+
+192
+00:23:49,910 --> 00:23:55,760
+دلتاوالـ delta هذه أصغر من أو ساوى delta واحد فهي
+
+193
+00:23:55,760 --> 00:23:58,200
+delta ال minimum الأصغر من delta واحد يعني ال
+
+194
+00:23:58,200 --> 00:24:02,260
+delta هنا من تعريف ال delta delta أصغر من أو ساوى
+
+195
+00:24:02,260 --> 00:24:05,520
+delta واحد وأصغر من أو ساوى delta minimum طيب لما
+
+196
+00:24:05,520 --> 00:24:08,820
+يكون ال absolute value ل X minus C أصغر من delta
+
+197
+00:24:08,820 --> 00:24:14,320
+واحد أصغر من delta واحد إذا by star بطلع absolute
+
+198
+00:24:14,320 --> 00:24:20,290
+F of X minus L أصغر من X على 2و كذلك الـ delta
+
+199
+00:24:20,290 --> 00:24:26,730
+اللي قلنا أصغر من أو يساوي delta two من تعريفها طب
+
+200
+00:24:26,730 --> 00:24:30,290
+لما يكون absolute x minus c أكبر من سفر و أصغر من
+
+201
+00:24:30,290 --> 00:24:38,970
+delta اتنين اذا by double star by
+
+202
+00:24:38,970 --> 00:24:44,550
+double star بيطلع absolute g of x minus m أصغر من
+
+203
+00:24:44,550 --> 00:24:51,340
+epsilon ع اتنين مجموعة بيطلع epsilonانلخص ايش
+
+204
+00:24:51,340 --> 00:24:57,560
+اثبتنا هنا اللي اثبتناه هو ما يريد for any epsilon
+
+205
+00:24:57,560 --> 00:25:01,240
+for any given epsilon اكبر من سفر there exists
+
+206
+00:25:01,240 --> 00:25:06,040
+دلتا positive number ويعتمد على ابسلون لان دلتا
+
+207
+00:25:06,040 --> 00:25:10,500
+واحد ودلتا اتنين يعتمدوا على ابسلون بحيث لكل x هي
+
+208
+00:25:10,500 --> 00:25:15,460
+ايه وabsolute x minus c اكبر من سفر اصغر من الدلتا
+
+209
+00:25:15,460 --> 00:25:17,700
+هذه طلع absolute
+
+210
+00:25:21,470 --> 00:25:25,930
+ال function تبعتي اللي هي مجموعة f و g minus ال
+
+211
+00:25:25,930 --> 00:25:32,150
+limit المقترحها أصغر من epsilon طبعا؟
+
+212
+00:25:32,150 --> 00:25:39,970
+اذا حسب التعريف بما أن هذا صحيح since epsilon أكبر
+
+213
+00:25:39,970 --> 00:25:46,070
+من سفر was arbitrary معناته احنا أثبتنا أن هذا
+
+214
+00:25:46,070 --> 00:25:51,360
+الكلام لكل epsilon موجبةفي delta تعطيني ال
+
+215
+00:25:51,360 --> 00:25:56,980
+implication تبقى تعريف ال limit وبالتالي by
+
+216
+00:25:56,980 --> 00:26:02,740
+definition by definition أو epsilon delta
+
+217
+00:26:02,740 --> 00:26:06,800
+definition of limit بتطلع عندي ال limit لل
+
+218
+00:26:06,800 --> 00:26:14,700
+function f plus g as x tends to c بتطلع بالساوية
+
+219
+00:26:14,700 --> 00:26:21,270
+لعدد L زاوية يعني وهو المطلوبتمام؟ لأن هنا
+
+220
+00:26:21,270 --> 00:26:26,150
+استخدمنا تعريف epsilon دلتا لإثبات أن limit مجموعة
+
+221
+00:26:26,150 --> 00:26:30,070
+two functions بيساوي مجموعة limits ل two functions
+
+222
+00:26:30,070 --> 00:26:36,350
+the limit of a sum is the sum of limits بالمثل
+
+223
+00:26:36,350 --> 00:26:43,630
+ممكن البرهن باقي أعزائي النظرية similarly or the
+
+224
+00:26:43,630 --> 00:26:47,410
+proof of
+
+225
+00:26:47,410 --> 00:26:48,390
+the other
+
+226
+00:26:52,480 --> 00:27:02,220
+parts is similar .. is similar لبرهان
+
+227
+00:27:02,220 --> 00:27:07,060
+اللي احنا اخدناه وبالتالي هسيبكم انتوا تكتبوا
+
+228
+00:27:14,260 --> 00:27:20,000
+ارجعوا لبرهان ال limit لحاصل ضرب two sequences
+
+229
+00:27:20,000 --> 00:27:24,180
+بساوي لحاصل ضرب ال limits و حاولوا تجلدوا البرهان
+
+230
+00:27:24,180 --> 00:27:28,220
+نفس الحاجة برضه هذا .. هذا طبعا corollary على
+
+231
+00:27:28,220 --> 00:27:32,460
+الجزء اللي قبله والتاني هناك limit خارج قسم ال two
+
+232
+00:27:32,460 --> 00:27:38,840
+sequences بساوي خارج قسم ال limits فممكن برضه نعمم
+
+233
+00:27:38,840 --> 00:27:42,910
+البرهان تبع ال sequence ل ال functionsفحاولوا
+
+234
+00:27:42,910 --> 00:27:47,470
+تكتبوا البراهين هذه لأنه ممكن كل بساطة انتحار نصف
+
+235
+00:27:47,470 --> 00:27:52,830
+التاني او النهائي هقولك مثلا برهنك ان ال limit
+
+236
+00:27:52,830 --> 00:27:55,890
+حاصل ضرب two functions بسبب حاصل ضرب ال limit
+
+237
+00:27:55,890 --> 00:28:03,850
+using epsilon delta definition حددلك الطريقة يجب
+
+238
+00:28:03,850 --> 00:28:08,750
+انكم تكتبوا براهين باق الأجزاء تمام؟
+
+239
+00:28:10,180 --> 00:28:14,320
+طيب هذا برهان الأول باستخدام epsilon delta
+
+240
+00:28:14,320 --> 00:28:19,160
+definition لكن في برهان تاني باستخدام ال
+
+241
+00:28:19,160 --> 00:28:22,700
+sequential criterion اللي أخدناه المرة اللي فاتت
+
+242
+00:28:22,700 --> 00:28:28,880
+فنشوف البرهان التاني
+
+243
+00:28:28,880 --> 00:28:36,940
+اذا
+
+244
+00:28:36,940 --> 00:28:38,400
+proof اتنين
+
+245
+00:29:03,440 --> 00:29:09,520
+فمثلا to show البرهن
+
+246
+00:29:09,520 --> 00:29:16,180
+جزء طبعا باج الأعزاء برهنها بالمثلالمرة هذه مثلا
+
+247
+00:29:16,180 --> 00:29:26,880
+لتبعت مثلا يزر بي to show limit ل f ضرب g of x as
+
+248
+00:29:26,880 --> 00:29:35,200
+x tends to c بساوي L times M هنستخدم الـ
+
+249
+00:29:35,200 --> 00:29:43,200
+sequential criterion ف let x in be sequence in A
+
+250
+00:29:45,300 --> 00:29:55,540
+وحدودها مختلفة عن الـ C such that limit x in as n
+
+251
+00:29:55,540 --> 00:30:04,580
+tends to infinity is 7C We must show
+
+252
+00:30:04,580 --> 00:30:09,620
+عشان أثبت limit حاصل ضرب دلت هنا الـ C موجودة
+
+253
+00:30:10,370 --> 00:30:14,950
+بتساوي L في N حسب الـ sequential criterion لازم
+
+254
+00:30:14,950 --> 00:30:18,370
+أثبت أنه لأي sequence حدودها مختلفة عن النقطة C
+
+255
+00:30:18,370 --> 00:30:26,450
+ونهايتها C لازم نهاية صورتها لازم نثبت نهاية
+
+256
+00:30:26,450 --> 00:30:35,130
+صورتها limit ال F ضرب G لسيكوينس X N as N times
+
+257
+00:30:35,130 --> 00:30:42,140
+infinity بساوي L ضرب Nففتنة ان ال limit صورة ال
+
+258
+00:30:42,140 --> 00:30:47,260
+sequence xn under the function f ضارف g بالساوية L
+
+259
+00:30:47,260 --> 00:30:51,740
+M لان حسب ال sequential criterion تطلع الدالة تبعت
+
+260
+00:30:51,740 --> 00:30:56,260
+ال limit لعنصر موجودة و بالساوية العدد L في M
+
+261
+00:30:56,260 --> 00:31:02,580
+تمام؟ لان باقى نثبت ان ال limit لل image of this
+
+262
+00:31:02,580 --> 00:31:06,760
+sequence أبقى عن العدد LM طيب
+
+263
+00:31:09,150 --> 00:31:17,010
+لبرهان ذلك .. Now .. تعالوا نثبت الكلام هذا Now ال
+
+264
+00:31:17,010 --> 00:31:26,770
+limit لف ضارب g of xn as n tends to infinity بساوي
+
+265
+00:31:26,770 --> 00:31:29,810
+ال
+
+266
+00:31:29,810 --> 00:31:36,280
+limitحاصل ضرب two functions عند أي x أو xn هذا
+
+267
+00:31:36,280 --> 00:31:44,720
+عبارة عن f of xn ضرب g of xn لأن هذا من تعريف حاصل
+
+268
+00:31:44,720 --> 00:31:50,120
+ضرب two functions الآن هذه عبارة عن sequence وهذه
+
+269
+00:31:50,120 --> 00:31:54,900
+عبارة عن sequenceو limit الـ sequence الأولى exist
+
+270
+00:31:54,900 --> 00:31:59,340
+و limit الـ sequence التانية exist إذا by limit
+
+271
+00:31:59,340 --> 00:32:03,460
+theorems لsequences إن ال limit حاصل الضرب بساوي
+
+272
+00:32:03,460 --> 00:32:11,300
+حاصل ضرب ال limits فهذا بساوي limit f of xn ضرب
+
+273
+00:32:12,730 --> 00:32:21,230
+limit g of xn as n times infinity as n times
+
+274
+00:32:21,230 --> 00:32:29,850
+infinity طب إيش اللي ضمنلي ماذا يضمن إن ال limit ل
+
+275
+00:32:29,850 --> 00:32:36,270
+f of xn موجودة و limit ل g of xn موجودة لأنه أنا
+
+276
+00:32:36,270 --> 00:32:48,500
+فارد since limitf of x as x tends to c exist لأن
+
+277
+00:32:48,500 --> 00:32:57,200
+هاد exist بساوي m exist and equals m and ال limit
+
+278
+00:32:57,200 --> 00:33:03,700
+لل function g of x as x tends to c exist and بساوي
+
+279
+00:33:03,700 --> 00:33:08,300
+العدد m وعندي
+
+280
+00:33:08,300 --> 00:33:19,540
+xmو limit xn بساوي c ف by sequential criterion اذا
+
+281
+00:33:19,540 --> 00:33:26,330
+limit صورة ال xn under f موجودة صح؟ اهو limit سورة
+
+282
+00:33:26,330 --> 00:33:32,150
+ال sequence xn under g موجودة okay إذا هذا مضمون
+
+283
+00:33:32,150 --> 00:33:37,630
+وجود ال limit هذه وجود ال limit هذه مضمون لأن
+
+284
+00:33:37,630 --> 00:33:40,930
+limit ال f عن c موجودة وبالتالي و limit ال
+
+285
+00:33:40,930 --> 00:33:45,610
+sequence xn بالساوى c إذا هذه موجودة من ال
+
+286
+00:33:45,610 --> 00:33:48,790
+sequential criterion و كذلك هذه من ال sequential
+
+287
+00:33:48,790 --> 00:33:49,330
+criterion
+
+288
+00:33:52,710 --> 00:33:59,930
+طيب ما هدى ال limit الأخيرة هدى بالساوي L by
+
+289
+00:33:59,930 --> 00:34:05,050
+sequential criterion اذا كانت limit f of x اما x
+
+290
+00:34:05,050 --> 00:34:08,890
+او لا c موجودة وبالساوي L واندي فيه sequence
+
+291
+00:34:08,890 --> 00:34:13,850
+نهايتها C فنهاية صورة ال sequence under F بالساوي
+
+292
+00:34:13,850 --> 00:34:20,550
+L وكذلكنفس الحاجة بما ان limit g and c موجودة و
+
+293
+00:34:20,550 --> 00:34:25,250
+بيساوي m و xn نهايتها c اذا by sequential
+
+294
+00:34:25,250 --> 00:34:32,550
+criterion limit g سورة ال xn under g بيساوي m اذا
+
+295
+00:34:32,550 --> 00:34:42,410
+هذا by sequential criterion وهذا
+
+296
+00:34:42,410 --> 00:34:43,650
+اللي احنا عايزين نثبته
+
+297
+00:34:48,110 --> 00:34:53,630
+عايزين نثبت ان limit f ضارب g او ال image لسيكوينس
+
+298
+00:34:53,630 --> 00:34:57,890
+x in under ال function f ضارب g بالساوية ال M هاي
+
+299
+00:34:57,890 --> 00:35:02,050
+بدينا ب limit ال image لسيكوينس x in under f g
+
+300
+00:35:02,050 --> 00:35:09,110
+وطلعت هي بالساوية ال M اذا by sequential
+
+301
+00:35:26,030 --> 00:35:34,050
+البرهين الأجزاء الأخرى مماثلة ال proof of the
+
+302
+00:35:34,050 --> 00:35:37,510
+other parts
+
+303
+00:35:40,000 --> 00:35:50,560
+is similar مشابه exercise it
+
+304
+00:35:50,560 --> 00:35:59,140
+اتمرن عليهم اتمرن علي كتابة البرهيم اكتبوهم okay
+
+305
+00:35:59,140 --> 00:36:03,340
+تمام إذا مافي عندي نهائي من القرآن لlimits
+
+306
+00:36:05,580 --> 00:36:10,940
+ناخد لنا بس مثال او اتنين او خلينا ناخد corollary
+
+307
+00:36:10,940 --> 00:36:19,420
+خليني
+
+308
+00:36:19,420 --> 00:36:23,600
+بس يعني اخدلي corollary سريع على النظرية هذه
+
+309
+00:36:39,870 --> 00:36:45,390
+Corollary 1 F
+
+310
+00:36:48,310 --> 00:36:58,610
+f1,f2 إلى fm are functions from a to r و c a
+
+311
+00:36:58,610 --> 00:37:02,190
+cluster point
+
+312
+00:37:02,190 --> 00:37:14,130
+of a and limit fk of x as x tends to c بالساوي الك
+
+313
+00:37:14,130 --> 00:37:17,750
+then
+
+314
+00:37:19,410 --> 00:37:29,030
+وطبعا هنا for all k بيساوي واحد اتنين الى M then
+
+315
+00:37:29,030 --> 00:37:42,870
+واحد limit ل F1 زي F2 زي وهاكذا زي Fn of X بيساوي
+
+316
+00:37:42,870 --> 00:38:02,630
+لما X تقول ل C هذا بيساوي N1 زي NL2 زائد زائد
+
+317
+00:38:02,630 --> 00:38:08,790
+LL اتنين limit لحاصل
+
+318
+00:38:08,790 --> 00:38:21,970
+ضرب F1 ضرب F2 ضرب FNF of X as X tends to C بساوي
+
+319
+00:38:21,970 --> 00:38:30,190
+L1 ضرب L2 ضرب و هكذا ضرب LL تلاتة
+
+320
+00:38:30,190 --> 00:38:34,130
+F
+
+321
+00:38:34,130 --> 00:38:41,090
+limit F of X as X tends to C بساوي L
+
+322
+00:38:45,730 --> 00:38:55,510
+ل F of X الكل أُس N as X tends to C بساوي L أُس N
+
+323
+00:38:55,510 --> 00:39:08,870
+لبرهان
+
+324
+00:39:08,870 --> 00:39:15,920
+ال Corollary هذافهذا الجزء الأول هو نتيجة على
+
+325
+00:39:15,920 --> 00:39:27,940
+الجزء A من النظرية زائد induction on M إذاً
+
+326
+00:39:27,940 --> 00:39:38,580
+to prove one and two use above theorem and
+
+327
+00:39:38,580 --> 00:39:39,280
+induction
+
+328
+00:39:42,280 --> 00:39:49,140
+invection on edge هنا
+
+329
+00:39:49,140 --> 00:39:56,840
+في اثبات الجزء التالت to prove تلاتة
+
+330
+00:40:00,320 --> 00:40:17,960
+أخذ F1 بساوي F2 بساوي Fn بساوي F in part اتنين to
+
+331
+00:40:17,960 --> 00:40:22,960
+get the
+
+332
+00:40:22,960 --> 00:40:23,420
+result
+
+333
+00:40:30,710 --> 00:40:35,510
+يعني لدرهان الكلام هذا إذا كانت limit F عن C بساوي
+
+334
+00:40:35,510 --> 00:40:42,330
+L فعوضي او خدي F واحد هي F و F اتنين F و كلهم
+
+335
+00:40:42,330 --> 00:40:50,570
+بساوي F فكأن إذا هنا limit F to the power N and X
+
+336
+00:40:50,570 --> 00:40:55,630
+هيطلع بساوي L مضروبة في نفس M المرات اللي هو إذا
+
+337
+00:40:55,630 --> 00:41:00,960
+هذا الجزء التالت corollary على الجزء التانيOkay
+
+338
+00:41:00,960 --> 00:41:04,040
+تمام؟
+
+339
+00:41:04,040 --> 00:41:08,220
+Okay إذا بنوقف هنا وإن شاء الله المرة الجاية هنأكل
+
+340
+00:41:08,220 --> 00:41:14,340
+أمثلة على المظريات هذه ونحاول
+
+341
+00:41:14,340 --> 00:41:19,260
+ناخد قصارى أخرى لل limits of functions إذا نكتفي
+
+342
+00:41:19,260 --> 00:41:23,600
+بهذا القدر ونشوفكم إن شاء الله يوم الأثنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a885fa262bc54b71807964c144efc91b1cfdbc40
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_postprocess.srt
@@ -0,0 +1,1400 @@
+1
+00:00:21,680 --> 00:00:30,360
+بسم الله الرحمن الرحيم في اللقاء هذا هحنحاول نكمل
+
+2
+00:00:30,360 --> 00:00:35,700
+ما بدأناه في section خمسة اتنين اخر نظرية أخدناها
+
+3
+00:00:35,700 --> 00:00:39,700
+في المحاضرة السابقة كانت ال boundedness theorem
+
+4
+00:00:39,700 --> 00:00:45,160
+بتقول لو كان في اندي اف function continuous على ال
+
+5
+00:00:45,160 --> 00:00:50,080
+interval Iو ال interval I is closed and bounded
+
+6
+00:00:50,080 --> 00:00:56,000
+فال function بتطلع bounded على الفترة I مبرهنة
+
+7
+00:00:56,000 --> 00:01:04,100
+نظرية في اللقاء السابق الآن في شوية ملاحظات في ال
+
+8
+00:01:04,100 --> 00:01:08,260
+boundedness theorem في النظرية هذه أي شرط من
+
+9
+00:01:08,260 --> 00:01:13,330
+الشروط ال boundedness و ال closedوالاتصال تبعد
+
+10
+00:01:13,330 --> 00:01:18,590
+دالة على الفترة I اي شرط لو اي شرط من الشروط هدولة
+
+11
+00:01:18,590 --> 00:01:24,890
+نجس او اختلف فالنظرية تفشل ماتبطل نظرية بصير
+
+12
+00:01:24,890 --> 00:01:33,790
+مالهاش معنى فعلى سبيل مثال لو كانت الفترة ماهياش
+
+13
+00:01:33,790 --> 00:01:35,710
+bounded الفترة I ماكانتش bounded
+
+14
+00:01:38,580 --> 00:01:45,820
+فالنظرية بتفشل و هاي مثال يوضح ذلك ناخد ال
+
+15
+00:01:45,820 --> 00:01:53,160
+identity function f و ناخد ال I هي الفترة المغلقة
+
+16
+00:01:53,160 --> 00:01:59,100
+من صفر إلى ملنهاي واضح ان الفترة I هذه ماهياش
+
+17
+00:01:59,100 --> 00:01:59,480
+bounded
+
+18
+00:02:05,080 --> 00:02:10,740
+وواضح ان الـ identity function F continuous على
+
+19
+00:02:10,740 --> 00:02:17,160
+الفترة I لأنها continuous على كل الـ R لكن الـ
+
+20
+00:02:17,160 --> 00:02:24,070
+function F ماهياش bounded على الفترة Iليست bounded
+
+21
+00:02:24,070 --> 00:02:32,490
+على I و البرهان لأي عدد موجب بقدر اختار XM X عنصر
+
+22
+00:02:32,490 --> 00:02:42,170
+يعتمد على M هذا ينتمي للفترة I و
+
+23
+00:02:42,170 --> 00:02:46,750
+ال absolute value ل F of XM تطلع absolute M زايد
+
+24
+00:02:46,750 --> 00:02:51,230
+واحد اللي هي M زايد واحد لأن هذا عدد موجب أكبر من
+
+25
+00:02:51,230 --> 00:02:57,750
+Mحسب الـ remark اللي هي بتكافئ تعريف الbounded
+
+26
+00:02:57,750 --> 00:03:03,430
+تعريف ال unboundedness ال function بتحقق شرط ال
+
+27
+00:03:03,430 --> 00:03:07,370
+unboundedness وبالتالي ال function unbounded on I
+
+28
+00:03:07,370 --> 00:03:15,630
+لأن هنا عشان الفترة ليست bounded النظرية فشلت في
+
+29
+00:03:15,630 --> 00:03:18,890
+الملاحظة التانية لو ال boundedness theorem is
+
+30
+00:03:18,890 --> 00:03:25,230
+false تفشلإذا كانت الـ interval ماهياش مغلقة فع
+
+31
+00:03:25,230 --> 00:03:29,730
+سبيل المثال consider الـ function واحد على X حيث X
+
+32
+00:03:29,730 --> 00:03:36,850
+أكبر من سفر و أصغر من أو ساول واحد فواضح
+
+33
+00:03:36,850 --> 00:03:43,490
+إن الفترة I هذه not closed ليست مغلقة و الدالة G
+
+34
+00:03:43,490 --> 00:03:48,650
+of X متصلة عليها لأنها لا تحتوى السفر لأن هذه عندي
+
+35
+00:03:48,650 --> 00:03:55,080
+function Gcontinuous على الفترة I الفترة I ماهياش
+
+36
+00:03:55,080 --> 00:04:00,460
+closed وبالتالي
+
+37
+00:04:00,460 --> 00:04:05,520
+وفي نفس الوجهة ال function هذه ليست bounded على
+
+38
+00:04:05,520 --> 00:04:10,040
+الفترة I احنا شفنا ان ال function 1 على X في بداية
+
+39
+00:04:10,040 --> 00:04:16,600
+اللقاء اللي فات ليست bounded على الفترة من صفر إلى
+
+40
+00:04:16,600 --> 00:04:23,910
+ملا نهايةو بالمثل ليست bounded على الفترة من صفر
+
+41
+00:04:23,910 --> 00:04:30,070
+إلى واحد ممكن اعطاء برهان مشابه، إذا هذا مثال على
+
+42
+00:04:30,070 --> 00:04:35,850
+function متصرة على فترة ولكنها ليست bounded على
+
+43
+00:04:35,850 --> 00:04:39,930
+نفس الفترة لأن الفترة هذه السبب أن الفترة هذه ليست
+
+44
+00:04:39,930 --> 00:04:44,130
+closed، تمام؟
+
+45
+00:04:44,130 --> 00:04:48,700
+إذا الفترة هذه لو ماكانتش closed، النظرية بتفشلطب
+
+46
+00:04:48,700 --> 00:04:54,060
+لو كانت closed و لا bounded، برضه بتفشل، لأن هذا
+
+47
+00:04:54,060 --> 00:05:01,060
+الشرط ضروري و هذا الشرط ضروري كذلك ال function
+
+48
+00:05:01,060 --> 00:05:04,760
+لازم تكون continuous على الفترة هذه، يعني لو كانت
+
+49
+00:05:04,760 --> 00:05:07,820
+الفترة هذه closed و bounded، فال function مش
+
+50
+00:05:07,820 --> 00:05:12,140
+continuous، فالنظرية تفشل، لأن هنا the boundedness
+
+51
+00:05:12,140 --> 00:05:12,640
+theorem
+
+52
+00:05:19,930 --> 00:05:31,430
+is false if الـ function لو كانت الدالة F is not
+
+53
+00:05:31,430 --> 00:05:34,690
+continuous
+
+54
+00:05:34,690 --> 00:05:39,910
+on الفترة I فعلى سبيل المثال
+
+55
+00:05:51,070 --> 00:05:58,870
+for example ممكن نعرف define H
+
+56
+00:05:58,870 --> 00:06:06,710
+من الفترة المغلقة من
+
+57
+00:06:06,710 --> 00:06:17,470
+سفر إلى واحد إلى R by H of X بالساوي
+
+58
+00:06:24,650 --> 00:06:34,070
+1 على x إذا كان x أكبر من 0 أصغر من أو ساوي 1
+
+59
+00:06:34,070 --> 00:06:46,730
+وخلّيني أخدها بالساوي 1 إذا كان x بالساوي 0 ال
+
+60
+00:06:46,730 --> 00:06:50,510
+function h of x 1 على x إذا كان x أكبر من 0 أصغر
+
+61
+00:06:50,510 --> 00:06:56,950
+من أو ساوي 1 و عند السفر خلّيها الساوي 1فطبعا ال
+
+62
+00:06:56,950 --> 00:07:10,430
+function h is not continuous at x بساوي سفر ليست
+
+63
+00:07:10,430 --> 00:07:18,070
+متصلة عند السفر لأن ال limit لل function h of x
+
+64
+00:07:18,070 --> 00:07:22,490
+لما x تقول إلى السفر
+
+65
+00:07:25,030 --> 00:07:33,670
+من اليمين بيساوي infinity لاتساوي
+
+66
+00:07:33,670 --> 00:07:41,690
+h عند الواحد عند السفر اللي هو بيساوي واحد او كون
+
+67
+00:07:41,690 --> 00:07:46,690
+ال limit بيساوي infinity اذا does not exist in R
+
+68
+00:07:46,690 --> 00:07:50,690
+وبالتالي هذا يكفي ان احنا نقول ان ال function
+
+69
+00:07:50,690 --> 00:08:00,080
+ماهياش continuous عند السفرفى نفس الواجهة Note
+
+70
+00:08:00,080 --> 00:08:04,360
+that الفترة
+
+71
+00:08:04,360 --> 00:08:15,740
+I بساوي سفر و واحد is closed and bounded يعني أنا
+
+72
+00:08:15,740 --> 00:08:22,180
+مافيش عند التصال عند السفر ال function هذه H طبعا
+
+73
+00:08:22,180 --> 00:08:29,870
+زى ما ال function هذيكNote also that
+
+74
+00:08:29,870 --> 00:08:35,890
+the function H is unbounded
+
+75
+00:08:35,890 --> 00:08:43,830
+على الفترة من صفر لواحد لأنها
+
+76
+00:08:43,830 --> 00:08:48,210
+unbounded على الفترة المفتوحة H بالساعة واحد على X
+
+77
+00:08:48,210 --> 00:08:52,590
+على الفترة نصف المفتوحة هذهو يقولنا إن الـ
+
+78
+00:08:52,590 --> 00:08:57,050
+function 1 على x ليست bounded 1 على x ماهياش
+
+79
+00:08:57,050 --> 00:09:03,570
+bounded على الفترة من 0 إلى 1 إذا
+
+80
+00:09:03,570 --> 00:09:07,850
+النظرية تفشل لأن الـ function تبعت هنا ليست
+
+81
+00:09:07,850 --> 00:09:12,750
+continuous عند السفر وبالتالي ليست continuous على
+
+82
+00:09:12,750 --> 00:09:19,150
+كل الفترة الواحد هنا H is not continuous على كل
+
+83
+00:09:19,150 --> 00:09:24,200
+الفترة Iلأنها مش continuous عند السفر اللي هي
+
+84
+00:09:24,200 --> 00:09:28,020
+answer في آية عشان تكون ال function continuous على
+
+85
+00:09:28,020 --> 00:09:31,840
+كل الفترة لازم تكون continuous عند كل نقطة في
+
+86
+00:09:31,840 --> 00:09:34,220
+الفترة لو كانت مش continuous عند نقطة واحدة في
+
+87
+00:09:34,220 --> 00:09:43,760
+الفترة فمايقدرش أقول continuous على الفترة تمام
+
+88
+00:09:43,760 --> 00:09:47,600
+إذا ناخد
+
+89
+00:09:54,630 --> 00:10:04,450
+تعيف definition a
+
+90
+00:10:04,450 --> 00:10:13,030
+function f from A to R has
+
+91
+00:10:13,030 --> 00:10:18,090
+an absolute
+
+92
+00:10:18,090 --> 00:10:19,110
+maximum
+
+93
+00:10:27,530 --> 00:10:39,550
+respectively على التوالي absolute minimum at
+
+94
+00:10:39,550 --> 00:10:47,110
+on
+
+95
+00:10:47,110 --> 00:10:56,610
+الفأس set A if الشرط التالي بيتحققthere exist x
+
+96
+00:10:56,610 --> 00:11:01,370
+super star respectively
+
+97
+00:11:01,370 --> 00:11:08,530
+x lower star تنتمي
+
+98
+00:11:08,530 --> 00:11:13,190
+إلى a بحيث
+
+99
+00:11:13,190 --> 00:11:22,650
+انه f of x أصغر من لو ساوي f of x super star
+
+100
+00:11:25,650 --> 00:11:33,030
+respectively على التوالي f of x lower star أصغر من
+
+101
+00:11:33,030 --> 00:11:42,630
+أو ساوي f of x لكل x تنتمي إلى a كمان
+
+102
+00:11:42,630 --> 00:11:48,030
+برا ال function f بيكون لها absolute maximum value
+
+103
+00:11:48,030 --> 00:11:54,170
+على المجموعة a إذا قدرنا نلاقي x super star ينتمي
+
+104
+00:11:54,170 --> 00:12:01,280
+ل aوقيم الدالة أصغر من أو ساوي قيمة الدالة عن ال X
+
+105
+00:12:01,280 --> 00:12:06,160
+Superstar لكل X في A فهذا ما نسميها Absolute
+
+106
+00:12:06,160 --> 00:12:11,100
+Maximum Value أكبر قيمة عظمة مطلقة و كذلك نستطيع
+
+107
+00:12:11,100 --> 00:12:13,900
+أن نعرف أن ال function لها Absolute Minimum Value
+
+108
+00:12:13,900 --> 00:12:21,640
+على المجموعة A إذا نجد X Lower Star عنصر في A بحيث
+
+109
+00:12:21,640 --> 00:12:26,290
+أنهقيمة الـ downland x lower star أصغر من أو ساوى
+
+110
+00:12:26,290 --> 00:12:36,250
+كل قيمها على المجموعة a in this case يا دي الحلقة
+
+111
+00:12:36,250 --> 00:12:46,950
+x super star respectively على التوالي x lower star
+
+112
+00:12:46,950 --> 00:12:50,710
+is called
+
+113
+00:12:57,680 --> 00:13:06,760
+and absolute maximum and absolute maximum point
+
+114
+00:13:06,760 --> 00:13:15,880
+maximum respectively على التوالي absolute minimum
+
+115
+00:13:15,880 --> 00:13:20,700
+point
+
+116
+00:13:20,700 --> 00:13:24,480
+of
+
+117
+00:13:25,260 --> 00:13:31,720
+الـ function f on a إذا
+
+118
+00:13:31,720 --> 00:13:34,800
+النقطة x super star اللي دالها عندها absolute
+
+119
+00:13:34,800 --> 00:13:41,210
+maximum بنسميها نقطة نهاية عظمة لدالة عالم جمعيةو
+
+120
+00:13:41,210 --> 00:13:44,970
+X lower star اللي عندها الدالة اللي لها absolute
+
+121
+00:13:44,970 --> 00:13:50,970
+minimum value بسميها نقطة absolute minimum point
+
+122
+00:13:50,970 --> 00:13:57,330
+of F على A نقطة نهاية صغيرة okay كلها مجرد تعريفات
+
+123
+00:13:57,330 --> 00:14:04,490
+الان في عندي ال maximum minimum theorem او ال
+
+124
+00:14:04,490 --> 00:14:07,770
+extreme value theorem مدرية دخلتها في calculus A
+
+125
+00:14:09,520 --> 00:14:17,420
+لكن أمرنا بقلبنا منكم برهانة فهو theorem max
+
+126
+00:14:17,420 --> 00:14:29,760
+minimum theorem نظرية
+
+127
+00:14:29,760 --> 00:14:33,420
+القيام القصوى أو القيام العظمى أو الصغرى
+
+128
+00:14:37,040 --> 00:14:48,300
+لت I بساوي closed ب .. ب closed and bounded
+
+129
+00:14:48,300 --> 00:14:56,120
+interval if
+
+130
+00:14:56,120 --> 00:15:04,400
+ال function F from I to R is continuous
+
+131
+00:15:06,950 --> 00:15:17,350
+on I then if attains its
+
+132
+00:15:17,350 --> 00:15:29,490
+maximum او its absolute maximum and absolute
+
+133
+00:15:29,490 --> 00:15:33,810
+minimum on I
+
+134
+00:15:37,830 --> 00:15:45,050
+that is هذا يعني there exist x upper star و x
+
+135
+00:15:45,050 --> 00:15:54,610
+lower star في I such that f of x أصغر أو ساوي f of
+
+136
+00:15:54,610 --> 00:16:06,090
+x super star لكل x في Iand F of X lower star أصغر
+
+137
+00:16:06,090 --> 00:16:13,130
+من أو ساوي F of X لكل X في I تمام؟ لأن هذه هي
+
+138
+00:16:13,130 --> 00:16:17,210
+نظرية القيامة القصوى لو كانت الدالة متصلة على
+
+139
+00:16:17,210 --> 00:16:20,590
+المجال تبعها والمجال تبعها closed bounded interval
+
+140
+00:16:20,590 --> 00:16:25,010
+فلابد أن كل الدالة قيمة عظمة مطلقة وقيمة صغيرة
+
+141
+00:16:25,010 --> 00:16:37,600
+مطلقة على هذه الفترةتمام؟ okay البرهان البرهان سهل
+
+142
+00:16:37,600 --> 00:16:44,660
+proof
+
+143
+00:16:44,660 --> 00:16:50,760
+note
+
+144
+00:16:50,760 --> 00:16:55,740
+first that
+
+145
+00:17:01,890 --> 00:17:08,190
+the non-empty set
+
+146
+00:17:08,190 --> 00:17:20,130
+اللي
+
+147
+00:17:20,130 --> 00:17:28,490
+هي F of I اللي هي كل ال F of X هي في X ينتمي ل I
+
+148
+00:17:29,720 --> 00:17:35,420
+الـ set is non-empty لأن الفترة I هنا طبعا ليست
+
+149
+00:17:35,420 --> 00:17:48,620
+فترة خالية، فيها على الأقل أناصر اشمال
+
+150
+00:17:48,620 --> 00:17:54,740
+الفترة هذه is bounded
+
+151
+00:17:58,460 --> 00:18:09,140
+by boundedness theorem احنا
+
+152
+00:18:09,140 --> 00:18:13,080
+أخدنا نظرية اللي جابي لهذه كانت في اللقاء الأول
+
+153
+00:18:13,080 --> 00:18:19,500
+بتقول لو كانت if function from I to R وكانت الدالة
+
+154
+00:18:19,500 --> 00:18:23,790
+هذه continuous و I closed bounded intervalفالـ
+
+155
+00:18:23,790 --> 00:18:28,550
+function f بتطلع bounded على الفترة I الـ function
+
+156
+00:18:28,550 --> 00:18:32,250
+bounded على الفترة I المعنية هو الـ range تبع الـ
+
+157
+00:18:32,250 --> 00:18:36,190
+function اللي هي ال set هذه is a bounded set okay
+
+158
+00:18:36,190 --> 00:18:40,630
+تمام هذا بنحصل عليه من ال boundedness ال theorem
+
+159
+00:18:40,630 --> 00:18:44,210
+طيب
+
+160
+00:18:44,210 --> 00:18:46,230
+وبالتالي
+
+161
+00:18:50,070 --> 00:18:53,570
+أحنا عايزين الأن نثبت ال claim .. بنا نثبت ال
+
+162
+00:18:53,570 --> 00:19:01,370
+claim التالي ال claim هذا بتكون من جزئين أنه يوجد
+
+163
+00:19:01,370 --> 00:19:09,350
+.. أه طيب كام ال claim hence مادام
+
+164
+00:19:09,350 --> 00:19:12,270
+الست هذي bounded .. إذا bounded above و bounded
+
+165
+00:19:12,270 --> 00:19:15,230
+below و بالتالي في إليها supremum و في إليها
+
+166
+00:19:15,230 --> 00:19:21,810
+infimum by supremum and infimum propertiesإذاً S
+
+167
+00:19:21,810 --> 00:19:27,910
+Superstar اللي هو بساوي ال supremum لست F of I
+
+168
+00:19:27,910 --> 00:19:36,310
+exist in R and S Lower Star اللي هو حاخده ال
+
+169
+00:19:36,310 --> 00:19:45,450
+infimum لست F of I برضه exist in R هذا by supremum
+
+170
+00:19:45,450 --> 00:20:09,180
+property وهذا by infimum propertyOkay تمام الان
+
+171
+00:20:09,180 --> 00:20:14,900
+بدي اثبت ال claim التالي ال claim هذا بتكون من
+
+172
+00:20:14,900 --> 00:20:24,610
+جزءينالجزء الأول انه there exist x superstar ينتمي
+
+173
+00:20:24,610 --> 00:20:33,510
+للفترة I بحيث انه F of X Superstar بساوي S
+
+174
+00:20:33,510 --> 00:20:34,490
+Superstar
+
+175
+00:20:37,010 --> 00:20:44,930
+and الجزء التاني ان يوجد x lower star عنصر ما في
+
+176
+00:20:44,930 --> 00:20:50,890
+الفترة i بحيث ان f ل x lower star بساوي s lower
+
+177
+00:20:50,890 --> 00:20:55,110
+star فلو
+
+178
+00:20:55,110 --> 00:21:01,710
+أثبتنا الجزء الأول معناته ال x star هذه نقطة
+
+179
+00:21:01,710 --> 00:21:06,530
+الدالة بتاخد قيمتها العظم المطلقة عندهاوهذا معناه
+
+180
+00:21:06,530 --> 00:21:11,790
+ان x lower star هي نقطة قيمة صغرى المطلقة لل
+
+181
+00:21:11,790 --> 00:21:17,710
+function f لأن هذه قيمة الصغرى المطلقة للfunction
+
+182
+00:21:17,710 --> 00:21:18,330
+f على i
+
+183
+00:21:32,020 --> 00:21:36,920
+فلو أثبتنا واحد واتنين يكون أثبتنا النظرية حنثبت
+
+184
+00:21:36,920 --> 00:21:41,260
+واحد وبرهان التاني بالمثل مشابه فحاسبكم أنتوا
+
+185
+00:21:41,260 --> 00:21:47,740
+تكتبوا هنا we prove one
+
+186
+00:21:47,740 --> 00:22:02,280
+and leave the proof of two for youزي ما قلنا
+
+187
+00:22:02,280 --> 00:22:10,500
+البرهان هذا تبع الجزء التاني مشابه للأول طيب
+
+188
+00:22:10,500 --> 00:22:25,160
+since S star بساوي ال supremum ل F of I then
+
+189
+00:22:25,160 --> 00:22:29,700
+for each N عدد طبيعي
+
+190
+00:22:32,660 --> 00:22:45,420
+S star minus one upon n is not upper bound of
+
+191
+00:22:45,420 --> 00:22:52,280
+set F of I لأن
+
+192
+00:22:52,280 --> 00:22:54,240
+هذا عبارة عن least upper bound
+
+193
+00:23:00,430 --> 00:23:07,710
+So وبالتالي there exists xn ينتمي للفترة I بحيث
+
+194
+00:23:07,710 --> 00:23:21,290
+أنه F of xn ده أكبر من S star minus واحد على Mهذا
+
+195
+00:23:21,290 --> 00:23:26,050
+العدد ليس upper bound لل set هذه طب ما إذا يوجد
+
+196
+00:23:26,050 --> 00:23:31,430
+عنصر في ال set هذه اللي هو صورة لعنصر في I وهذا
+
+197
+00:23:31,430 --> 00:23:37,640
+العنصر أكبر من SSR ماس واحدةالان S*) هذا عبارة عن
+
+198
+00:23:37,640 --> 00:23:42,420
+upper bound للset F of I وهذا العنصر ينتمي ل F of
+
+199
+00:23:42,420 --> 00:23:46,180
+I، إذن هذا أصغر من أو ساوي ال upper bound لكل
+
+200
+00:23:46,180 --> 00:23:50,400
+المجموعة اللي بتحتوي العناصر اللي زي هذا، إذن هذا
+
+201
+00:23:50,400 --> 00:23:59,240
+أصغر من أو ساوي S*)، okay؟ وهذا طبعا صحيح لكل N في
+
+202
+00:23:59,240 --> 00:24:09,550
+N، كل N في Nتمام اذا ان انا في ياندي المتباينة هذه
+
+203
+00:24:09,550 --> 00:24:13,010
+بما
+
+204
+00:24:13,010 --> 00:24:25,010
+انه ال .. ال set since
+
+205
+00:24:25,010 --> 00:24:30,710
+الفترة I is closed and bounded is bounded
+
+206
+00:24:34,570 --> 00:24:41,410
+و X in .. لاحظوا X in ال sequence هذه contained in
+
+207
+00:24:41,410 --> 00:24:53,230
+I then ال sequence X in is bounded so
+
+208
+00:24:53,230 --> 00:25:00,530
+by Bolzano Weierstrass theorem
+
+209
+00:25:03,780 --> 00:25:07,160
+النظرية بولزانو فايرستراسي بتقول لكل bounded
+
+210
+00:25:07,160 --> 00:25:11,940
+sequence في إلها convergent subsequence إذا there
+
+211
+00:25:11,940 --> 00:25:23,980
+exist a subsequence x in k of sequence x in which
+
+212
+00:25:23,980 --> 00:25:33,710
+converges بتكون convergentsay يعني اننا دعينا نسمي
+
+213
+00:25:33,710 --> 00:25:37,950
+ال limit تبع ال subsequence اللي احنا قلنا انها
+
+214
+00:25:37,950 --> 00:25:45,050
+convergent دعينا نسمي ال limit تبعتها x وهذا ينتمي
+
+215
+00:25:45,050 --> 00:25:55,620
+الى r تمام؟ كمان مرة sinceالـ subsequence X, N, K
+
+216
+00:25:55,620 --> 00:26:03,260
+كل عناصرها موجودة في I اللي هي الفترة المغلقة A و
+
+217
+00:26:03,260 --> 00:26:08,980
+B و ال subsequence هي convergent إذا حسب نظرية في
+
+218
+00:26:08,980 --> 00:26:13,280
+chapter 3 إذا كانت ال sequence عناصرها محصورة بين
+
+219
+00:26:13,280 --> 00:26:17,380
+A و B و convergent فنهايتها هتكون محصورة بين A و B
+
+220
+00:26:17,380 --> 00:26:27,050
+إذا ال X تنتمي للفترة A و Bاللي هي ال I طيب بما
+
+221
+00:26:27,050 --> 00:26:34,070
+انه ال F continuous ال function F continuous على
+
+222
+00:26:34,070 --> 00:26:39,590
+الفترة I و
+
+223
+00:26:39,590 --> 00:26:47,110
+X خلينا نسمي ال X هذا X star عشان بس يكون ايه
+
+224
+00:26:47,110 --> 00:26:51,350
+نتمشي مع ايه مع النص خلينا نسمي ال X هذا X
+
+225
+00:26:51,350 --> 00:26:59,970
+Superstarإذا by hypothesis احنا فرضين ان ال
+
+226
+00:26:59,970 --> 00:27:04,270
+function F متصل على الفترة I و X Superstar عنصر في
+
+227
+00:27:04,270 --> 00:27:14,470
+I إذا F is continuous at X Superstar اللي هو عنصر
+
+228
+00:27:14,470 --> 00:27:19,610
+في I تمام؟و في عندي ال sequence او ال subsequence
+
+229
+00:27:19,610 --> 00:27:23,850
+هى دى convergent ل x star و if continuous at
+
+230
+00:27:23,850 --> 00:27:28,490
+continuous at x star اذا by sequential criterion
+
+231
+00:27:28,490 --> 00:27:35,190
+for continuous function hence by sequential
+
+232
+00:27:35,190 --> 00:27:41,610
+criterion for continuous functions بطلع ال limit
+
+233
+00:27:43,100 --> 00:27:48,160
+للـ image للـ convergence sequence اللي هي X in K
+
+234
+00:27:48,160 --> 00:27:59,560
+K تقوى لإنفينتي بتساوي ال image ل X Superstar تمام
+
+235
+00:27:59,560 --> 00:28:05,320
+إذا أنا في عندي ال image لل sequence أو لل
+
+236
+00:28:05,320 --> 00:28:09,240
+subsequence X in K تطلع convergence
+
+237
+00:28:13,340 --> 00:28:24,480
+و ال ..
+
+238
+00:28:24,480 --> 00:28:29,460
+و متحقق طبعا ..
+
+239
+00:28:29,460 --> 00:28:33,020
+نعم
+
+240
+00:28:33,020 --> 00:28:36,940
+من
+
+241
+00:28:36,940 --> 00:28:41,520
+هنا .. من هنا نسمي هذه قصة
+
+242
+00:28:46,300 --> 00:29:03,200
+by star we have f بدل x in ب x in k فهذا أصغر من
+
+243
+00:29:03,200 --> 00:29:09,900
+أو ساوي s star و أكبر من أو ساوي s super star
+
+244
+00:29:09,900 --> 00:29:13,340
+minus واحد على nk
+
+245
+00:29:19,300 --> 00:29:27,860
+وهذا صحيح لكل K ينتمي إلى N، بظبط؟ الآن خلّي ال K
+
+246
+00:29:27,860 --> 00:29:34,760
+تقوى ل Infinity فإذا 1 على NK لما K تقوى ل
+
+247
+00:29:34,760 --> 00:29:40,720
+Infinity بطلع السفر وبالتالي هذا بروح ل S*) وهذا
+
+248
+00:29:40,720 --> 00:29:47,070
+ثابت لما K تقوى ل Infinityهذه سيكوانس الحد اللي
+
+249
+00:29:47,070 --> 00:29:52,870
+عام تبعها ثابت بتروح ل S Star So
+
+250
+00:29:52,870 --> 00:29:57,890
+by Squeeze Theorem
+
+251
+00:29:57,890 --> 00:30:08,670
+بتطلع عند ال limit ل F of X in K as K till
+
+252
+00:30:08,670 --> 00:30:15,050
+infinity بساوي S Super Star Hence
+
+253
+00:30:17,550 --> 00:30:23,450
+بنسمي هذه double star this
+
+254
+00:30:23,450 --> 00:30:30,170
+and double star المعادلة
+
+255
+00:30:30,170 --> 00:30:39,030
+الأخيرة هو double star yield بيعطوني التالي ان f
+
+256
+00:30:39,030 --> 00:30:53,120
+of x super star بساوي ال limitلـ f of x in k لما k
+
+257
+00:30:53,120 --> 00:31:02,260
+تقول لإنفينيتي و limit من هنا limit f of x in k
+
+258
+00:31:02,260 --> 00:31:07,520
+بساوي ال super star وهذا اللي بدنا يهو هو المطلوب
+
+259
+00:31:09,430 --> 00:31:17,250
+البرهان هذا اثبتنا ان يوجد X Superstar في R و X
+
+260
+00:31:17,250 --> 00:31:23,880
+Superstar هذا طلع في الفترة I موجود في Iبحيث أن
+
+261
+00:31:23,880 --> 00:31:29,360
+صورة الـ F عند X Superstar بساوي S Superstar، اللي
+
+262
+00:31:29,360 --> 00:31:32,540
+هو الـ Supremum لـ Range الـ Function F، اللي هي
+
+263
+00:31:32,540 --> 00:31:37,400
+القيمة العظمى المطلقة، إذن هنا هيوجد نقطة X
+
+264
+00:31:37,400 --> 00:31:40,920
+Superstar في I عندها الـ Function تأخذ قيمتها
+
+265
+00:31:40,920 --> 00:31:45,560
+العظمى المطلقة، بالمثل ممكن نبرهن الجزء التاني،
+
+266
+00:31:45,560 --> 00:31:51,320
+وهذا بكملبرهان النظرية تمام؟ إذا البرهان مش صعب
+
+267
+00:31:51,320 --> 00:31:56,860
+يمكن صحيح طويل شوية لكن يعني ممكن أي واحد يعني
+
+268
+00:31:56,860 --> 00:32:09,040
+يبرهنه لو فهمه الفهم الصحيح النظرية هذه طبعا
+
+269
+00:32:09,040 --> 00:32:14,600
+في يعني ملاحظات أنه يعني المفروض نتطرقلهم
+
+270
+00:32:20,990 --> 00:32:24,830
+إنه لازم عشان نظريةها تكون صحيحة لازم الفترة I
+
+271
+00:32:24,830 --> 00:32:30,750
+تكون closed و bounded يعني لو كانت closed ماهياش
+
+272
+00:32:30,750 --> 00:32:35,330
+bounded مش ممكن ال function تكون نظرية صحيحة و لو
+
+273
+00:32:35,330 --> 00:32:39,190
+كانت bounded و مش closed مش ممكن تكون نظرية صحيحة
+
+274
+00:32:39,190 --> 00:32:44,070
+لو كان مجال الدالة هذه الفترة I closed و bounded
+
+275
+00:32:44,070 --> 00:32:49,470
+زي ما هو مطلوب لكن الدالة مش متصلةفممكن تكون لها
+
+276
+00:32:49,470 --> 00:32:51,990
+absolute maximum او absolute minimum على الأقل
+
+277
+00:32:51,990 --> 00:33:03,430
+واحدة منهم بتكون مش موجودة okay فمثلا
+
+278
+00:33:03,430 --> 00:33:11,190
+marks
+
+279
+00:33:11,190 --> 00:33:15,810
+واحد
+
+280
+00:33:15,810 --> 00:33:21,330
+a functionاو a continuous function .. a continuous
+
+281
+00:33:21,330 --> 00:33:35,230
+function on a set .. on a set A may
+
+282
+00:33:35,230 --> 00:33:39,070
+not have
+
+283
+00:33:39,070 --> 00:33:46,870
+an absolute maximum or absolute
+
+284
+00:33:51,480 --> 00:34:01,680
+minimum on a فعلى سبيل المثال consider خد ال
+
+285
+00:34:01,680 --> 00:34:12,240
+function f of x بساوي واحد على x و x ينتمي للفترة
+
+286
+00:34:12,240 --> 00:34:14,020
+المفتوحة من سفر إلى ملن
+
+287
+00:34:17,550 --> 00:34:25,970
+الـ function هذه if has neither
+
+288
+00:34:25,970 --> 00:34:39,150
+absolute maximum nor absolute minimum on a اللي هي
+
+289
+00:34:39,150 --> 00:34:41,950
+الفترة المفتوحة من صفر إلى نهاية
+
+290
+00:34:44,130 --> 00:34:50,190
+رغم أن الدالة متصلة رغم أن الدالة متصلة السبب أن
+
+291
+00:34:50,190 --> 00:34:57,370
+الفترة هذه ليست مغلقة وليست مفتوحة وبالتالي ماقدرش
+
+292
+00:34:57,370 --> 00:35:04,990
+أدمن نتيجة النظرية، النظرية هذه ماتتطبقش وهذا
+
+293
+00:35:04,990 --> 00:35:11,510
+واضح من الرسم، هذه ال function تبعتي هي واحد على X
+
+294
+00:35:12,820 --> 00:35:18,600
+F of X بيساوي واحد على X و X أكبر من السفر هذا هي
+
+295
+00:35:18,600 --> 00:35:25,520
+فال function هذه مالهاش قيمة صغيرة مافيش X lower
+
+296
+00:35:25,520 --> 00:35:32,320
+star لاحظوا انتوا ان ال infimum ل F of A هنا
+
+297
+00:35:32,320 --> 00:35:35,260
+بيساوي سفر
+
+298
+00:35:40,900 --> 00:35:47,580
+القيمة الـ infimum تبعها بساوي سفر لكن الدالة
+
+299
+00:35:47,580 --> 00:35:52,880
+مالهاش السفر ليس absolute minimum لل function هذه
+
+300
+00:35:52,880 --> 00:36:01,460
+لأن مافيش x lower star عنده
+
+301
+00:36:01,460 --> 00:36:06,240
+قيمة ال function بساوي سفر مافيش في
+
+302
+00:36:07,510 --> 00:36:14,370
+أي x star ينتمي لإيه للفترة هذه وعنده الدالة بساوي
+
+303
+00:36:14,370 --> 00:36:20,270
+سفر مافيش كذلك الدالة هذه مالهاش قيمة عظمى الدالة
+
+304
+00:36:20,270 --> 00:36:26,190
+هذه unbounded from above يعني مافيش
+
+305
+00:36:26,190 --> 00:36:34,550
+أي x superstar عنده الدالة بتساوي أكبر قيمة okay
+
+306
+00:36:36,570 --> 00:36:40,170
+إذا إدّالة ممكن مايكونش إلا لأ قيمة صغيرة مطلقة
+
+307
+00:36:40,170 --> 00:36:51,250
+ولا قيمة عظمة مطلقة كذلك ال ..
+
+308
+00:36:51,250 --> 00:36:57,970
+ملاحظة تانية ال continuous function
+
+309
+00:37:08,760 --> 00:37:15,340
+إذا كانت المفهوم لديه
+
+310
+00:37:15,340 --> 00:37:25,960
+نقطة أكتر فاكتر فاكتر
+
+311
+00:37:25,960 --> 00:37:31,640
+فاكتر فاكتر فاكتر فاكتر فاكتر فاكتر فاكتر فاكتر
+
+312
+00:37:31,640 --> 00:37:37,020
+فاكتر فاكتر فاكتر فاكترthis point is not
+
+313
+00:37:37,020 --> 00:37:40,440
+necessarily
+
+314
+00:37:40,440 --> 00:37:52,180
+مش من الضروري not necessarily uniquely determined
+
+315
+00:38:03,370 --> 00:38:06,750
+يعني لو كانت دالة لها absolute maximum أو لها
+
+316
+00:38:06,750 --> 00:38:13,450
+absolute minimum فالقيمة هذه الأغمق أو الصغرة مش
+
+317
+00:38:13,450 --> 00:38:18,950
+شرط تكون يعني مش شرط ان احنا نحصل عليها عند نقطة
+
+318
+00:38:18,950 --> 00:38:23,150
+واحدة ممكن دالة يكون لها absolute maximum او
+
+319
+00:38:23,150 --> 00:38:26,850
+absolute minimum عند اكثر من نقطة في الدمية تبعها
+
+320
+00:38:26,850 --> 00:38:32,110
+for example على سبيل المثال consider
+
+321
+00:38:33,800 --> 00:38:43,480
+اعتبري الـ function f of x بساوي x تربية وطبعا هنا
+
+322
+00:38:43,480 --> 00:38:48,480
+x ينتمي إلى R ده للتربعية المجال تبع كل العداد
+
+323
+00:38:48,480 --> 00:38:55,000
+الحقيقية خلينا ناخد الـ x بس في الفترة المغلقة
+
+324
+00:38:55,000 --> 00:39:02,420
+والمحدودة من سالب واحد إلى واحد تمام؟
+
+325
+00:39:03,700 --> 00:39:08,380
+فال function f is
+
+326
+00:39:08,380 --> 00:39:16,480
+continuous on I and بنلاحظ
+
+327
+00:39:16,480 --> 00:39:24,740
+أن f عن سلب واحد بساوي f عن واحد بساوي واحد is an
+
+328
+00:39:24,740 --> 00:39:28,260
+absolute maximum
+
+329
+00:39:33,020 --> 00:39:42,000
+at x بساوي سالب واحد واحد إذا سالب واحد واحد عبارة
+
+330
+00:39:42,000 --> 00:39:45,260
+عن absolute maximum points مظبوط هاي الدالة
+
+331
+00:39:45,260 --> 00:39:50,840
+التربية أنا بس ماخد المجال تبعها اللي هو الفترة
+
+332
+00:39:50,840 --> 00:39:54,900
+المغلقة والمحدودة من سالب واحد إلى واحد فطبعا
+
+333
+00:39:54,900 --> 00:39:56,120
+رسمتها زي هيك
+
+334
+00:40:00,870 --> 00:40:05,170
+هذه الدالة التربعية ان المجال تبعها الفترة المغلقة
+
+335
+00:40:05,170 --> 00:40:10,230
+هذه فواضح انه هيقلها absolute maximum اللي هي
+
+336
+00:40:10,230 --> 00:40:15,630
+الواحد وال absolute maximum هذا حصلنا عليه عن مقطه
+
+337
+00:40:15,630 --> 00:40:24,870
+تالى سالب واحد واحد نعم مظبوط صحيح وطبعا هنا
+
+338
+00:40:24,870 --> 00:40:29,980
+فيقلها absolute minimum واحدة اللي هي السفرواضح
+
+339
+00:40:29,980 --> 00:40:34,940
+هنا أن دالة هنا المجال تبعها فترة مغلقة ومحدودة
+
+340
+00:40:34,940 --> 00:40:38,900
+اذا by maximum minimum theorem اكيد فيه اللي هي
+
+341
+00:40:38,900 --> 00:40:41,980
+absolute maximum وفيه اللي هي absolute minimum ال
+
+342
+00:40:41,980 --> 00:40:45,640
+absolute minimum point هي السفرالـ absolute
+
+343
+00:40:45,640 --> 00:40:50,380
+maximum point هنا في نقطتين النظرية مجرد يعني مجرد
+
+344
+00:40:50,380 --> 00:40:53,820
+يوجد على الأقل نقطة واحدة في end absolute maximum
+
+345
+00:40:53,820 --> 00:41:03,500
+فلو كانوا تنتين لا ضرر لا بأس تمام okay إذن يعني
+
+346
+00:41:03,500 --> 00:41:11,440
+هذه بعض الملاحظات على ال maximum theorem ال
+
+347
+00:41:11,440 --> 00:41:11,820
+..
+
+348
+00:41:22,530 --> 00:41:29,810
+أعتقد أن احنا هنكتفي بهذا القدر و ان شاء الله
+
+349
+00:41:29,810 --> 00:41:36,850
+بنكمل ال section هذا في المحاضرة القادمة فاشكركم
+
+350
+00:41:36,850 --> 00:41:44,390
+لحسن استماعكم و ان شاء الله نلتقي يوم الأتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a885fa262bc54b71807964c144efc91b1cfdbc40
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tKghhKfPOf0_raw.srt
@@ -0,0 +1,1400 @@
+1
+00:00:21,680 --> 00:00:30,360
+بسم الله الرحمن الرحيم في اللقاء هذا هحنحاول نكمل
+
+2
+00:00:30,360 --> 00:00:35,700
+ما بدأناه في section خمسة اتنين اخر نظرية أخدناها
+
+3
+00:00:35,700 --> 00:00:39,700
+في المحاضرة السابقة كانت ال boundedness theorem
+
+4
+00:00:39,700 --> 00:00:45,160
+بتقول لو كان في اندي اف function continuous على ال
+
+5
+00:00:45,160 --> 00:00:50,080
+interval Iو ال interval I is closed and bounded
+
+6
+00:00:50,080 --> 00:00:56,000
+فال function بتطلع bounded على الفترة I مبرهنة
+
+7
+00:00:56,000 --> 00:01:04,100
+نظرية في اللقاء السابق الآن في شوية ملاحظات في ال
+
+8
+00:01:04,100 --> 00:01:08,260
+boundedness theorem في النظرية هذه أي شرط من
+
+9
+00:01:08,260 --> 00:01:13,330
+الشروط ال boundedness و ال closedوالاتصال تبعد
+
+10
+00:01:13,330 --> 00:01:18,590
+دالة على الفترة I اي شرط لو اي شرط من الشروط هدولة
+
+11
+00:01:18,590 --> 00:01:24,890
+نجس او اختلف فالنظرية تفشل ماتبطل نظرية بصير
+
+12
+00:01:24,890 --> 00:01:33,790
+مالهاش معنى فعلى سبيل مثال لو كانت الفترة ماهياش
+
+13
+00:01:33,790 --> 00:01:35,710
+bounded الفترة I ماكانتش bounded
+
+14
+00:01:38,580 --> 00:01:45,820
+فالنظرية بتفشل و هاي مثال يوضح ذلك ناخد ال
+
+15
+00:01:45,820 --> 00:01:53,160
+identity function f و ناخد ال I هي الفترة المغلقة
+
+16
+00:01:53,160 --> 00:01:59,100
+من صفر إلى ملنهاي واضح ان الفترة I هذه ماهياش
+
+17
+00:01:59,100 --> 00:01:59,480
+bounded
+
+18
+00:02:05,080 --> 00:02:10,740
+وواضح ان الـ identity function F continuous على
+
+19
+00:02:10,740 --> 00:02:17,160
+الفترة I لأنها continuous على كل الـ R لكن الـ
+
+20
+00:02:17,160 --> 00:02:24,070
+function F ماهياش bounded على الفترة Iليست bounded
+
+21
+00:02:24,070 --> 00:02:32,490
+على I و البرهان لأي عدد موجب بقدر اختار XM X عنصر
+
+22
+00:02:32,490 --> 00:02:42,170
+يعتمد على M هذا ينتمي للفترة I و
+
+23
+00:02:42,170 --> 00:02:46,750
+ال absolute value ل F of XM تطلع absolute M زايد
+
+24
+00:02:46,750 --> 00:02:51,230
+واحد اللي هي M زايد واحد لأن هذا عدد موجب أكبر من
+
+25
+00:02:51,230 --> 00:02:57,750
+Mحسب الـ remark اللي هي بتكافئ تعريف الbounded
+
+26
+00:02:57,750 --> 00:03:03,430
+تعريف ال unboundedness ال function بتحقق شرط ال
+
+27
+00:03:03,430 --> 00:03:07,370
+unboundedness وبالتالي ال function unbounded on I
+
+28
+00:03:07,370 --> 00:03:15,630
+لأن هنا عشان الفترة ليست bounded النظرية فشلت في
+
+29
+00:03:15,630 --> 00:03:18,890
+الملاحظة التانية لو ال boundedness theorem is
+
+30
+00:03:18,890 --> 00:03:25,230
+false تفشلإذا كانت الـ interval ماهياش مغلقة فع
+
+31
+00:03:25,230 --> 00:03:29,730
+سبيل المثال consider الـ function واحد على X حيث X
+
+32
+00:03:29,730 --> 00:03:36,850
+أكبر من سفر و أصغر من أو ساول واحد فواضح
+
+33
+00:03:36,850 --> 00:03:43,490
+إن الفترة I هذه not closed ليست مغلقة و الدالة G
+
+34
+00:03:43,490 --> 00:03:48,650
+of X متصلة عليها لأنها لا تحتوى السفر لأن هذه عندي
+
+35
+00:03:48,650 --> 00:03:55,080
+function Gcontinuous على الفترة I الفترة I ماهياش
+
+36
+00:03:55,080 --> 00:04:00,460
+closed وبالتالي
+
+37
+00:04:00,460 --> 00:04:05,520
+وفي نفس الوجهة ال function هذه ليست bounded على
+
+38
+00:04:05,520 --> 00:04:10,040
+الفترة I احنا شفنا ان ال function 1 على X في بداية
+
+39
+00:04:10,040 --> 00:04:16,600
+اللقاء اللي فات ليست bounded على الفترة من صفر إلى
+
+40
+00:04:16,600 --> 00:04:23,910
+ملا نهايةو بالمثل ليست bounded على الفترة من صفر
+
+41
+00:04:23,910 --> 00:04:30,070
+إلى واحد ممكن اعطاء برهان مشابه، إذا هذا مثال على
+
+42
+00:04:30,070 --> 00:04:35,850
+function متصرة على فترة ولكنها ليست bounded على
+
+43
+00:04:35,850 --> 00:04:39,930
+نفس الفترة لأن الفترة هذه السبب أن الفترة هذه ليست
+
+44
+00:04:39,930 --> 00:04:44,130
+closed، تمام؟
+
+45
+00:04:44,130 --> 00:04:48,700
+إذا الفترة هذه لو ماكانتش closed، النظرية بتفشلطب
+
+46
+00:04:48,700 --> 00:04:54,060
+لو كانت closed و لا bounded، برضه بتفشل، لأن هذا
+
+47
+00:04:54,060 --> 00:05:01,060
+الشرط ضروري و هذا الشرط ضروري كذلك ال function
+
+48
+00:05:01,060 --> 00:05:04,760
+لازم تكون continuous على الفترة هذه، يعني لو كانت
+
+49
+00:05:04,760 --> 00:05:07,820
+الفترة هذه closed و bounded، فال function مش
+
+50
+00:05:07,820 --> 00:05:12,140
+continuous، فالنظرية تفشل، لأن هنا the boundedness
+
+51
+00:05:12,140 --> 00:05:12,640
+theorem
+
+52
+00:05:19,930 --> 00:05:31,430
+is false if الـ function لو كانت الدالة F is not
+
+53
+00:05:31,430 --> 00:05:34,690
+continuous
+
+54
+00:05:34,690 --> 00:05:39,910
+on الفترة I فعلى سبيل المثال
+
+55
+00:05:51,070 --> 00:05:58,870
+for example ممكن نعرف define H
+
+56
+00:05:58,870 --> 00:06:06,710
+من الفترة المغلقة من
+
+57
+00:06:06,710 --> 00:06:17,470
+سفر إلى واحد إلى R by H of X بالساوي
+
+58
+00:06:24,650 --> 00:06:34,070
+1 على x إذا كان x أكبر من 0 أصغر من أو ساوي 1
+
+59
+00:06:34,070 --> 00:06:46,730
+وخلّيني أخدها بالساوي 1 إذا كان x بالساوي 0 ال
+
+60
+00:06:46,730 --> 00:06:50,510
+function h of x 1 على x إذا كان x أكبر من 0 أصغر
+
+61
+00:06:50,510 --> 00:06:56,950
+من أو ساوي 1 و عند السفر خلّيها الساوي 1فطبعا ال
+
+62
+00:06:56,950 --> 00:07:10,430
+function h is not continuous at x بساوي سفر ليست
+
+63
+00:07:10,430 --> 00:07:18,070
+متصلة عند السفر لأن ال limit لل function h of x
+
+64
+00:07:18,070 --> 00:07:22,490
+لما x تقول إلى السفر
+
+65
+00:07:25,030 --> 00:07:33,670
+من اليمين بيساوي infinity لاتساوي
+
+66
+00:07:33,670 --> 00:07:41,690
+h عند الواحد عند السفر اللي هو بيساوي واحد او كون
+
+67
+00:07:41,690 --> 00:07:46,690
+ال limit بيساوي infinity اذا does not exist in R
+
+68
+00:07:46,690 --> 00:07:50,690
+وبالتالي هذا يكفي ان احنا نقول ان ال function
+
+69
+00:07:50,690 --> 00:08:00,080
+ماهياش continuous عند السفرفى نفس الواجهة Note
+
+70
+00:08:00,080 --> 00:08:04,360
+that الفترة
+
+71
+00:08:04,360 --> 00:08:15,740
+I بساوي سفر و واحد is closed and bounded يعني أنا
+
+72
+00:08:15,740 --> 00:08:22,180
+مافيش عند التصال عند السفر ال function هذه H طبعا
+
+73
+00:08:22,180 --> 00:08:29,870
+زى ما ال function هذيكNote also that
+
+74
+00:08:29,870 --> 00:08:35,890
+the function H is unbounded
+
+75
+00:08:35,890 --> 00:08:43,830
+على الفترة من صفر لواحد لأنها
+
+76
+00:08:43,830 --> 00:08:48,210
+unbounded على الفترة المفتوحة H بالساعة واحد على X
+
+77
+00:08:48,210 --> 00:08:52,590
+على الفترة نصف المفتوحة هذهو يقولنا إن الـ
+
+78
+00:08:52,590 --> 00:08:57,050
+function 1 على x ليست bounded 1 على x ماهياش
+
+79
+00:08:57,050 --> 00:09:03,570
+bounded على الفترة من 0 إلى 1 إذا
+
+80
+00:09:03,570 --> 00:09:07,850
+النظرية تفشل لأن الـ function تبعت هنا ليست
+
+81
+00:09:07,850 --> 00:09:12,750
+continuous عند السفر وبالتالي ليست continuous على
+
+82
+00:09:12,750 --> 00:09:19,150
+كل الفترة الواحد هنا H is not continuous على كل
+
+83
+00:09:19,150 --> 00:09:24,200
+الفترة Iلأنها مش continuous عند السفر اللي هي
+
+84
+00:09:24,200 --> 00:09:28,020
+answer في آية عشان تكون ال function continuous على
+
+85
+00:09:28,020 --> 00:09:31,840
+كل الفترة لازم تكون continuous عند كل نقطة في
+
+86
+00:09:31,840 --> 00:09:34,220
+الفترة لو كانت مش continuous عند نقطة واحدة في
+
+87
+00:09:34,220 --> 00:09:43,760
+الفترة فمايقدرش أقول continuous على الفترة تمام
+
+88
+00:09:43,760 --> 00:09:47,600
+إذا ناخد
+
+89
+00:09:54,630 --> 00:10:04,450
+تعيف definition a
+
+90
+00:10:04,450 --> 00:10:13,030
+function f from A to R has
+
+91
+00:10:13,030 --> 00:10:18,090
+an absolute
+
+92
+00:10:18,090 --> 00:10:19,110
+maximum
+
+93
+00:10:27,530 --> 00:10:39,550
+respectively على التوالي absolute minimum at
+
+94
+00:10:39,550 --> 00:10:47,110
+on
+
+95
+00:10:47,110 --> 00:10:56,610
+الفأس set A if الشرط التالي بيتحققthere exist x
+
+96
+00:10:56,610 --> 00:11:01,370
+super star respectively
+
+97
+00:11:01,370 --> 00:11:08,530
+x lower star تنتمي
+
+98
+00:11:08,530 --> 00:11:13,190
+إلى a بحيث
+
+99
+00:11:13,190 --> 00:11:22,650
+انه f of x أصغر من لو ساوي f of x super star
+
+100
+00:11:25,650 --> 00:11:33,030
+respectively على التوالي f of x lower star أصغر من
+
+101
+00:11:33,030 --> 00:11:42,630
+أو ساوي f of x لكل x تنتمي إلى a كمان
+
+102
+00:11:42,630 --> 00:11:48,030
+برا ال function f بيكون لها absolute maximum value
+
+103
+00:11:48,030 --> 00:11:54,170
+على المجموعة a إذا قدرنا نلاقي x super star ينتمي
+
+104
+00:11:54,170 --> 00:12:01,280
+ل aوقيم الدالة أصغر من أو ساوي قيمة الدالة عن ال X
+
+105
+00:12:01,280 --> 00:12:06,160
+Superstar لكل X في A فهذا ما نسميها Absolute
+
+106
+00:12:06,160 --> 00:12:11,100
+Maximum Value أكبر قيمة عظمة مطلقة و كذلك نستطيع
+
+107
+00:12:11,100 --> 00:12:13,900
+أن نعرف أن ال function لها Absolute Minimum Value
+
+108
+00:12:13,900 --> 00:12:21,640
+على المجموعة A إذا نجد X Lower Star عنصر في A بحيث
+
+109
+00:12:21,640 --> 00:12:26,290
+أنهقيمة الـ downland x lower star أصغر من أو ساوى
+
+110
+00:12:26,290 --> 00:12:36,250
+كل قيمها على المجموعة a in this case يا دي الحلقة
+
+111
+00:12:36,250 --> 00:12:46,950
+x super star respectively على التوالي x lower star
+
+112
+00:12:46,950 --> 00:12:50,710
+is called
+
+113
+00:12:57,680 --> 00:13:06,760
+and absolute maximum and absolute maximum point
+
+114
+00:13:06,760 --> 00:13:15,880
+maximum respectively على التوالي absolute minimum
+
+115
+00:13:15,880 --> 00:13:20,700
+point
+
+116
+00:13:20,700 --> 00:13:24,480
+of
+
+117
+00:13:25,260 --> 00:13:31,720
+الـ function f on a إذا
+
+118
+00:13:31,720 --> 00:13:34,800
+النقطة x super star اللي دالها عندها absolute
+
+119
+00:13:34,800 --> 00:13:41,210
+maximum بنسميها نقطة نهاية عظمة لدالة عالم جمعيةو
+
+120
+00:13:41,210 --> 00:13:44,970
+X lower star اللي عندها الدالة اللي لها absolute
+
+121
+00:13:44,970 --> 00:13:50,970
+minimum value بسميها نقطة absolute minimum point
+
+122
+00:13:50,970 --> 00:13:57,330
+of F على A نقطة نهاية صغيرة okay كلها مجرد تعريفات
+
+123
+00:13:57,330 --> 00:14:04,490
+الان في عندي ال maximum minimum theorem او ال
+
+124
+00:14:04,490 --> 00:14:07,770
+extreme value theorem مدرية دخلتها في calculus A
+
+125
+00:14:09,520 --> 00:14:17,420
+لكن أمرنا بقلبنا منكم برهانة فهو theorem max
+
+126
+00:14:17,420 --> 00:14:29,760
+minimum theorem نظرية
+
+127
+00:14:29,760 --> 00:14:33,420
+القيام القصوى أو القيام العظمى أو الصغرى
+
+128
+00:14:37,040 --> 00:14:48,300
+لت I بساوي closed ب .. ب closed and bounded
+
+129
+00:14:48,300 --> 00:14:56,120
+interval if
+
+130
+00:14:56,120 --> 00:15:04,400
+ال function F from I to R is continuous
+
+131
+00:15:06,950 --> 00:15:17,350
+on I then if attains its
+
+132
+00:15:17,350 --> 00:15:29,490
+maximum او its absolute maximum and absolute
+
+133
+00:15:29,490 --> 00:15:33,810
+minimum on I
+
+134
+00:15:37,830 --> 00:15:45,050
+that is هذا يعني there exist x upper star و x
+
+135
+00:15:45,050 --> 00:15:54,610
+lower star في I such that f of x أصغر أو ساوي f of
+
+136
+00:15:54,610 --> 00:16:06,090
+x super star لكل x في Iand F of X lower star أصغر
+
+137
+00:16:06,090 --> 00:16:13,130
+من أو ساوي F of X لكل X في I تمام؟ لأن هذه هي
+
+138
+00:16:13,130 --> 00:16:17,210
+نظرية القيامة القصوى لو كانت الدالة متصلة على
+
+139
+00:16:17,210 --> 00:16:20,590
+المجال تبعها والمجال تبعها closed bounded interval
+
+140
+00:16:20,590 --> 00:16:25,010
+فلابد أن كل الدالة قيمة عظمة مطلقة وقيمة صغيرة
+
+141
+00:16:25,010 --> 00:16:37,600
+مطلقة على هذه الفترةتمام؟ okay البرهان البرهان سهل
+
+142
+00:16:37,600 --> 00:16:44,660
+proof
+
+143
+00:16:44,660 --> 00:16:50,760
+note
+
+144
+00:16:50,760 --> 00:16:55,740
+first that
+
+145
+00:17:01,890 --> 00:17:08,190
+the non-empty set
+
+146
+00:17:08,190 --> 00:17:20,130
+اللي
+
+147
+00:17:20,130 --> 00:17:28,490
+هي F of I اللي هي كل ال F of X هي في X ينتمي ل I
+
+148
+00:17:29,720 --> 00:17:35,420
+الـ set is non-empty لأن الفترة I هنا طبعا ليست
+
+149
+00:17:35,420 --> 00:17:48,620
+فترة خالية، فيها على الأقل أناصر اشمال
+
+150
+00:17:48,620 --> 00:17:54,740
+الفترة هذه is bounded
+
+151
+00:17:58,460 --> 00:18:09,140
+by boundedness theorem احنا
+
+152
+00:18:09,140 --> 00:18:13,080
+أخدنا نظرية اللي جابي لهذه كانت في اللقاء الأول
+
+153
+00:18:13,080 --> 00:18:19,500
+بتقول لو كانت if function from I to R وكانت الدالة
+
+154
+00:18:19,500 --> 00:18:23,790
+هذه continuous و I closed bounded intervalفالـ
+
+155
+00:18:23,790 --> 00:18:28,550
+function f بتطلع bounded على الفترة I الـ function
+
+156
+00:18:28,550 --> 00:18:32,250
+bounded على الفترة I المعنية هو الـ range تبع الـ
+
+157
+00:18:32,250 --> 00:18:36,190
+function اللي هي ال set هذه is a bounded set okay
+
+158
+00:18:36,190 --> 00:18:40,630
+تمام هذا بنحصل عليه من ال boundedness ال theorem
+
+159
+00:18:40,630 --> 00:18:44,210
+طيب
+
+160
+00:18:44,210 --> 00:18:46,230
+وبالتالي
+
+161
+00:18:50,070 --> 00:18:53,570
+أحنا عايزين الأن نثبت ال claim .. بنا نثبت ال
+
+162
+00:18:53,570 --> 00:19:01,370
+claim التالي ال claim هذا بتكون من جزئين أنه يوجد
+
+163
+00:19:01,370 --> 00:19:09,350
+.. أه طيب كام ال claim hence مادام
+
+164
+00:19:09,350 --> 00:19:12,270
+الست هذي bounded .. إذا bounded above و bounded
+
+165
+00:19:12,270 --> 00:19:15,230
+below و بالتالي في إليها supremum و في إليها
+
+166
+00:19:15,230 --> 00:19:21,810
+infimum by supremum and infimum propertiesإذاً S
+
+167
+00:19:21,810 --> 00:19:27,910
+Superstar اللي هو بساوي ال supremum لست F of I
+
+168
+00:19:27,910 --> 00:19:36,310
+exist in R and S Lower Star اللي هو حاخده ال
+
+169
+00:19:36,310 --> 00:19:45,450
+infimum لست F of I برضه exist in R هذا by supremum
+
+170
+00:19:45,450 --> 00:20:09,180
+property وهذا by infimum propertyOkay تمام الان
+
+171
+00:20:09,180 --> 00:20:14,900
+بدي اثبت ال claim التالي ال claim هذا بتكون من
+
+172
+00:20:14,900 --> 00:20:24,610
+جزءينالجزء الأول انه there exist x superstar ينتمي
+
+173
+00:20:24,610 --> 00:20:33,510
+للفترة I بحيث انه F of X Superstar بساوي S
+
+174
+00:20:33,510 --> 00:20:34,490
+Superstar
+
+175
+00:20:37,010 --> 00:20:44,930
+and الجزء التاني ان يوجد x lower star عنصر ما في
+
+176
+00:20:44,930 --> 00:20:50,890
+الفترة i بحيث ان f ل x lower star بساوي s lower
+
+177
+00:20:50,890 --> 00:20:55,110
+star فلو
+
+178
+00:20:55,110 --> 00:21:01,710
+أثبتنا الجزء الأول معناته ال x star هذه نقطة
+
+179
+00:21:01,710 --> 00:21:06,530
+الدالة بتاخد قيمتها العظم المطلقة عندهاوهذا معناه
+
+180
+00:21:06,530 --> 00:21:11,790
+ان x lower star هي نقطة قيمة صغرى المطلقة لل
+
+181
+00:21:11,790 --> 00:21:17,710
+function f لأن هذه قيمة الصغرى المطلقة للfunction
+
+182
+00:21:17,710 --> 00:21:18,330
+f على i
+
+183
+00:21:32,020 --> 00:21:36,920
+فلو أثبتنا واحد واتنين يكون أثبتنا النظرية حنثبت
+
+184
+00:21:36,920 --> 00:21:41,260
+واحد وبرهان التاني بالمثل مشابه فحاسبكم أنتوا
+
+185
+00:21:41,260 --> 00:21:47,740
+تكتبوا هنا we prove one
+
+186
+00:21:47,740 --> 00:22:02,280
+and leave the proof of two for youزي ما قلنا
+
+187
+00:22:02,280 --> 00:22:10,500
+البرهان هذا تبع الجزء التاني مشابه للأول طيب
+
+188
+00:22:10,500 --> 00:22:25,160
+since S star بساوي ال supremum ل F of I then
+
+189
+00:22:25,160 --> 00:22:29,700
+for each N عدد طبيعي
+
+190
+00:22:32,660 --> 00:22:45,420
+S star minus one upon n is not upper bound of
+
+191
+00:22:45,420 --> 00:22:52,280
+set F of I لأن
+
+192
+00:22:52,280 --> 00:22:54,240
+هذا عبارة عن least upper bound
+
+193
+00:23:00,430 --> 00:23:07,710
+So وبالتالي there exists xn ينتمي للفترة I بحيث
+
+194
+00:23:07,710 --> 00:23:21,290
+أنه F of xn ده أكبر من S star minus واحد على Mهذا
+
+195
+00:23:21,290 --> 00:23:26,050
+العدد ليس upper bound لل set هذه طب ما إذا يوجد
+
+196
+00:23:26,050 --> 00:23:31,430
+عنصر في ال set هذه اللي هو صورة لعنصر في I وهذا
+
+197
+00:23:31,430 --> 00:23:37,640
+العنصر أكبر من SSR ماس واحدةالان S*) هذا عبارة عن
+
+198
+00:23:37,640 --> 00:23:42,420
+upper bound للset F of I وهذا العنصر ينتمي ل F of
+
+199
+00:23:42,420 --> 00:23:46,180
+I، إذن هذا أصغر من أو ساوي ال upper bound لكل
+
+200
+00:23:46,180 --> 00:23:50,400
+المجموعة اللي بتحتوي العناصر اللي زي هذا، إذن هذا
+
+201
+00:23:50,400 --> 00:23:59,240
+أصغر من أو ساوي S*)، okay؟ وهذا طبعا صحيح لكل N في
+
+202
+00:23:59,240 --> 00:24:09,550
+N، كل N في Nتمام اذا ان انا في ياندي المتباينة هذه
+
+203
+00:24:09,550 --> 00:24:13,010
+بما
+
+204
+00:24:13,010 --> 00:24:25,010
+انه ال .. ال set since
+
+205
+00:24:25,010 --> 00:24:30,710
+الفترة I is closed and bounded is bounded
+
+206
+00:24:34,570 --> 00:24:41,410
+و X in .. لاحظوا X in ال sequence هذه contained in
+
+207
+00:24:41,410 --> 00:24:53,230
+I then ال sequence X in is bounded so
+
+208
+00:24:53,230 --> 00:25:00,530
+by Bolzano Weierstrass theorem
+
+209
+00:25:03,780 --> 00:25:07,160
+النظرية بولزانو فايرستراسي بتقول لكل bounded
+
+210
+00:25:07,160 --> 00:25:11,940
+sequence في إلها convergent subsequence إذا there
+
+211
+00:25:11,940 --> 00:25:23,980
+exist a subsequence x in k of sequence x in which
+
+212
+00:25:23,980 --> 00:25:33,710
+converges بتكون convergentsay يعني اننا دعينا نسمي
+
+213
+00:25:33,710 --> 00:25:37,950
+ال limit تبع ال subsequence اللي احنا قلنا انها
+
+214
+00:25:37,950 --> 00:25:45,050
+convergent دعينا نسمي ال limit تبعتها x وهذا ينتمي
+
+215
+00:25:45,050 --> 00:25:55,620
+الى r تمام؟ كمان مرة sinceالـ subsequence X, N, K
+
+216
+00:25:55,620 --> 00:26:03,260
+كل عناصرها موجودة في I اللي هي الفترة المغلقة A و
+
+217
+00:26:03,260 --> 00:26:08,980
+B و ال subsequence هي convergent إذا حسب نظرية في
+
+218
+00:26:08,980 --> 00:26:13,280
+chapter 3 إذا كانت ال sequence عناصرها محصورة بين
+
+219
+00:26:13,280 --> 00:26:17,380
+A و B و convergent فنهايتها هتكون محصورة بين A و B
+
+220
+00:26:17,380 --> 00:26:27,050
+إذا ال X تنتمي للفترة A و Bاللي هي ال I طيب بما
+
+221
+00:26:27,050 --> 00:26:34,070
+انه ال F continuous ال function F continuous على
+
+222
+00:26:34,070 --> 00:26:39,590
+الفترة I و
+
+223
+00:26:39,590 --> 00:26:47,110
+X خلينا نسمي ال X هذا X star عشان بس يكون ايه
+
+224
+00:26:47,110 --> 00:26:51,350
+نتمشي مع ايه مع النص خلينا نسمي ال X هذا X
+
+225
+00:26:51,350 --> 00:26:59,970
+Superstarإذا by hypothesis احنا فرضين ان ال
+
+226
+00:26:59,970 --> 00:27:04,270
+function F متصل على الفترة I و X Superstar عنصر في
+
+227
+00:27:04,270 --> 00:27:14,470
+I إذا F is continuous at X Superstar اللي هو عنصر
+
+228
+00:27:14,470 --> 00:27:19,610
+في I تمام؟و في عندي ال sequence او ال subsequence
+
+229
+00:27:19,610 --> 00:27:23,850
+هى دى convergent ل x star و if continuous at
+
+230
+00:27:23,850 --> 00:27:28,490
+continuous at x star اذا by sequential criterion
+
+231
+00:27:28,490 --> 00:27:35,190
+for continuous function hence by sequential
+
+232
+00:27:35,190 --> 00:27:41,610
+criterion for continuous functions بطلع ال limit
+
+233
+00:27:43,100 --> 00:27:48,160
+للـ image للـ convergence sequence اللي هي X in K
+
+234
+00:27:48,160 --> 00:27:59,560
+K تقوى لإنفينتي بتساوي ال image ل X Superstar تمام
+
+235
+00:27:59,560 --> 00:28:05,320
+إذا أنا في عندي ال image لل sequence أو لل
+
+236
+00:28:05,320 --> 00:28:09,240
+subsequence X in K تطلع convergence
+
+237
+00:28:13,340 --> 00:28:24,480
+و ال ..
+
+238
+00:28:24,480 --> 00:28:29,460
+و متحقق طبعا ..
+
+239
+00:28:29,460 --> 00:28:33,020
+نعم
+
+240
+00:28:33,020 --> 00:28:36,940
+من
+
+241
+00:28:36,940 --> 00:28:41,520
+هنا .. من هنا نسمي هذه قصة
+
+242
+00:28:46,300 --> 00:29:03,200
+by star we have f بدل x in ب x in k فهذا أصغر من
+
+243
+00:29:03,200 --> 00:29:09,900
+أو ساوي s star و أكبر من أو ساوي s super star
+
+244
+00:29:09,900 --> 00:29:13,340
+minus واحد على nk
+
+245
+00:29:19,300 --> 00:29:27,860
+وهذا صحيح لكل K ينتمي إلى N، بظبط؟ الآن خلّي ال K
+
+246
+00:29:27,860 --> 00:29:34,760
+تقوى ل Infinity فإذا 1 على NK لما K تقوى ل
+
+247
+00:29:34,760 --> 00:29:40,720
+Infinity بطلع السفر وبالتالي هذا بروح ل S*) وهذا
+
+248
+00:29:40,720 --> 00:29:47,070
+ثابت لما K تقوى ل Infinityهذه سيكوانس الحد اللي
+
+249
+00:29:47,070 --> 00:29:52,870
+عام تبعها ثابت بتروح ل S Star So
+
+250
+00:29:52,870 --> 00:29:57,890
+by Squeeze Theorem
+
+251
+00:29:57,890 --> 00:30:08,670
+بتطلع عند ال limit ل F of X in K as K till
+
+252
+00:30:08,670 --> 00:30:15,050
+infinity بساوي S Super Star Hence
+
+253
+00:30:17,550 --> 00:30:23,450
+بنسمي هذه double star this
+
+254
+00:30:23,450 --> 00:30:30,170
+and double star المعادلة
+
+255
+00:30:30,170 --> 00:30:39,030
+الأخيرة هو double star yield بيعطوني التالي ان f
+
+256
+00:30:39,030 --> 00:30:53,120
+of x super star بساوي ال limitلـ f of x in k لما k
+
+257
+00:30:53,120 --> 00:31:02,260
+تقول لإنفينيتي و limit من هنا limit f of x in k
+
+258
+00:31:02,260 --> 00:31:07,520
+بساوي ال super star وهذا اللي بدنا يهو هو المطلوب
+
+259
+00:31:09,430 --> 00:31:17,250
+البرهان هذا اثبتنا ان يوجد X Superstar في R و X
+
+260
+00:31:17,250 --> 00:31:23,880
+Superstar هذا طلع في الفترة I موجود في Iبحيث أن
+
+261
+00:31:23,880 --> 00:31:29,360
+صورة الـ F عند X Superstar بساوي S Superstar، اللي
+
+262
+00:31:29,360 --> 00:31:32,540
+هو الـ Supremum لـ Range الـ Function F، اللي هي
+
+263
+00:31:32,540 --> 00:31:37,400
+القيمة العظمى المطلقة، إذن هنا هيوجد نقطة X
+
+264
+00:31:37,400 --> 00:31:40,920
+Superstar في I عندها الـ Function تأخذ قيمتها
+
+265
+00:31:40,920 --> 00:31:45,560
+العظمى المطلقة، بالمثل ممكن نبرهن الجزء التاني،
+
+266
+00:31:45,560 --> 00:31:51,320
+وهذا بكملبرهان النظرية تمام؟ إذا البرهان مش صعب
+
+267
+00:31:51,320 --> 00:31:56,860
+يمكن صحيح طويل شوية لكن يعني ممكن أي واحد يعني
+
+268
+00:31:56,860 --> 00:32:09,040
+يبرهنه لو فهمه الفهم الصحيح النظرية هذه طبعا
+
+269
+00:32:09,040 --> 00:32:14,600
+في يعني ملاحظات أنه يعني المفروض نتطرقلهم
+
+270
+00:32:20,990 --> 00:32:24,830
+إنه لازم عشان نظريةها تكون صحيحة لازم الفترة I
+
+271
+00:32:24,830 --> 00:32:30,750
+تكون closed و bounded يعني لو كانت closed ماهياش
+
+272
+00:32:30,750 --> 00:32:35,330
+bounded مش ممكن ال function تكون نظرية صحيحة و لو
+
+273
+00:32:35,330 --> 00:32:39,190
+كانت bounded و مش closed مش ممكن تكون نظرية صحيحة
+
+274
+00:32:39,190 --> 00:32:44,070
+لو كان مجال الدالة هذه الفترة I closed و bounded
+
+275
+00:32:44,070 --> 00:32:49,470
+زي ما هو مطلوب لكن الدالة مش متصلةفممكن تكون لها
+
+276
+00:32:49,470 --> 00:32:51,990
+absolute maximum او absolute minimum على الأقل
+
+277
+00:32:51,990 --> 00:33:03,430
+واحدة منهم بتكون مش موجودة okay فمثلا
+
+278
+00:33:03,430 --> 00:33:11,190
+marks
+
+279
+00:33:11,190 --> 00:33:15,810
+واحد
+
+280
+00:33:15,810 --> 00:33:21,330
+a functionاو a continuous function .. a continuous
+
+281
+00:33:21,330 --> 00:33:35,230
+function on a set .. on a set A may
+
+282
+00:33:35,230 --> 00:33:39,070
+not have
+
+283
+00:33:39,070 --> 00:33:46,870
+an absolute maximum or absolute
+
+284
+00:33:51,480 --> 00:34:01,680
+minimum on a فعلى سبيل المثال consider خد ال
+
+285
+00:34:01,680 --> 00:34:12,240
+function f of x بساوي واحد على x و x ينتمي للفترة
+
+286
+00:34:12,240 --> 00:34:14,020
+المفتوحة من سفر إلى ملن
+
+287
+00:34:17,550 --> 00:34:25,970
+الـ function هذه if has neither
+
+288
+00:34:25,970 --> 00:34:39,150
+absolute maximum nor absolute minimum on a اللي هي
+
+289
+00:34:39,150 --> 00:34:41,950
+الفترة المفتوحة من صفر إلى نهاية
+
+290
+00:34:44,130 --> 00:34:50,190
+رغم أن الدالة متصلة رغم أن الدالة متصلة السبب أن
+
+291
+00:34:50,190 --> 00:34:57,370
+الفترة هذه ليست مغلقة وليست مفتوحة وبالتالي ماقدرش
+
+292
+00:34:57,370 --> 00:35:04,990
+أدمن نتيجة النظرية، النظرية هذه ماتتطبقش وهذا
+
+293
+00:35:04,990 --> 00:35:11,510
+واضح من الرسم، هذه ال function تبعتي هي واحد على X
+
+294
+00:35:12,820 --> 00:35:18,600
+F of X بيساوي واحد على X و X أكبر من السفر هذا هي
+
+295
+00:35:18,600 --> 00:35:25,520
+فال function هذه مالهاش قيمة صغيرة مافيش X lower
+
+296
+00:35:25,520 --> 00:35:32,320
+star لاحظوا انتوا ان ال infimum ل F of A هنا
+
+297
+00:35:32,320 --> 00:35:35,260
+بيساوي سفر
+
+298
+00:35:40,900 --> 00:35:47,580
+القيمة الـ infimum تبعها بساوي سفر لكن الدالة
+
+299
+00:35:47,580 --> 00:35:52,880
+مالهاش السفر ليس absolute minimum لل function هذه
+
+300
+00:35:52,880 --> 00:36:01,460
+لأن مافيش x lower star عنده
+
+301
+00:36:01,460 --> 00:36:06,240
+قيمة ال function بساوي سفر مافيش في
+
+302
+00:36:07,510 --> 00:36:14,370
+أي x star ينتمي لإيه للفترة هذه وعنده الدالة بساوي
+
+303
+00:36:14,370 --> 00:36:20,270
+سفر مافيش كذلك الدالة هذه مالهاش قيمة عظمى الدالة
+
+304
+00:36:20,270 --> 00:36:26,190
+هذه unbounded from above يعني مافيش
+
+305
+00:36:26,190 --> 00:36:34,550
+أي x superstar عنده الدالة بتساوي أكبر قيمة okay
+
+306
+00:36:36,570 --> 00:36:40,170
+إذا إدّالة ممكن مايكونش إلا لأ قيمة صغيرة مطلقة
+
+307
+00:36:40,170 --> 00:36:51,250
+ولا قيمة عظمة مطلقة كذلك ال ..
+
+308
+00:36:51,250 --> 00:36:57,970
+ملاحظة تانية ال continuous function
+
+309
+00:37:08,760 --> 00:37:15,340
+إذا كانت المفهوم لديه
+
+310
+00:37:15,340 --> 00:37:25,960
+نقطة أكتر فاكتر فاكتر
+
+311
+00:37:25,960 --> 00:37:31,640
+فاكتر فاكتر فاكتر فاكتر فاكتر فاكتر فاكتر فاكتر
+
+312
+00:37:31,640 --> 00:37:37,020
+فاكتر فاكتر فاكتر فاكترthis point is not
+
+313
+00:37:37,020 --> 00:37:40,440
+necessarily
+
+314
+00:37:40,440 --> 00:37:52,180
+مش من الضروري not necessarily uniquely determined
+
+315
+00:38:03,370 --> 00:38:06,750
+يعني لو كانت دالة لها absolute maximum أو لها
+
+316
+00:38:06,750 --> 00:38:13,450
+absolute minimum فالقيمة هذه الأغمق أو الصغرة مش
+
+317
+00:38:13,450 --> 00:38:18,950
+شرط تكون يعني مش شرط ان احنا نحصل عليها عند نقطة
+
+318
+00:38:18,950 --> 00:38:23,150
+واحدة ممكن دالة يكون لها absolute maximum او
+
+319
+00:38:23,150 --> 00:38:26,850
+absolute minimum عند اكثر من نقطة في الدمية تبعها
+
+320
+00:38:26,850 --> 00:38:32,110
+for example على سبيل المثال consider
+
+321
+00:38:33,800 --> 00:38:43,480
+اعتبري الـ function f of x بساوي x تربية وطبعا هنا
+
+322
+00:38:43,480 --> 00:38:48,480
+x ينتمي إلى R ده للتربعية المجال تبع كل العداد
+
+323
+00:38:48,480 --> 00:38:55,000
+الحقيقية خلينا ناخد الـ x بس في الفترة المغلقة
+
+324
+00:38:55,000 --> 00:39:02,420
+والمحدودة من سالب واحد إلى واحد تمام؟
+
+325
+00:39:03,700 --> 00:39:08,380
+فال function f is
+
+326
+00:39:08,380 --> 00:39:16,480
+continuous on I and بنلاحظ
+
+327
+00:39:16,480 --> 00:39:24,740
+أن f عن سلب واحد بساوي f عن واحد بساوي واحد is an
+
+328
+00:39:24,740 --> 00:39:28,260
+absolute maximum
+
+329
+00:39:33,020 --> 00:39:42,000
+at x بساوي سالب واحد واحد إذا سالب واحد واحد عبارة
+
+330
+00:39:42,000 --> 00:39:45,260
+عن absolute maximum points مظبوط هاي الدالة
+
+331
+00:39:45,260 --> 00:39:50,840
+التربية أنا بس ماخد المجال تبعها اللي هو الفترة
+
+332
+00:39:50,840 --> 00:39:54,900
+المغلقة والمحدودة من سالب واحد إلى واحد فطبعا
+
+333
+00:39:54,900 --> 00:39:56,120
+رسمتها زي هيك
+
+334
+00:40:00,870 --> 00:40:05,170
+هذه الدالة التربعية ان المجال تبعها الفترة المغلقة
+
+335
+00:40:05,170 --> 00:40:10,230
+هذه فواضح انه هيقلها absolute maximum اللي هي
+
+336
+00:40:10,230 --> 00:40:15,630
+الواحد وال absolute maximum هذا حصلنا عليه عن مقطه
+
+337
+00:40:15,630 --> 00:40:24,870
+تالى سالب واحد واحد نعم مظبوط صحيح وطبعا هنا
+
+338
+00:40:24,870 --> 00:40:29,980
+فيقلها absolute minimum واحدة اللي هي السفرواضح
+
+339
+00:40:29,980 --> 00:40:34,940
+هنا أن دالة هنا المجال تبعها فترة مغلقة ومحدودة
+
+340
+00:40:34,940 --> 00:40:38,900
+اذا by maximum minimum theorem اكيد فيه اللي هي
+
+341
+00:40:38,900 --> 00:40:41,980
+absolute maximum وفيه اللي هي absolute minimum ال
+
+342
+00:40:41,980 --> 00:40:45,640
+absolute minimum point هي السفرالـ absolute
+
+343
+00:40:45,640 --> 00:40:50,380
+maximum point هنا في نقطتين النظرية مجرد يعني مجرد
+
+344
+00:40:50,380 --> 00:40:53,820
+يوجد على الأقل نقطة واحدة في end absolute maximum
+
+345
+00:40:53,820 --> 00:41:03,500
+فلو كانوا تنتين لا ضرر لا بأس تمام okay إذن يعني
+
+346
+00:41:03,500 --> 00:41:11,440
+هذه بعض الملاحظات على ال maximum theorem ال
+
+347
+00:41:11,440 --> 00:41:11,820
+..
+
+348
+00:41:22,530 --> 00:41:29,810
+أعتقد أن احنا هنكتفي بهذا القدر و ان شاء الله
+
+349
+00:41:29,810 --> 00:41:36,850
+بنكمل ال section هذا في المحاضرة القادمة فاشكركم
+
+350
+00:41:36,850 --> 00:41:44,390
+لحسن استماعكم و ان شاء الله نلتقي يوم الأتنين
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tY_G9XVDP9s.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tY_G9XVDP9s.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a133c28a2723546896ad4f355e6822643cdbf0d3
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/tY_G9XVDP9s.srt
@@ -0,0 +1,959 @@
+1
+00:00:23,300 --> 00:00:28,680
+الـ completeness property of R عشان نعرف إيه ال
+
+2
+00:00:28,680 --> 00:00:33,540
+completeness property of R فبنحتاج لبعض التعريفات
+
+3
+00:00:33,540 --> 00:00:39,380
+فأول تعريف ما معناه أن عدد حقيقي يكون upper bound
+
+4
+00:00:39,380 --> 00:00:44,460
+للمجموعة S معناه أن الـ U أكبر من أو يساوي كل عناصر
+
+5
+00:00:44,460 --> 00:00:50,800
+الـ S ما معناه أن عدد حقيقي w يكون lower bound لـ S
+
+6
+00:00:50,800 --> 00:00:58,580
+معناه أن الـ w أصغر من أو يساوي كل عناصر الـ set S و
+
+7
+00:00:58,580 --> 00:01:06,600
+من السهل أن احنا من التعريف
+
+8
+00:01:06,600 --> 00:01:12,760
+هذا ممكن نلاحظ الملاحظات التالية أن أي set ممكن
+
+9
+00:01:13,520 --> 00:01:16,740
+يكون لها upper bound وممكن ما يكون لهاش upper bound
+
+10
+00:01:16,740 --> 00:01:23,380
+فعلى سبيل المثال مجموعة الأعداد الحقيقية هذه ليس لها
+
+11
+00:01:23,380 --> 00:01:29,740
+upper bound أو أي lower bound في المقابل الفترة
+
+12
+00:01:29,740 --> 00:01:38,520
+المغلقة S من صفر لواحد أو الفترة المغلقة المفتوحة
+
+13
+00:01:38,520 --> 00:01:44,480
+من صفر لواحد هذه لها lower bounds الصفر أو أي عدد
+
+14
+00:01:44,480 --> 00:01:50,540
+أصغر من أو يساوي الصفر و كذلك لها upper bounds الواحد
+
+15
+00:01:50,540 --> 00:01:55,540
+أو أي عدد أكبر من أو يساوي الواحد بالمثل للمجموعة
+
+16
+00:01:55,540 --> 00:02:00,920
+اللي تحت كذلك لو المجموعة
+
+17
+00:02:02,480 --> 00:02:08,060
+لها upper bound U فبكون لها infinitely many upper
+
+18
+00:02:08,060 --> 00:02:12,700
+bounds إذا كان U upper bound لمجموعة زي هذه فكل
+
+19
+00:02:12,700 --> 00:02:16,200
+عدد أكبر من الـ upper bound هو upper bound آخر
+
+20
+00:02:16,200 --> 00:02:20,600
+بالمثل بالنسبة لل lower bounds إذا المجموعة لها
+
+21
+00:02:20,600 --> 00:02:24,660
+lower bound واحد فممكن نجد عدد لانهائي من ال lower
+
+22
+00:02:24,660 --> 00:02:30,770
+bounds لو أخدنا المجموعة S بالساوية فاي فأي عدد
+
+23
+00:02:30,770 --> 00:02:35,170
+حقيقي بيكون upper bound وفي نفس الوقت بيكون lower
+
+24
+00:02:35,170 --> 00:02:40,330
+bound للمجموعة هذه و البرهان by contradiction نفترض
+
+25
+00:02:40,330 --> 00:02:46,470
+على النقيض أن يوجد R0 عدد حقيقي و هذا العدد
+
+26
+00:02:46,470 --> 00:02:52,200
+الحقيقي ما هو upper bound معناه إذا كان R0 ليس
+
+27
+00:02:52,200 --> 00:02:55,940
+upper bound لـ S معناه بقدر ألاقي عنصر S في
+
+28
+00:02:55,940 --> 00:03:02,820
+المجموعة capital S أكبر من الـ R0 وهذا بتديني
+
+29
+00:03:02,820 --> 00:03:08,960
+التناقض لأن المجموعة S لا يوجد بها عناصر فرض أنه
+
+30
+00:03:08,960 --> 00:03:17,960
+يوجد فرض 
+
+31
+00:03:17,960 --> 00:03:24,460
+أنه يوجد عدد حقيقي وما هو upper bound هذا خطأ هذا
+
+32
+00:03:24,460 --> 00:03:29,360
+معناه أن every real number is upper bound بالمثل
+
+33
+00:03:29,360 --> 00:03:33,160
+ممكن إثبات أن every real number is lower bound
+
+34
+00:03:33,160 --> 00:03:35,840
+للمجموعة الخالية five
+
+35
+00:03:56,010 --> 00:04:01,510
+بنقول إن المجموعة أي مجموعة من R bounded above إذا 
+
+36
+00:04:01,510 --> 00:04:06,770
+كان لها upper bound وبنقول إنها bounded below إذا 
+
+37
+00:04:06,770 --> 00:04:15,310
+كانت إذا كان لها lower bound إذا كانت المجموعة
+
+38
+00:04:15,310 --> 00:04:20,810
+بنسميها bounded إذا كانت bounded above and bounded
+
+39
+00:04:20,810 --> 00:04:26,720
+below في نفس الوقت وبنسميها unbounded إذا كان
+
+40
+00:04:26,720 --> 00:04:32,220
+ينقصها أنها تكون bounded above أو بينقصها أنها
+
+41
+00:04:32,220 --> 00:04:36,780
+تكون bounded below مثال على unbounded set اللي هو
+
+42
+00:04:36,780 --> 00:04:41,100
+الأعداد الحقيقية، هذه الـ set unbounded غير محدودة،
+
+43
+00:04:41,100 --> 00:04:47,180
+غير محصورة، لأنه ليس لها upper bounds أو lower
+
+44
+00:04:47,180 --> 00:04:51,620
+bounds مثال على bounded sets الـ set هذه و الـ set هذه
+
+45
+00:04:51,620 --> 00:04:57,740
+الـ sets هذول كل منهم bounded above and bounded
+
+46
+00:04:57,740 --> 00:05:07,080
+below وبالتالي bounded المجموعة
+
+47
+00:05:07,080 --> 00:05:11,940
+أي subset من S، ده التعريف definition أي subset S
+
+48
+00:05:11,940 --> 00:05:14,260
+من R، بنسميه bounded above
+
+49
+00:05:22,060 --> 00:05:26,400
+بنسميها bounded below إذا كان في لها upper bound
+
+50
+00:05:26,400 --> 00:05:27,860
+إذا كان في لها lower bound
+
+51
+00:05:36,000 --> 00:05:41,960
+المجموعة bounded if كانت كان لها upper bound و
+
+52
+00:05:41,960 --> 00:05:46,140
+lower bound أو كانت bounded above و bounded below
+
+53
+00:05:46,140 --> 00:05:52,720
+إذا الـ set بنسميها bounded if كانت bounded above و
+
+54
+00:05:52,720 --> 00:05:57,520
+bounded below المجموعة بتكون محدودة أو محصورة إذا
+
+55
+00:05:57,520 --> 00:06:02,740
+كانت محصورة من الأعلى و من الأسفل في نفس الوقت أما
+
+56
+00:06:04,210 --> 00:06:08,550
+if S lacks either an upper bound or lower bound
+
+57
+00:06:08,550 --> 00:06:15,810
+فبنسميها unbounded S بنسميها unbounded إذا
+
+58
+00:06:15,810 --> 00:06:19,750
+المجموعة لو كانت بنخصها upper bound يعني ما لهاش أي
+
+59
+00:06:19,750 --> 00:06:26,250
+upper bound ولها lower bounds بتكون unbounded أو
+
+60
+00:06:26,250 --> 00:06:32,250
+لو كانت المجموعة bounded below ما لهاش upper bounds
+
+61
+00:06:32,250 --> 00:06:38,670
+أو not bounded above برضه بنسميها unbounded إذا الـ
+
+62
+00:06:38,670 --> 00:06:41,890
+bounded معناه bounded above و below في نفس الوجه
+
+63
+00:06:41,890 --> 00:06:48,710
+إذا واحد مش متحقق بنسميها unbounded أعزائي
+
+64
+00:06:48,710 --> 00:06:53,690
+المثال مجموعة الأعداد الحقيقية are and is 
+
+65
+00:06:53,690 --> 00:06:58,350
+unbounded لأنها ليست bounded above ولا bounded
+
+66
+00:06:58,350 --> 00:06:58,710
+below
+
+67
+00:07:01,720 --> 00:07:08,900
+كذلك المجموعة هذه لو
+
+68
+00:07:08,900 --> 00:07:14,260
+أخذنا مجموعة الأعداد الحقيقية الغير سالبة
+
+69
+00:07:14,260 --> 00:07:17,660
+فالمجموعة
+
+70
+00:07:17,660 --> 00:07:20,920
+هذه is bounded below، لها lower bound، الصفر أو أي
+
+71
+00:07:20,920 --> 00:07:26,060
+عدد أصغر من أو يساوي الصفر لكن it is not bounded above،
+
+72
+00:07:26,060 --> 00:07:28,240
+إذا المجموعة هذه unbounded
+
+73
+00:07:32,210 --> 00:07:40,410
+it is unbounded المجموعة هذي هذي bounded لأنها
+
+74
+00:07:40,410 --> 00:07:45,450
+bounded above and bounded below طبعا إذا فهمنا شو
+
+75
+00:07:45,450 --> 00:07:54,890
+معناه bounded و unbounded ناخد تعريف ثاني لو
+
+76
+00:07:54,890 --> 00:08:01,410
+كانت S bounded above فبنعرف الـ supremum أو الـ least
+
+77
+00:08:01,410 --> 00:08:07,730
+upper bound للـ set S على أنه العدد الحقيقي U اللي
+
+78
+00:08:07,730 --> 00:08:12,910
+بيحقق الشرطين هذول الشرط الأول أن العدد U is an
+
+79
+00:08:12,910 --> 00:08:18,610
+upper bound of S U أكبر من أو يساوي كل عناصر S يعني
+
+80
+00:08:18,610 --> 00:08:24,450
+U upper bound لـ S اثنين لو أخدت أي upper bound V
+
+81
+00:08:26,010 --> 00:08:35,150
+فلازم الـ U يطلع أصغر من أو يساوي تمام؟
+
+82
+00:08:35,150 --> 00:08:43,250
+فإذا الـ .. الـ .. الـ supremum هو أصغر upper bound
+
+83
+00:08:43,250 --> 00:08:48,490
+للمجموعة هو أصغر upper bound الـ supremum هو least
+
+84
+00:08:48,490 --> 00:08:54,640
+.. الـ least upper bound أصغر حد أعلى إذا إيش هو
+
+85
+00:08:54,640 --> 00:08:59,960
+أصغر حد أعلى؟ متى بيكون U هو أصغر حد أعلى للمجموعة
+
+86
+00:08:59,960 --> 00:09:08,500
+إذا كان أول شيء upper bound للمجموعة اثنين U أصغر
+
+87
+00:09:08,500 --> 00:09:14,580
+من أو يساوي أي upper bound آخر الـ
+
+88
+00:09:14,580 --> 00:09:20,360
+supremum أو أصغر حد أعلى نرمز له بالرمز supremum S
+
+89
+00:09:20,360 --> 00:09:22,800
+أو least upper bound لـ S
+
+90
+00:09:25,630 --> 00:09:32,390
+بالمثل ممكن نعرف الـ minimum أو أكبر حد أدنى أكبر
+
+91
+00:09:32,390 --> 00:09:35,330
+حد أدنى للمجموعة
+
+92
+00:09:40,350 --> 00:09:45,970
+لازم تكون الـ set bounded below عشان أقدر أعرف الـ
+
+93
+00:09:45,970 --> 00:09:49,990
+minimum أو أكبر حد أدنى لازم تكون الـ set bounded
+
+94
+00:09:49,990 --> 00:09:55,230
+below فلو كانت الـ set bounded below فأي عدد حقيقي
+
+95
+00:09:55,230 --> 00:10:02,110
+w بيكون الـ minimum للـ set أو بنسميه greatest lower
+
+96
+00:10:02,110 --> 00:10:08,180
+bound للـ set إذا حقق شرطين الشرط الأول أن الـ W
+
+97
+00:10:08,180 --> 00:10:14,120
+أبرًا lower bound لـ S الشرط الثاني أن الـ W أكبر
+
+98
+00:10:14,120 --> 00:10:20,940
+من أو يساوي أي lower bound Z أي lower bound Z لـ S
+
+99
+00:10:22,820 --> 00:10:27,200
+وبالتالي من الشرطين هذول .. الشرطين هذول معناهما مع
+
+100
+00:10:27,200 --> 00:10:33,320
+بعض أن الـ minimum أو greatest lower bound للـ S هو
+
+101
+00:10:33,320 --> 00:10:38,240
+أكبر حد أدنى .. أكبر حد أدنى أو greatest lower
+
+102
+00:10:38,240 --> 00:10:45,280
+bound للـ S نرمز للـ infimum أو greatest lower
+
+103
+00:10:45,280 --> 00:10:52,560
+bound بالرمز هذا inf S أو glb لـ S هذا اختصار
+
+104
+00:10:52,560 --> 00:10:59,400
+greatest lower bound of S تمام ففي
+
+105
+00:10:59,400 --> 00:11:07,080
+المثال هذا اللي هنا في
+
+106
+00:11:07,080 --> 00:11:11,900
+المثال اللي فوق هذا بنلاحظ
+
+107
+00:11:11,900 --> 00:11:19,080
+أن الصفر بساوي الـ infimum لـ S اللي هي الفترة
+
+108
+00:11:19,080 --> 00:11:24,180
+المغلقة من صفر لواحد لأن
+
+109
+00:11:24,180 --> 00:11:28,620
+الصفر عبارة
+
+110
+00:11:28,620 --> 00:11:34,160
+عن lower bound أصغر من أو يساوي كل عناصر الـ set S
+
+111
+00:11:34,160 --> 00:11:40,340
+كمان لو أخدت أي lower bound للـ set S لازم هذا الـ
+
+112
+00:11:40,340 --> 00:11:43,560
+lower bound يكون أصغر من أو يساوي الصفر
+
+113
+00:11:46,660 --> 00:11:50,360
+وبالتالي الصفر أكبر من أو يساوي أي lower bound هذا
+
+114
+00:11:50,360 --> 00:12:01,280
+واضح طيب كذلك الواحد بيساوي الـ supremum للـ S اللي
+
+115
+00:12:01,280 --> 00:12:08,170
+هي الفترة المغلقة من صفر لواحد ليه؟ لأن الواحد
+
+116
+00:12:08,170 --> 00:12:12,470
+أبرًا upper bound للـ set هذه أكبر من أو يساوي كل
+
+117
+00:12:12,470 --> 00:12:20,650
+عناصر الفترة و لو أخدت أي V أكبر من أو يساوي كل
+
+118
+00:12:20,650 --> 00:12:24,950
+عناصر الفترة لكل
+
+119
+00:12:24,950 --> 00:12:28,630
+S في الفترة من 0 ل1
+
+120
+00:12:32,170 --> 00:12:37,050
+فبيطلع عندي V أكبر من أو يساوي الواحد لأن الواحد
+
+121
+00:12:37,050 --> 00:12:43,130
+عنصر في الـ set و V أكبر من أو يساوي كل عناصر الـ set فـ V
+
+122
+00:12:43,130 --> 00:12:49,130
+أكبر من أو يساوي الواحد وبالتالي V أو الواحد أصغر
+
+123
+00:12:49,130 --> 00:12:52,610
+من أو يساوي أي حد أعلى آخر
+
+124
+00:12:56,370 --> 00:13:02,410
+إذا الواحد هو عبارة عن الـ supremum للـ set صفر و واحد
+
+125
+00:13:02,410 --> 00:13:08,190
+الـ
+
+126
+00:13:08,190 --> 00:13:16,650
+.. عندي شوية ملاحظات هنا نشوفهم
+
+127
+00:13:16,650 --> 00:13:17,190
+مع بعض
+
+128
+00:13:30,420 --> 00:13:36,600
+لو الـ supremum تبع الـ S موجود لو
+
+129
+00:13:36,600 --> 00:13:41,240
+الـ S الـ supremum تبعها exists يعني لها supremum
+
+130
+00:13:41,240 --> 00:13:47,180
+زي الـ S هذه فلازم الـ supremum هذا يكون unique وحيد
+
+131
+00:13:47,180 --> 00:13:52,540
+يعني ما فيش أكثر من supremum فلبرهان
+
+132
+00:13:52,540 --> 00:13:53,060
+ذلك
+
+133
+00:13:59,930 --> 00:14:08,490
+اللي برهان ذلك نفرض أن الـ S لها أكثر
+
+134
+00:14:08,490 --> 00:14:20,750
+من supremum U1 supremum و U2 supremum ثاني بدنا
+
+135
+00:14:20,750 --> 00:14:29,710
+نثبت أن U1 بدنا نثبت أن U1 بساوي U2 وبالتالي الـ two
+
+136
+00:14:29,710 --> 00:14:32,990
+supremums بنطبقوا على بعض يعني الـ supremum واحد
+
+137
+00:14:32,990 --> 00:14:41,590
+طيب أنا فيه أندي U1 upper bound
+
+138
+00:14:46,410 --> 00:14:52,110
+لـ set S ليه لأن U واحد عبارة عن supremum و الـ
+
+139
+00:14:52,110 --> 00:14:55,990
+supremum بيكون upper كل supremum هو upper bound من
+
+140
+00:14:55,990 --> 00:15:03,510
+تعريفه صح الشرط الأول و U اثنين بساوي الـ supremum 
+
+141
+00:15:03,510 --> 00:15:13,770
+اللي هو least upper bound لـ S هو أصغر حد أعلى
+
+142
+00:15:16,780 --> 00:15:25,220
+و هذا حد أعلى إذا .. إذا بيطلع عندي هذا أصغر حد 
+
+143
+00:15:25,220 --> 00:15:30,360
+أعلى لـ S وهذا حد أعلى فمن تعريف الـ supremum أو
+
+144
+00:15:30,360 --> 00:15:36,720
+أصغر حد أعلى إذا U2 بيطلع أصغر من أو يساوي U1
+
+145
+00:15:36,720 --> 00:15:43,640
+نستنتج أن U2 أصغر من أو يساوي 1 نسمي المتباينة هذه
+
+146
+00:15:43,640 --> 00:15:53,050
+1 كذلك أنا عندي U1 أو U2 المرة هذه هاخد U2 هذا
+
+147
+00:15:53,050 --> 00:16:03,770
+عبارة عن upper bound لـ S ليه؟ لأنه الـ
+
+148
+00:16:03,770 --> 00:16:09,530
+supremum لـ S وكل supremum هو upper bound وعندي
+
+149
+00:16:09,530 --> 00:16:13,930
+كمان U1 عبارة عن
+
+150
+00:16:16,610 --> 00:16:24,390
+الـ least upper bound لـ S يعني U1 هو أكبر upper 
+
+151
+00:16:24,390 --> 00:16:30,170
+bound لـ S و U2 upper bound فمن تعريف الـ least
+
+152
+00:16:30,170 --> 00:16:37,030
+upper bound بيطلع عندي أن الـ U1 لازم يكون أصغر من 
+
+153
+00:16:37,030 --> 00:16:45,010
+أو يساوي U2 فبنسمي المتباينة هذه اتنين الآن من
+
+154
+00:16:45,010 --> 00:16:46,050
+واحد و اثنين
+
+155
+00:16:49,530 --> 00:16:57,290
+إذا from واحد and اثنين نستنتج
+
+156
+00:16:57,290 --> 00:17:02,250
+أن يو اثنين أصغر من أو يساوي يو واحد و يو واحد
+
+157
+00:17:02,250 --> 00:17:03,850
+أصغر من أو يساوي يو اثنين
+
+158
+00:17:09,610 --> 00:17:15,170
+وبالتالي إذا هيك لو سمحت ما تتكلميش معي هنا بشرح
+
+159
+00:17:15,170 --> 00:17:16,910
+اللي بتتكلم ما تتكلمش
+
+160
+00:17:20,750 --> 00:17:25,170
+هين أثبتنا أنه لو الست إلها أكثر من supremum فالـ
+
+161
+00:17:25,170 --> 00:17:28,410
+supremums بيساويوا بعض وبالتالي في عندي الـ
+
+162
+00:17:28,410 --> 00:17:34,050
+supremum is unique بالمثل ممكن أثبت أنه لو كانت
+
+163
+00:17:34,050 --> 00:17:37,730
+الست S إلها infimum يعني الـ infimum تبعها exist
+
+164
+00:17:37,730 --> 00:17:44,990
+فأيضا بيطلع unique هو البرهان مثل لبرهان أن الـ 
+
+165
+00:17:44,990 --> 00:17:49,490
+supreme م unique تمام؟ فحاسبكم الجزء الثاني هذا
+
+166
+00:17:49,490 --> 00:17:55,530
+تكتب البرهان تبعه كتمرين أو كـ exercise في أي سؤال
+
+167
+00:17:55,530 --> 00:18:02,030
+واضح بالبرهان؟ في أي استفسار على البرهان؟
+
+168
+00:18:02,030 --> 00:18:08,430
+طيب الـ .. في عندي لمة واحدة
+
+169
+00:18:08,430 --> 00:18:23,560
+اثنين لمة كثير مهمة واللمة هذه تعتبر
+
+170
+00:18:23,560 --> 00:18:28,080
+يعني بتكافئ بتعريف الـ supremum يعني ما معنى لو
+
+171
+00:18:28,080 --> 00:18:31,900
+أخدت مجموعة غير خالية مجموعة جزئية من الأعداد
+
+172
+00:18:31,900 --> 00:18:38,680
+الحقيقية غير خالية وفرضت أن U is upper bound لها
+
+173
+00:18:40,090 --> 00:18:44,790
+فالسؤال متى الـ upper bound U لست هذه S بيكون هو الـ
+
+174
+00:18:44,790 --> 00:18:48,890
+upper bound الجواب
+
+175
+00:18:48,890 --> 00:18:54,430
+في اللمة أن if and only if إذا كانت الـ U بتحقق
+
+176
+00:18:54,430 --> 00:19:01,330
+الشرط هذا طب نشوف الشرط هذا نحاول نشوف الشرط هذا
+
+177
+00:19:01,330 --> 00:19:04,810
+نوضحه على رسمة
+
+178
+00:19:11,720 --> 00:19:19,300
+هذه مجموعة الأعداد الحقيقية وهذه
+
+179
+00:19:19,300 --> 00:19:30,040
+مجموعة S وهذه مجموعة جزئية من R وغير خالية تمام؟
+
+180
+00:19:30,040 --> 00:19:39,120
+وهذه U upper bound للـ S اللي لما بتقول عشان الـ
+
+181
+00:19:39,120 --> 00:19:44,180
+upper bound هذا يكون هو الـ supremum لازم يتحقق أنه
+
+182
+00:19:44,180 --> 00:19:50,200
+لأي epsilon أكبر من الصفر لازم ألاقي .. لازم ألاقي
+
+183
+00:19:50,200 --> 00:20:00,960
+هنا في عنصر نسميه S epsilon ينتمي لـ S بحيث أنه لو
+
+184
+00:20:00,960 --> 00:20:02,280
+طرحت من الـ U
+
+185
+00:20:05,140 --> 00:20:12,320
+لو طرحت من الـ U إبسلون فلازم ألاقي عنصر S إبسلون
+
+186
+00:20:12,320 --> 00:20:17,820
+يعتمد على إبسلون في S وهذا العنصر أكبر من U سالب
+
+187
+00:20:17,820 --> 00:20:23,840
+إبسلون زي ما هو في الرسم مرة
+
+188
+00:20:23,840 --> 00:20:29,380
+ثانية عشان الـ upper bound هذا يكون supremum لـ S هو
+
+189
+00:20:29,380 --> 00:20:35,240
+أصغر حد أعلى لازم لأي إبسلون أكبر من الصفر لو طرحت
+
+190
+00:20:35,240 --> 00:20:39,840
+من الـ U الأبسلون العدد الموجب فلازم ألاقي عنصر في
+
+191
+00:20:39,840 --> 00:20:46,300
+الـ set أكبر من U سالب إبسلون نشوف البرهان طبعا هذا
+
+192
+00:20:46,300 --> 00:20:50,670
+by conditional statement في الـ logic أو في
+
+193
+00:20:50,670 --> 00:20:56,250
+البراهين في الـ only if part و الـ if part من جزء
+
+194
+00:20:56,250 --> 00:21:00,690
+البرهان إلى جزئين هذا بيسميه الـ if part و هذا الـ
+
+195
+00:21:00,690 --> 00:21:06,870
+only if part نثبت الاتجاه هذا الآن هفرض أن الشرط
+
+196
+00:21:06,870 --> 00:21:14,550
+هذا متحقق وأثبت أن u هو الـ supremum هو نعم مش s
+
+197
+00:21:14,550 --> 00:21:19,630
+epsilon هي عبارة عن ماذا بتقول؟ neighborhood لأ لأ
+
+198
+00:21:19,630 --> 00:21:24,050
+ما حدش حاكى حاجة عن neighborhood لأ لأ دا كـ U
+
+199
+00:21:24,050 --> 00:21:31,090
+capital U و بيكون يعني موضح في السياق ما فيش هنا
+
+200
+00:21:31,090 --> 00:21:35,730
+أي حاجة تتعلق بالـ neighborhood U عدد حقيقي إذا U
+
+201
+00:21:35,730 --> 00:21:41,000
+هنا بس عبارة عن upper bound لـ S عشان يكون هو الـ
+
+202
+00:21:41,000 --> 00:21:44,140
+supremum هو أصغر upper bound لازم يتحقق الشرط هذا
+
+203
+00:21:44,140 --> 00:21:52,200
+فنشوف افرض أن الشرط هذا يتحقق وبينا نثبت الـ U هو
+
+204
+00:21:52,200 --> 00:21:59,340
+الـ supremum فلإثبات الـ U هو الـ supremum أحنا فرضين
+
+205
+00:21:59,340 --> 00:22:03,560
+أن الـ U upper bound ارجع لتعريف الـ supremum الـ U
+
+206
+00:22:03,560 --> 00:22:07,980
+بيكون supremum إذا كان واحد upper bound وهذا ما
+
+207
+00:22:07,980 --> 00:22:11,780
+اتحقق من الفرض فرضينه upper bound الشرط الثاني لو
+
+208
+00:22:11,780 --> 00:22:16,520
+أخدنا أي upper bound بيه فلازم نثبت أن الـ U أصغر
+
+209
+00:22:16,520 --> 00:22:20,880
+من أو يساوي فتعالوا نثبت الشرط الثاني من شروط الـ
+
+210
+00:22:20,880 --> 00:22:28,570
+supremum let V be any upper bound لـ S لو كان الـ U
+
+211
+00:22:28,570 --> 00:22:33,670
+بيساوي الـ V فبيطلع U أصغر من أو يساوي V وهو المطلوب،
+
+212
+00:22:33,670 --> 00:22:40,610
+إذا بفرض أن V لا يساوي الـ U، إذا بدي أثبت أن U
+
+213
+00:22:40,610 --> 00:22:47,850
+أصغر من V فإذا كان الآن v و u أعداد حقيقية مختلفة
+
+214
+00:22:47,850 --> 00:22:51,350
+إذا الخاصية الثلاثية trichotomy property بتقول
+
+215
+00:22:51,350 --> 00:22:56,550
+إما u أصغر من .. إما v أصغر من u أو u أصغر من v
+
+216
+00:22:56,550 --> 00:23:01,370
+أنا بدي u أثبت u أصغر من v تعالوا ناخد الاحتمال
+
+217
+00:23:01,370 --> 00:23:07,410
+الأول ونشوف إن هذا مستحيل فـ assume
+
+218
+00:23:10,700 --> 00:23:15,960
+إذا هنا أخذنا الاحتمال الأول وفرضنا أنه هو الصح
+
+219
+00:23:15,960 --> 00:23:23,560
+الآن الفرق بين U و V بيطلع عدد موجب لأن U أكبر من V
+
+220
+00:23:24,640 --> 00:23:29,820
+الآن سمى هذا الفرق إبسلون إذا الآن إبسلون اللي هو
+
+221
+00:23:29,820 --> 00:23:35,660
+الفرق هذا عدد موجب طب من الفرض من الفرض هذا هذا هو
+
+222
+00:23:35,660 --> 00:23:41,320
+لأي إبسلون زي هذه لأي عدد موجب إبسلون هو ليكن هذا
+
+223
+00:23:41,320 --> 00:23:50,840
+احنا فرضين أنه يوجد يوجد S إبسلون في S وهذا الـ S Y
+
+224
+00:23:50,840 --> 00:23:57,820
+أكبر من U سالب Y مغلق؟ هذا من الفرض الآن من هنا
+
+225
+00:23:57,820 --> 00:24:06,140
+من تعريف الـ V الـ U سالب V بيساوي V إذا هنا عندي
+
+226
+00:24:06,140 --> 00:24:13,980
+أصبح V أصغر من S Y وهذا بتناقض مع كون V upper
+
+227
+00:24:13,980 --> 00:24:19,460
+bound احنا فرضين أن V upper bound لـ S فكيف V upper
+
+228
+00:24:19,460 --> 00:24:25,260
+bound وفي نفس الوقت V أصغر من عنصر M في المجموعة
+
+229
+00:24:25,260 --> 00:24:31,080
+S هذا بتناقض مع تعريف الـ upper bound إذا هذا
+
+230
+00:24:31,080 --> 00:24:37,920
+التناقض بيقول لي إن الاحتمال هذا غلط أو مستحيل طيب
+
+231
+00:24:37,920 --> 00:24:44,550
+أنا عندي احنا استنتجنا من الـ hypothesis أن الاحتمال
+
+232
+00:24:44,550 --> 00:24:50,790
+هذا أو هذا صح الآن هذا مش صح إذا يبقى أن هذا صح
+
+233
+00:24:50,790 --> 00:24:56,730
+إذا we are left يعني الآن احنا طريقنا بأن U أصغر
+
+234
+00:24:56,730 --> 00:25:01,050
+من V وبالتالي هذا بيكمل برهان الشرط الثاني من الـ
+
+235
+00:25:01,050 --> 00:25:05,750
+supremum يعني U هو الـ supremum إذن هنا أثبتنا إنه
+
+236
+00:25:05,750 --> 00:25:10,470
+لو الشرط هذا اتحقق فالـ upper bound U هذا بيطلع هو
+
+237
+00:25:10,470 --> 00:25:15,090
+الـ supremum هو أصغر upper bound اللي يعني بدنا نثبت
+
+238
+00:25:15,090 --> 00:25:23,110
+العكس .. نثبت العكس .. نثبت العكس لو كان .. لو ..
+
+239
+00:25:23,110 --> 00:25:27,450
+لو كان U هو الـ supremum لو فرضنا U هو الـ supremum
+
+240
+00:25:27,450 --> 00:25:31,550
+بدنا نثبت إن الـ U بيحقق الشرط اللي على اليمين 
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/u5C7kmY6Ltw_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/u5C7kmY6Ltw_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..2757e1f9cef4bbf4f135526b6a7d8af28e916510
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/u5C7kmY6Ltw_raw.srt
@@ -0,0 +1,1516 @@
+1
+00:00:20,100 --> 00:00:29,020
+بسم الله الرحمن الرحيم ناخد اليومالجزء المتعلق بال
+
+2
+00:00:29,020 --> 00:00:33,400
+composition of continuous functions لو كان في عندي
+
+3
+00:00:33,400 --> 00:00:38,960
+اقترانين متصلين فهنشوف ان ال composition تبعهم
+
+4
+00:00:38,960 --> 00:00:47,360
+بيطلع ايضا متصل على المجال الدالة الاولى وبعد هيك
+
+5
+00:00:47,360 --> 00:00:54,360
+بعد ما نقلص هنبدأ ب section خمسة تلاتةاللي بتحدث
+
+6
+00:00:54,360 --> 00:00:58,100
+عن ال continuous functions على ال intervals أو على
+
+7
+00:00:58,100 --> 00:01:06,420
+الفترات الأول بناخد النظرية اللي خاصة بال
+
+8
+00:01:06,420 --> 00:01:18,240
+composition of continuous functions النظرية
+
+9
+00:01:18,240 --> 00:01:19,360
+هذه بتقول
+
+10
+00:01:22,140 --> 00:01:34,600
+Let a و b be subsets of R و if from a to b is
+
+11
+00:01:34,600 --> 00:01:38,960
+continuous is
+
+12
+00:01:38,960 --> 00:01:47,700
+continuous at c تنتمي إلى a and g function
+
+13
+00:01:51,030 --> 00:01:59,250
+from a to r هنا و g function from b to r إذا مجال
+
+14
+00:01:59,250 --> 00:02:05,010
+ال function g هو ال set b مجال ال function f هو ال
+
+15
+00:02:05,010 --> 00:02:10,370
+set a be
+
+16
+00:02:10,370 --> 00:02:14,070
+continuous at
+
+17
+00:02:17,610 --> 00:02:27,310
+B بساوي F of C تنتمي الى
+
+18
+00:02:27,310 --> 00:02:32,530
+الـ B where
+
+19
+00:02:32,530 --> 00:02:41,410
+طبعا ال range لل function F is contained in Bعشان
+
+20
+00:02:41,410 --> 00:02:45,990
+يكون ال composition لل two functions f و g يكون
+
+21
+00:02:45,990 --> 00:02:50,590
+معرف لازم range ال f يكون subset من domain ال g
+
+22
+00:02:50,590 --> 00:03:00,410
+then في الحالة هذه the composite function the
+
+23
+00:03:00,410 --> 00:03:10,680
+composite function f circle g او g circle fG
+
+24
+00:03:10,680 --> 00:03:15,660
+Circle F from
+
+25
+00:03:15,660 --> 00:03:26,360
+A to R is continuous تطلع continuous at C
+
+26
+00:03:39,140 --> 00:03:45,580
+أنا في عندي function يعني المجال تبعها A و
+
+27
+00:03:45,580 --> 00:03:57,140
+F function من A ل R و كنتيل يوسع عند النقطة C و في
+
+28
+00:03:57,140 --> 00:04:05,180
+عندي function تانية يعني عندي ال 6 هذه F of A
+
+29
+00:04:09,920 --> 00:04:17,400
+F of A subset من الـ B المجموعة هذه نسميها B وهي F
+
+30
+00:04:17,400 --> 00:04:24,140
+of A subset فالان وفي function G continuous
+
+31
+00:04:24,140 --> 00:04:28,840
+function G
+
+32
+00:04:28,840 --> 00:04:31,140
+is continuous عند النقطة
+
+33
+00:04:36,550 --> 00:04:43,070
+النقطة f of c بيبساوي f of c اذا ال function g
+
+34
+00:04:43,070 --> 00:04:48,450
+فاندي f continuous عند c تنتمي ل a و g continuous
+
+35
+00:04:48,450 --> 00:04:57,110
+عند b اللي هي صورة ال f and c اذا في المحصلة تطلع
+
+36
+00:04:57,110 --> 00:05:04,420
+عندي ال function طبعا هي المفروض تطلع g of bففي
+
+37
+00:05:04,420 --> 00:05:07,220
+المحصلة بيطلع عند الـ function G circle F
+
+38
+00:05:07,220 --> 00:05:13,620
+continuous تطلع continuous عند النقطة C طبعا عشان
+
+39
+00:05:13,620 --> 00:05:17,400
+ال composition هذا يكون معرف لازم range ال F اللي
+
+40
+00:05:17,400 --> 00:05:22,020
+هو ده هو F of A يكون subset من ال B اللي هو domain
+
+41
+00:05:22,020 --> 00:05:29,160
+ال function G وهذا طبعا احنا فرضينهلبرهان ذلك
+
+42
+00:05:29,160 --> 00:05:33,380
+هنستخدم الـ neighborhood definition of continuity
+
+43
+00:05:33,380 --> 00:05:48,380
+at a point ف let W be an epsilon neighborhood of G
+
+44
+00:05:48,380 --> 00:06:02,050
+circle F ل C اللي هي G ل F of Cاللي هي جي او بي
+
+45
+00:06:02,050 --> 00:06:10,710
+صح؟ وبنثبت أنه يوجد Delta neighborhood للـC اللي
+
+46
+00:06:10,710 --> 00:06:15,210
+هي تنتمي لـA بحيث أنه صورة ال neighborhood هذا
+
+47
+00:06:15,210 --> 00:06:19,110
+subset من الـW neighborhood هنا تعريف ال
+
+48
+00:06:19,110 --> 00:06:22,730
+neighborhood definition للاتصال عن النقطة طيب
+
+49
+00:06:22,730 --> 00:06:25,650
+since
+
+50
+00:06:28,450 --> 00:06:40,530
+الفنشن الـ g is continuous at b اللي هي بالساوي f
+
+51
+00:06:40,530 --> 00:06:47,430
+of c تنتمي ل b وهي في عندي w neighborhood w is
+
+52
+00:06:47,430 --> 00:06:52,810
+epsilon neighborhood ل g of b إذا من تعريف الاتصال
+
+53
+00:06:52,810 --> 00:07:04,400
+عن النقطة يوجدdelta neighborhood نسميه V of ال
+
+54
+00:07:04,400 --> 00:07:12,120
+B بحيث انه لكل
+
+55
+00:07:12,120 --> 00:07:24,000
+Y ينتمي إلى B تقاطع ال V لازم يطلع عندى صورة ال YG
+
+56
+00:07:24,000 --> 00:07:35,680
+of Y تنتمي لـ W الـ
+
+57
+00:07:35,680 --> 00:07:39,460
+function
+
+58
+00:07:39,460 --> 00:07:45,700
+G continuous عند B في مجالها وبالتالي حسب الاتصال
+
+59
+00:07:45,700 --> 00:07:53,520
+لأي epsilon neighborhood لصورة Bيوجد delta
+
+60
+00:07:53,520 --> 00:07:59,820
+neighborhood V للـ B بحيث انه لكل Y في ال
+
+61
+00:07:59,820 --> 00:08:04,520
+neighborhood و في مجال الدالة صورته تطلع موجودة في
+
+62
+00:08:04,520 --> 00:08:08,720
+ال epsilon neighborhood نسمي هذه ال implication
+
+63
+00:08:08,720 --> 00:08:19,400
+star طيب since انا
+
+64
+00:08:19,400 --> 00:08:20,620
+عندي ال function
+
+65
+00:08:25,730 --> 00:08:34,370
+Since الـ function F is continuous at
+
+66
+00:08:34,370 --> 00:08:43,550
+C و هي NDV عبارة عن delta neighborhood للـ B اللي
+
+67
+00:08:43,550 --> 00:08:47,810
+هي F of C إذا
+
+68
+00:08:47,810 --> 00:08:57,710
+يوجد Gammaneighborhood نسميه U of
+
+69
+00:08:57,710 --> 00:09:01,570
+..
+
+70
+00:09:01,570 --> 00:09:14,250
+of C بحيث أنه لكل X ينتمي إلى A تقاطع ال U بطلع
+
+71
+00:09:14,250 --> 00:09:20,410
+عندي صورة ال X تنتمي ل ال U
+
+72
+00:09:24,690 --> 00:09:43,130
+بنسمي هذه double star اذا
+
+73
+00:09:43,130 --> 00:09:59,970
+now from star and double starأثبتنا إنه يوجد Gamma
+
+74
+00:09:59,970 --> 00:10:08,110
+neighborhood اللي هو U of C تنتمي إلى A بحيث إنه
+
+75
+00:10:08,110 --> 00:10:20,830
+كل X ينتمي إلى A تخاطى U هذا بيؤدي إنه F
+
+76
+00:10:20,830 --> 00:10:21,510
+of X
+
+77
+00:10:26,740 --> 00:10:42,500
+تنتمي الى لأ
+
+78
+00:10:42,500 --> 00:10:47,120
+هذا المفروض يطلع تنتمي
+
+79
+00:10:47,120 --> 00:10:52,100
+يعني
+
+80
+00:10:52,100 --> 00:10:53,980
+المفروض تنتمي الى D
+
+81
+00:10:58,230 --> 00:11:05,750
+الـ F continuous and الـ C و V neighborhood ل B
+
+82
+00:11:05,750 --> 00:11:07,110
+اللي هي صورة الـ C
+
+83
+00:11:10,240 --> 00:11:14,920
+Gamma neighbourhood U للـC بحيث لكل X مجال الدالة
+
+84
+00:11:14,920 --> 00:11:19,740
+وفي الـGamma neighbourhood صورة الـX هتطلع موجودة
+
+85
+00:11:19,740 --> 00:11:24,560
+في الـV neighbourhood الـneighbourhood لF of C V
+
+86
+00:11:24,560 --> 00:11:29,140
+اللي هو neighbourhood لمين of B اللي هي بالساوي F
+
+87
+00:11:29,140 --> 00:11:32,380
+of C إذن هذا الكلام صحيح
+
+88
+00:11:39,160 --> 00:11:49,360
+الان من star هذا بيقدي ان f of x ينتمي الى f of a
+
+89
+00:11:49,360 --> 00:11:57,340
+تقاطع .. تقاطع
+
+90
+00:11:57,340 --> 00:11:58,860
+f of u
+
+91
+00:12:06,740 --> 00:12:13,500
+وعندي هذا ال F of A من الفرض subset من B و ال F of
+
+92
+00:12:13,500 --> 00:12:17,120
+U او
+
+93
+00:12:17,120 --> 00:12:23,120
+لكل X ينتمي ل U F of X ينتمي ل V هذا موجود في V
+
+94
+00:12:23,120 --> 00:12:25,640
+وبالتالي التقاطع هذا
+
+95
+00:12:32,220 --> 00:12:41,340
+أذا هذا بيقدي أن f of x تنتمي إلى b تقاطع ال v هذا
+
+96
+00:12:41,340 --> 00:12:51,520
+باستخدام double star انه
+
+97
+00:12:51,520 --> 00:12:59,560
+لكل x في a تقاطع u f of x تنتمي لf of u اللي هي
+
+98
+00:12:59,560 --> 00:13:01,120
+subset من b
+
+99
+00:13:10,220 --> 00:13:17,340
+أو ممكن حتى كمان هنا بلاش هذه ممكن نحط هنا V لأن
+
+100
+00:13:17,340 --> 00:13:23,840
+هي لكل X في A تقاطة U F of X ينتمي ل B و لكل X
+
+101
+00:13:23,840 --> 00:13:29,620
+ينتمي إلى A تقاطة U ما ده subset من A فبطلع F of X
+
+102
+00:13:29,620 --> 00:13:37,660
+ينتمي ل F of A إذا F of X ينتمي ل F of A و ينتمي ل
+
+103
+00:13:37,660 --> 00:13:46,030
+ال Bطيب الآن من star لما
+
+104
+00:13:46,030 --> 00:13:53,010
+اكون عندي لو سمي هذه y لما اكون عندي y تنتمي ل b
+
+105
+00:13:53,010 --> 00:13:59,830
+تقاطع بيه فبطلع عندي g of y اللي هي g ل f of x
+
+106
+00:13:59,830 --> 00:14:07,730
+تنتمي ل w طب ما هذه عبارة عن g circle f ل x
+
+107
+00:14:12,350 --> 00:14:24,070
+الان since w was an arbitrary epsilon neighborhood
+
+108
+00:14:24,070 --> 00:14:29,110
+of g
+
+109
+00:14:29,110 --> 00:14:37,270
+circle f of c we
+
+110
+00:14:37,270 --> 00:14:49,290
+getfrom the neighborhood definition of
+
+111
+00:14:49,290 --> 00:14:57,770
+continuity that G
+
+112
+00:14:57,770 --> 00:15:03,650
+circle F is continuous at
+
+113
+00:15:03,650 --> 00:15:12,870
+C وهو المصممكمان مرة مامعنى ان الـ function هذه
+
+114
+00:15:12,870 --> 00:15:16,590
+مامعنى ان الـ composite function اللى هى G circle
+
+115
+00:15:16,590 --> 00:15:21,530
+if it's continuous in C هذا حسب التعريف التعريف
+
+116
+00:15:21,530 --> 00:15:27,370
+الجوار للاتصال and نقطة هذا معناه لأى W نبرهود
+
+117
+00:15:27,370 --> 00:15:30,810
+لصورة الـ C يوجد
+
+118
+00:15:32,310 --> 00:15:40,450
+Gamma neighborhood U لـ C بحيث لكل X في مجال A
+
+119
+00:15:40,450 --> 00:15:48,430
+تقاطع Gamma neighborhood تطلع صورة X هذه تنتمي لـ
+
+120
+00:15:48,430 --> 00:15:52,830
+Y neighborhood اللي بدأنا فيه وبالتالي حسب التعريف
+
+121
+00:15:52,830 --> 00:15:57,550
+أن الـ function هي continuous and اسي okay إذا هذا
+
+122
+00:15:57,550 --> 00:16:04,270
+برهن النظرية تمامفي اي سؤال في اي صفارة على
+
+123
+00:16:04,270 --> 00:16:08,350
+البرهان ناخد
+
+124
+00:16:08,350 --> 00:16:14,830
+امثلة او تطبقات على نظرية هذه نظرية هذه كتير قوية
+
+125
+00:16:14,830 --> 00:16:19,210
+لان هنشوف الجهتنا بتبرهن انها نظريات سابقة برهنها
+
+126
+00:16:19,210 --> 00:16:28,330
+سابقا فمثلا
+
+127
+00:16:28,330 --> 00:16:30,070
+عندي ال function
+
+128
+00:16:38,720 --> 00:16:47,700
+بالمناسبة النظرية دي ممكن طبعا انعممها و بدل
+
+129
+00:16:47,700 --> 00:16:52,720
+ما تكون صحيحة على الاتصال عند نقطة بصي صحيح على
+
+130
+00:16:52,720 --> 00:16:57,420
+الاتصال على مجموعة يعني لو كانت if continuous على
+
+131
+00:16:57,420 --> 00:17:02,680
+كل مجموعة a on
+
+132
+00:17:02,680 --> 00:17:04,700
+a and g is continuous
+
+133
+00:17:06,910 --> 00:17:13,690
+على كل المجال تبعها on b then the composite
+
+134
+00:17:13,690 --> 00:17:21,690
+function is continuous على كل ال a وهو نتيجة عن
+
+135
+00:17:21,690 --> 00:17:27,890
+نظرية السابقة الان
+
+136
+00:17:27,890 --> 00:17:37,550
+لو أخدنا letجي وان او اكس بساوي اكسل يوت اكس و اكس
+
+137
+00:17:37,550 --> 00:17:42,950
+ينتمي الار claim
+
+138
+00:17:42,950 --> 00:17:46,910
+جي
+
+139
+00:17:46,910 --> 00:17:54,750
+is continuous on R let
+
+140
+00:17:54,750 --> 00:18:00,710
+ابسلون اكبر من السفر be given choose
+
+141
+00:18:06,010 --> 00:18:11,910
+دلتا بساوي إبسلون، عدد موجة بيعتمد على إبسلون،
+
+142
+00:18:11,910 --> 00:18:20,410
+then لو كان X ينتمي إلى مجال الدالة اللي هو R و
+
+143
+00:18:20,410 --> 00:18:24,930
+absolute X minus C
+
+144
+00:18:36,770 --> 00:18:42,650
+خلّيني أقول هنا fix C
+
+145
+00:18:42,650 --> 00:18:52,290
+ينتمي إلى R and let epsilon
+
+146
+00:18:52,290 --> 00:18:56,770
+أكبر من السفر be given أنا بدأ أثبت أن الـ
+
+147
+00:18:56,770 --> 00:19:01,990
+function G continuous على الـ R فبأثبت الـ C ينتمي
+
+148
+00:19:01,990 --> 00:19:05,450
+إلى R و أثبت أن الـ function continuous عند النقطة
+
+149
+00:19:05,450 --> 00:19:10,720
+Cالأول بعدين بقول بما أنه C was arbitrary إذا ال
+
+150
+00:19:10,720 --> 00:19:15,600
+function continuous على كل ال R فلأي إبسلون أكبر
+
+151
+00:19:15,600 --> 00:19:20,580
+من السفر ت choose دلتا بساوي إبسلون then لكل X في
+
+152
+00:19:20,580 --> 00:19:25,580
+R و absolute X minus C أصغر من دلتا هذا بيقدر أنه
+
+153
+00:19:25,580 --> 00:19:34,900
+absolute G1 of Xg1 of x minus g1 of c اللي هو
+
+154
+00:19:34,900 --> 00:19:41,700
+بيساوي absolute absolute x minus absolute c طبعا
+
+155
+00:19:41,700 --> 00:19:44,800
+هذا by triangle inequality أصغر من أو ساوي
+
+156
+00:19:44,800 --> 00:19:54,440
+absolute x minus c ومن الفرض ومن
+
+157
+00:19:54,440 --> 00:19:58,960
+الفرض أنا ماخد ال x لحياتي اللي مسافة بينها وبين
+
+158
+00:19:58,960 --> 00:20:02,660
+ال c أصغر من دلتاعشان هيك انا اختارت الـ delta
+
+159
+00:20:02,660 --> 00:20:07,080
+بالساوية epsilon عشان يطلع absolute g1 ل x minus
+
+160
+00:20:07,080 --> 00:20:14,520
+g1 ل c أصغر من epsilon و بما انه since epsilon
+
+161
+00:20:14,520 --> 00:20:23,160
+أكبر من سفر was arbitrary اذا g1 is continuous
+
+162
+00:20:27,330 --> 00:20:29,730
+بما أن الـ C ينتبه إلى الـ R كان عبارة عن الـ
+
+163
+00:20:29,730 --> 00:20:50,630
+Arbitrary فالفنشن G1 مستمر على كل الـ R تمام فهو
+
+164
+00:20:50,630 --> 00:21:03,110
+فنشن من A إلى Ris continuous is continuous on
+
+165
+00:21:03,110 --> 00:21:18,650
+a then by composition theorem بيطلع
+
+166
+00:21:18,650 --> 00:21:31,550
+عندي g و g onecircle F absolute F is continuous on
+
+167
+00:21:31,550 --> 00:21:37,770
+A هاي ال composition theorem او النتيجة تبعتها انا
+
+168
+00:21:37,770 --> 00:21:46,190
+لسه اثبات ان ال G1 function from R to R is
+
+169
+00:21:46,190 --> 00:21:50,730
+continuous وفي
+
+170
+00:21:50,730 --> 00:21:51,830
+عندى ال
+
+171
+00:21:54,980 --> 00:22:06,640
+عندي f function from a to r f
+
+172
+00:22:06,640 --> 00:22:10,500
+function from a to r continuous انا فرض ان f is
+
+173
+00:22:10,500 --> 00:22:20,000
+continuous على a و ال function التاني g1 g1
+
+174
+00:22:20,000 --> 00:22:24,680
+function from r to rأيضا أثبتنا أنها continuous
+
+175
+00:22:24,680 --> 00:22:28,980
+فلو كانت هذه continuous من A ل B فال composition
+
+176
+00:22:28,980 --> 00:22:33,960
+تبعهم G1 circle F اللي بيطلع absolute F تطلع
+
+177
+00:22:33,960 --> 00:22:37,080
+continuous على A وهذه نظرية أثبتناها المرة اللي
+
+178
+00:22:37,080 --> 00:22:45,400
+فاتت فهي برهان أخر لهذه النظرية تمام مثال
+
+179
+00:22:45,400 --> 00:22:50,560
+تاني let G2
+
+180
+00:22:57,490 --> 00:23:04,270
+let let مثال تاني let g2 of x function بيساوي
+
+181
+00:23:04,270 --> 00:23:08,810
+square root of x و x أكبر من أو بيساوي سفر طبعا ده
+
+182
+00:23:08,810 --> 00:23:11,910
+اللي تجد التربيه بس معرفة للأعداد الحقيقية غير
+
+183
+00:23:11,910 --> 00:23:19,510
+السالبة claim g2 أثبتنا احنا ملاش claim by
+
+184
+00:23:19,510 --> 00:23:24,800
+previous exerciseأجيبنالكم في الامتحان حتى مثال في
+
+185
+00:23:24,800 --> 00:23:37,120
+الامتحان by a previous exercise g2
+
+186
+00:23:37,120 --> 00:23:43,020
+is continuous تطلع continuous على كل فترة من السفر
+
+187
+00:23:43,020 --> 00:23:47,140
+إلى ما نهيها هذا السؤال في الامتحان المناسب أخر
+
+188
+00:23:47,140 --> 00:23:51,300
+سؤال في الامتحان now
+
+189
+00:23:53,630 --> 00:24:02,570
+إذا .. إذا من a إلى r هو
+
+190
+00:24:02,570 --> 00:24:08,890
+مثلًا إن f من
+
+191
+00:24:08,890 --> 00:24:19,050
+x أكبر من أو ساقو سفر لكل x تنتمي إلى a و f
+
+192
+00:24:19,050 --> 00:24:20,130
+مستمر
+
+193
+00:24:23,230 --> 00:24:29,110
+on A then
+
+194
+00:24:29,110 --> 00:24:33,810
+by composition composition
+
+195
+00:24:33,810 --> 00:24:40,130
+theorem بطلع
+
+196
+00:24:40,130 --> 00:24:48,210
+G to circle F اللي هي بسوى جذر ال F is continuous
+
+197
+00:24:48,210 --> 00:24:57,810
+على المجموعة Aوهذا بيعطي برهان آخر لنظرية أخدناها
+
+198
+00:24:57,810 --> 00:25:02,570
+في آخر محاضرة بتقول لو كانت F دل غير سالب على
+
+199
+00:25:02,570 --> 00:25:07,630
+المجال تبعها ومتصلة على مجالها ف ال square root لل
+
+200
+00:25:07,630 --> 00:25:12,010
+function بتطلع أيضا متصلة على مجالها صح؟ وهذا
+
+201
+00:25:12,010 --> 00:25:17,170
+بيعتبر برهان آخر غير عن البرهان اللي أخدناه في
+
+202
+00:25:17,170 --> 00:25:19,210
+المحاضرة السابقة okay تمام؟
+
+203
+00:25:29,990 --> 00:25:35,230
+بثال تلاتة we
+
+204
+00:25:35,230 --> 00:25:41,310
+know ان ال function f
+
+205
+00:25:41,310 --> 00:25:49,690
+of x بساوي sin x is continuous اثبتنا ان الدالي
+
+206
+00:25:49,690 --> 00:25:53,490
+هاد continuous on R صح؟
+
+207
+00:25:58,290 --> 00:26:04,870
+خلّيني أسميها g of x وفي
+
+208
+00:26:04,870 --> 00:26:18,670
+عندي f of x بساوي واحد على x و
+
+209
+00:26:18,670 --> 00:26:23,050
+x لا يساوي سفر is continuous
+
+210
+00:26:29,880 --> 00:26:34,600
+for all x لا يساوي السفر ده لما اتصل معدى عند
+
+211
+00:26:34,600 --> 00:26:41,940
+السفر صح is continuous on a اللي هي بساوي r معدى
+
+212
+00:26:41,940 --> 00:26:46,520
+السفر
+
+213
+00:26:46,520 --> 00:26:55,960
+hence by composition theorem
+
+214
+00:26:55,960 --> 00:27:04,070
+اللي هو النتيجة هذهبطلع عندي g circle f of x اللي
+
+215
+00:27:04,070 --> 00:27:14,350
+هو g ل f of x اللي هي بتطلع بالساوي sin واحد على x
+
+216
+00:27:14,350 --> 00:27:21,390
+is continuous على
+
+217
+00:27:21,390 --> 00:27:29,730
+كل الأعداد الحقيقية مع عدد السفرOkay فالـ function
+
+218
+00:27:29,730 --> 00:27:37,410
+هذه متصلة مع ذا عند السفر وهي البرهان تبعها ان هذه
+
+219
+00:27:37,410 --> 00:27:43,230
+كلها تطبيقات على ال composition theorem فهذا
+
+220
+00:27:43,230 --> 00:27:49,390
+بورجيكم قوة ال composition theorem تمام؟ هيك ممكن
+
+221
+00:27:49,390 --> 00:27:54,610
+خلصنا section اتنين خمسة اتنين خلينا نبدأ في
+
+222
+00:27:54,610 --> 00:27:56,350
+section خمسة تلاتة
+
+223
+00:28:15,560 --> 00:28:30,180
+continuous functions on
+
+224
+00:28:30,180 --> 00:28:40,780
+intervals الدول
+
+225
+00:28:40,780 --> 00:28:47,270
+المتصلة على الفتراتقبل ان نبدأ في هذا الموضوع
+
+226
+00:28:47,270 --> 00:28:58,450
+نحتاج التعريف التالي a function f from A to R is
+
+227
+00:28:58,450 --> 00:29:04,630
+bounded on
+
+228
+00:29:04,630 --> 00:29:14,650
+المجموعة A إذا تحققالشرط التالي يوجد M عدد موجب
+
+229
+00:29:14,650 --> 00:29:22,090
+بحيث أنه absolute F of X أصغر بيكون أصغر من أو
+
+230
+00:29:22,090 --> 00:29:31,670
+ساوي M لكل X ينتمي إلى A that
+
+231
+00:29:31,670 --> 00:29:39,030
+is هذا معناه أنه the range
+
+232
+00:29:42,830 --> 00:29:52,470
+The range of F which is the set F of A which is
+
+233
+00:29:52,470 --> 00:29:57,950
+the whole F of X حيث X ينتبه إلى A
+
+234
+00:30:12,380 --> 00:30:18,740
+is a bounded subset
+
+235
+00:30:18,740 --> 00:30:23,320
+of
+
+236
+00:30:23,320 --> 00:30:28,240
+R الـ
+
+237
+00:30:28,240 --> 00:30:31,400
+function بتكون bounded function إذا كان ال range
+
+238
+00:30:31,400 --> 00:30:35,520
+تبعها كمجموعة بتطلع bounded set إيه يعني bounded
+
+239
+00:30:35,520 --> 00:30:40,080
+set يعني يوجد عدد موجة بحيث القيمة المطلقة لكل
+
+240
+00:30:40,080 --> 00:30:55,220
+عناصر ال set هذهأصغر من أو يساوي الـ M طيب
+
+241
+00:30:55,220 --> 00:30:59,660
+لو بدى أقول ما معنى أن تكون الدالة unbounded على
+
+242
+00:30:59,660 --> 00:31:09,660
+المجموعة A لأن هنا F from A to R is unbounded
+
+243
+00:31:13,140 --> 00:31:18,900
+is unbounded on
+
+244
+00:31:18,900 --> 00:31:26,580
+a if الشرط
+
+245
+00:31:26,580 --> 00:31:36,020
+التالي اتحقق خلّينا
+
+246
+00:31:36,020 --> 00:31:37,220
+ننفي الشرط هذا
+
+247
+00:31:41,360 --> 00:31:47,660
+ننفي الشرق بقى الـ bounded فيوجد M عدد موجب لكل
+
+248
+00:31:47,660 --> 00:31:57,700
+بيصير لكل M عدد موجب يوجد X يوجد X يعتمد على M في
+
+249
+00:31:57,700 --> 00:32:03,600
+مجال الدولة بحيث انه نفي المتبين هذا يتحقق اللي هو
+
+250
+00:32:03,600 --> 00:32:17,160
+absolute F of Xبطلع أكبر من أو ساوي أكبر
+
+251
+00:32:17,160 --> 00:32:22,500
+من أو ساوي ال M أكبر من أو ساوي ال M هذا معناه أن
+
+252
+00:32:22,500 --> 00:32:27,700
+الدالة ما تكونش bounded على المجال تبعها معناه أنه
+
+253
+00:32:27,700 --> 00:32:31,920
+لأي عدد موجب بقدر ألاقي عنصر يعتمد على عدد الموجب
+
+254
+00:32:31,920 --> 00:32:35,520
+هذا في مجال الدالة والقيمة المطلقة
+
+255
+00:32:38,330 --> 00:32:44,270
+قيمة الدالة عند العنصر أكبر من أو يساوي ال M فمثال
+
+256
+00:32:44,270 --> 00:32:51,010
+على ذلك هاي
+
+257
+00:32:51,010 --> 00:33:00,450
+مثال example لو
+
+258
+00:33:00,450 --> 00:33:14,520
+أخدت ال function f of xshow that f of x بيساوي
+
+259
+00:33:14,520 --> 00:33:22,240
+واحد على x و x أكبر من سفر is unbounded
+
+260
+00:33:22,240 --> 00:33:26,500
+على
+
+261
+00:33:26,500 --> 00:33:31,680
+المجموعة a اللي هي الفترة المفتوحة من سفر إلى مانا
+
+262
+00:33:31,680 --> 00:33:32,320
+نهائي
+
+263
+00:33:49,080 --> 00:33:57,820
+proof حسب تعريف ال unboundedness بدي اخد ابدأ let
+
+264
+00:33:57,820 --> 00:34:06,960
+m be an arbitrarily positive number بيه given بدي
+
+265
+00:34:06,960 --> 00:34:14,440
+ارد عليه باكس اعتمد xm بحيث ان المتباين هذا تتحقق
+
+266
+00:34:14,440 --> 00:34:24,160
+ف chooseإذاً لهذا العدد الموجب هختار XM X يعتمد
+
+267
+00:34:24,160 --> 00:34:31,920
+على M على إنه واحد على M زائد واحد فهذا بالتأكيد
+
+268
+00:34:31,920 --> 00:34:39,440
+عدد موجب لأن M عدد موجب صح؟ فبالتالي هذا ينتبه
+
+269
+00:34:39,440 --> 00:34:44,180
+للفترة المفتوحة من صفر إلى ملد يعني اللي هي ال A
+
+270
+00:34:47,860 --> 00:34:53,380
+يعني هاي لأي عدد موجب يعني وجدت there exist xm
+
+271
+00:34:53,380 --> 00:34:59,800
+ينتمي للمجال تبع الدالة وهذا يثبت أنه بحقق
+
+272
+00:34:59,800 --> 00:35:04,340
+المتباين هذا then
+
+273
+00:35:04,340 --> 00:35:15,540
+absolute f of xm بساوي absolute مقلوب
+
+274
+00:35:17,150 --> 00:35:26,330
+الـ Xm هو absolute واحد على Xm وهذا بيساوي M زياد
+
+275
+00:35:26,330 --> 00:35:32,290
+واحد هو مفروض اكتب absolute M زياد واحد اللي هو M
+
+276
+00:35:32,290 --> 00:35:39,450
+زياد واحد لأن M عدد موجة وهذا أكبر من M لأن هذه
+
+277
+00:35:39,450 --> 00:35:47,910
+المتباينة اتحققت أكبر من Mلأن نفي أصغر من أوي
+
+278
+00:35:47,910 --> 00:35:53,330
+الساوي مفروض يكون أكبر وهي أثبتت لأي عدد موجة
+
+279
+00:35:53,330 --> 00:35:58,570
+بيوجد XM في مجال الدالة وAbsolute صورة ال XM أكبر
+
+280
+00:35:58,570 --> 00:36:05,470
+من أمي okay تمام حسب التعريف أو ال remark بنطلع ال
+
+281
+00:36:05,470 --> 00:36:09,830
+function تبعتي unbounded على الفترة المفتوحة من
+
+282
+00:36:09,830 --> 00:36:12,630
+صفر إلى ملا نهائي تمام
+
+283
+00:36:15,080 --> 00:36:28,840
+الان في عندي boundedness theorem في
+
+284
+00:36:28,840 --> 00:36:41,220
+نظرية مهمة اسمها boundedness theorem boundedness
+
+285
+00:36:47,400 --> 00:36:54,560
+bound this theorem let
+
+286
+00:36:54,560 --> 00:37:04,580
+خلينا ناخد I let I be closed and bounded interval
+
+287
+00:37:04,580 --> 00:37:10,740
+be closed and
+
+288
+00:37:10,740 --> 00:37:16,500
+bounded interval
+
+289
+00:37:18,720 --> 00:37:27,740
+لو كانت F F from I to R is continuous is
+
+290
+00:37:27,740 --> 00:37:41,980
+continuous on I then F is bounded on
+
+291
+00:37:41,980 --> 00:37:49,110
+I لإن لو كان مجلة ده close bounded intervalوكانت
+
+292
+00:37:49,110 --> 00:37:53,950
+ال function continuous على I فلابد انها تطلع
+
+293
+00:37:53,950 --> 00:38:05,750
+bounded على الفترة I ولبرهان ذلك البرهان سهل proof
+
+294
+00:38:05,750 --> 00:38:13,590
+برهان بالتناقض assume on
+
+295
+00:38:13,590 --> 00:38:14,330
+contrary
+
+296
+00:38:18,360 --> 00:38:23,540
+إذا كان الـ F
+
+297
+00:38:23,540 --> 00:38:29,060
+غير مجموعة على
+
+298
+00:38:29,060 --> 00:38:34,620
+I فبعد
+
+299
+00:38:34,620 --> 00:38:35,780
+ذلك بمعالجة أعلى
+
+300
+00:38:43,910 --> 00:38:46,770
+by above remark اللي هي ال remark اللي مسحناها
+
+301
+00:38:46,770 --> 00:38:55,790
+اللي هي تعطيني تعريف ال unboundedness لكل n for
+
+302
+00:38:55,790 --> 00:39:10,890
+every n عدد طبيعي يوجد x يعتمد على n في الفترة I
+
+303
+00:39:10,890 --> 00:39:12,250
+في مجال الدالة
+
+304
+00:39:15,350 --> 00:39:24,050
+بحيث أنه بحيث
+
+305
+00:39:24,050 --> 00:39:35,750
+أنه صورة ال XN أكبر من .. أوي أكبر من LN
+
+306
+00:39:35,750 --> 00:39:45,140
+أثبت ال function F is unbounded on Iهذا معناه لكل
+
+307
+00:39:45,140 --> 00:39:50,140
+n عدد موجب وبالتالي لكل n عدد طبيعي لإن العداد
+
+308
+00:39:50,140 --> 00:39:56,180
+الطبيعي كلها موجبة يوجد x يعتمد على n في مجال الدل
+
+309
+00:39:56,180 --> 00:40:02,240
+بحيث f of x n يكون أكبر من n
+
+310
+00:40:11,940 --> 00:40:17,960
+طيب ما هذا بتديني طبعا
+
+311
+00:40:17,960 --> 00:40:22,340
+هنا في absolute value احنا نسين ال absolute value
+
+312
+00:40:22,340 --> 00:40:29,060
+اذا يوجد هيك بنكون كوّننا so there exist a
+
+313
+00:40:29,060 --> 00:40:38,180
+sequence x in contained in I such
+
+314
+00:40:38,180 --> 00:40:38,740
+that
+
+315
+00:40:42,110 --> 00:40:48,930
+absolute f of x in أكبر من n for all n belong to n
+
+316
+00:40:48,930 --> 00:40:55,690
+خليني أسميها دستار طيب
+
+317
+00:40:55,690 --> 00:41:03,770
+since ال sequence x in contained
+
+318
+00:41:03,770 --> 00:41:07,990
+in
+
+319
+00:41:07,990 --> 00:41:08,530
+I
+
+320
+00:41:11,610 --> 00:41:18,170
+والـ I عبارة عن الفترة المغلقة من A ل B إذا X in
+
+321
+00:41:18,170 --> 00:41:26,690
+is bounded لأن كل حدود ال sequence محصورة X in
+
+322
+00:41:26,690 --> 00:41:32,070
+أكبر من أو ساوي A أصفر من أو ساوي B لكل M هذه ال
+
+323
+00:41:32,070 --> 00:41:35,350
+sequence محدودة من أسفل ومحدودة من أعلى كل حدودها
+
+324
+00:41:35,350 --> 00:41:39,630
+محصورة بين A وB إذا ال sequence هذه bounded صح؟
+
+325
+00:41:40,270 --> 00:41:44,810
+وبالتالي So
+
+326
+00:41:44,810 --> 00:41:52,470
+by Bolzano-Weierstrass theorem نظرية Bolzano
+
+327
+00:41:52,470 --> 00:41:55,410
+-Weierstrass بتقول كل bounded sequence has a
+
+328
+00:41:55,410 --> 00:41:59,790
+convergent subsequence لإن هنا there exists a
+
+329
+00:41:59,790 --> 00:42:08,950
+subsequence سمية X in R of a sequence X in
+
+330
+00:42:12,270 --> 00:42:19,590
+which converges وهذه الـ subsequence converges say
+
+331
+00:42:19,590 --> 00:42:27,550
+خلّينا نفترض إنه limit لـ subsequence x in R as R
+
+332
+00:42:27,550 --> 00:42:36,790
+tends to infinity بساوي x تمام؟ since
+
+333
+00:42:38,330 --> 00:42:45,570
+الـ X in R أكبر من أو ساوي A أصغر من أو ساوي B لكل
+
+334
+00:42:45,570 --> 00:42:51,190
+R في N الـ sequence X in contained in I وهذه
+
+335
+00:42:51,190 --> 00:42:55,090
+subsequence منها لأن علاصتها موجودة في I بالتالي
+
+336
+00:42:55,090 --> 00:43:01,610
+محصورة بين A وB والsequence هذه converge ل Xإذا
+
+337
+00:43:01,610 --> 00:43:06,010
+حسب نظرية سابقة لما تكون ال sequence محصورة بين a
+
+338
+00:43:06,010 --> 00:43:13,670
+و b وconvergent فنهايتها أيضا we get from previous
+
+339
+00:43:13,670 --> 00:43:22,850
+theorem أن x تنتمي إلى I أو
+
+340
+00:43:22,850 --> 00:43:28,310
+تنتمي للفترة المغلقة من a إلى b اللي هي I تمام
+
+341
+00:43:32,130 --> 00:43:39,430
+طيب بما أن F continuous على I hence F is
+
+342
+00:43:39,430 --> 00:43:46,710
+continuous at X لأن F continuous على كل الفترة I و
+
+343
+00:43:46,710 --> 00:43:55,890
+X تنتمي ل I لأن F continuous at X طيب so by
+
+344
+00:43:55,890 --> 00:43:57,910
+sequential criterion
+
+345
+00:43:59,960 --> 00:44:04,080
+بسيطة لشركات مستمرين هيفي عند سيكوينس كونتينيوس ل
+
+346
+00:44:04,080 --> 00:44:12,200
+X و F continuous عند ال X إذا بسيطة لشركات مستمرين
+
+347
+00:44:12,200 --> 00:44:19,640
+نحصل على انه limit ال image لل subsequence X in R
+
+348
+00:44:19,640 --> 00:44:25,380
+as R tends to infinity بساوي F of X
+
+349
+00:44:30,470 --> 00:44:36,850
+طبما هذا معناه ان ال sequence هذه او ال image لل
+
+350
+00:44:36,850 --> 00:44:40,330
+subsequence اللي هي sequence اخرى convergent
+
+351
+00:44:40,330 --> 00:44:53,750
+وبالتالي فهي bounded okay تمام اذا it follows thus
+
+352
+00:44:53,750 --> 00:44:57,210
+the sequence
+
+353
+00:45:00,200 --> 00:45:08,660
+اللي هي f of x in R من R بالساوية واحدة لإنفينيتي
+
+354
+00:45:08,660 --> 00:45:15,980
+is bounded لإن أنا convergence كل convergence
+
+355
+00:45:15,980 --> 00:45:22,380
+sequence is bounded حسب نظرية السابقة وبالتالي so
+
+356
+00:45:22,380 --> 00:45:31,870
+there existSo there exist m عدد موجب بحيث انه
+
+357
+00:45:31,870 --> 00:45:44,450
+absolute
+
+358
+00:45:44,450 --> 00:45:52,730
+f of x in R تطلع اصغر من او ساوي m لكل
+
+359
+00:46:07,550 --> 00:46:19,170
+by our chameleon by our
+
+360
+00:46:19,170 --> 00:46:30,390
+chameleon propertyيوجد R ينتمي إلى N أو خلّيني
+
+361
+00:46:30,390 --> 00:46:41,390
+أسميه R0 ينتمي إلى N بحيث أنه R0 أكبر من M صح؟
+
+362
+00:46:41,390 --> 00:46:45,690
+لأي عدد حقيقي سواء موجب أو سالب فيه دائما عدد
+
+363
+00:46:45,690 --> 00:46:58,760
+طبيعي أكبر منه now by starand double star we have
+
+364
+00:46:58,760 --> 00:47:10,740
+لدينا انه ال M أكبر من أو يساوي absolute F of X in
+
+365
+00:47:10,740 --> 00:47:18,320
+R0 صح؟ هاي من هنا خدي R بساوي R0 فهذا الكلام صحيح
+
+366
+00:47:18,320 --> 00:47:21,500
+هذا من double star
+
+367
+00:47:24,010 --> 00:47:32,970
+و من star هذا أكبر من .. إذا من star f of x in R0
+
+368
+00:47:32,970 --> 00:47:41,450
+أكبر من in R0 هذا صحيح لكل عناصر ال sequence x in
+
+369
+00:47:41,450 --> 00:47:50,330
+وبالتالي صحيح لكل عناصر ال subsequence x in Rو in
+
+370
+00:47:50,330 --> 00:47:54,410
+R0 من تعريف الـ subsequence دائما أكبر من أو ساوي
+
+371
+00:47:54,410 --> 00:48:01,570
+R0 و من هنا R0 أكبر من M فنحن
+
+372
+00:48:01,570 --> 00:48:10,210
+نحصل على M أكبر من M Contradiction فالتناقض هذا
+
+373
+00:48:10,210 --> 00:48:14,610
+يقول لي أن ال assumption تبعي ال assumption تبعنا
+
+374
+00:48:16,810 --> 00:48:22,790
+إن ال function if is unbounded on I كان assumption
+
+375
+00:48:22,790 --> 00:48:26,870
+خاطئ، إذا ليس صح إن ال function تطلع bounded على I
+
+376
+00:48:26,870 --> 00:48:32,850
+وهذا بيكمل برهان أنبضرية، okay تمام؟ واضح؟
+
+377
+00:48:47,090 --> 00:48:53,650
+طيب ناخد احنا شوية break نكتفي
+
+378
+00:48:53,650 --> 00:49:00,710
+بهذا القدر و هناخد ان شاء الله break و نواصل في
+
+379
+00:49:00,710 --> 00:49:01,650
+اللقاء القادم
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d5dd44db5ce8d397d86f177d9bff266576c4c124
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_postprocess.srt
@@ -0,0 +1,1656 @@
+1
+00:00:20,650 --> 00:00:27,710
+السلام عليكم اليوم ان شاء الله هنكمل ما ابتدأناه
+
+2
+00:00:27,710 --> 00:00:35,620
+سابقا بخصوص موضوع الكوشي sequencesأخر نظرية
+
+3
+00:00:35,620 --> 00:00:41,060
+هناخدها بالنسبة لهذا الموضوع هتكون نظرية التالية
+
+4
+00:00:41,060 --> 00:00:45,580
+لكن في الأول خلينا نراجع بس تعريف الـ Cauchy
+
+5
+00:00:45,580 --> 00:00:48,600
+sequence فطبعا تعريف الـ Cauchy sequence زي ما
+
+6
+00:00:48,600 --> 00:00:52,560
+انتوا شايفين sequence of real numbers is Cauchy if
+
+7
+00:00:52,560 --> 00:00:55,500
+and only if for every epsilon فيه capital N
+
+8
+00:00:55,500 --> 00:01:00,200
+natural number بحيث for every N و M bigger than or
+
+9
+00:01:00,200 --> 00:01:06,260
+equal capital Nالمقارنة بين xn و xm أقل من إبسلون
+
+10
+00:01:06,260 --> 00:01:12,980
+رمارك في ملاحظة هنا it can be easily shown يعني من
+
+11
+00:01:12,980 --> 00:01:17,540
+السهل اثبات أنه sequence of real numbers is Cauchy
+
+12
+00:01:17,540 --> 00:01:23,420
+if and only if limit the distance between xn و xm
+
+13
+00:01:23,420 --> 00:01:29,700
+بالساوي سفر whenever n و m tends to infinityيعني
+
+14
+00:01:29,700 --> 00:01:34,060
+المعنى أخر لما N و M تكون large، و N و M large
+
+15
+00:01:34,060 --> 00:01:39,660
+enough، ال distance between XN و XM بتكون very
+
+16
+00:01:39,660 --> 00:01:44,860
+small، تقول لصفر، فهو نفس المعنى تقريبا، you can
+
+17
+00:01:44,860 --> 00:01:50,260
+write a proof for this remark و it is easy زي ما
+
+18
+00:01:50,260 --> 00:01:54,310
+هو مذكورفي الان definition a sequence of real
+
+19
+00:01:54,310 --> 00:02:02,570
+numbers بنسميها contractive إذا وجد ثابت c عدد بين
+
+20
+00:02:02,570 --> 00:02:08,630
+صفر و واحد عدد موجب بحيث أنه المسافة بين xn plus
+
+21
+00:02:08,630 --> 00:02:14,830
+two و xn plus one أصغر من أو ساوى ثابت c في
+
+22
+00:02:14,830 --> 00:02:20,140
+المسافة بين xn plus one minus xnوهذا الكلام طبعا
+
+23
+00:02:20,140 --> 00:02:24,920
+بيكون متحقق for every natural number in الثابت C
+
+24
+00:02:24,920 --> 00:02:33,760
+هذا بيسمي the constant of the contractive sequence
+
+25
+00:02:33,760 --> 00:02:39,140
+الان النظرية اللي حكيت عنها في المقدمة هي النظرية
+
+26
+00:02:39,140 --> 00:02:46,000
+التالية theorem رقم
+
+27
+00:02:46,000 --> 00:02:47,860
+خمسة و عشرين
+
+28
+00:02:51,670 --> 00:02:59,830
+النظرية هذه بتقول every every contractive every
+
+29
+00:02:59,830 --> 00:03:08,050
+contractive sequence is cauchy كل contractive
+
+30
+00:03:08,050 --> 00:03:15,230
+sequence بتكون cauchy البرهان مش صعب proof for
+
+31
+00:03:15,230 --> 00:03:20,170
+every natural number in apply
+
+32
+00:03:22,820 --> 00:03:28,900
+the defining .. apply
+
+33
+00:03:28,900 --> 00:03:34,980
+the defining condition ال
+
+34
+00:03:34,980 --> 00:03:39,120
+defining condition اللي هو هذا الشرط تبع ال
+
+35
+00:03:39,120 --> 00:03:42,720
+contractive of contractive sequence
+
+36
+00:03:46,850 --> 00:03:52,790
+of contractive sequence لان هنطبق التعريف هذا to
+
+37
+00:03:52,790 --> 00:03:56,830
+get لنحصل
+
+38
+00:03:56,830 --> 00:03:57,450
+على mainly
+
+39
+00:04:01,950 --> 00:04:08,910
+هي عندي absolute xn من التعريف هي absolute xn plus
+
+40
+00:04:08,910 --> 00:04:16,110
+two minus xn plus one less than or equal to c في
+
+41
+00:04:16,110 --> 00:04:25,890
+absolute xn plus one negative xn الآن
+
+42
+00:04:25,890 --> 00:04:35,390
+من نفس التعريف هذا بدل n هناكب N سالب واحد فبصير
+
+43
+00:04:35,390 --> 00:04:40,010
+الطرف الشمال هكذا وهذا بطلع اصغر من ال absolute
+
+44
+00:04:40,010 --> 00:04:45,070
+value هذي اصغر من او ساوي C وفي اندي C تانية فبصير
+
+45
+00:04:45,070 --> 00:04:51,470
+C تربيه في absolute X X
+
+46
+00:04:51,470 --> 00:04:58,190
+N minus X N minus واحد تمام؟
+
+47
+00:04:59,310 --> 00:05:05,730
+والان ممكن من ال defining condition هذا اطبخه على
+
+48
+00:05:05,730 --> 00:05:11,670
+ال absolute value هذه يعني ابدل n هناك ب n سالب
+
+49
+00:05:11,670 --> 00:05:17,410
+اتنين فبصير n بصير الطرف الشمال absolute xn minus
+
+50
+00:05:17,410 --> 00:05:22,540
+xn minus واحدهذا هيطلع أصغر من أوي ساوي C في
+
+51
+00:05:22,540 --> 00:05:27,860
+absolute value تانية وفي عندي C تربية فهيصير عندي
+
+52
+00:05:27,860 --> 00:05:34,960
+C تكايم في absolute xn minus one minus xn minus
+
+53
+00:05:34,960 --> 00:05:42,000
+two وطبعا لو استمرنا على هذا النمط هنصل في الآخر
+
+54
+00:05:42,000 --> 00:05:47,620
+خالص إلىأصغر من أو يساوي c os n في absolute x2
+
+55
+00:05:47,620 --> 00:05:52,980
+negative x1 و بعد هيك بنوقف خلاص لأن هنكرر تطبيق
+
+56
+00:05:52,980 --> 00:05:57,460
+ال defining condition هذا n من المرات و بعد هيك
+
+57
+00:05:57,460 --> 00:06:02,240
+هنوقف لأنه خلاص هنصل ل absolute x2 minus x1 خلصت
+
+58
+00:06:02,240 --> 00:06:04,600
+الحدود أظبط؟
+
+59
+00:06:06,830 --> 00:06:12,490
+تبام اذا انا اندي طلع absolute xn plus two minus
+
+60
+00:06:12,490 --> 00:06:17,730
+xn plus one less than or equal c to n في absolute
+
+61
+00:06:17,730 --> 00:06:23,150
+x two negative x one الكلام هذا صحيح for every n
+
+62
+00:06:23,150 --> 00:06:31,250
+كل الأعداد الطبيعية n الان من ال inequality هذه
+
+63
+00:06:34,380 --> 00:06:39,140
+this inequality and the triangle inequality
+
+64
+00:06:39,140 --> 00:06:50,600
+متباينة المثلث بيؤدوا and geometric progression
+
+65
+00:07:04,750 --> 00:07:11,290
+imply انه .. بيقده انه for .. لو أخدت M أكبر من N
+
+66
+00:07:11,290 --> 00:07:14,870
+فبطلع
+
+67
+00:07:14,870 --> 00:07:23,930
+عندي absolute XM minus XN هذا هيكون باستخدام
+
+68
+00:07:23,930 --> 00:07:36,720
+متبينة المثلث هترح من XMلو طرحت من XM XM-1 ورجعتها
+
+69
+00:07:36,720 --> 00:07:50,860
+ورجعتها و بعدين هطرح XM-2 ورجعها إلى
+
+70
+00:07:50,860 --> 00:08:02,450
+أن أصل إلىxn زاد واحد minus xm شو اللي عملته هنا
+
+71
+00:08:02,450 --> 00:08:09,790
+انا طرحت من xm xm minus one و رجعتها بعدين طرحت xm
+
+72
+00:08:09,790 --> 00:08:14,790
+سالب اتنين و رجعتها و هكذا طرحت xn زاد واحد و
+
+73
+00:08:14,790 --> 00:08:19,550
+رجعتها و طبعا هاي سالب xn هو بوقف لانه بعد كده
+
+74
+00:08:19,550 --> 00:08:30,360
+خلاصالان من ال .. يسمى المتباينة بال star فمن
+
+75
+00:08:30,360 --> 00:08:36,460
+ال star this inequality اللي هي ال star لو
+
+76
+00:08:36,460 --> 00:08:46,240
+أخدت هنا M بدلت N زاد 2 بدلتها ب M معناته N بساوي
+
+77
+00:08:46,240 --> 00:08:56,130
+M سالد 2 صح؟وبالتالي هذا بيصير أصغر
+
+78
+00:08:56,130 --> 00:09:03,190
+من أو ساوي C أُس M اللي هي M سالب اتنين في
+
+79
+00:09:03,190 --> 00:09:09,750
+absolute X2 نيجاتيب X1 إذن هذا باستخدام المتباين
+
+80
+00:09:09,750 --> 00:09:14,290
+أسطار حيث ال M هذه أخدتها بساوي N زي اتنين
+
+81
+00:09:14,290 --> 00:09:19,390
+وبالتالي إذا ال N بساوي M نيجاتيب Twoفال absolute
+
+82
+00:09:19,390 --> 00:09:23,350
+value الأولانية أصغر حسب ال star أصغر من أو ساوي c
+
+83
+00:09:23,350 --> 00:09:27,650
+to m minus 2 في absolute x2 minus 1 Similarly
+
+84
+00:09:27,650 --> 00:09:31,730
+باستخدام ال star المتباين ال star ال absolute
+
+85
+00:09:31,730 --> 00:09:36,490
+value التاني هذه أو second term هذا بيطلع أصغر من
+
+86
+00:09:36,490 --> 00:09:43,530
+أو ساوي c to m minus 3 في absolute x2 minus x1 و
+
+87
+00:09:43,530 --> 00:09:44,130
+هكذا
+
+88
+00:09:46,560 --> 00:09:51,660
+إلى أن نصل آخر حد هيطلع عند you c to n minus واحد
+
+89
+00:09:51,660 --> 00:09:58,380
+في absolute x2 minus x1 طيب
+
+90
+00:09:58,380 --> 00:10:01,880
+ناخد الآن عامل مشترك
+
+91
+00:10:04,360 --> 00:10:11,340
+هذا بيساوي c to n negative two زائد c to n
+
+92
+00:10:11,340 --> 00:10:17,800
+negative three زائد و هكذا to c to n negative one
+
+93
+00:10:17,800 --> 00:10:23,460
+كل هذا مضروب في العامل المشترك absolute x two
+
+94
+00:10:23,460 --> 00:10:28,540
+minus x one المجموع
+
+95
+00:10:28,540 --> 00:10:36,720
+هذا هاخد عامل مشترك c to n minus واحد منهفهيبقى
+
+96
+00:10:36,720 --> 00:10:46,020
+لدي هنا c to m minus n minus 1 وهنا هاخد لو جسمت
+
+97
+00:10:46,020 --> 00:10:52,420
+had على c to n minus 1 هيطلع c to m minus n minus
+
+98
+00:10:52,420 --> 00:10:55,800
+2
+
+99
+00:10:55,800 --> 00:11:06,070
+وهكذا إلى أخر حد هيكون 1و طبعا كل هذا مضروب في
+
+100
+00:11:06,070 --> 00:11:17,890
+absolute x2 minus x1 y ساوي c to n negative one
+
+101
+00:11:17,890 --> 00:11:22,450
+الان هذه عبارة عن geometric progression متوالية
+
+102
+00:11:22,450 --> 00:11:30,230
+هندسية الحد الأول فيها واحد والأساس تبعها cفمجموعة
+
+103
+00:11:30,230 --> 00:11:33,930
+المتوالي الهندسية أو الـ geometric progression
+
+104
+00:11:33,930 --> 00:11:40,210
+مجموعة بساوي الحد الأول واحد سالب الحد الأخير
+
+105
+00:11:40,210 --> 00:11:47,170
+مضروب في الأساس اللي هو C فبطلع C to M negative N
+
+106
+00:11:47,170 --> 00:11:54,070
+على واحد minus الأساسإذن هذا مجموع الـ geometric
+
+107
+00:11:54,070 --> 00:11:58,910
+progression كل هذا مضروب في absolute X2 negative
+
+108
+00:11:58,910 --> 00:12:04,610
+X1 طيب
+
+109
+00:12:04,610 --> 00:12:10,330
+أنا عندي الـC أنا
+
+110
+00:12:10,330 --> 00:12:15,630
+عندي الـC الـC
+
+111
+00:12:15,630 --> 00:12:22,980
+عدد بين 0 و 1وبالتالي c to m minus n بيبقى العدد
+
+112
+00:12:22,980 --> 00:12:32,020
+بين سفر واحد وبالتالي اذا واحد سالب c اكيد هيطلع
+
+113
+00:12:32,020 --> 00:12:37,880
+اكبر من سفر اصغر
+
+114
+00:12:37,880 --> 00:12:38,440
+من واحد
+
+115
+00:12:41,670 --> 00:12:47,450
+أذا هاي أنا عندي هذا هستبدله بأصغر من هاي c to n
+
+116
+00:12:47,450 --> 00:12:53,570
+negative one الآن هذا الكسر ال bus تبعه واحد سالب
+
+117
+00:12:53,570 --> 00:12:59,970
+c to n minus n أصغر من واحد فهذا الكسر أصغر من
+
+118
+00:12:59,970 --> 00:13:06,150
+واحد على واحد سالب c أصغر من واحد على واحد سالب c
+
+119
+00:13:06,150 --> 00:13:10,510
+في absolute x to negative x one
+
+120
+00:13:14,990 --> 00:13:23,230
+طيب انا اندي برضه اخدنا قبل هيك مثال بيقول اذا كان
+
+121
+00:13:23,230 --> 00:13:29,790
+C أكبر من صفر اصغر من واحد هذا بيقدي ان ال limit ل
+
+122
+00:13:29,790 --> 00:13:35,290
+C to N او C to N سالب واحد لما N تقل ل infinity
+
+123
+00:13:35,290 --> 00:13:39,410
+بيساوي صفر فبالاستخدام المثال هذا اللي اثبتناه قبل
+
+124
+00:13:39,410 --> 00:13:47,130
+هيكبنلاحظ ان c to n سالب واحد هنا هذا ثابت وهذا
+
+125
+00:13:47,130 --> 00:13:54,710
+ثابت ف ال c to n سالب واحد تقول للسفر في ثابت تقول
+
+126
+00:13:54,710 --> 00:13:59,730
+لثابت في سفر يعني سفر إذا هذا المقدار كله tends to
+
+127
+00:13:59,730 --> 00:14:03,590
+zero as n tends to infinity
+
+128
+00:14:07,000 --> 00:14:12,180
+وبالتالي اذا هيك احنا اثبتنا ان ال limit ل
+
+129
+00:14:12,180 --> 00:14:19,740
+absolute xm minus xn لما ال M تقول ل infinity
+
+130
+00:14:19,740 --> 00:14:23,680
+بتساوي
+
+131
+00:14:23,680 --> 00:14:32,280
+سفر وطبعا ال M انا ماخدها هنا ال M ماخدها M اكبر
+
+132
+00:14:32,280 --> 00:14:36,860
+من N فلما ال M تقول ل infinity ال M ايضاتقول
+
+133
+00:14:36,860 --> 00:14:42,000
+لإنفينيتي إذا هنا ممكن أحط and M تقول لإنفينيتي
+
+134
+00:14:42,000 --> 00:14:46,220
+إذا هنا أثبتنا إن ال limit ل absolute XM minus XN
+
+135
+00:14:46,220 --> 00:14:51,540
+as N and M both tends to infinity بساوي سفر
+
+136
+00:14:51,540 --> 00:14:56,800
+وبالتالي إذا by above remark حسب ال remark اللي
+
+137
+00:14:56,800 --> 00:15:01,200
+فوق ال sequence XM is Cauchy
+
+138
+00:15:04,210 --> 00:15:12,090
+و هذا بيكمل برهان النظرية تمام؟ واضح؟
+
+139
+00:15:12,090 --> 00:15:16,690
+في أي استفسار؟ في أي سؤال على البرهان؟ في أي قطة
+
+140
+00:15:16,690 --> 00:15:23,570
+مش واضحة؟ is there any question؟
+
+141
+00:15:23,570 --> 00:15:27,790
+okay then this ends
+
+142
+00:15:30,870 --> 00:15:37,390
+section تلاتة خمسة and we are going to start a new
+
+143
+00:15:37,390 --> 00:15:44,490
+section هنبدأ section جديد وهذا ال section بتحدث
+
+144
+00:15:44,490 --> 00:15:55,870
+عن موضوع properly divergent sequences section
+
+145
+00:15:55,870 --> 00:16:00,450
+three point six
+
+146
+00:16:02,990 --> 00:16:11,770
+properly .. properly .. divergent .. divergent ..
+
+147
+00:16:11,770 --> 00:16:12,710
+sequences
+
+148
+00:16:17,180 --> 00:16:21,080
+أي يعني properly .. properly divergence sequence
+
+149
+00:16:21,080 --> 00:16:27,140
+احنا جفنا قبل هيك أمثلة examples about divergence
+
+150
+00:16:27,140 --> 00:16:31,200
+sequences من ال .. ال divergence sequences هذه كان
+
+151
+00:16:31,200 --> 00:16:36,480
+negative one to n بيقولوا أنه هذه has a divergence
+
+152
+00:16:36,480 --> 00:16:40,640
+برضه
+
+153
+00:16:40,640 --> 00:16:48,480
+sequence nDivergent means infinity وبالتالي
+
+154
+00:16:48,480 --> 00:16:51,480
+Divergent ال sequence negative and intense
+
+155
+00:16:51,480 --> 00:16:59,480
+وبالتالي Divergent و هكذا في كتير sequences مر
+
+156
+00:16:59,480 --> 00:17:04,520
+علينا sequences على الأقل هدول إذا ماكانش أكتر
+
+157
+00:17:04,520 --> 00:17:11,930
+وشوفنا ان كل ال sequences هذي are divergentهناك
+
+158
+00:17:11,930 --> 00:17:17,130
+لحد الآن ماتحدثناش عن تفريق التفريق في ال
+
+159
+00:17:17,130 --> 00:17:22,070
+divergence كنا نقول إن إذا كانت ال sequence
+
+160
+00:17:22,070 --> 00:17:26,550
+ماليهاش limit أو تقول ل infinity أو سالب infinity
+
+161
+00:17:26,550 --> 00:17:30,130
+فكنا نقول ال sequence is divergent أو not
+
+162
+00:17:30,130 --> 00:17:34,210
+convergent اليوم ال divergence sequences هنجزقهم
+
+163
+00:17:34,210 --> 00:17:40,450
+إلى نوعين في sequences هنقول عنهم divergentو في
+
+164
+00:17:40,450 --> 00:17:44,850
+نوع معين من الـ divergence sequences هنسميهم
+
+165
+00:17:44,850 --> 00:17:49,810
+properly divergent إذن الـ sequences اللي زي هدول
+
+166
+00:17:49,810 --> 00:17:54,890
+اللي ال limit بتاعتهم إما infinity أو negative
+
+167
+00:17:54,890 --> 00:17:59,110
+infinity طبعا هدول ال sequences are divergent لكن
+
+168
+00:17:59,110 --> 00:18:02,170
+هذا النوع من ال divergence sequences هنسميه
+
+169
+00:18:02,170 --> 00:18:06,410
+properly divergent متباعدة تباعدا صحيحا
+
+170
+00:18:09,530 --> 00:18:13,590
+Okay إذا ال .. ال sequences التلاتي هدول all of
+
+171
+00:18:13,590 --> 00:18:17,690
+them are divergent كلهم ممكن نقول عنهم divergent
+
+172
+00:18:17,690 --> 00:18:23,710
+لكن التنتين هدول الأخرانيين ممكن نقول عنهم أيضا
+
+173
+00:18:23,710 --> 00:18:27,490
+properly divergent أما هذه مقدرش أقول عنها
+
+174
+00:18:27,490 --> 00:18:32,450
+properly divergent مجرد divergent okay تمام إذا
+
+175
+00:18:32,450 --> 00:18:38,270
+نكتب التعريفات هذه definition
+
+176
+00:18:42,170 --> 00:18:49,570
+اتنين ستة وعشرين let
+
+177
+00:18:49,570 --> 00:18:52,590
+x in be sequence of real numbers
+
+178
+00:18:56,340 --> 00:19:05,960
+نقول إن xn tends to infinity أو limit xn as n
+
+179
+00:19:05,960 --> 00:19:14,500
+tends to infinity بساوي infinity اذا تحقق الشرط
+
+180
+00:19:14,500 --> 00:19:15,220
+التالي
+
+181
+00:19:23,170 --> 00:19:31,350
+for every real number alpha there exists capital N
+
+182
+00:19:31,350 --> 00:19:41,150
+depends on alpha natural number such that xn such
+
+183
+00:19:41,150 --> 00:19:47,270
+that لو كان N أكبر
+
+184
+00:19:47,270 --> 00:19:51,150
+من أو ساوي capital N هذا بقدر أن xn أكبر من alpha
+
+185
+00:19:53,550 --> 00:19:56,650
+اللي بتتكلموا لو سمحتوا ماتتكلمش امنع الكلام
+
+186
+00:19:59,910 --> 00:20:04,290
+إذا ما معنى الـ sequence Xn tends to infinity أو
+
+187
+00:20:04,290 --> 00:20:09,410
+limit لها بالساوية infinity معناه لأي number ألفة
+
+188
+00:20:09,410 --> 00:20:12,670
+نقدر نجد ال number الناترال نقدر نجد ال number
+
+189
+00:20:12,670 --> 00:20:13,090
+الناترال نقدر نجد ال number الناترال نقدر نجد ال
+
+190
+00:20:13,090 --> 00:20:14,290
+number الناترال نقدر نجد ال number الناترال نقدر
+
+191
+00:20:14,290 --> 00:20:14,970
+نجد ال number الناترال نقدر نجد ال number الناترال
+
+192
+00:20:14,970 --> 00:20:21,010
+نقدر نجد ال number الناترال نقدر نجد ال number
+
+193
+00:20:21,010 --> 00:20:25,370
+الناترال نقدر نجد ال number الناترال نقدر نجد ال
+
+194
+00:20:25,370 --> 00:20:26,370
+number الناترال نقدر نجد ال number الناترال نقدر
+
+195
+00:20:26,370 --> 00:20:27,990
+نجد ال number الناترال نقدر نجد ال number الناترال
+
+196
+00:20:27,990 --> 00:20:32,450
+نقدر نجد ال numberمن capital N أو كل حدودها for
+
+197
+00:20:32,450 --> 00:20:37,490
+large N أكبر من أي عدد Alpha أي عدد حقيقي Alpha
+
+198
+00:20:37,490 --> 00:20:41,810
+عشوائي okay إذا قدرنا نخلي حدود ال sequence أكبر
+
+199
+00:20:41,810 --> 00:20:46,330
+من أي عدد حقيقي مهما كان فال sequence معناته
+
+200
+00:20:46,330 --> 00:20:54,110
+نهايتها infinity بالمثل ممكن نعرف نقول x n tends
+
+201
+00:20:54,110 --> 00:21:01,710
+to negative infinity أو limitxn as n tends to
+
+202
+00:21:01,710 --> 00:21:07,450
+infinity بساوي negative infinity إذا تحقق الشرط
+
+203
+00:21:07,450 --> 00:21:17,190
+التالي for every beta real numberيوجد capital N
+
+204
+00:21:17,190 --> 00:21:24,390
+يعتمد على ال beta natural number such that لكل N
+
+205
+00:21:24,390 --> 00:21:29,750
+أكبر من أو ساوي capital N بطلع اندي xn أصغر من
+
+206
+00:21:29,750 --> 00:21:34,770
+beta إذا لو قدرت أخلي حدود ال sequence أصغر من أي
+
+207
+00:21:34,770 --> 00:21:40,640
+عدد حقيقي betafor large N فكل .. فالـ sequence هذه
+
+208
+00:21:40,640 --> 00:21:44,780
+بيقول إنها tends to negative infinity أو ال limit
+
+209
+00:21:44,780 --> 00:21:51,020
+بتاعتها is negative infinity هاي أمثلة examples
+
+210
+00:21:51,020 --> 00:22:03,600
+show
+
+211
+00:22:03,600 --> 00:22:06,180
+that limit
+
+212
+00:22:07,910 --> 00:22:13,670
+الـ sequence n بساوي infinity احنا كلنا عارفين
+
+213
+00:22:13,670 --> 00:22:18,790
+that the sequence of natural number ال limit
+
+214
+00:22:18,790 --> 00:22:26,490
+تبعتها infinity لكن ممكن نثبت الان هذا باستخدام
+
+215
+00:22:26,490 --> 00:22:31,590
+التعريف فنشوف
+
+216
+00:22:31,590 --> 00:22:37,560
+هاي البرهان proofحسب التعريف بدنا نبدأ بقول let
+
+217
+00:22:37,560 --> 00:22:45,180
+alpha belonging to R be given ناخد
+
+218
+00:22:45,180 --> 00:22:52,520
+alpha عدد حقيقي عشوائي by
+
+219
+00:22:52,520 --> 00:23:00,260
+Archimedean property حسب خاصية Archimedes
+
+220
+00:23:04,680 --> 00:23:13,220
+يوجد capital N عدد طبيعي بحيث انه capital N أكبر
+
+221
+00:23:13,220 --> 00:23:19,740
+من ال alpha نظبوت؟ هذا حسب ال Archimedean property
+
+222
+00:23:19,740 --> 00:23:22,240
+now
+
+223
+00:23:24,690 --> 00:23:28,990
+لو أخدت N bigger than or equal capital N هذا هيقدي
+
+224
+00:23:28,990 --> 00:23:35,330
+ان XN ال sequence هذه لحد العام XN تبعها ايش
+
+225
+00:23:35,330 --> 00:23:42,730
+بيساوي بيساوي N الان ال N هذه small n أكبر منه
+
+226
+00:23:42,730 --> 00:23:47,010
+يساوي capital N وانا بيختار capital N by
+
+227
+00:23:47,010 --> 00:23:49,350
+Archimedean property أكبر من Alpha
+
+228
+00:23:55,340 --> 00:24:01,740
+وبالتالي إذا ال .. ال .. ال implication هذه تتحقق
+
+229
+00:24:01,740 --> 00:24:08,220
+هنا أثبتنا for any Alpha تنتمي ل R there exists
+
+230
+00:24:08,220 --> 00:24:12,600
+natural number يعتمد على Alpha هيوجه .. يعتمد على
+
+231
+00:24:12,600 --> 00:24:18,160
+Alpha N مرتبطة بAlpha بحيث لكل N أكبر من أوساو
+
+232
+00:24:18,160 --> 00:24:22,740
+capital N طول عندي Xn أكبر من Alpha therefore
+
+233
+00:24:27,120 --> 00:24:31,940
+Alpha capital N definition of limit
+
+234
+00:24:38,380 --> 00:24:43,960
+أو alpha capital N definition بطلع عندي limit xn
+
+235
+00:24:43,960 --> 00:24:50,940
+بساوي infinity طبعا هنا xn مقصود فيها الحد العام
+
+236
+00:24:50,940 --> 00:24:56,440
+لل sequence N يعني xn بساوي N okay تمام واضح
+
+237
+00:24:56,440 --> 00:25:04,360
+البرهان طب هاي مثال تاني show thatإنه limit الـ
+
+238
+00:25:04,360 --> 00:25:06,800
+sequence اللي الحد اللي عام تبقىها negative
+
+239
+00:25:06,800 --> 00:25:16,080
+Interbia equals negative infinity هنطبق
+
+240
+00:25:16,080 --> 00:25:23,660
+التعريف في الجزء التاني ونشوف كيف ممكن نثبت إن ال
+
+241
+00:25:23,660 --> 00:25:28,940
+implication كيف لأي Beta بقدر ألاقي capital N
+
+242
+00:25:28,940 --> 00:25:35,040
+يعتمد على Betaبحيث ان ال implication هذه هي تتحقق
+
+243
+00:25:35,040 --> 00:25:40,320
+هاي
+
+244
+00:25:40,320 --> 00:25:45,800
+البرهان proof بالمناسبة
+
+245
+00:25:45,800 --> 00:25:51,980
+انا ماعنديش يعني عصة سحرية عشان اعرف مسبقا لأي
+
+246
+00:25:51,980 --> 00:25:58,440
+beta كيف اختار ال N ماعيش عصة سحرية فبنعمل
+
+247
+00:25:58,440 --> 00:26:05,060
+analysis تحليلوبنكتشف كيف نختار الـ capital N لأي
+
+248
+00:26:05,060 --> 00:26:12,500
+given Beta إذا هنا حد البرهان بـ let Beta بأي real
+
+249
+00:26:12,500 --> 00:26:16,800
+number بـ given طبعا
+
+250
+00:26:16,800 --> 00:26:21,540
+حسب التعريف عشان أثبت أنه ال sequence الحد العام
+
+251
+00:26:21,540 --> 00:26:22,760
+تبعها XN
+
+252
+00:26:25,210 --> 00:26:29,610
+بساوي سالب n تربية عشان اثبت ان ال limit لل
+
+253
+00:26:29,610 --> 00:26:33,830
+sequence هذه بساوي negative infinity بدي اثبت بدي
+
+254
+00:26:33,830 --> 00:26:37,470
+ارد على ال given beta هذه بcapital N تعتمد عليها
+
+255
+00:26:37,470 --> 00:26:42,510
+بحيث ان ال implication هذه تتحقق فبنيجي بنعمل زي
+
+256
+00:26:42,510 --> 00:26:47,510
+ما عملنا في تعريف epsilon capital N للنهايات انا
+
+257
+00:26:47,510 --> 00:26:53,340
+بقول من الآخر انا عايز ان xnاللي هي سالب enter
+
+258
+00:26:53,340 --> 00:26:58,840
+بيها بدي هذه في النهاية for any given beta real
+
+259
+00:26:58,840 --> 00:27:05,520
+number بدي x in اللي هي negative n squared بديها
+
+260
+00:27:05,520 --> 00:27:12,100
+less than beta تمام؟
+
+261
+00:27:13,650 --> 00:27:19,790
+و طبعا هذا لكل N أكبر من أو ساوي capital N فما هي
+
+262
+00:27:19,790 --> 00:27:27,090
+ال N أنا بدي أجيب ال N اللي بتخلي هذا الكلام صحيح
+
+263
+00:27:27,090 --> 00:27:31,270
+أنا
+
+264
+00:27:31,270 --> 00:27:36,030
+بعرف أنه لازم ال N تكون أكبر من أو ساوي capital N
+
+265
+00:27:36,030 --> 00:27:38,630
+وبالتالي
+
+266
+00:27:40,310 --> 00:27:47,610
+هذا بيقدّي أن N تربية أكبر من أو يساوي N صح؟ أكبر
+
+267
+00:27:47,610 --> 00:27:52,550
+من أو يساوي capital N وهذا
+
+268
+00:27:52,550 --> 00:27:59,530
+بيقدّي أن سالب N تربية أصغر من أو يساوي سالب
+
+269
+00:27:59,530 --> 00:28:00,290
+capital N
+
+270
+00:28:05,790 --> 00:28:13,770
+وانا بدي هذا يطلع أصغر من ال beta عشان يطلع xn
+
+271
+00:28:13,770 --> 00:28:22,870
+أصغر من beta صح؟ إذا بسأل نفسي متى هذا سالب n
+
+272
+00:28:22,870 --> 00:28:27,670
+تربيه اللي هو xn أصغر من beta لإنما negative
+
+273
+00:28:27,670 --> 00:28:33,880
+capital N أصغر من beta إذا for anyfor any beta
+
+274
+00:28:33,880 --> 00:28:40,360
+belonging to R يقول هنا for any beta belonging to
+
+275
+00:28:40,360 --> 00:28:42,640
+R يقول هنا for any beta belonging to R يقول هنا
+
+276
+00:28:42,640 --> 00:28:44,360
+for any beta belonging to R يقول هنا for any beta
+
+277
+00:28:44,360 --> 00:28:44,580
+belonging to R يقول هنا for any beta belonging to
+
+278
+00:28:44,580 --> 00:28:44,600
+for any beta belonging to R يقول هنا for any beta
+
+279
+00:28:44,600 --> 00:28:44,700
+belonging to R يقول هنا for any beta belonging to
+
+280
+00:28:44,700 --> 00:28:44,720
+R يقول هنا for any beta belonging to R يقول هنا
+
+281
+00:28:44,720 --> 00:28:44,740
+for any beta belonging to R يقول هنا for any beta
+
+282
+00:28:44,740 --> 00:28:50,520
+R يقول هنا for any beta belonging to R يقول هنا
+
+283
+00:28:50,520 --> 00:28:54,400
+for any beta belonging to R يقول هنا for any beta
+
+284
+00:28:54,400 --> 00:29:04,050
+belonging to R يقولوهذا بقدر اختاره by Archimedean
+
+285
+00:29:04,050 --> 00:29:11,170
+propertyby Archimedean property لأي real number
+
+286
+00:29:11,170 --> 00:29:15,610
+beta سالب beta is real number و بقدر ألاقي capital
+
+287
+00:29:15,610 --> 00:29:20,410
+N أكبر من أي real number by Archimedean property
+
+288
+00:29:20,410 --> 00:29:23,810
+اذا capital N اللي انا عايزها أكبر من ال given
+
+289
+00:29:23,810 --> 00:29:29,270
+بسالب ال given beta أكبر من سالب ال given beta اذا
+
+290
+00:29:29,270 --> 00:29:36,450
+هنا باجي بقول let beta be given it choose using
+
+291
+00:29:38,220 --> 00:29:44,220
+الـ Archimedean property capital
+
+292
+00:29:44,220 --> 00:29:50,640
+N عدد طبيعي natural number such that capital N
+
+293
+00:29:50,640 --> 00:29:56,660
+أكبر من negative beta وهذا مقدر أعمله by
+
+294
+00:29:56,660 --> 00:30:03,340
+Archimedean propertyالان تعالى نشوف اذا انا لأي
+
+295
+00:30:03,340 --> 00:30:08,660
+beta وجدت عدد طبيعي هيه بيعتمد على beta هاي
+
+296
+00:30:08,660 --> 00:30:12,420
+capital M تعتمد على beta مرتبطة فيها بالمتباينة
+
+297
+00:30:12,420 --> 00:30:19,340
+هذهالان فاضل باقي أثبت أنه لو أخدت أي small n أكبر
+
+298
+00:30:19,340 --> 00:30:25,540
+من أو ساوي capital N بدي أثبت أن هذا بيقدي أن xn
+
+299
+00:30:25,540 --> 00:30:30,820
+أصغر من beta طيب هاي n أكبر من أو ساوي capital N
+
+300
+00:30:30,820 --> 00:30:35,400
+شوف أن هذا بيقدي أن n تربية أكبر من أو ساوي small
+
+301
+00:30:35,400 --> 00:30:41,760
+n لأي عداد طبيعي طيب small n أكبر من أو ساوي
+
+302
+00:30:41,760 --> 00:30:42,520
+capital N
+
+303
+00:30:45,500 --> 00:30:51,940
+و capital N أكبر من negative beta حسب اختيارنا اذا
+
+304
+00:30:51,940 --> 00:30:56,480
+هذا بيقدي اضرب في سالب واحد اذا هذا بيقدي ان xn
+
+305
+00:30:56,480 --> 00:31:01,520
+اللي هي سالب او negative n تربيه اضرب في سالب واحد
+
+306
+00:31:01,520 --> 00:31:07,760
+هذا بيصير اصغر من beta وهذه
+
+307
+00:31:07,760 --> 00:31:12,340
+هي ال implication اللي انا عايز احققها صح؟ اذا انا
+
+308
+00:31:12,340 --> 00:31:20,580
+هيندي صارby definition حققت انه for any beta يوجد
+
+309
+00:31:20,580 --> 00:31:26,100
+capital N عدد طبيعي يعتمد على ال beta بحيث لكل N
+
+310
+00:31:26,100 --> 00:31:32,640
+أكبر من أو يساوي capital N هذا بيقدي ان Xn أصغر من
+
+311
+00:31:32,640 --> 00:31:35,060
+beta therefore by definition
+
+312
+00:31:37,910 --> 00:31:44,330
+حسب التعريف التاني بطلع عندي limit xn equals
+
+313
+00:31:44,330 --> 00:31:53,950
+negative infinity وهو المطلوب تمام هيك؟ في أي
+
+314
+00:31:53,950 --> 00:31:59,250
+سؤال؟ في أي استفسار؟
+
+315
+00:31:59,250 --> 00:32:04,730
+طيب ناخد الآن نظرية
+
+316
+00:32:19,760 --> 00:32:32,580
+خلّيني أغير الجلم theorem
+
+317
+00:32:32,580 --> 00:32:40,260
+رقمها تمانية عشرين هذه
+
+318
+00:32:40,260 --> 00:32:49,040
+تعتبر monotone convergence theorem for properly
+
+319
+00:32:55,550 --> 00:33:03,530
+divergent sequences monotone
+
+320
+00:33:03,530 --> 00:33:06,630
+convergence theorem for properly divergent
+
+321
+00:33:06,630 --> 00:33:15,410
+sequences النظرية هذه بتنص على ان a monotone .. a
+
+322
+00:33:15,410 --> 00:33:19,450
+monotone sequence
+
+323
+00:33:21,930 --> 00:33:35,070
+in R is properly is properly divergent if
+
+324
+00:33:35,070 --> 00:33:41,270
+and only if it is unbounded
+
+325
+00:33:41,270 --> 00:33:46,230
+in
+
+326
+00:33:46,230 --> 00:33:50,950
+fact في حقيقة الأمر in fact
+
+327
+00:33:54,620 --> 00:34:00,520
+إذا XN غير
+
+328
+00:34:00,520 --> 00:34:06,580
+مجموعة ومزيد
+
+329
+00:34:06,580 --> 00:34:09,740
+ومزيد
+
+330
+00:34:09,740 --> 00:34:13,280
+ثم
+
+331
+00:34:13,280 --> 00:34:18,400
+XN يتنقل إلى الانفصال
+
+332
+00:34:27,110 --> 00:34:34,510
+وإذا كان Xn غير مجموعة ومتناخصة
+
+333
+00:34:34,510 --> 00:34:38,090
+فإن
+
+334
+00:34:38,090 --> 00:34:43,330
+سيكوان Xn يتنقص إلى نقاط نقاط نقاط نقاط نقاط نقاط
+
+335
+00:34:43,330 --> 00:34:45,210
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+336
+00:34:45,210 --> 00:34:50,090
+نقاط
+
+337
+00:34:50,090 --> 00:34:50,850
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+338
+00:34:50,850 --> 00:34:50,990
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+339
+00:34:50,990 --> 00:34:51,050
+نقاط نقاط نقاط نقاط نقاط نق
+
+340
+00:35:00,740 --> 00:35:12,340
+الأول ..الأول مصادر يتبع من
+
+341
+00:35:12,340 --> 00:35:16,960
+الـ monotone convergence theorem
+
+342
+00:35:21,340 --> 00:35:24,060
+الـ first statement اللي هو بيقول a monotone
+
+343
+00:35:24,060 --> 00:35:29,100
+sequence of real numbers is properly divergent if
+
+344
+00:35:29,100 --> 00:35:32,300
+and only if it is unbounded هذا نتيجة على الـ
+
+345
+00:35:32,300 --> 00:35:35,740
+monotone convergence theorem لأن الـ monotone
+
+346
+00:35:35,740 --> 00:35:40,340
+convergence theorem اللي أخدناها قبل هيك بتقول a
+
+347
+00:35:40,340 --> 00:35:45,600
+monotone sequence is convergent if and only if it
+
+348
+00:35:45,600 --> 00:35:51,040
+is boundedفمن نفس النظرية ومن نفس النص ممكن نقول
+
+349
+00:35:51,040 --> 00:35:54,960
+الـ monotone sequence is not convergent أو
+
+350
+00:35:54,960 --> 00:36:00,580
+divergent if and only if it is unbounded بغض النظر
+
+351
+00:36:00,580 --> 00:36:04,800
+ال divergence هنا شو نوعه okay إذا ال face
+
+352
+00:36:04,800 --> 00:36:10,000
+statement العبارة الأولى هذه لحد هنانتيجة على او
+
+353
+00:36:10,000 --> 00:36:15,900
+corollary to the monotone convergence theorem الآن
+
+354
+00:36:15,900 --> 00:36:22,400
+بدنا نثبت الأجزاء واحد واثنين الآن خلّينا نثبت
+
+355
+00:36:22,400 --> 00:36:28,260
+الجزء الأول والتاني برهانه بالمثل similar to one
+
+356
+00:36:28,260 --> 00:36:32,660
+إذا هنا نثبت الجزء الأول assume
+
+357
+00:36:35,940 --> 00:36:45,260
+إن XIN is a sequence of real numbers is unbounded
+
+358
+00:36:45,260 --> 00:36:48,540
+and
+
+359
+00:36:48,540 --> 00:36:51,560
+increasing
+
+360
+00:37:02,370 --> 00:37:09,050
+بنثبت ان الـ sequence xn properly
+
+361
+00:37:09,050 --> 00:37:10,890
+divergent to infinity
+
+362
+00:37:28,250 --> 00:37:33,650
+طيب بس نستذكر هنا في هذه المناسبة خلينا نستذكر
+
+363
+00:37:33,650 --> 00:37:39,950
+تعريف ال bounded sequence definition a sequence x
+
+364
+00:37:39,950 --> 00:37:47,230
+in contained in R is bounded is
+
+365
+00:37:47,230 --> 00:37:54,550
+bounded if and only if there exists positive real
+
+366
+00:37:54,550 --> 00:38:04,160
+numberأو حتى عدد حقيقي M بحيث أنه absolute X N
+
+367
+00:38:04,160 --> 00:38:13,300
+أصغر من أو يساوي M for every N ينتمي ل N مش هيك
+
+368
+00:38:13,300 --> 00:38:20,010
+تعريف ال bounded sequence؟وطبعا دايما لأي sequence
+
+369
+00:38:20,010 --> 00:38:24,510
+دايما ال x in بالمناسبة أصغر من أو ساوى القيمة
+
+370
+00:38:24,510 --> 00:38:28,130
+المطلقة تبعته أي real number is less than or equal
+
+371
+00:38:28,130 --> 00:38:34,630
+its absolute value هذا مافيش فيها شكل الان تعالوا
+
+372
+00:38:34,630 --> 00:38:39,390
+نعمل negation لهذا مامعنى ان هنا هقول in a sense
+
+373
+00:38:39,390 --> 00:38:47,030
+ال sequence x in is unboundedأحنا فرضين أن ال
+
+374
+00:38:47,030 --> 00:38:51,770
+sequence تبعتي unbounded و increasing بما أن ال
+
+375
+00:38:51,770 --> 00:38:57,530
+sequence x in is unbounded
+
+376
+00:38:57,530 --> 00:39:03,150
+فلكل
+
+377
+00:39:03,150 --> 00:39:10,730
+in then
+
+378
+00:39:10,730 --> 00:39:24,330
+for every alphafor every alpha عدد حقيقي يوجد
+
+379
+00:39:24,330 --> 00:39:29,210
+by
+
+380
+00:39:29,210 --> 00:39:40,080
+Archimedean propertyلأ يوجد N يعتمد على Alpha عدب
+
+381
+00:39:40,080 --> 00:39:48,060
+طبيعي بحيث ان ال X رقم capital N Alpha هذا بيطلع
+
+382
+00:39:48,060 --> 00:39:55,700
+أكبر من Alpha يعني
+
+383
+00:39:55,700 --> 00:40:00,360
+لو كان هذا ال M أسمنها Alpha
+
+384
+00:40:03,420 --> 00:40:07,900
+معناه أن الـ sequence هذه تكون bounded فما معناه
+
+385
+00:40:07,900 --> 00:40:10,860
+أن الـ sequence هذه تكون unbounded معناه أننا بدنا
+
+386
+00:40:10,860 --> 00:40:15,480
+ننفي الشرط هذا عشان أنفي الشرط هذا هذا معناه أن
+
+387
+00:40:15,480 --> 00:40:20,920
+بدل يوجد alpha موجبة لكل alpha سواء موجبة أو سالبة
+
+388
+00:40:20,920 --> 00:40:25,980
+لكل alpha for any alpha يوجد
+
+389
+00:40:28,130 --> 00:40:34,230
+بدل لكل n عدد طبيعي يوجد عدد طبيعي capital N يوجد
+
+390
+00:40:34,230 --> 00:40:41,740
+واحد capital N عدد طبيعي مافي لكل يوجدبحيث أن هذا
+
+391
+00:40:41,740 --> 00:40:48,720
+الـ xn بدل
+
+392
+00:40:48,720 --> 00:40:52,880
+ما هي أصغر من أو ساوي Alpha نفي xn أصغر من أو ساوي
+
+393
+00:40:52,880 --> 00:41:00,880
+Alpha هو أكبر من Alpha تمام؟ إذن هذا هو نفي الشرط
+
+394
+00:41:00,880 --> 00:41:04,940
+هذا أو نفي boundednessإذا الـ sequence unbounded
+
+395
+00:41:04,940 --> 00:41:09,400
+معناه لأي عدد حقيقي Alpha فهي عدد طبيعي يعتمد على
+
+396
+00:41:09,400 --> 00:41:14,680
+Alpha بحيث ان ال X المؤشر تبعه capital N بيطلع
+
+397
+00:41:14,680 --> 00:41:19,340
+أكبر من Alpha طيب الآن بما أن ال sequence
+
+398
+00:41:19,340 --> 00:41:27,540
+increasing as ال sequence XN is increasing احنا
+
+399
+00:41:27,540 --> 00:41:35,200
+فرضين انها متزايدةWe get نحصل على لو كان n أكبر من
+
+400
+00:41:35,200 --> 00:41:41,620
+أو ساوي n of alpha فهذا بالتأكيد هيقدّي أن x
+
+401
+00:41:41,620 --> 00:41:47,060
+المؤشر تبعها small n أكبر من أو ساوي x المؤشر
+
+402
+00:41:47,060 --> 00:41:52,840
+تبعها n of alpha وهذا من هنا أكبر من alpha
+
+403
+00:41:56,150 --> 00:42:02,150
+أذن خلّيني ألخّص شو عملنا أحنا أثبتنا الآن أن for
+
+404
+00:42:02,150 --> 00:42:06,570
+every alpha real number there exists a natural
+
+405
+00:42:06,570 --> 00:42:14,310
+number depends on alpha هذا هو بحيث أنه لكل N أكبر
+
+406
+00:42:14,310 --> 00:42:20,530
+من أو ساوي capital N طلع ندي xn أكبر من alphaهذا
+
+407
+00:42:20,530 --> 00:42:25,310
+اذا حسب التعريف الأولاني حسب تعريف one by
+
+408
+00:42:25,310 --> 00:42:32,350
+definition one طبعا هذا معناه ان limit x in بساوي
+
+409
+00:42:32,350 --> 00:42:39,170
+infinity وهو المطلوب okay برهان الجزء التاني the
+
+410
+00:42:39,170 --> 00:42:46,290
+proof of part اتنين is similar
+
+411
+00:42:49,840 --> 00:42:55,180
+two one فحاسبكم انتوا تكتبوا البرهان تبعه okay
+
+412
+00:42:55,180 --> 00:43:01,640
+تمام و هيك بنكون كملنا النظرية اذا هنوقف هنا و
+
+413
+00:43:01,640 --> 00:43:10,420
+بنكمل ان شاء الله الموضوع هذا المرة القادمة فشوفكم
+
+414
+00:43:10,420 --> 00:43:11,720
+ان شاء الله المرة الجاية على فيه
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..0f6e92c840c467f447e6f582f407469e240d403b
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 3229, "start": 20.65, "end": 32.29, "text": "السلام عليكم اليوم ان شاء الله هنكمل ما ابتدأناه سابقا بخصوص موضوع الكوشي sequences", "tokens": [6027, 3794, 37440, 25894, 24793, 45595, 20498, 16472, 13412, 16606, 21984, 8032, 1863, 24793, 1211, 19446, 48127, 2655, 3215, 10721, 8315, 3224, 8608, 16758, 4587, 995, 4724, 9778, 9381, 2407, 9381, 3714, 2407, 11242, 45367, 33251, 2407, 8592, 1829, 22978], "avg_logprob": -0.12595274099489537, "compression_ratio": 1.2972972972972974, "no_speech_prob": 0.0, "words": [{"start": 20.65, "end": 21.29, "word": "السلام", "probability": 0.97412109375}, {"start": 21.29, "end": 22.01, "word": " عليكم", "probability": 0.94140625}, {"start": 22.01, "end": 23.35, "word": " اليوم", "probability": 0.770751953125}, {"start": 23.35, "end": 23.49, "word": " ان", "probability": 0.70947265625}, {"start": 23.49, "end": 23.71, "word": " شاء", "probability": 0.98193359375}, {"start": 23.71, "end": 23.81, "word": " الله", "probability": 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"probability": 0.95947265625}, {"start": 108.12, "end": 108.62, "word": " remark", "probability": 0.98095703125}, {"start": 108.62, "end": 108.96, "word": " و", "probability": 0.491943359375}, {"start": 108.96, "end": 109.24, "word": " it", "probability": 0.95654296875}, {"start": 109.24, "end": 109.46, "word": " is", "probability": 0.935546875}, {"start": 109.46, "end": 109.88, "word": " easy", "probability": 0.93603515625}, {"start": 109.88, "end": 110.18, "word": " زي", "probability": 0.9716796875}, {"start": 110.18, "end": 110.26, "word": " ما", "probability": 0.97119140625}, {"start": 110.26, "end": 110.42, "word": " هو", "probability": 0.97509765625}, {"start": 110.42, "end": 110.86, "word": " مذكور", "probability": 0.9017333984375}], "temperature": 1.0}, {"id": 5, "seek": 13849, "start": 111.85, "end": 138.49, "text": "في الان definition a sequence of real numbers بنسميها contractive إذا وجد ثابت c عدد بين صفر و واحد عدد موجب بحيث أنه المسافة بين xn plus two و xn plus one أصغر 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{"start": 621.02, "end": 621.4, "word": " العامل", "probability": 0.9404296875}, {"start": 621.4, "end": 621.94, "word": " المشترك", "probability": 0.9681396484375}, {"start": 621.94, "end": 622.72, "word": " absolute", "probability": 0.86328125}, {"start": 622.72, "end": 623.24, "word": " x", "probability": 0.916015625}, {"start": 623.24, "end": 623.46, "word": " two", "probability": 0.449951171875}, {"start": 623.46, "end": 623.8, "word": " minus", "probability": 0.984375}, {"start": 623.8, "end": 624.12, "word": " x", "probability": 0.99169921875}, {"start": 624.12, "end": 624.42, "word": " one", "probability": 0.95166015625}, {"start": 624.42, "end": 628.54, "word": " المجموع", "probability": 0.901611328125}, {"start": 628.54, "end": 628.84, "word": " هذا", "probability": 0.87451171875}, {"start": 628.84, "end": 629.3, "word": " هاخد", "probability": 0.8387044270833334}, {"start": 629.3, "end": 629.62, "word": " عامل", "probability": 0.9931640625}, {"start": 629.62, "end": 630.16, 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{"start": 641.24, "end": 643.36, "word": " وهنا", "probability": 0.7159423828125}, {"start": 643.36, "end": 644.1, "word": " هاخد", "probability": 0.7347005208333334}, {"start": 644.1, "end": 645.4, "word": " لو", "probability": 0.615234375}, {"start": 645.4, "end": 646.02, "word": " جسمت", "probability": 0.8307291666666666}, {"start": 646.02, "end": 646.26, "word": " had", "probability": 0.15673828125}, {"start": 646.26, "end": 646.46, "word": " على", "probability": 0.669921875}, {"start": 646.46, "end": 646.78, "word": " c", "probability": 0.8671875}, {"start": 646.78, "end": 646.98, "word": " to", "probability": 0.93310546875}, {"start": 646.98, "end": 647.16, "word": " n", "probability": 0.75048828125}, {"start": 647.16, "end": 647.52, "word": " minus", "probability": 0.9677734375}, {"start": 647.52, "end": 647.88, "word": " 1", "probability": 0.89453125}, {"start": 647.88, "end": 648.38, "word": " هيطلع", "probability": 0.83564453125}, {"start": 648.38, "end": 648.84, "word": " c", "probability": 0.73681640625}, {"start": 648.84, "end": 649.82, "word": " to", "probability": 0.9501953125}, {"start": 649.82, "end": 650.16, "word": " m", "probability": 0.93505859375}, {"start": 650.16, "end": 650.88, "word": " minus", "probability": 0.99365234375}, {"start": 650.88, "end": 651.38, "word": " n", "probability": 0.91064453125}, {"start": 651.38, "end": 652.42, "word": " minus", "probability": 0.98974609375}, {"start": 652.42, "end": 655.8, "word": " 2", "probability": 0.83447265625}, {"start": 655.8, "end": 658.74, "word": " وهكذا", "probability": 0.8740234375}, {"start": 658.74, "end": 659.4, "word": " إلى", "probability": 0.85205078125}, {"start": 659.4, "end": 660.74, "word": " أخر", "probability": 0.7003173828125}, {"start": 660.74, "end": 661.12, "word": " حد", "probability": 0.992919921875}, {"start": 661.12, "end": 661.86, "word": " هيكون", "probability": 0.9072265625}, {"start": 661.86, "end": 662.22, "word": " 1", "probability": 0.67822265625}], "temperature": 1.0}, {"id": 26, "seek": 68821, "start": 664.15, "end": 688.21, "text": "و طبعا كل هذا مضروب في absolute x2 minus x1 y ساوي c to n negative one الان هذه عبارة عن geometric progression متوالية هندسية الحد الأول فيها واحد والأساس تبعها c", "tokens": [2407, 23032, 3555, 3615, 995, 28242, 23758, 3714, 11242, 32887, 3555, 8978, 8236, 2031, 17, 3175, 2031, 16, 288, 8608, 995, 45865, 269, 281, 297, 3671, 472, 2423, 7649, 29538, 6225, 3555, 9640, 3660, 18871, 33246, 18733, 44650, 2407, 6027, 10632, 8032, 41260, 3794, 10632, 21542, 3215, 16247, 12610, 8978, 11296, 36764, 24401, 16070, 10721, 3794, 32277, 6055, 3555, 3615, 11296, 269], "avg_logprob": -0.22891864511701795, "compression_ratio": 1.3277777777777777, "no_speech_prob": 0.0, "words": [{"start": 664.15, "end": 664.33, "word": "و", "probability": 0.591796875}, {"start": 664.33, "end": 664.65, "word": " طبعا", "probability": 0.83074951171875}, {"start": 664.65, "end": 664.87, "word": " كل", "probability": 0.80126953125}, {"start": 664.87, "end": 665.09, "word": " هذا", "probability": 0.931640625}, {"start": 665.09, "end": 665.73, "word": " مضروب", "probability": 0.96826171875}, {"start": 665.73, "end": 666.07, "word": " في", "probability": 0.94140625}, {"start": 666.07, "end": 666.93, "word": " absolute", "probability": 0.759765625}, {"start": 666.93, "end": 667.91, "word": " x2", "probability": 0.774658203125}, {"start": 667.91, "end": 668.99, "word": " minus", "probability": 0.2147216796875}, {"start": 668.99, "end": 669.81, "word": " x1", "probability": 0.9912109375}, {"start": 669.81, "end": 672.45, "word": " y", "probability": 0.178466796875}, {"start": 672.45, "end": 673.11, "word": " ساوي", "probability": 0.6130574544270834}, {"start": 673.11, "end": 675.71, "word": " c", "probability": 0.291259765625}, {"start": 675.71, "end": 676.73, "word": " to", "probability": 0.477783203125}, {"start": 676.73, "end": 677.01, "word": " n", "probability": 0.84716796875}, {"start": 677.01, "end": 677.41, "word": " negative", "probability": 0.6953125}, {"start": 677.41, "end": 677.89, "word": " one", "probability": 0.77734375}, {"start": 677.89, "end": 678.77, "word": " الان", "probability": 0.6070556640625}, {"start": 678.77, "end": 679.59, "word": " هذه", "probability": 0.51416015625}, {"start": 679.59, "end": 679.95, "word": " عبارة", "probability": 0.980224609375}, {"start": 679.95, "end": 680.17, "word": " عن", "probability": 0.986328125}, {"start": 680.17, "end": 680.63, "word": " geometric", "probability": 0.66455078125}, {"start": 680.63, "end": 681.47, "word": " progression", "probability": 0.94482421875}, {"start": 681.47, "end": 682.45, "word": " متوالية", "probability": 0.976318359375}, {"start": 682.45, "end": 683.89, "word": " هندسية", "probability": 0.9849853515625}, {"start": 683.89, "end": 684.85, "word": " الحد", "probability": 0.965087890625}, {"start": 684.85, "end": 685.23, "word": " الأول", "probability": 0.855224609375}, {"start": 685.23, "end": 685.61, "word": " فيها", "probability": 0.984619140625}, {"start": 685.61, "end": 686.15, "word": " واحد", "probability": 0.9365234375}, {"start": 686.15, "end": 687.33, "word": " والأساس", "probability": 0.793212890625}, {"start": 687.33, "end": 687.87, "word": " تبعها", "probability": 0.861083984375}, {"start": 687.87, "end": 688.21, "word": " c", "probability": 0.791015625}], "temperature": 1.0}, {"id": 27, "seek": 71017, "start": 689.41, "end": 710.17, "text": "فمجموعة المتوالي الهندسية أو الـ geometric progression مجموعة بساوي الحد الأول واحد سالب الحد الأخير مضروب في الأساس اللي هو C فبطلع C to M negative N على واحد minus الأساس", "tokens": [5172, 2304, 7435, 2304, 2407, 27884, 9673, 2655, 2407, 6027, 1829, 2423, 3224, 41260, 3794, 10632, 34051, 2423, 39184, 33246, 18733, 3714, 7435, 2304, 2407, 27884, 4724, 3794, 995, 45865, 21542, 3215, 16247, 12610, 36764, 24401, 8608, 6027, 3555, 21542, 3215, 16247, 9778, 13546, 3714, 11242, 32887, 3555, 8978, 16247, 3794, 32277, 13672, 1829, 31439, 383, 6156, 3555, 9566, 1211, 3615, 383, 281, 376, 3671, 426, 15844, 36764, 24401, 3175, 16247, 3794, 32277], "avg_logprob": -0.14558699646511594, "compression_ratio": 1.5988372093023255, "no_speech_prob": 0.0, "words": [{"start": 689.41, "end": 690.23, "word": "فمجموعة", "probability": 0.8053385416666666}, {"start": 690.23, "end": 690.95, "word": " المتوالي", "probability": 0.9212890625}, {"start": 690.95, "end": 691.55, "word": " الهندسية", "probability": 0.95078125}, {"start": 691.55, "end": 691.77, "word": " أو", "probability": 0.46240234375}, {"start": 691.77, "end": 691.91, "word": " الـ", "probability": 0.805908203125}, {"start": 691.91, "end": 692.35, "word": " geometric", "probability": 0.498046875}, {"start": 692.35, "end": 693.93, "word": " progression", "probability": 0.9150390625}, {"start": 693.93, "end": 694.59, "word": " مجموعة", "probability": 0.9220703125}, {"start": 694.59, "end": 695.33, "word": " بساوي", "probability": 0.864013671875}, {"start": 695.33, "end": 696.57, "word": " الحد", "probability": 0.93896484375}, {"start": 696.57, "end": 697.05, "word": " الأول", "probability": 0.961669921875}, {"start": 697.05, "end": 697.65, "word": " واحد", "probability": 0.654296875}, {"start": 697.65, "end": 698.31, "word": " سالب", "probability": 0.9500325520833334}, {"start": 698.31, "end": 699.55, "word": " الحد", "probability": 0.988037109375}, {"start": 699.55, "end": 700.21, "word": " الأخير", "probability": 0.8699544270833334}, {"start": 700.21, "end": 701.21, "word": " مضروب", "probability": 0.95361328125}, {"start": 701.21, "end": 701.33, "word": " في", "probability": 0.98291015625}, {"start": 701.33, "end": 701.97, "word": " الأساس", "probability": 0.9685872395833334}, {"start": 701.97, "end": 702.95, "word": " اللي", "probability": 0.913330078125}, {"start": 702.95, "end": 703.27, "word": " هو", "probability": 0.99169921875}, {"start": 703.27, "end": 703.63, "word": " C", "probability": 0.58984375}, {"start": 703.63, "end": 704.41, "word": " فبطلع", "probability": 0.85703125}, {"start": 704.41, "end": 704.85, "word": " C", "probability": 0.6298828125}, {"start": 704.85, "end": 705.29, "word": " to", "probability": 0.6142578125}, {"start": 705.29, "end": 705.61, "word": " M", "probability": 0.951171875}, {"start": 705.61, "end": 706.67, "word": " negative", "probability": 0.7802734375}, {"start": 706.67, "end": 707.17, "word": " N", "probability": 0.8955078125}, {"start": 707.17, "end": 707.53, "word": " على", "probability": 0.8193359375}, {"start": 707.53, "end": 709.01, "word": " واحد", "probability": 0.950927734375}, {"start": 709.01, "end": 709.47, "word": " minus", "probability": 0.9267578125}, {"start": 709.47, "end": 710.17, "word": " الأساس", "probability": 0.9698893229166666}], "temperature": 1.0}, {"id": 28, "seek": 73749, "start": 712.07, "end": 737.49, "text": "إذن هذا مجموع الـ geometric progression كل هذا مضروب في absolute X2 negative X1 طيب أنا عندي الـC أنا عندي الـC الـC عدد بين 0 و 1", "tokens": [28814, 8848, 1863, 23758, 3714, 7435, 2304, 45367, 2423, 39184, 33246, 18733, 28242, 23758, 3714, 11242, 32887, 3555, 8978, 8236, 1783, 17, 3671, 1783, 16, 23032, 1829, 3555, 41850, 18871, 16254, 2423, 39184, 34, 41850, 18871, 16254, 2423, 39184, 34, 2423, 39184, 34, 6225, 3215, 3215, 49374, 1958, 4032, 502], "avg_logprob": -0.2594975572006375, "compression_ratio": 1.3216783216783217, "no_speech_prob": 0.0, "words": [{"start": 712.07, "end": 712.41, "word": "إذن", "probability": 0.5281575520833334}, {"start": 712.41, "end": 712.73, "word": " هذا", "probability": 0.80859375}, {"start": 712.73, "end": 713.41, "word": " مجموع", "probability": 0.900634765625}, {"start": 713.41, "end": 713.75, "word": " الـ", "probability": 0.821533203125}, {"start": 713.75, "end": 714.07, "word": " geometric", "probability": 0.386962890625}, {"start": 714.07, "end": 714.87, "word": " progression", "probability": 0.93115234375}, {"start": 714.87, "end": 715.95, "word": " كل", "probability": 0.5791015625}, {"start": 715.95, "end": 716.13, "word": " هذا", "probability": 0.96923828125}, {"start": 716.13, "end": 716.77, "word": " مضروب", "probability": 0.978271484375}, {"start": 716.77, "end": 717.01, "word": " في", "probability": 0.96435546875}, {"start": 717.01, "end": 717.51, "word": " absolute", "probability": 0.7412109375}, {"start": 717.51, "end": 718.47, "word": " X2", "probability": 0.6168212890625}, {"start": 718.47, "end": 718.91, "word": " negative", "probability": 0.187744140625}, {"start": 718.91, "end": 720.21, "word": " X1", "probability": 0.981689453125}, {"start": 720.21, "end": 724.61, "word": " طيب", "probability": 0.87548828125}, {"start": 724.61, "end": 724.81, "word": " أنا", "probability": 0.611328125}, {"start": 724.81, "end": 725.19, "word": " عندي", "probability": 0.88330078125}, {"start": 725.19, "end": 725.79, "word": " الـC", "probability": 0.599853515625}, {"start": 725.79, "end": 730.33, "word": " أنا", "probability": 0.6083984375}, {"start": 730.33, "end": 730.81, "word": " عندي", "probability": 0.96484375}, {"start": 730.81, "end": 732.21, "word": " الـC", "probability": 0.78515625}, {"start": 732.21, "end": 735.63, "word": " الـC", "probability": 0.9182942708333334}, {"start": 735.63, "end": 736.21, "word": " عدد", "probability": 0.9762369791666666}, {"start": 736.21, "end": 736.51, "word": " بين", "probability": 0.75146484375}, {"start": 736.51, "end": 736.89, "word": " 0", "probability": 0.53173828125}, {"start": 736.89, "end": 737.15, "word": " و", "probability": 0.91845703125}, {"start": 737.15, "end": 737.49, "word": " 1", "probability": 0.720703125}], "temperature": 1.0}, {"id": 29, "seek": 75844, "start": 738.96, "end": 758.44, "text": "وبالتالي c to m minus n بيبقى العدد بين سفر واحد وبالتالي اذا واحد سالب c اكيد هيطلع اكبر من سفر اصغر من واحد", "tokens": [37746, 6027, 2655, 6027, 1829, 269, 281, 275, 3175, 297, 4724, 1829, 3555, 4587, 7578, 18863, 3215, 3215, 49374, 8608, 5172, 2288, 36764, 24401, 46599, 6027, 2655, 6027, 1829, 1975, 15730, 36764, 24401, 8608, 6027, 3555, 269, 1975, 4117, 25708, 39896, 9566, 1211, 3615, 1975, 4117, 26890, 9154, 8608, 5172, 2288, 1975, 9381, 17082, 2288, 9154, 36764, 24401], "avg_logprob": -0.2004766994613712, "compression_ratio": 1.5206611570247934, "no_speech_prob": 0.0, "words": [{"start": 738.96, "end": 739.98, "word": "وبالتالي", "probability": 0.863671875}, {"start": 739.98, "end": 740.5, "word": " c", "probability": 0.28173828125}, {"start": 740.5, "end": 740.9, "word": " to", "probability": 0.260009765625}, {"start": 740.9, "end": 741.22, "word": " m", "probability": 0.76904296875}, {"start": 741.22, "end": 741.8, "word": " minus", "probability": 0.6962890625}, {"start": 741.8, "end": 742.18, "word": " n", "probability": 0.962890625}, {"start": 742.18, "end": 742.54, "word": " بيبقى", "probability": 0.721728515625}, {"start": 742.54, "end": 742.98, "word": " العدد", "probability": 0.90625}, {"start": 742.98, "end": 743.26, "word": " بين", "probability": 0.86279296875}, {"start": 743.26, "end": 743.76, "word": " سفر", "probability": 0.75732421875}, {"start": 743.76, "end": 744.98, "word": " واحد", "probability": 0.869384765625}, {"start": 744.98, "end": 747.12, "word": " وبالتالي", "probability": 0.87529296875}, {"start": 747.12, "end": 747.54, "word": " اذا", "probability": 0.703369140625}, {"start": 747.54, "end": 748.56, "word": " واحد", "probability": 0.928466796875}, {"start": 748.56, "end": 749.14, "word": " سالب", "probability": 0.78173828125}, {"start": 749.14, "end": 749.5, "word": " c", "probability": 0.7939453125}, {"start": 749.5, "end": 751.42, "word": " اكيد", "probability": 0.896484375}, {"start": 751.42, "end": 752.02, "word": " هيطلع", "probability": 0.9390869140625}, {"start": 752.02, "end": 752.62, "word": " اكبر", "probability": 0.95361328125}, {"start": 752.62, "end": 752.86, "word": " من", "probability": 0.99609375}, {"start": 752.86, "end": 754.5, "word": " سفر", "probability": 0.9827473958333334}, {"start": 754.5, "end": 757.88, "word": " اصغر", "probability": 0.876220703125}, {"start": 757.88, "end": 758.06, "word": " من", "probability": 0.9970703125}, {"start": 758.06, "end": 758.44, "word": " واحد", "probability": 0.989013671875}], "temperature": 1.0}, {"id": 30, "seek": 79051, "start": 761.67, "end": 790.51, "text": "أذا هاي أنا عندي هذا هستبدله بأصغر من هاي c to n negative one الآن هذا الكسر ال bus تبعه واحد سالب c to n minus n أصغر من واحد فهذا الكسر أصغر من واحد على واحد سالب c أصغر من واحد على واحد سالب c في absolute x to negative x one", "tokens": [10721, 15730, 8032, 47302, 41850, 18871, 16254, 23758, 8032, 14851, 44510, 1211, 3224, 4724, 10721, 9381, 17082, 2288, 9154, 8032, 47302, 269, 281, 297, 3671, 472, 6024, 48506, 23758, 33251, 3794, 2288, 2423, 1255, 6055, 3555, 3615, 3224, 36764, 24401, 8608, 6027, 3555, 269, 281, 297, 3175, 297, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 6156, 3224, 15730, 33251, 3794, 2288, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 15844, 36764, 24401, 8608, 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859.16, "end": 859.74, "word": " infinity", "probability": 0.6533203125}, {"start": 859.74, "end": 863.68, "word": " بتساوي", "probability": 0.9072265625}, {"start": 863.68, "end": 864.26, "word": " سفر", "probability": 0.8225911458333334}, {"start": 864.26, "end": 865.9, "word": " وطبعا", "probability": 0.8822265625}, {"start": 865.9, "end": 866.06, "word": " ال", "probability": 0.95751953125}, {"start": 866.06, "end": 866.32, "word": " M", "probability": 0.9873046875}, {"start": 866.32, "end": 866.56, "word": " انا", "probability": 0.825927734375}, {"start": 866.56, "end": 867.06, "word": " ماخدها", "probability": 0.9156494140625}, {"start": 867.06, "end": 867.32, "word": " هنا", "probability": 0.978515625}, {"start": 867.32, "end": 867.74, "word": " ال", "probability": 0.673828125}, {"start": 867.74, "end": 868.0, "word": " M", "probability": 0.978515625}, {"start": 868.0, "end": 869.06, "word": " ماخدها", "probability": 0.880126953125}, {"start": 869.06, "end": 871.72, "word": " M", "probability": 0.295654296875}, {"start": 871.72, "end": 872.28, "word": " اكبر", "probability": 0.7869466145833334}, {"start": 872.28, "end": 872.54, "word": " من", "probability": 0.98828125}, {"start": 872.54, "end": 872.8, "word": " N", "probability": 0.87109375}, {"start": 872.8, "end": 873.34, "word": " فلما", "probability": 0.92333984375}, {"start": 873.34, "end": 873.5, "word": " ال", "probability": 0.9794921875}, {"start": 873.5, "end": 873.64, "word": " M", "probability": 0.9365234375}, {"start": 873.64, "end": 874.02, "word": " تقول", "probability": 0.9697265625}, {"start": 874.02, "end": 874.16, "word": " ل", "probability": 0.96337890625}, {"start": 874.16, "end": 874.62, "word": " infinity", "probability": 0.853515625}, {"start": 874.62, "end": 874.92, "word": " ال", "probability": 0.81640625}, {"start": 874.92, "end": 875.1, "word": " M", "probability": 0.98974609375}, {"start": 875.1, "end": 875.6, "word": " ايضا", "probability": 0.918701171875}], "temperature": 1.0}, {"id": 34, "seek": 90120, "start": 876.32, "end": 901.2, "text": "تقول لإنفينيتي إذا هنا ممكن أحط and M تقول لإنفينيتي إذا هنا أثبتنا إن ال limit ل absolute XM minus XN as N and M both tends to infinity بساوي سفر وبالتالي إذا by above remark حسب ال remark اللي فوق ال sequence XM is Cauchy", "tokens": [2655, 4587, 12610, 5296, 28814, 1863, 5172, 9957, 1829, 31371, 11933, 15730, 34105, 3714, 43020, 5551, 5016, 9566, 293, 376, 6055, 4587, 12610, 5296, 28814, 1863, 5172, 9957, 1829, 31371, 11933, 15730, 34105, 5551, 12984, 3555, 2655, 8315, 36145, 2423, 4948, 5296, 8236, 1783, 44, 3175, 1783, 45, 382, 426, 293, 376, 1293, 12258, 281, 13202, 4724, 3794, 995, 45865, 8608, 5172, 2288, 46599, 6027, 2655, 6027, 1829, 11933, 15730, 538, 3673, 7942, 11331, 35457, 2423, 7942, 13672, 1829, 6156, 30543, 2423, 8310, 1783, 44, 307, 7544, 625, 88], "avg_logprob": -0.22256944295432832, "compression_ratio": 1.5294117647058822, "no_speech_prob": 0.0, "words": [{"start": 876.32, "end": 876.86, "word": "تقول", "probability": 0.3841552734375}, {"start": 876.86, "end": 877.54, "word": " لإنفينيتي", "probability": 0.7158900669642857}, {"start": 877.54, "end": 878.74, "word": " إذا", "probability": 0.40167236328125}, {"start": 878.74, "end": 878.94, "word": " هنا", "probability": 0.75244140625}, {"start": 878.94, "end": 879.26, "word": " ممكن", "probability": 0.962158203125}, {"start": 879.26, "end": 879.68, "word": " أحط", "probability": 0.86767578125}, {"start": 879.68, "end": 880.24, "word": " and", "probability": 0.85302734375}, {"start": 880.24, "end": 880.54, "word": " M", "probability": 0.65771484375}, {"start": 880.54, "end": 881.02, "word": " تقول", "probability": 0.8914388020833334}, {"start": 881.02, "end": 882.0, "word": " لإنفينيتي", "probability": 0.9834681919642857}, {"start": 882.0, "end": 882.76, "word": " إذا", "probability": 0.749267578125}, {"start": 882.76, "end": 882.98, "word": " هنا", "probability": 0.90576171875}, {"start": 882.98, "end": 883.46, "word": " أثبتنا", "probability": 0.9712890625}, {"start": 883.46, "end": 883.62, "word": " إن", "probability": 0.442138671875}, {"start": 883.62, "end": 883.78, "word": " ال", "probability": 0.66259765625}, {"start": 883.78, "end": 884.0, "word": " limit", "probability": 0.73291015625}, {"start": 884.0, "end": 884.16, "word": " ل", "probability": 0.9638671875}, {"start": 884.16, "end": 884.48, "word": " absolute", "probability": 0.388916015625}, {"start": 884.48, "end": 885.14, "word": " XM", "probability": 0.703857421875}, {"start": 885.14, "end": 885.5, "word": " minus", "probability": 0.46435546875}, {"start": 885.5, "end": 886.22, "word": " XN", "probability": 0.956787109375}, {"start": 886.22, "end": 887.12, "word": " as", "probability": 0.81787109375}, {"start": 887.12, "end": 887.6, "word": " N", "probability": 0.83837890625}, {"start": 887.6, "end": 888.04, "word": " and", "probability": 0.921875}, {"start": 888.04, "end": 888.4, "word": " M", "probability": 0.99658203125}, {"start": 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"probability": 0.92333984375}, {"start": 896.3, "end": 896.58, "word": " remark", "probability": 0.916015625}, {"start": 896.58, "end": 896.8, "word": " اللي", "probability": 0.953125}, {"start": 896.8, "end": 897.2, "word": " فوق", "probability": 0.975341796875}, {"start": 897.2, "end": 897.88, "word": " ال", "probability": 0.81787109375}, {"start": 897.88, "end": 898.36, "word": " sequence", "probability": 0.974609375}, {"start": 898.36, "end": 899.28, "word": " XM", "probability": 0.788818359375}, {"start": 899.28, "end": 900.62, "word": " is", "probability": 0.94677734375}, {"start": 900.62, "end": 901.2, "word": " Cauchy", "probability": 0.8741861979166666}], "temperature": 1.0}, {"id": 35, "seek": 92779, "start": 904.21, "end": 927.79, "text": "و هذا بيكمل برهان النظرية تمام؟ واضح؟ في أي استفسار؟ في أي سؤال على البرهان؟ في أي قطة مش واضحة؟ is there any question؟ okay then this ends", "tokens": [2407, 23758, 4724, 1829, 24793, 1211, 4724, 2288, 3224, 7649, 28239, 19913, 2288, 10632, 46811, 10943, 22807, 4032, 46958, 5016, 22807, 8978, 36632, 44713, 36178, 9640, 22807, 8978, 36632, 8608, 33604, 6027, 15844, 2423, 26890, 3224, 7649, 22807, 8978, 36632, 12174, 9566, 3660, 37893, 4032, 46958, 5016, 3660, 22807, 307, 456, 604, 1168, 22807, 1392, 550, 341, 5314], "avg_logprob": -0.15240995813224276, "compression_ratio": 1.4466666666666668, "no_speech_prob": 0.0, "words": [{"start": 904.21, "end": 904.57, "word": "و", "probability": 0.9716796875}, {"start": 904.57, "end": 905.11, "word": " هذا", "probability": 0.7333984375}, {"start": 905.11, "end": 905.73, "word": " بيكمل", "probability": 0.8359375}, {"start": 905.73, "end": 906.35, "word": " برهان", "probability": 0.9322509765625}, {"start": 906.35, "end": 907.13, "word": " النظرية", "probability": 0.950927734375}, {"start": 907.13, "end": 908.67, "word": " تمام؟", "probability": 0.6263020833333334}, {"start": 908.67, "end": 912.09, "word": " واضح؟", "probability": 0.963134765625}, {"start": 912.09, "end": 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0.97548828125}, {"start": 918.83, "end": 919.09, "word": " is", "probability": 0.73095703125}, {"start": 919.09, "end": 919.33, "word": " there", "probability": 0.9384765625}, {"start": 919.33, "end": 919.53, "word": " any", "probability": 0.90966796875}, {"start": 919.53, "end": 923.57, "word": " question؟", "probability": 0.815673828125}, {"start": 923.57, "end": 924.27, "word": " okay", "probability": 0.84716796875}, {"start": 924.27, "end": 924.83, "word": " then", "probability": 0.81982421875}, {"start": 924.83, "end": 927.05, "word": " this", "probability": 0.814453125}, {"start": 927.05, "end": 927.79, "word": " ends", "probability": 0.86279296875}], "temperature": 1.0}, {"id": 36, "seek": 96045, "start": 930.87, "end": 960.45, "text": "section تلاتة خمسة and we are going to start a new section هنبدأ section جديد وهذا ال section بتحدث عن موضوع properly divergent sequences section three point six", "tokens": [11963, 6055, 1211, 9307, 3660, 16490, 2304, 3794, 3660, 293, 321, 366, 516, 281, 722, 257, 777, 3541, 8032, 1863, 44510, 10721, 3541, 10874, 16254, 3215, 37037, 15730, 2423, 3541, 39894, 24401, 12984, 18871, 3714, 2407, 11242, 45367, 6108, 18558, 6930, 22978, 3541, 1045, 935, 2309], "avg_logprob": -0.2544880268421579, "compression_ratio": 1.2960526315789473, "no_speech_prob": 0.0, "words": [{"start": 930.87, "end": 931.91, "word": "section", "probability": 0.1112060546875}, {"start": 931.91, "end": 932.43, "word": " تلاتة", "probability": 0.8427734375}, {"start": 932.43, "end": 933.05, "word": " خمسة", "probability": 0.9151611328125}, {"start": 933.05, "end": 935.13, "word": " and", "probability": 0.268798828125}, {"start": 935.13, "end": 935.35, "word": " we", "probability": 0.9208984375}, {"start": 935.35, "end": 935.53, "word": " are", "probability": 0.91650390625}, {"start": 935.53, "end": 935.95, "word": " going", "probability": 0.95849609375}, {"start": 935.95, "end": 936.41, "word": " to", "probability": 0.9716796875}, {"start": 936.41, "end": 937.05, "word": " start", "probability": 0.96728515625}, {"start": 937.05, "end": 937.21, "word": " a", "probability": 0.84228515625}, {"start": 937.21, "end": 937.39, "word": " new", "probability": 0.9228515625}, {"start": 937.39, "end": 937.85, "word": " section", "probability": 0.86474609375}, {"start": 937.85, "end": 938.99, "word": " هنبدأ", "probability": 0.8724365234375}, {"start": 938.99, "end": 939.35, "word": " section", "probability": 0.63427734375}, {"start": 939.35, "end": 941.13, "word": " جديد", "probability": 0.9674479166666666}, {"start": 941.13, "end": 943.25, "word": " وهذا", "probability": 0.767333984375}, {"start": 943.25, "end": 943.39, "word": " ال", "probability": 0.90087890625}, {"start": 943.39, "end": 943.67, "word": " section", "probability": 0.63916015625}, {"start": 943.67, "end": 944.49, "word": " بتحدث", "probability": 0.9109700520833334}, {"start": 944.49, "end": 945.05, "word": " عن", "probability": 0.990234375}, {"start": 945.05, "end": 946.75, "word": " موضوع", "probability": 0.9620361328125}, {"start": 946.75, "end": 947.81, "word": " properly", "probability": 0.302490234375}, {"start": 947.81, "end": 949.83, "word": " divergent", "probability": 0.6123046875}, {"start": 949.83, "end": 950.69, "word": " sequences", "probability": 0.88916015625}, {"start": 950.69, "end": 955.87, "word": " section", "probability": 0.5087890625}, {"start": 955.87, "end": 958.77, "word": " three", "probability": 0.49072265625}, {"start": 958.77, "end": 959.77, "word": " point", "probability": 0.962890625}, {"start": 959.77, "end": 960.45, "word": " six", "probability": 0.9658203125}], "temperature": 1.0}, {"id": 37, "seek": 97271, "start": 962.99, "end": 972.71, "text": "properly .. properly .. divergent .. divergent .. sequences", "tokens": [4318, 610, 356, 4386, 6108, 4386, 18558, 6930, 4386, 18558, 6930, 4386, 22978], "avg_logprob": -0.44531251064368654, "compression_ratio": 1.4047619047619047, "no_speech_prob": 0.0, "words": [{"start": 962.99, "end": 964.01, "word": "properly", "probability": 0.7095540364583334}, {"start": 964.01, "end": 964.15, "word": " ..", "probability": 0.2349853515625}, {"start": 964.15, "end": 966.57, "word": " properly", "probability": 0.483642578125}, {"start": 966.57, "end": 966.91, "word": " ..", "probability": 0.529296875}, {"start": 966.91, "end": 967.85, "word": " divergent", "probability": 0.867919921875}, {"start": 967.85, "end": 970.47, "word": " ..", "probability": 0.77392578125}, {"start": 970.47, "end": 971.41, "word": " divergent", "probability": 0.921875}, {"start": 971.41, "end": 971.77, "word": " ..", "probability": 0.59521484375}, {"start": 971.77, "end": 972.71, "word": " sequences", "probability": 0.8994140625}], "temperature": 1.0}, {"id": 38, "seek": 100260, "start": 977.18, "end": 1002.6, "text": "أي يعني properly .. properly divergence sequence احنا جفنا قبل هيك أمثلة examples about divergence sequences من ال .. ال divergence sequences هذه كان negative one to n بيقولوا أنه هذه has a divergence برضه sequence n", "tokens": [10721, 1829, 37495, 22653, 6108, 4386, 6108, 47387, 8310, 1975, 5016, 8315, 10874, 5172, 8315, 12174, 36150, 39896, 4117, 5551, 2304, 12984, 37977, 5110, 466, 47387, 22978, 9154, 2423, 4386, 2423, 47387, 22978, 29538, 25961, 3671, 472, 281, 297, 4724, 1829, 39648, 14407, 14739, 3224, 29538, 575, 257, 47387, 4724, 43042, 3224, 8310, 297], "avg_logprob": -0.39602273810993543, "compression_ratio": 1.560693641618497, "no_speech_prob": 0.0, "words": [{"start": 977.18, "end": 977.46, "word": "أي", "probability": 0.456512451171875}, {"start": 977.46, "end": 977.72, "word": " يعني", "probability": 0.8134765625}, {"start": 977.72, "end": 978.32, "word": " properly", "probability": 0.666015625}, {"start": 978.32, "end": 978.32, "word": " ..", "probability": 0.07891845703125}, {"start": 978.32, "end": 978.9, "word": " properly", "probability": 0.6083984375}, {"start": 978.9, "end": 980.44, "word": " divergence", "probability": 0.32373046875}, {"start": 980.44, "end": 981.08, "word": " sequence", "probability": 0.94677734375}, {"start": 981.08, "end": 981.28, "word": " احنا", "probability": 0.7984212239583334}, {"start": 981.28, "end": 981.64, "word": " جفنا", "probability": 0.6903483072916666}, {"start": 981.64, "end": 982.02, "word": " قبل", "probability": 0.938232421875}, {"start": 982.02, "end": 982.98, "word": " هيك", "probability": 0.75146484375}, {"start": 982.98, "end": 983.98, "word": " أمثلة", "probability": 0.9224853515625}, {"start": 983.98, "end": 985.32, "word": " examples", "probability": 0.63232421875}, {"start": 985.32, "end": 986.34, "word": " about", "probability": 0.9287109375}, {"start": 986.34, "end": 987.14, "word": " divergence", "probability": 0.86328125}, {"start": 987.14, "end": 987.92, "word": " sequences", "probability": 0.943359375}, {"start": 987.92, "end": 988.38, "word": " من", "probability": 0.9619140625}, {"start": 988.38, "end": 988.84, "word": " ال", "probability": 0.9892578125}, {"start": 988.84, "end": 989.02, "word": " ..", "probability": 0.27392578125}, {"start": 989.02, "end": 989.52, "word": " ال", "probability": 0.97314453125}, {"start": 989.52, "end": 989.98, "word": " divergence", "probability": 0.931640625}, {"start": 989.98, "end": 990.66, "word": " sequences", "probability": 0.94873046875}, {"start": 990.66, "end": 990.9, "word": " هذه", "probability": 0.52978515625}, {"start": 990.9, "end": 991.2, "word": " كان", "probability": 0.98486328125}, {"start": 991.2, "end": 991.56, "word": " negative", "probability": 0.65673828125}, {"start": 991.56, "end": 991.94, "word": " one", "probability": 0.75927734375}, {"start": 991.94, "end": 992.12, "word": " to", "probability": 0.92822265625}, {"start": 992.12, "end": 992.54, "word": " n", "probability": 0.60986328125}, {"start": 992.54, "end": 993.74, "word": " بيقولوا", "probability": 0.464141845703125}, {"start": 993.74, "end": 994.0, "word": " أنه", "probability": 0.382568359375}, {"start": 994.0, "end": 994.38, "word": " هذه", "probability": 0.69091796875}, {"start": 994.38, "end": 995.4, "word": " has", "probability": 0.869140625}, {"start": 995.4, "end": 995.76, "word": " a", "probability": 0.8720703125}, {"start": 995.76, "end": 996.48, "word": " divergence", "probability": 0.89111328125}, {"start": 996.48, "end": 1000.64, "word": " برضه", "probability": 0.970703125}, {"start": 1000.64, "end": 1001.38, "word": " sequence", "probability": 0.9775390625}, {"start": 1001.38, "end": 1002.6, "word": " n", "probability": 0.5615234375}], "temperature": 1.0}, {"id": 39, "seek": 102916, "start": 1004.3, "end": 1029.16, "text": "Divergent means infinity وبالتالي Divergent ال sequence negative and intense وبالتالي Divergent و هكذا في كتير sequences مر علينا sequences على الأقل هدول إذا ماكانش أكتر وشوفنا ان كل ال sequences هذي are divergent", "tokens": [35, 1837, 6930, 1355, 13202, 46599, 6027, 2655, 6027, 1829, 413, 1837, 6930, 2423, 8310, 3671, 293, 9447, 46599, 6027, 2655, 6027, 1829, 413, 1837, 6930, 4032, 8032, 4117, 15730, 8978, 9122, 2655, 13546, 22978, 3714, 2288, 25894, 8315, 22978, 15844, 16247, 4587, 1211, 8032, 3215, 12610, 11933, 15730, 19446, 41361, 8592, 5551, 4117, 2655, 2288, 4032, 8592, 38688, 8315, 16472, 28242, 2423, 22978, 8032, 8848, 1829, 366, 18558, 6930], "avg_logprob": -0.35299295690697685, "compression_ratio": 1.6201117318435754, "no_speech_prob": 0.0, "words": [{"start": 1004.3, "end": 1005.28, "word": "Divergent", "probability": 0.6449381510416666}, {"start": 1005.28, "end": 1006.04, "word": " means", "probability": 0.0811767578125}, {"start": 1006.04, "end": 1006.8, "word": " infinity", "probability": 0.65283203125}, {"start": 1006.8, "end": 1008.48, "word": " وبالتالي", "probability": 0.8139892578125}, {"start": 1008.48, "end": 1009.32, "word": " Divergent", "probability": 0.6936848958333334}, {"start": 1009.32, "end": 1009.58, "word": " ال", "probability": 0.4599609375}, {"start": 1009.58, "end": 1009.88, "word": " sequence", "probability": 0.401611328125}, {"start": 1009.88, "end": 1010.52, "word": " negative", "probability": 0.63671875}, {"start": 1010.52, "end": 1011.0, "word": " and", "probability": 0.2088623046875}, {"start": 1011.0, "end": 1011.48, "word": " intense", "probability": 0.46142578125}, {"start": 1011.48, "end": 1013.6, "word": " وبالتالي", "probability": 0.894921875}, {"start": 1013.6, "end": 1014.52, "word": " Divergent", "probability": 0.916015625}, {"start": 1014.52, "end": 1014.7, "word": " و", "probability": 0.59765625}, {"start": 1014.7, "end": 1015.2, "word": " هكذا", "probability": 0.8062337239583334}, {"start": 1015.2, "end": 1015.96, "word": " في", "probability": 0.457275390625}, {"start": 1015.96, "end": 1016.36, "word": " كتير", "probability": 0.8917643229166666}, {"start": 1016.36, "end": 1017.16, "word": " sequences", "probability": 0.81396484375}, {"start": 1017.16, "end": 1019.48, "word": " مر", "probability": 0.594482421875}, {"start": 1019.48, "end": 1019.8, "word": " علينا", "probability": 0.985107421875}, {"start": 1019.8, "end": 1020.64, "word": " sequences", "probability": 0.82470703125}, {"start": 1020.64, "end": 1022.1, "word": " على", "probability": 0.91943359375}, {"start": 1022.1, "end": 1022.56, "word": " الأقل", "probability": 0.98681640625}, {"start": 1022.56, "end": 1023.02, "word": " هدول", "probability": 0.86474609375}, {"start": 1023.02, "end": 1023.64, "word": " إذا", "probability": 0.77099609375}, {"start": 1023.64, "end": 1024.1, "word": " ماكانش", "probability": 0.7195638020833334}, {"start": 1024.1, "end": 1024.52, "word": " أكتر", "probability": 0.9486083984375}, {"start": 1024.52, "end": 1026.78, "word": " وشوفنا", "probability": 0.8721923828125}, {"start": 1026.78, "end": 1026.9, "word": " ان", "probability": 0.391845703125}, {"start": 1026.9, "end": 1027.18, "word": " كل", "probability": 0.9189453125}, {"start": 1027.18, "end": 1027.3, "word": " ال", "probability": 0.91552734375}, {"start": 1027.3, "end": 1027.76, "word": " sequences", "probability": 0.919921875}, {"start": 1027.76, "end": 1028.22, "word": " هذي", "probability": 0.7205403645833334}, {"start": 1028.22, "end": 1028.48, "word": " are", "probability": 0.80078125}, {"start": 1028.48, "end": 1029.16, "word": " divergent", "probability": 0.824951171875}], "temperature": 1.0}, {"id": 40, "seek": 105843, "start": 1031.01, "end": 1058.43, "text": "هناك لحد الآن ماتحدثناش عن تفريق التفريق في ال divergence كنا نقول إن إذا كانت ال sequence ماليهاش limit أو تقول ل infinity أو سالب infinity فكنا نقول ال sequence is divergent أو not convergent اليوم ال divergence sequences هنجزقهم إلى نوعين في sequences هنقول عنهم divergent", "tokens": [3224, 8315, 4117, 5296, 24401, 6024, 48506, 19446, 2655, 24401, 12984, 1863, 33599, 18871, 6055, 5172, 16572, 4587, 16712, 5172, 16572, 4587, 8978, 2423, 47387, 9122, 8315, 8717, 39648, 36145, 11933, 15730, 25961, 2655, 2423, 8310, 19446, 20292, 3224, 33599, 4948, 34051, 6055, 4587, 12610, 5296, 13202, 34051, 8608, 6027, 3555, 13202, 6156, 4117, 8315, 8717, 39648, 2423, 8310, 307, 18558, 6930, 34051, 406, 9652, 6930, 45595, 20498, 2423, 47387, 22978, 8032, 1863, 7435, 11622, 4587, 16095, 30731, 8717, 45367, 9957, 8978, 22978, 8032, 1863, 39648, 18871, 16095, 18558, 6930], "avg_logprob": -0.20793269787515914, "compression_ratio": 1.9130434782608696, "no_speech_prob": 0.0, "words": [{"start": 1031.01, "end": 1031.93, "word": "هناك", "probability": 0.6357421875}, {"start": 1031.93, "end": 1032.65, "word": " لحد", "probability": 0.66845703125}, {"start": 1032.65, "end": 1033.03, "word": " الآن", "probability": 0.683349609375}, {"start": 1033.03, "end": 1034.01, "word": " ماتحدثناش", "probability": 0.8203125}, {"start": 1034.01, "end": 1034.91, "word": " عن", "probability": 0.97314453125}, {"start": 1034.91, "end": 1035.67, "word": " تفريق", "probability": 0.9591064453125}, {"start": 1035.67, "end": 1036.89, "word": " التفريق", "probability": 0.9061279296875}, {"start": 1036.89, "end": 1037.05, "word": " في", "probability": 0.46728515625}, {"start": 1037.05, "end": 1037.13, "word": " ال", "probability": 0.75927734375}, {"start": 1037.13, "end": 1037.63, "word": " divergence", "probability": 0.9755859375}, {"start": 1037.63, "end": 1040.29, "word": " كنا", "probability": 0.8974609375}, {"start": 1040.29, "end": 1040.59, "word": " نقول", "probability": 0.92919921875}, {"start": 1040.59, "end": 1040.73, "word": " إن", "probability": 0.53662109375}, {"start": 1040.73, "end": 1041.27, "word": " إذا", "probability": 0.6319580078125}, {"start": 1041.27, "end": 1041.63, "word": " كانت", "probability": 0.977783203125}, {"start": 1041.63, "end": 1041.73, "word": " ال", "probability": 0.8603515625}, {"start": 1041.73, "end": 1042.07, "word": " sequence", "probability": 0.96435546875}, {"start": 1042.07, "end": 1042.53, "word": " ماليهاش", "probability": 0.72076416015625}, {"start": 1042.53, "end": 1042.95, "word": " limit", "probability": 0.9677734375}, {"start": 1042.95, "end": 1043.91, "word": " أو", "probability": 0.80517578125}, {"start": 1043.91, "end": 1045.09, "word": " تقول", "probability": 0.6642252604166666}, {"start": 1045.09, "end": 1045.21, "word": " ل", "probability": 0.8447265625}, {"start": 1045.21, "end": 1045.63, "word": " infinity", "probability": 0.572265625}, {"start": 1045.63, "end": 1045.83, "word": " أو", "probability": 0.7890625}, {"start": 1045.83, "end": 1046.15, "word": " سالب", "probability": 0.8235677083333334}, {"start": 1046.15, "end": 1046.55, "word": " infinity", "probability": 0.78173828125}, {"start": 1046.55, "end": 1047.01, "word": " فكنا", "probability": 0.9093424479166666}, {"start": 1047.01, "end": 1047.33, "word": " نقول", "probability": 0.97509765625}, {"start": 1047.33, "end": 1047.55, "word": " ال", "probability": 0.81298828125}, {"start": 1047.55, "end": 1048.01, "word": " sequence", "probability": 0.9794921875}, {"start": 1048.01, "end": 1048.43, "word": " is", "probability": 0.94091796875}, {"start": 1048.43, "end": 1049.67, "word": " divergent", "probability": 0.863525390625}, {"start": 1049.67, "end": 1049.91, "word": " أو", "probability": 0.822265625}, {"start": 1049.91, "end": 1050.13, "word": " not", "probability": 0.89453125}, {"start": 1050.13, "end": 1050.89, "word": " convergent", "probability": 0.97314453125}, {"start": 1050.89, "end": 1052.07, "word": " اليوم", "probability": 0.912353515625}, {"start": 1052.07, "end": 1052.37, "word": " ال", "probability": 0.986328125}, {"start": 1052.37, "end": 1052.85, "word": " divergence", "probability": 0.83837890625}, {"start": 1052.85, "end": 1053.47, "word": " sequences", "probability": 0.87109375}, {"start": 1053.47, "end": 1054.21, "word": " هنجزقهم", "probability": 0.8734537760416666}, {"start": 1054.21, "end": 1054.41, "word": " إلى", "probability": 0.9453125}, {"start": 1054.41, "end": 1054.81, "word": " نوعين", "probability": 0.970703125}, {"start": 1054.81, "end": 1055.73, "word": " في", "probability": 0.69091796875}, {"start": 1055.73, "end": 1056.41, "word": " sequences", "probability": 0.7451171875}, {"start": 1056.41, "end": 1057.35, "word": " هنقول", "probability": 0.9544270833333334}, {"start": 1057.35, "end": 1057.63, "word": " عنهم", "probability": 0.974609375}, {"start": 1057.63, "end": 1058.43, "word": " divergent", "probability": 0.78271484375}], "temperature": 1.0}, {"id": 41, "seek": 108641, "start": 1059.71, "end": 1086.41, "text": "و في نوع معين من الـ divergence sequences هنسميهم properly divergent إذن الـ sequences اللي زي هدول اللي ال limit بتاعتهم إما infinity أو negative infinity طبعا هدول ال sequences are divergent لكن هذا النوع من ال divergence sequences هنسميه properly divergent متباعدة تباعدا صحيحا", "tokens": [2407, 8978, 8717, 45367, 20449, 9957, 9154, 2423, 39184, 47387, 22978, 8032, 1863, 38251, 1829, 16095, 6108, 18558, 6930, 11933, 8848, 1863, 2423, 39184, 22978, 13672, 1829, 30767, 1829, 8032, 3215, 12610, 13672, 1829, 2423, 4948, 39894, 995, 34268, 16095, 11933, 15042, 13202, 34051, 3671, 13202, 23032, 3555, 3615, 995, 8032, 3215, 12610, 2423, 22978, 366, 18558, 6930, 44381, 23758, 28239, 45367, 9154, 2423, 47387, 22978, 8032, 1863, 38251, 1829, 3224, 6108, 18558, 6930, 44650, 3555, 995, 22488, 3660, 6055, 3555, 995, 22488, 995, 20328, 5016, 1829, 5016, 995], "avg_logprob": -0.16779513756434122, "compression_ratio": 1.8872549019607843, "no_speech_prob": 0.0, "words": [{"start": 1059.71, "end": 1060.21, "word": "و", "probability": 0.5146484375}, {"start": 1060.21, "end": 1060.45, "word": " في", "probability": 0.233154296875}, {"start": 1060.45, "end": 1060.91, "word": " نوع", "probability": 0.94580078125}, {"start": 1060.91, "end": 1061.43, "word": " معين", "probability": 0.9775390625}, {"start": 1061.43, "end": 1061.65, "word": " من", "probability": 0.978515625}, {"start": 1061.65, "end": 1061.83, "word": " الـ", "probability": 0.832275390625}, {"start": 1061.83, "end": 1062.27, "word": " divergence", "probability": 0.54736328125}, {"start": 1062.27, "end": 1063.13, "word": " sequences", "probability": 0.92041015625}, {"start": 1063.13, "end": 1064.85, "word": " هنسميهم", "probability": 0.88310546875}, {"start": 1064.85, "end": 1065.49, "word": " properly", "probability": 0.87548828125}, {"start": 1065.49, "end": 1066.35, "word": " divergent", "probability": 0.8564453125}, {"start": 1066.35, "end": 1067.57, "word": " إذن", "probability": 0.5359700520833334}, {"start": 1067.57, "end": 1067.85, "word": " الـ", "probability": 0.643310546875}, {"start": 1067.85, "end": 1068.73, "word": " sequences", "probability": 0.85302734375}, {"start": 1068.73, "end": 1069.05, "word": " اللي", "probability": 0.934326171875}, {"start": 1069.05, "end": 1069.31, "word": " زي", "probability": 0.9853515625}, {"start": 1069.31, "end": 1069.81, "word": " هدول", "probability": 0.9454752604166666}, {"start": 1069.81, "end": 1072.47, "word": " اللي", "probability": 0.72509765625}, {"start": 1072.47, "end": 1072.67, "word": " ال", "probability": 0.73974609375}, {"start": 1072.67, "end": 1072.93, "word": " limit", "probability": 0.904296875}, {"start": 1072.93, "end": 1073.45, "word": " بتاعتهم", "probability": 0.8731689453125}, {"start": 1073.45, "end": 1073.77, "word": " إما", "probability": 0.7369384765625}, {"start": 1073.77, "end": 1074.27, "word": " infinity", "probability": 0.84228515625}, {"start": 1074.27, "end": 1074.49, "word": " أو", "probability": 0.88330078125}, {"start": 1074.49, "end": 1074.89, "word": " negative", "probability": 0.9482421875}, {"start": 1074.89, "end": 1075.59, "word": " infinity", "probability": 0.93896484375}, {"start": 1075.59, "end": 1076.65, "word": " طبعا", "probability": 0.97509765625}, {"start": 1076.65, "end": 1076.89, "word": " هدول", "probability": 0.919921875}, {"start": 1076.89, "end": 1077.01, "word": " ال", "probability": 0.9814453125}, {"start": 1077.01, "end": 1077.43, "word": " sequences", "probability": 0.88134765625}, {"start": 1077.43, "end": 1077.75, "word": " are", "probability": 0.8837890625}, {"start": 1077.75, "end": 1078.45, "word": " divergent", "probability": 0.7373046875}, {"start": 1078.45, "end": 1079.11, "word": " لكن", "probability": 0.89013671875}, {"start": 1079.11, "end": 1079.45, "word": " هذا", "probability": 0.53662109375}, {"start": 1079.45, "end": 1079.75, "word": " النوع", "probability": 0.990966796875}, {"start": 1079.75, "end": 1079.93, "word": " من", "probability": 0.994140625}, {"start": 1079.93, "end": 1080.09, "word": " ال", "probability": 0.9921875}, {"start": 1080.09, "end": 1080.53, "word": " divergence", "probability": 0.9228515625}, {"start": 1080.53, "end": 1081.11, "word": " sequences", "probability": 0.96435546875}, {"start": 1081.11, "end": 1082.17, "word": " هنسميه", "probability": 0.93701171875}, {"start": 1082.17, "end": 1082.83, "word": " properly", "probability": 0.94873046875}, {"start": 1082.83, "end": 1083.79, "word": " divergent", "probability": 0.94384765625}, {"start": 1083.79, "end": 1085.19, "word": " متباعدة", "probability": 0.86416015625}, {"start": 1085.19, "end": 1085.81, "word": " تباعدا", "probability": 0.93623046875}, {"start": 1085.81, "end": 1086.41, "word": " صحيحا", "probability": 0.98115234375}], "temperature": 1.0}, {"id": 42, "seek": 111827, "start": 1089.53, "end": 1118.27, "text": "Okay إذا ال .. ال sequences التلاتي هدول all of them are divergent كلهم ممكن نقول عنهم divergent لكن التنتين هدول الأخرانيين ممكن نقول عنهم أيضا properly divergent أما هذه مقدرش أقول عنها properly divergent مجرد divergent okay تمام إذا نكتب التعريفات هذه definition", "tokens": [8297, 11933, 15730, 2423, 4386, 2423, 22978, 16712, 1211, 9307, 1829, 8032, 3215, 12610, 439, 295, 552, 366, 18558, 6930, 28242, 16095, 3714, 43020, 8717, 39648, 18871, 16095, 18558, 6930, 44381, 16712, 29399, 9957, 8032, 3215, 12610, 16247, 34740, 7649, 1829, 9957, 3714, 43020, 8717, 39648, 18871, 16095, 36632, 11242, 995, 6108, 18558, 6930, 5551, 15042, 29538, 3714, 28543, 2288, 8592, 5551, 39648, 18871, 11296, 6108, 18558, 6930, 3714, 7435, 2288, 3215, 18558, 6930, 1392, 46811, 10943, 11933, 15730, 8717, 4117, 2655, 3555, 16712, 3615, 16572, 5172, 9307, 29538, 7123], "avg_logprob": -0.17530906331408155, "compression_ratio": 1.7660550458715596, "no_speech_prob": 0.0, "words": [{"start": 1089.53, "end": 1089.97, "word": "Okay", "probability": 0.55029296875}, {"start": 1089.97, "end": 1090.93, "word": " إذا", "probability": 0.55279541015625}, {"start": 1090.93, "end": 1091.37, "word": " ال", "probability": 0.82177734375}, {"start": 1091.37, "end": 1091.43, "word": " ..", "probability": 0.42919921875}, {"start": 1091.43, "end": 1091.71, "word": " ال", "probability": 0.8525390625}, {"start": 1091.71, "end": 1092.13, "word": " sequences", "probability": 0.68017578125}, {"start": 1092.13, "end": 1092.71, "word": " التلاتي", "probability": 0.8367919921875}, {"start": 1092.71, "end": 1093.09, "word": " هدول", "probability": 0.87744140625}, {"start": 1093.09, "end": 1093.43, "word": " all", "probability": 0.4453125}, 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"seek": 113258, "start": 1122.17, "end": 1132.59, "text": "اتنين ستة وعشرين let x in be sequence of real numbers", "tokens": [995, 2655, 1863, 9957, 8608, 2655, 3660, 4032, 3615, 46309, 9957, 718, 2031, 294, 312, 8310, 295, 957, 3547], "avg_logprob": -0.4578125089406967, "compression_ratio": 0.9054054054054054, "no_speech_prob": 0.0, "words": [{"start": 1122.17, "end": 1122.99, "word": "اتنين", "probability": 0.539947509765625}, {"start": 1122.99, "end": 1123.95, "word": " ستة", "probability": 0.9013671875}, {"start": 1123.95, "end": 1124.57, "word": " وعشرين", "probability": 0.78515625}, {"start": 1124.57, "end": 1129.57, "word": " let", "probability": 0.29248046875}, {"start": 1129.57, "end": 1130.03, "word": " x", "probability": 0.716796875}, {"start": 1130.03, "end": 1130.43, "word": " in", "probability": 0.453369140625}, {"start": 1130.43, "end": 1130.83, "word": " be", "probability": 0.86376953125}, {"start": 1130.83, "end": 1131.33, "word": " sequence", "probability": 0.501953125}, {"start": 1131.33, "end": 1131.81, "word": " of", "probability": 0.97509765625}, {"start": 1131.81, "end": 1132.07, "word": " real", "probability": 0.96875}, {"start": 1132.07, "end": 1132.59, "word": " numbers", "probability": 0.787109375}], "temperature": 1.0}, {"id": 44, "seek": 115522, "start": 1136.34, "end": 1155.22, "text": "نقول إن xn tends to infinity أو limit xn as n tends to infinity بساوي infinity اذا تحقق الشرط التالي", "tokens": [1863, 39648, 36145, 2031, 77, 12258, 281, 13202, 34051, 4948, 2031, 77, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 13202, 1975, 15730, 6055, 5016, 4587, 4587, 25124, 2288, 9566, 16712, 6027, 1829], "avg_logprob": -0.34196427890232634, "compression_ratio": 1.2843137254901962, "no_speech_prob": 0.0, "words": [{"start": 1136.34, "end": 1137.74, "word": "نقول", "probability": 0.45758056640625}, {"start": 1137.74, "end": 1138.06, "word": " إن", "probability": 0.38720703125}, {"start": 1138.06, "end": 1138.74, "word": " xn", 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"temperature": 1.0}, {"id": 46, "seek": 119665, "start": 1193.55, "end": 1196.65, "text": "اللي بتتكلموا لو سمحتوا ماتتكلمش امنع الكلام", "tokens": [6027, 20292, 39894, 2655, 4117, 19528, 14407, 45164, 8608, 2304, 33753, 14407, 3714, 9307, 2655, 4117, 19528, 8592, 1975, 27842, 3615, 2423, 28820, 10943], "avg_logprob": -0.18015625476837158, "compression_ratio": 1.2238805970149254, "no_speech_prob": 0.0, "words": [{"start": 1193.55, "end": 1193.91, "word": "اللي", "probability": 0.964111328125}, {"start": 1193.91, "end": 1194.51, "word": " بتتكلموا", "probability": 0.95087890625}, {"start": 1194.51, "end": 1194.59, "word": " لو", "probability": 0.9130859375}, {"start": 1194.59, "end": 1195.13, "word": " سمحتوا", "probability": 0.9332275390625}, {"start": 1195.13, "end": 1195.91, "word": " ماتتكلمش", "probability": 0.8371988932291666}, {"start": 1195.91, "end": 1196.23, "word": " امنع", "probability": 0.5477701822916666}, {"start": 1196.23, "end": 1196.65, "word": " الكلام", "probability": 0.9484049479166666}], "temperature": 1.0}, {"id": 47, "seek": 122847, "start": 1199.91, "end": 1228.47, "text": "إذا ما معنى الـ sequence Xn tends to infinity أو limit لها بالساوية infinity معناه لأي number ألفة نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number الناترال نقدر نجد ال number", "tokens": [28814, 15730, 19446, 20449, 1863, 7578, 2423, 39184, 8310, 1783, 77, 12258, 281, 13202, 34051, 4948, 5296, 11296, 20666, 3794, 995, 2407, 10632, 13202, 20449, 8315, 3224, 5296, 10721, 1829, 1230, 5551, 46538, 3660, 8717, 28543, 2288, 8717, 7435, 3215, 2423, 1230, 2423, 8315, 2655, 2288, 6027, 8717, 28543, 2288, 8717, 7435, 3215, 2423, 1230, 2423, 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"word": " أو", "probability": 0.8291015625}, {"start": 1257.23, "end": 1257.73, "word": " limit", "probability": 0.96630859375}], "temperature": 1.0}, {"id": 49, "seek": 127347, "start": 1259.55, "end": 1273.47, "text": "xn as n tends to infinity بساوي negative infinity إذا تحقق الشرط التالي for every beta real number", "tokens": [87, 77, 382, 297, 12258, 281, 13202, 4724, 3794, 995, 45865, 3671, 13202, 11933, 15730, 6055, 5016, 4587, 4587, 25124, 2288, 9566, 16712, 6027, 1829, 337, 633, 9861, 957, 1230], "avg_logprob": -0.2566784331875463, "compression_ratio": 1.1, "no_speech_prob": 0.0, "words": [{"start": 1259.55, "end": 1260.55, "word": "xn", "probability": 0.16058349609375}, {"start": 1260.55, "end": 1260.99, "word": " as", "probability": 0.2098388671875}, {"start": 1260.99, "end": 1261.31, "word": " n", "probability": 0.81689453125}, {"start": 1261.31, "end": 1261.57, "word": " tends", "probability": 0.548828125}, {"start": 1261.57, "end": 1261.71, "word": " to", "probability": 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حقيقي", "probability": 0.9908854166666666}, {"start": 1295.67, "end": 1296.01, "word": " beta", "probability": 0.734375}], "temperature": 1.0}, {"id": 51, "seek": 132618, "start": 1297.38, "end": 1326.18, "text": "for large N فكل .. فالـ sequence هذه بيقول إنها tends to negative infinity أو ال limit بتاعتها is negative infinity هاي أمثلة examples show that limit", "tokens": [2994, 2416, 426, 6156, 28820, 4386, 6156, 6027, 39184, 8310, 29538, 4724, 1829, 39648, 36145, 11296, 12258, 281, 3671, 13202, 34051, 2423, 4948, 39894, 995, 34268, 11296, 307, 3671, 13202, 8032, 47302, 5551, 2304, 12984, 37977, 5110, 855, 300, 4948], "avg_logprob": -0.32641006679069706, "compression_ratio": 1.2533333333333334, "no_speech_prob": 0.0, "words": [{"start": 1297.38, "end": 1297.72, "word": "for", "probability": 0.209228515625}, {"start": 1297.72, "end": 1298.12, "word": " large", "probability": 0.86376953125}, {"start": 1298.12, "end": 1298.52, "word": " N", "probability": 0.4609375}, {"start": 1298.52, 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1304.58, "word": " ال", "probability": 0.430419921875}, {"start": 1304.58, "end": 1304.78, "word": " limit", "probability": 0.7314453125}, {"start": 1304.78, "end": 1305.38, "word": " بتاعتها", "probability": 0.946533203125}, {"start": 1305.38, "end": 1306.16, "word": " is", "probability": 0.9072265625}, {"start": 1306.16, "end": 1306.66, "word": " negative", "probability": 0.978515625}, {"start": 1306.66, "end": 1307.52, "word": " infinity", "probability": 0.90478515625}, {"start": 1307.52, "end": 1308.7, "word": " هاي", "probability": 0.6605224609375}, {"start": 1308.7, "end": 1309.26, "word": " أمثلة", "probability": 0.9862060546875}, {"start": 1309.26, "end": 1311.02, "word": " examples", "probability": 0.60595703125}, {"start": 1311.02, "end": 1323.6, "word": " show", "probability": 0.841796875}, {"start": 1323.6, "end": 1324.06, "word": " that", "probability": 0.9501953125}, {"start": 1324.06, "end": 1326.18, "word": " limit", "probability": 0.9658203125}], "temperature": 1.0}, {"id": 52, "seek": 135287, "start": 1327.91, "end": 1352.87, "text": "الـ sequence n بساوي infinity احنا كلنا عارفين that the sequence of natural number ال limit تبعتها infinity لكن ممكن نثبت الان هذا باستخدام التعريف فنشوف هاي البرهان proof", "tokens": [6027, 39184, 8310, 297, 4724, 3794, 995, 45865, 13202, 1975, 5016, 8315, 28242, 8315, 6225, 9640, 5172, 9957, 300, 264, 8310, 295, 3303, 1230, 2423, 4948, 6055, 3555, 34268, 11296, 13202, 44381, 3714, 43020, 8717, 12984, 3555, 2655, 2423, 7649, 23758, 4724, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 6156, 1863, 8592, 38688, 8032, 47302, 2423, 26890, 3224, 7649, 8177], "avg_logprob": -0.26041666761277216, "compression_ratio": 1.3910614525139664, "no_speech_prob": 0.0, "words": [{"start": 1327.91, "end": 1328.33, "word": "الـ", "probability": 0.3599853515625}, {"start": 1328.33, "end": 1328.97, "word": " sequence", "probability": 0.64501953125}, {"start": 1328.97, "end": 1329.49, "word": " n", "probability": 0.45849609375}, {"start": 1329.49, "end": 1331.63, "word": " بساوي", "probability": 0.72967529296875}, {"start": 1331.63, "end": 1332.23, "word": " infinity", "probability": 0.689453125}, {"start": 1332.23, "end": 1332.83, "word": " احنا", "probability": 0.7610677083333334}, {"start": 1332.83, "end": 1333.17, "word": " كلنا", "probability": 0.93212890625}, {"start": 1333.17, "end": 1333.67, "word": " عارفين", "probability": 0.9844970703125}, {"start": 1333.67, "end": 1335.55, "word": " that", "probability": 0.1265869140625}, {"start": 1335.55, "end": 1336.17, "word": " the", "probability": 0.488525390625}, {"start": 1336.17, "end": 1336.59, "word": " sequence", "probability": 0.966796875}, {"start": 1336.59, "end": 1336.79, "word": " of", "probability": 0.9599609375}, {"start": 1336.79, "end": 1337.19, "word": " natural", "probability": 0.80517578125}, {"start": 1337.19, "end": 1337.71, "word": " number", "probability": 0.732421875}, {"start": 1337.71, "end": 1338.55, "word": " ال", "probability": 0.50732421875}, {"start": 1338.55, "end": 1338.79, "word": " limit", "probability": 0.689453125}, {"start": 1338.79, "end": 1339.43, "word": " تبعتها", "probability": 0.895751953125}, {"start": 1339.43, "end": 1339.97, "word": " infinity", "probability": 0.857421875}, {"start": 1339.97, "end": 1341.41, "word": " لكن", "probability": 0.81689453125}, {"start": 1341.41, "end": 1342.53, "word": " ممكن", "probability": 0.9052734375}, {"start": 1342.53, "end": 1343.07, "word": " نثبت", "probability": 0.99169921875}, {"start": 1343.07, "end": 1343.43, "word": " الان", "probability": 0.4637451171875}, {"start": 1343.43, "end": 1344.01, "word": " هذا", "probability": 0.90234375}, {"start": 1344.01, "end": 1346.49, "word": " باستخدام", "probability": 0.973388671875}, {"start": 1346.49, "end": 1347.59, "word": " التعريف", "probability": 0.989990234375}, {"start": 1347.59, "end": 1351.59, "word": " فنشوف", "probability": 0.9412841796875}, {"start": 1351.59, "end": 1351.83, "word": " هاي", "probability": 0.530029296875}, {"start": 1351.83, "end": 1352.39, "word": " البرهان", "probability": 0.902587890625}, {"start": 1352.39, "end": 1352.87, "word": " proof", "probability": 0.467529296875}], "temperature": 1.0}, {"id": 53, "seek": 138026, "start": 1354.96, "end": 1380.26, "text": "حسب التعريف بدنا نبدأ بقول let alpha belonging to R be given ناخد alpha عدد حقيقي عشوائي by Archimedean property حسب خاصية Archimedes", "tokens": [5016, 35457, 16712, 3615, 16572, 5172, 47525, 8315, 8717, 44510, 10721, 4724, 39648, 718, 8961, 22957, 281, 497, 312, 2212, 8717, 47283, 3215, 8961, 6225, 3215, 3215, 11331, 38436, 38436, 6225, 8592, 2407, 16373, 1829, 538, 10984, 332, 4858, 282, 4707, 11331, 35457, 16490, 33546, 10632, 10984, 332, 48490], "avg_logprob": -0.14875000208616257, "compression_ratio": 1.2066666666666668, "no_speech_prob": 0.0, "words": [{"start": 1354.96, "end": 1355.36, "word": "حسب", "probability": 0.830078125}, {"start": 1355.36, "end": 1355.88, "word": " التعريف", "probability": 0.982177734375}, {"start": 1355.88, "end": 1356.18, "word": " بدنا", "probability": 0.55670166015625}, {"start": 1356.18, "end": 1356.7, "word": " نبدأ", "probability": 0.9755859375}, {"start": 1356.7, "end": 1357.2, "word": " بقول", "probability": 0.870361328125}, {"start": 1357.2, "end": 1357.56, "word": " let", "probability": 0.63525390625}, {"start": 1357.56, "end": 1358.16, "word": " alpha", "probability": 0.7412109375}, {"start": 1358.16, "end": 1359.74, "word": " belonging", "probability": 0.42041015625}, {"start": 1359.74, "end": 1360.02, "word": " to", "probability": 0.9853515625}, {"start": 1360.02, "end": 1360.34, "word": " R", "probability": 0.8310546875}, {"start": 1360.34, "end": 1360.54, "word": " be", "probability": 0.82421875}, {"start": 1360.54, "end": 1361.0, "word": " given", "probability": 0.89794921875}, {"start": 1361.0, "end": 1365.18, "word": " ناخد", "probability": 0.9010416666666666}, {"start": 1365.18, "end": 1365.6, "word": " alpha", "probability": 0.650390625}, {"start": 1365.6, "end": 1366.26, "word": " عدد", "probability": 0.96875}, {"start": 1366.26, "end": 1367.06, "word": " حقيقي", "probability": 0.96435546875}, {"start": 1367.06, "end": 1368.12, "word": " عشوائي", "probability": 0.97255859375}, {"start": 1368.12, "end": 1372.52, "word": " by", "probability": 0.71484375}, {"start": 1372.52, "end": 1373.88, "word": " Archimedean", "probability": 0.8546142578125}, {"start": 1373.88, "end": 1376.46, "word": " property", "probability": 0.93798828125}, {"start": 1376.46, "end": 1377.72, "word": " حسب", "probability": 0.941650390625}, {"start": 1377.72, "end": 1378.5, "word": " خاصية", "probability": 0.9109700520833334}, {"start": 1378.5, "end": 1380.26, "word": " Archimedes", "probability": 0.9169921875}], "temperature": 1.0}, {"id": 54, "seek": 140224, "start": 1384.68, "end": 1402.24, "text": "يوجد capital N عدد طبيعي بحيث انه capital N أكبر من ال alpha نظبوت؟ هذا حسب ال Archimedean property now", "tokens": [1829, 29245, 3215, 4238, 426, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 16472, 3224, 4238, 426, 5551, 4117, 26890, 9154, 2423, 8961, 8717, 19913, 3555, 35473, 22807, 23758, 11331, 35457, 2423, 10984, 332, 4858, 282, 4707, 586], "avg_logprob": -0.28143601261434104, "compression_ratio": 1.1162790697674418, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1384.68, "end": 1385.58, "word": "يوجد", "probability": 0.8777669270833334}, {"start": 1385.58, "end": 1386.6, "word": " capital", "probability": 0.59423828125}, {"start": 1386.6, "end": 1386.96, "word": " N", "probability": 0.63427734375}, {"start": 1386.96, "end": 1387.44, "word": " عدد", "probability": 0.98291015625}, {"start": 1387.44, "end": 1388.14, "word": " طبيعي", "probability": 0.9630126953125}, {"start": 1388.14, "end": 1390.18, "word": " بحيث", "probability": 0.952392578125}, {"start": 1390.18, "end": 1390.58, "word": " انه", "probability": 0.72412109375}, {"start": 1390.58, "end": 1390.96, "word": " capital", "probability": 0.783203125}, {"start": 1390.96, "end": 1391.58, "word": " N", "probability": 0.9306640625}, {"start": 1391.58, "end": 1393.22, "word": " أكبر", "probability": 0.8712565104166666}, {"start": 1393.22, "end": 1393.62, "word": " من", "probability": 0.99267578125}, {"start": 1393.62, "end": 1393.92, "word": " ال", "probability": 0.728515625}, {"start": 1393.92, "end": 1394.28, "word": " alpha", "probability": 0.51025390625}, {"start": 1394.28, "end": 1396.78, "word": " نظبوت؟", "probability": 0.55899658203125}, {"start": 1396.78, "end": 1397.02, "word": " هذا", "probability": 0.64208984375}, {"start": 1397.02, "end": 1397.36, "word": " حسب", "probability": 0.98681640625}, {"start": 1397.36, "end": 1397.5, "word": " ال", "probability": 0.9443359375}, {"start": 1397.5, "end": 1398.38, "word": " Archimedean", "probability": 0.75958251953125}, {"start": 1398.38, "end": 1399.74, "word": " property", "probability": 0.88037109375}, {"start": 1399.74, "end": 1402.24, "word": " now", "probability": 0.493896484375}], "temperature": 1.0}, {"id": 55, "seek": 142935, "start": 1404.69, "end": 1429.35, "text": "لو أخدت N bigger than or equal capital N هذا هيقدي ان XN ال sequence هذه لحد العام XN تبعها ايش بيساوي بيساوي N الان ال N هذه small n أكبر منه يساوي capital N وانا بيختار capital N by Archimedean property أكبر من Alpha", "tokens": [1211, 2407, 5551, 9778, 3215, 2655, 426, 3801, 813, 420, 2681, 4238, 426, 23758, 39896, 4587, 16254, 16472, 1783, 45, 2423, 8310, 29538, 5296, 24401, 18863, 10943, 1783, 45, 6055, 3555, 3615, 11296, 1975, 1829, 8592, 4724, 1829, 3794, 995, 45865, 4724, 1829, 3794, 995, 45865, 426, 2423, 7649, 2423, 426, 29538, 1359, 297, 5551, 4117, 26890, 9154, 3224, 7251, 3794, 995, 45865, 4238, 426, 4032, 7649, 995, 4724, 1829, 46456, 9640, 4238, 426, 538, 10984, 332, 4858, 282, 4707, 5551, 4117, 26890, 9154, 20588], "avg_logprob": -0.29614824368510134, "compression_ratio": 1.5353535353535352, "no_speech_prob": 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"probability": 0.5732421875}, {"start": 1410.45, "end": 1411.37, "word": " XN", "probability": 0.483154296875}, {"start": 1411.37, "end": 1412.55, "word": " ال", "probability": 0.318603515625}, {"start": 1412.55, "end": 1412.97, "word": " sequence", "probability": 0.55615234375}, {"start": 1412.97, "end": 1413.35, "word": " هذه", "probability": 0.79638671875}, {"start": 1413.35, "end": 1413.65, "word": " لحد", "probability": 0.8427734375}, {"start": 1413.65, "end": 1414.01, "word": " العام", "probability": 0.96484375}, {"start": 1414.01, "end": 1414.49, "word": " XN", "probability": 0.916748046875}, {"start": 1414.49, "end": 1415.03, "word": " تبعها", "probability": 0.916748046875}, {"start": 1415.03, "end": 1415.33, "word": " ايش", "probability": 0.7500813802083334}, {"start": 1415.33, "end": 1415.87, "word": " بيساوي", "probability": 0.80234375}, {"start": 1415.87, "end": 1417.07, "word": " بيساوي", "probability": 0.868798828125}, {"start": 1417.07, "end": 1417.35, "word": " N", "probability": 0.91650390625}, {"start": 1417.35, "end": 1418.67, "word": " الان", "probability": 0.5712890625}, {"start": 1418.67, "end": 1418.87, "word": " ال", "probability": 0.93017578125}, {"start": 1418.87, "end": 1419.15, "word": " N", "probability": 0.85205078125}, {"start": 1419.15, "end": 1419.77, "word": " هذه", "probability": 0.64404296875}, {"start": 1419.77, "end": 1420.37, "word": " small", "probability": 0.6806640625}, {"start": 1420.37, "end": 1420.89, "word": " n", "probability": 0.62939453125}, {"start": 1420.89, "end": 1422.25, "word": " أكبر", "probability": 0.8404947916666666}, {"start": 1422.25, "end": 1422.73, "word": " منه", "probability": 0.6502685546875}, {"start": 1422.73, "end": 1423.51, "word": " يساوي", "probability": 0.8826904296875}, {"start": 1423.51, "end": 1424.11, "word": " capital", "probability": 0.75537109375}, {"start": 1424.11, "end": 1424.43, "word": " N", "probability": 0.9814453125}, {"start": 1424.43, "end": 1425.55, "word": " وانا", "probability": 0.736328125}, {"start": 1425.55, "end": 1425.99, "word": " بيختار", "probability": 0.7755126953125}, {"start": 1425.99, "end": 1426.45, "word": " capital", "probability": 0.88330078125}, {"start": 1426.45, "end": 1426.73, "word": " N", "probability": 0.9775390625}, {"start": 1426.73, "end": 1427.01, "word": " by", "probability": 0.86962890625}, {"start": 1427.01, "end": 1427.71, "word": " Archimedean", "probability": 0.9259033203125}, {"start": 1427.71, "end": 1428.37, "word": " property", "probability": 0.9013671875}, {"start": 1428.37, "end": 1428.81, "word": " أكبر", "probability": 0.9296875}, {"start": 1428.81, "end": 1429.01, "word": " من", "probability": 0.994140625}, {"start": 1429.01, "end": 1429.35, "word": " Alpha", "probability": 0.63525390625}], "temperature": 1.0}, {"id": 56, "seek": 146274, "start": 1435.34, "end": 1462.74, "text": "وبالتالي إذا ال .. ال .. ال implication هذه تتحقق هنا أثبتنا for any Alpha تنتمي ل R there exists natural number يعتمد على Alpha هيوجه .. يعتمد على Alpha N مرتبطة بAlpha بحيث لكل N أكبر من أوساو capital N طول عندي Xn أكبر من Alpha therefore", "tokens": [37746, 6027, 2655, 6027, 1829, 11933, 15730, 2423, 4386, 2423, 4386, 2423, 37814, 29538, 6055, 2655, 5016, 4587, 4587, 34105, 5551, 12984, 3555, 2655, 8315, 337, 604, 20588, 6055, 29399, 2304, 1829, 5296, 497, 456, 8198, 3303, 1230, 7251, 34268, 2304, 3215, 15844, 20588, 39896, 2407, 7435, 3224, 4386, 7251, 34268, 2304, 3215, 15844, 20588, 426, 3714, 43500, 3555, 9566, 3660, 4724, 9171, 7211, 4724, 5016, 1829, 12984, 5296, 28820, 426, 5551, 4117, 26890, 9154, 34051, 3794, 995, 2407, 4238, 426, 23032, 12610, 18871, 16254, 1783, 77, 5551, 4117, 26890, 9154, 20588, 4412], "avg_logprob": -0.2639627795904241, "compression_ratio": 1.5066666666666666, "no_speech_prob": 0.0, "words": [{"start": 1435.34, "end": 1436.22, "word": "وبالتالي", "probability": 0.925390625}, {"start": 1436.22, "end": 1436.52, "word": " إذا", "probability": 0.642578125}, {"start": 1436.52, 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التعريف عشان أثبت أنه ال sequence الحد العام تبعها XN", "tokens": [37746, 1863, 4117, 2655, 8592, 5172, 9122, 33911, 8717, 46456, 9640, 2423, 39184, 4238, 426, 5296, 10721, 1829, 2212, 33286, 11933, 15730, 34105, 11331, 3215, 2423, 26890, 3224, 7649, 4724, 39184, 718, 33286, 4724, 10721, 1829, 957, 1230, 4724, 39184, 2212, 23032, 3555, 3615, 995, 11331, 35457, 16712, 3615, 16572, 5172, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 14739, 3224, 2423, 8310, 21542, 3215, 18863, 10943, 6055, 3555, 3615, 11296, 1783, 45], "avg_logprob": -0.228595885511947, "compression_ratio": 1.398876404494382, "no_speech_prob": 0.0, "words": [{"start": 1560.8200000000002, "end": 1561.66, "word": "وبنكتشف", "probability": 0.8936360677083334}, {"start": 1561.66, "end": 1562.5, "word": " كيف", "probability": 0.9755859375}, {"start": 1562.5, "end": 1563.02, "word": " نختار", "probability": 0.9820963541666666}, {"start": 1563.02, "end": 1563.24, "word": " الـ", "probability": 0.4591064453125}, {"start": 1563.24, 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165863, "start": 1633.65, "end": 1658.63, "text": "و طبعا هذا لكل N أكبر من أو ساوي capital N فما هي ال N أنا بدي أجيب ال N اللي بتخلي هذا الكلام صحيح أنا بعرف أنه لازم ال N تكون أكبر من أو ساوي capital N وبالتالي", "tokens": [2407, 23032, 3555, 3615, 995, 23758, 5296, 28820, 426, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 6156, 15042, 39896, 2423, 426, 41850, 4724, 16254, 5551, 7435, 1829, 3555, 2423, 426, 13672, 1829, 39894, 9778, 20292, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 41850, 45030, 28480, 14739, 3224, 5296, 31377, 2304, 2423, 426, 6055, 30544, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 46599, 6027, 2655, 6027, 1829], "avg_logprob": -0.1471706052889695, "compression_ratio": 1.75, "no_speech_prob": 0.0, "words": [{"start": 1633.65, "end": 1633.87, "word": "و", "probability": 0.92578125}, {"start": 1633.87, "end": 1634.19, "word": " طبعا", "probability": 0.892578125}, {"start": 1634.19, "end": 1634.53, "word": " هذا", 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"end": 2190.62, "word": " الأول", "probability": 0.9833984375}, {"start": 2190.62, "end": 2192.66, "word": " assume", "probability": 0.80859375}], "temperature": 1.0}, {"id": 85, "seek": 221156, "start": 2195.94, "end": 2211.56, "text": "إن XIN is a sequence of real numbers is unbounded and increasing", "tokens": [28814, 1863, 1783, 1464, 307, 257, 8310, 295, 957, 3547, 307, 517, 18767, 292, 293, 5662], "avg_logprob": -0.29411764705882354, "compression_ratio": 0.9295774647887324, "no_speech_prob": 0.0, "words": [{"start": 2195.94, "end": 2196.26, "word": "إن", "probability": 0.6488037109375}, {"start": 2196.26, "end": 2197.1, "word": " XIN", "probability": 0.511474609375}, {"start": 2197.1, "end": 2198.22, "word": " is", "probability": 0.50341796875}, {"start": 2198.22, "end": 2198.4, "word": " a", "probability": 0.83984375}, {"start": 2198.4, "end": 2198.88, "word": " sequence", "probability": 0.974609375}, {"start": 2198.88, "end": 2199.14, "word": " of", "probability": 0.96875}, {"start": 2199.14, "end": 2199.36, "word": " real", "probability": 0.96630859375}, {"start": 2199.36, "end": 2199.94, "word": " numbers", "probability": 0.88330078125}, {"start": 2199.94, "end": 2201.66, "word": " is", "probability": 0.669921875}, {"start": 2201.66, "end": 2205.26, "word": " unbounded", "probability": 0.9407552083333334}, {"start": 2205.26, "end": 2208.54, "word": " and", "probability": 0.9296875}, {"start": 2208.54, "end": 2211.56, "word": " increasing", "probability": 0.6318359375}], "temperature": 1.0}, {"id": 86, "seek": 223089, "start": 2222.37, "end": 2230.89, "text": "بنثبت ان الـ sequence xn properly divergent to infinity", "tokens": [3555, 1863, 12984, 3555, 2655, 16472, 2423, 39184, 8310, 2031, 77, 6108, 18558, 6930, 281, 13202], "avg_logprob": -0.5909926470588235, "compression_ratio": 0.9027777777777778, "no_speech_prob": 0.0, "words": [{"start": 2222.37, "end": 2223.29, "word": "بنثبت", "probability": 0.6771728515625}, {"start": 2223.29, "end": 2223.51, 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" in", "probability": 0.65283203125}, {"start": 2332.67, "end": 2333.01, "word": " is", "probability": 0.8876953125}, {"start": 2333.01, "end": 2337.53, "word": " unbounded", "probability": 0.962890625}, {"start": 2337.53, "end": 2343.15, "word": " فلكل", "probability": 0.7576497395833334}, {"start": 2343.15, "end": 2343.51, "word": " in", "probability": 0.58251953125}, {"start": 2343.51, "end": 2350.73, "word": " then", "probability": 0.66162109375}, {"start": 2350.73, "end": 2352.85, "word": " for", "probability": 0.947265625}, {"start": 2352.85, "end": 2353.33, "word": " every", "probability": 0.81982421875}, {"start": 2353.33, "end": 2353.83, "word": " alpha", "probability": 0.79443359375}], "temperature": 1.0}, {"id": 91, "seek": 237105, "start": 2355.43, "end": 2371.05, "text": "for every alpha عدد حقيقي يوجد by Archimedean property", "tokens": [2994, 633, 8961, 6225, 3215, 3215, 11331, 38436, 38436, 7251, 29245, 3215, 538, 10984, 332, 4858, 282, 4707], "avg_logprob": 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ان ال X رقم capital N Alpha هذا بيطلع أكبر من Alpha يعني لو كان هذا ال M أسمنها Alpha", "tokens": [1211, 10721, 7251, 29245, 3215, 426, 7251, 34268, 2304, 3215, 15844, 20588, 6225, 3215, 3555, 23032, 21292, 3615, 1829, 4724, 5016, 1829, 12984, 16472, 2423, 1783, 12602, 4587, 2304, 4238, 426, 20588, 23758, 4724, 1829, 9566, 1211, 3615, 5551, 4117, 26890, 9154, 20588, 37495, 22653, 45164, 25961, 23758, 2423, 376, 5551, 3794, 27842, 11296, 20588], "avg_logprob": -0.21205357408949307, "compression_ratio": 1.3496503496503496, "no_speech_prob": 0.0, "words": [{"start": 2372.44, "end": 2372.88, "word": "لأ", "probability": 0.93359375}, {"start": 2372.88, "end": 2373.52, "word": " يوجد", "probability": 0.9527994791666666}, {"start": 2373.52, "end": 2375.36, "word": " N", "probability": 0.45361328125}, {"start": 2375.36, "end": 2378.2, "word": " يعتمد", "probability": 0.8714599609375}, {"start": 2378.2, "end": 2378.42, "word": " على", "probability": 0.857421875}, {"start": 2378.42, "end": 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"word": " بيطلع", "probability": 0.785546875}, {"start": 2388.06, "end": 2388.74, "word": " أكبر", "probability": 0.8499348958333334}, {"start": 2388.74, "end": 2389.0, "word": " من", "probability": 0.99267578125}, {"start": 2389.0, "end": 2390.08, "word": " Alpha", "probability": 0.93798828125}, {"start": 2390.08, "end": 2395.7, "word": " يعني", "probability": 0.92138671875}, {"start": 2395.7, "end": 2395.92, "word": " لو", "probability": 0.97314453125}, {"start": 2395.92, "end": 2396.24, "word": " كان", "probability": 0.990234375}, {"start": 2396.24, "end": 2396.7, "word": " هذا", "probability": 0.90234375}, {"start": 2396.7, "end": 2397.76, "word": " ال", "probability": 0.80712890625}, {"start": 2397.76, "end": 2399.2, "word": " M", "probability": 0.7470703125}, {"start": 2399.2, "end": 2399.96, "word": " أسمنها", "probability": 0.7176513671875}, {"start": 2399.96, "end": 2400.36, "word": " Alpha", "probability": 0.95361328125}], "temperature": 1.0}, {"id": 93, "seek": 242598, "start": 2403.42, "end": 2425.98, "text": "معناه أن الـ sequence هذه تكون bounded فما معناه أن الـ sequence هذه تكون unbounded معناه أننا بدنا ننفي الشرط هذا عشان أنفي الشرط هذا هذا معناه أن بدل يوجد alpha موجبة لكل alpha سواء موجبة أو سالبة لكل alpha for any alpha يوجد", "tokens": [2304, 3615, 8315, 3224, 14739, 2423, 39184, 8310, 29538, 6055, 30544, 37498, 6156, 15042, 20449, 8315, 3224, 14739, 2423, 39184, 8310, 29538, 6055, 30544, 517, 18767, 292, 20449, 8315, 3224, 14739, 8315, 47525, 8315, 8717, 1863, 41185, 25124, 2288, 9566, 23758, 6225, 8592, 7649, 14739, 41185, 25124, 2288, 9566, 23758, 23758, 20449, 8315, 3224, 14739, 47525, 1211, 7251, 29245, 3215, 8961, 3714, 29245, 49401, 5296, 28820, 8961, 8608, 2407, 16606, 3714, 29245, 49401, 34051, 8608, 6027, 49401, 5296, 28820, 8961, 337, 604, 8961, 7251, 29245, 3215], "avg_logprob": -0.2828663779401231, "compression_ratio": 2.0764705882352943, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2403.4199999999996, "end": 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{"start": 2413.28, "end": 2413.58, "word": " هذا", "probability": 0.94091796875}, {"start": 2413.58, "end": 2414.34, "word": " هذا", "probability": 0.55078125}, {"start": 2414.34, "end": 2414.96, "word": " معناه", "probability": 0.970703125}, {"start": 2414.96, "end": 2415.48, "word": " أن", "probability": 0.68115234375}, {"start": 2415.48, "end": 2415.92, "word": " بدل", "probability": 0.760986328125}, {"start": 2415.92, "end": 2416.4, "word": " يوجد", "probability": 0.8312174479166666}, {"start": 2416.4, "end": 2416.74, "word": " alpha", "probability": 0.499755859375}, {"start": 2416.74, "end": 2417.38, "word": " موجبة", "probability": 0.7841796875}, {"start": 2417.38, "end": 2418.38, "word": " لكل", "probability": 0.95458984375}, {"start": 2418.38, "end": 2418.92, "word": " alpha", "probability": 0.826171875}, {"start": 2418.92, "end": 2419.76, "word": " سواء", "probability": 0.9384765625}, {"start": 2419.76, "end": 2420.22, "word": " موجبة", "probability": 0.9713541666666666}, {"start": 2420.22, "end": 2420.4, "word": " أو", "probability": 0.90966796875}, {"start": 2420.4, "end": 2420.92, "word": " سالبة", "probability": 0.9542643229166666}, {"start": 2420.92, "end": 2421.88, "word": " لكل", "probability": 0.912353515625}, {"start": 2421.88, "end": 2422.32, "word": " alpha", "probability": 0.81884765625}, {"start": 2422.32, "end": 2422.6, "word": " for", "probability": 0.79150390625}, {"start": 2422.6, "end": 2422.88, "word": " any", "probability": 0.90283203125}, {"start": 2422.88, "end": 2423.3, "word": " alpha", "probability": 0.92724609375}, {"start": 2423.3, "end": 2425.98, "word": " يوجد", "probability": 0.9158528645833334}], "temperature": 1.0}, {"id": 94, "seek": 243887, "start": 2428.13, "end": 2438.87, "text": "بدل لكل n عدد طبيعي يوجد عدد طبيعي capital N يوجد واحد capital N عدد طبيعي مافي لكل يوجد", "tokens": [3555, 3215, 1211, 5296, 28820, 297, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 7251, 29245, 3215, 6225, 3215, 3215, 23032, 21292, 3615, 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"probability": 0.68359375}, {"start": 2432.97, "end": 2433.31, "word": " N", "probability": 0.5888671875}, {"start": 2433.31, "end": 2434.23, "word": " يوجد", "probability": 0.93310546875}, {"start": 2434.23, "end": 2434.89, "word": " واحد", "probability": 0.895263671875}, {"start": 2434.89, "end": 2435.79, "word": " capital", "probability": 0.6357421875}, {"start": 2435.79, "end": 2436.21, "word": " N", "probability": 0.93505859375}, {"start": 2436.21, "end": 2436.59, "word": " عدد", "probability": 0.98876953125}, {"start": 2436.59, "end": 2437.31, "word": " طبيعي", "probability": 0.989013671875}, {"start": 2437.31, "end": 2437.87, "word": " مافي", "probability": 0.63232421875}, {"start": 2437.87, "end": 2438.31, "word": " لكل", "probability": 0.95556640625}, {"start": 2438.31, "end": 2438.87, "word": " يوجد", "probability": 0.9715169270833334}], "temperature": 1.0}, {"id": 95, "seek": 246273, "start": 2440.3, "end": 2462.74, "text": "بحيث أن هذا الـ xn بدل ما هي أصغر من أو ساوي Alpha نفي xn أصغر من أو ساوي Alpha هو أكبر من Alpha تمام؟ إذن هذا هو نفي الشرط هذا أو نفي boundedness", "tokens": [3555, 5016, 1829, 12984, 14739, 23758, 2423, 39184, 2031, 77, 4724, 3215, 1211, 19446, 39896, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 20588, 8717, 41185, 2031, 77, 5551, 9381, 17082, 2288, 9154, 34051, 8608, 995, 45865, 20588, 31439, 5551, 4117, 26890, 9154, 20588, 46811, 10943, 22807, 11933, 8848, 1863, 23758, 31439, 8717, 41185, 25124, 2288, 9566, 23758, 34051, 8717, 41185, 37498, 1287], "avg_logprob": -0.2935096153846154, "compression_ratio": 1.5793103448275863, "no_speech_prob": 0.0, "words": [{"start": 2440.3, "end": 2441.02, "word": "بحيث", "probability": 0.82012939453125}, {"start": 2441.02, "end": 2441.28, "word": " أن", "probability": 0.4189453125}, {"start": 2441.28, "end": 2441.74, "word": " هذا", "probability": 0.427001953125}, {"start": 2441.74, "end": 2442.74, "word": " الـ", "probability": 0.4429931640625}, {"start": 2442.74, "end": 2443.48, 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نفي", "probability": 0.83349609375}, {"start": 2457.8, "end": 2460.88, "word": " الشرط", "probability": 0.80224609375}, {"start": 2460.88, "end": 2461.22, "word": " هذا", "probability": 0.8076171875}, {"start": 2461.22, "end": 2461.6, "word": " أو", "probability": 0.8564453125}, {"start": 2461.6, "end": 2461.88, "word": " نفي", "probability": 0.92236328125}, {"start": 2461.88, "end": 2462.74, "word": " boundedness", "probability": 0.6884765625}], "temperature": 1.0}, {"id": 96, "seek": 248898, "start": 2463.36, "end": 2488.98, "text": "إذا الـ sequence unbounded معناه لأي عدد حقيقي Alpha فهي عدد طبيعي يعتمد على Alpha بحيث ان ال X المؤشر تبعه capital N بيطلع أكبر من Alpha طيب الآن بما أن ال sequence increasing as ال sequence XN is increasing احنا فرضين انها متزايدة", "tokens": [28814, 15730, 2423, 39184, 8310, 517, 18767, 292, 20449, 8315, 3224, 5296, 10721, 1829, 6225, 3215, 3215, 11331, 38436, 38436, 20588, 6156, 3224, 1829, 6225, 3215, 3215, 23032, 21292, 3615, 1829, 7251, 34268, 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"probability": 0.59716796875}, {"start": 2471.44, "end": 2471.84, "word": " X", "probability": 0.424560546875}, {"start": 2471.84, "end": 2472.72, "word": " المؤشر", "probability": 0.7171630859375}, {"start": 2472.72, "end": 2473.16, "word": " تبعه", "probability": 0.740081787109375}, {"start": 2473.16, "end": 2473.42, "word": " capital", "probability": 0.57275390625}, {"start": 2473.42, "end": 2473.78, "word": " N", "probability": 0.9228515625}, {"start": 2473.78, "end": 2474.68, "word": " بيطلع", "probability": 0.76611328125}, {"start": 2474.68, "end": 2475.26, "word": " أكبر", "probability": 0.92919921875}, {"start": 2475.26, "end": 2475.5, "word": " من", "probability": 0.990234375}, {"start": 2475.5, "end": 2475.88, "word": " Alpha", "probability": 0.97412109375}, {"start": 2475.88, "end": 2478.08, "word": " طيب", "probability": 0.94091796875}, {"start": 2478.08, "end": 2478.42, "word": " الآن", "probability": 0.59332275390625}, {"start": 2478.42, "end": 2478.7, "word": " بما", "probability": 0.952392578125}, {"start": 2478.7, "end": 2478.82, "word": " أن", "probability": 0.626953125}, {"start": 2478.82, "end": 2478.94, "word": " ال", "probability": 0.51025390625}, {"start": 2478.94, "end": 2479.34, "word": " sequence", "probability": 0.9443359375}, {"start": 2479.34, "end": 2480.3, "word": " increasing", "probability": 0.94140625}, {"start": 2480.3, "end": 2483.3, "word": " as", "probability": 0.7568359375}, {"start": 2483.3, "end": 2483.84, "word": " ال", "probability": 0.64404296875}, {"start": 2483.84, "end": 2484.28, "word": " sequence", "probability": 0.97216796875}, {"start": 2484.28, "end": 2485.12, "word": " XN", "probability": 0.547607421875}, {"start": 2485.12, "end": 2486.34, "word": " is", "probability": 0.92626953125}, {"start": 2486.34, "end": 2487.2, "word": " increasing", "probability": 0.97607421875}, {"start": 2487.2, "end": 2487.54, "word": " احنا", "probability": 0.819091796875}, {"start": 2487.54, "end": 2487.9, "word": " فرضين", "probability": 0.9580078125}, {"start": 2487.9, "end": 2488.06, "word": " انها", "probability": 0.709228515625}, {"start": 2488.06, "end": 2488.98, "word": " متزايدة", "probability": 0.984375}], "temperature": 1.0}, {"id": 97, "seek": 251284, "start": 2490.46, "end": 2512.84, "text": "We get نحصل على لو كان n أكبر من أو ساوي n of alpha فهذا بالتأكيد هيقدّي أن x المؤشر تبعها small n أكبر من أو ساوي x المؤشر تبعها n of alpha وهذا من هنا أكبر من alpha", "tokens": [4360, 483, 8717, 5016, 36520, 15844, 45164, 25961, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 297, 295, 8961, 6156, 3224, 15730, 20666, 2655, 10721, 4117, 25708, 39896, 28543, 11703, 1829, 14739, 2031, 9673, 33604, 46309, 6055, 3555, 3615, 11296, 1359, 297, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 2031, 9673, 33604, 46309, 6055, 3555, 3615, 11296, 297, 295, 8961, 37037, 15730, 9154, 34105, 5551, 4117, 26890, 9154, 8961], "avg_logprob": -0.20876735573013624, "compression_ratio": 1.7039473684210527, "no_speech_prob": 0.0, "words": [{"start": 2490.46, "end": 2490.8, "word": "We", "probability": 0.036834716796875}, {"start": 2490.8, "end": 2491.3, "word": " get", "probability": 0.90087890625}, {"start": 2491.3, "end": 2492.1, "word": " نحصل", "probability": 0.8619791666666666}, {"start": 2492.1, "end": 2492.44, "word": " على", "probability": 0.7900390625}, {"start": 2492.44, "end": 2493.5, "word": " لو", "probability": 0.457763671875}, {"start": 2493.5, "end": 2494.04, "word": " كان", "probability": 0.966796875}, {"start": 2494.04, "end": 2494.4, "word": " n", "probability": 0.443115234375}, {"start": 2494.4, "end": 2495.0, "word": " أكبر", "probability": 0.9401041666666666}, {"start": 2495.0, "end": 2495.2, "word": " من", "probability": 0.89990234375}, {"start": 2495.2, "end": 2495.38, "word": " أو", "probability": 0.97705078125}, {"start": 2495.38, "end": 2495.96, "word": " ساوي", "probability": 0.9488932291666666}, {"start": 2495.96, "end": 2496.34, "word": " n", "probability": 0.484375}, {"start": 2496.34, "end": 2496.56, "word": " of", "probability": 0.87109375}, {"start": 2496.56, "end": 2497.02, "word": " alpha", "probability": 0.681640625}, {"start": 2497.02, "end": 2498.64, "word": " فهذا", "probability": 0.9378255208333334}, {"start": 2498.64, "end": 2499.44, "word": " بالتأكيد", "probability": 0.992578125}, {"start": 2499.44, "end": 2500.3, "word": " هيقدّي", "probability": 0.66424560546875}, {"start": 2500.3, "end": 2501.02, "word": " أن", "probability": 0.383056640625}, {"start": 2501.02, "end": 2501.62, "word": " x", "probability": 0.705078125}, {"start": 2501.62, "end": 2503.0, "word": " المؤشر", "probability": 0.8678385416666666}, {"start": 2503.0, "end": 2503.44, "word": " تبعها", "probability": 0.8978271484375}, {"start": 2503.44, "end": 2503.78, "word": " small", "probability": 0.890625}, {"start": 2503.78, "end": 2504.16, "word": " n", "probability": 0.90625}, {"start": 2504.16, "end": 2504.66, "word": " أكبر", "probability": 0.9778645833333334}, {"start": 2504.66, "end": 2504.8, "word": " من", "probability": 0.98779296875}, {"start": 2504.8, "end": 2504.94, "word": " أو", "probability": 0.99267578125}, {"start": 2504.94, "end": 2505.38, "word": " ساوي", "probability": 0.9615885416666666}, {"start": 2505.38, "end": 2505.84, "word": " x", "probability": 0.5419921875}, {"start": 2505.84, "end": 2507.06, "word": " المؤشر", "probability": 0.9596354166666666}, {"start": 2507.06, "end": 2507.64, "word": " تبعها", "probability": 0.9248046875}, {"start": 2507.64, "end": 2508.08, "word": " n", "probability": 0.93359375}, {"start": 2508.08, "end": 2508.28, "word": " of", "probability": 0.97265625}, {"start": 2508.28, "end": 2508.7, "word": " alpha", "probability": 0.94580078125}, {"start": 2508.7, "end": 2510.86, "word": " وهذا", "probability": 0.705322265625}, {"start": 2510.86, "end": 2511.04, "word": " من", "probability": 0.767578125}, {"start": 2511.04, "end": 2511.38, "word": " هنا", "probability": 0.79541015625}, {"start": 2511.38, "end": 2512.28, "word": " أكبر", "probability": 0.96826171875}, {"start": 2512.28, "end": 2512.48, "word": " من", "probability": 0.99267578125}, {"start": 2512.48, "end": 2512.84, "word": " alpha", "probability": 0.599609375}], "temperature": 1.0}, {"id": 98, "seek": 253875, "start": 2516.15, "end": 2538.75, "text": "أذن خلّيني ألخّص شو عملنا أحنا أثبتنا الآن أن for every alpha real number there exists a natural number depends on alpha هذا هو بحيث أنه لكل N أكبر من أو ساوي capital N طلع ندي xn أكبر من alpha", "tokens": [10721, 8848, 1863, 16490, 1211, 11703, 9957, 1829, 5551, 1211, 9778, 11703, 9381, 13412, 2407, 6225, 42213, 8315, 5551, 5016, 8315, 5551, 12984, 3555, 2655, 8315, 6024, 48506, 14739, 337, 633, 8961, 957, 1230, 456, 8198, 257, 3303, 1230, 5946, 322, 8961, 23758, 31439, 4724, 5016, 1829, 12984, 14739, 3224, 5296, 28820, 426, 5551, 4117, 26890, 9154, 34051, 8608, 995, 45865, 4238, 426, 23032, 1211, 3615, 8717, 16254, 2031, 77, 5551, 4117, 26890, 9154, 8961], "avg_logprob": -0.2796052611972156, "compression_ratio": 1.4083769633507854, "no_speech_prob": 0.0, "words": [{"start": 2516.15, "end": 2516.69, "word": "أذن", "probability": 0.5875651041666666}, {"start": 2516.69, "end": 2517.53, "word": " خلّيني", "probability": 0.86875}, {"start": 2517.53, "end": 2518.13, "word": " ألخّص", "probability": 0.7404296875}, {"start": 2518.13, "end": 2518.37, "word": " شو", "probability": 0.66162109375}, {"start": 2518.37, "end": 2518.75, "word": " عملنا", "probability": 0.9768880208333334}, {"start": 2518.75, "end": 2519.15, "word": " أحنا", "probability": 0.8162434895833334}, {"start": 2519.15, "end": 2521.07, "word": " أثبتنا", "probability": 0.9267578125}, {"start": 2521.07, "end": 2521.41, "word": " الآن", "probability": 0.6082763671875}, {"start": 2521.41, "end": 2521.69, "word": " أن", "probability": 0.59912109375}, {"start": 2521.69, "end": 2522.15, "word": " for", "probability": 0.552734375}, {"start": 2522.15, "end": 2522.79, "word": " every", "probability": 0.79541015625}, {"start": 2522.79, "end": 2523.35, "word": " alpha", "probability": 0.5576171875}, {"start": 2523.35, "end": 2524.19, "word": " real", "probability": 0.828125}, {"start": 2524.19, "end": 2524.73, "word": " number", "probability": 0.9814453125}, {"start": 2524.73, "end": 2525.59, "word": " there", "probability": 0.70458984375}, {"start": 2525.59, "end": 2526.03, "word": " exists", "probability": 0.6259765625}, {"start": 2526.03, "end": 2526.23, "word": " a", "probability": 0.98779296875}, {"start": 2526.23, "end": 2526.57, "word": " natural", "probability": 0.9228515625}, {"start": 2526.57, "end": 2527.17, "word": " number", "probability": 0.96630859375}, {"start": 2527.17, "end": 2527.73, "word": " depends", "probability": 0.66845703125}, {"start": 2527.73, "end": 2528.11, "word": " on", "probability": 0.955078125}, {"start": 2528.11, "end": 2528.47, "word": " alpha", "probability": 0.6708984375}, {"start": 2528.47, "end": 2528.71, "word": " هذا", "probability": 0.1851806640625}, {"start": 2528.71, "end": 2529.05, "word": " هو", "probability": 0.99169921875}, {"start": 2529.05, "end": 2531.07, "word": " بحيث", "probability": 0.947021484375}, {"start": 2531.07, "end": 2532.13, "word": " أنه", "probability": 0.85791015625}, {"start": 2532.13, "end": 2533.17, "word": " لكل", "probability": 0.990478515625}, {"start": 2533.17, "end": 2533.57, "word": " N", "probability": 0.55615234375}, {"start": 2533.57, "end": 2534.31, "word": " أكبر", "probability": 0.9615885416666666}, {"start": 2534.31, "end": 2534.49, "word": " من", "probability": 0.7333984375}, {"start": 2534.49, "end": 2534.65, "word": " أو", "probability": 0.84375}, {"start": 2534.65, "end": 2535.01, "word": " ساوي", "probability": 0.7918294270833334}, {"start": 2535.01, "end": 2535.37, "word": " capital", "probability": 0.578125}, {"start": 2535.37, "end": 2535.77, "word": " N", "probability": 0.96826171875}, {"start": 2535.77, "end": 2536.57, "word": " طلع", "probability": 0.8238932291666666}, {"start": 2536.57, "end": 2536.87, "word": " ندي", "probability": 0.4677734375}, {"start": 2536.87, "end": 2537.55, "word": " xn", "probability": 0.3756103515625}, {"start": 2537.55, "end": 2538.19, "word": " أكبر", "probability": 0.9777018229166666}, {"start": 2538.19, "end": 2538.39, "word": " من", "probability": 0.99365234375}, {"start": 2538.39, "end": 2538.75, "word": " alpha", "probability": 0.611328125}], "temperature": 1.0}, {"id": 99, "seek": 256629, "start": 2540.01, "end": 2566.29, "text": "هذا اذا حسب التعريف الأولاني حسب تعريف one by definition one طبعا هذا معناه ان limit x in بساوي infinity وهو المطلوب okay برهان الجزء التاني the proof of part اتنين is similar", "tokens": [3224, 15730, 1975, 15730, 11331, 35457, 16712, 3615, 16572, 5172, 16247, 12610, 7649, 1829, 11331, 35457, 37279, 16572, 5172, 472, 538, 7123, 472, 23032, 3555, 3615, 995, 23758, 20449, 8315, 3224, 16472, 4948, 2031, 294, 4724, 3794, 995, 45865, 13202, 37037, 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"probability": 0.9423828125}], "temperature": 1.0}, {"id": 100, "seek": 259171, "start": 2569.84, "end": 2591.72, "text": "two one فحاسبكم انتوا تكتبوا البرهان تبعه okay تمام و هيك بنكون كملنا النظرية اذا هنوقف هنا و بنكمل ان شاء الله الموضوع هذا المرة القادمة فشوفكم ان شاء الله المرة الجاية على فيه", "tokens": [20534, 472, 6156, 5016, 32277, 3555, 24793, 16472, 2655, 14407, 6055, 4117, 2655, 3555, 14407, 2423, 26890, 3224, 7649, 6055, 3555, 3615, 3224, 1392, 46811, 10943, 4032, 39896, 4117, 44945, 30544, 9122, 42213, 8315, 28239, 19913, 2288, 10632, 1975, 15730, 8032, 1863, 30543, 5172, 34105, 4032, 44945, 24793, 1211, 16472, 13412, 16606, 21984, 9673, 2407, 11242, 45367, 23758, 9673, 25720, 25062, 18513, 46007, 6156, 8592, 38688, 24793, 16472, 13412, 16606, 21984, 9673, 25720, 25724, 995, 10632, 15844, 8978, 3224], "avg_logprob": -0.19003906473517418, "compression_ratio": 1.718232044198895, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 2569.84, "end": 2570.22, "word": "two", "probability": 0.290283203125}, {"start": 2570.22, "end": 2570.68, "word": " one", "probability": 0.7080078125}, {"start": 2570.68, "end": 2572.94, "word": " فحاسبكم", "probability": 0.82646484375}, {"start": 2572.94, "end": 2573.26, "word": " انتوا", "probability": 0.660888671875}, {"start": 2573.26, "end": 2573.64, "word": " تكتبوا", "probability": 0.9923828125}, {"start": 2573.64, "end": 2574.08, "word": " البرهان", "probability": 0.752685546875}, {"start": 2574.08, "end": 2574.54, "word": " تبعه", "probability": 0.8629150390625}, {"start": 2574.54, "end": 2575.18, "word": " okay", "probability": 0.6884765625}, {"start": 2575.18, "end": 2576.26, "word": " تمام", "probability": 0.864501953125}, {"start": 2576.26, "end": 2576.84, "word": " و", "probability": 0.465087890625}, {"start": 2576.84, "end": 2577.08, "word": " هيك", "probability": 0.972900390625}, {"start": 2577.08, "end": 2577.44, "word": " بنكون", "probability": 0.8134765625}, {"start": 2577.44, "end": 2578.22, "word": " كملنا", "probability": 0.8712565104166666}, {"start": 2578.22, "end": 2579.06, "word": " النظرية", "probability": 0.955810546875}, {"start": 2579.06, "end": 2579.74, "word": " اذا", "probability": 0.7188720703125}, {"start": 2579.74, "end": 2580.68, "word": " هنوقف", "probability": 0.8128662109375}, {"start": 2580.68, "end": 2581.1, "word": " هنا", "probability": 0.9833984375}, {"start": 2581.1, "end": 2581.64, "word": " و", "probability": 0.92236328125}, {"start": 2581.64, "end": 2583.9, "word": " بنكمل", "probability": 0.9280598958333334}, {"start": 2583.9, "end": 2584.08, "word": " ان", "probability": 0.86376953125}, {"start": 2584.08, "end": 2584.26, "word": " شاء", "probability": 0.97802734375}, {"start": 2584.26, "end": 2584.38, "word": " الله", "probability": 0.9521484375}, {"start": 2584.38, "end": 2584.92, "word": " الموضوع", "probability": 0.9952392578125}, {"start": 2584.92, "end": 2585.34, "word": " هذا", "probability": 0.83544921875}, {"start": 2585.34, "end": 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8f42e056203ece51447c975a121affac350dbb2f
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/uTGQ5_eGie4_raw.srt
@@ -0,0 +1,1696 @@
+1
+00:00:20,650 --> 00:00:27,710
+السلام عليكم اليوم ان شاء الله هنكمل ما ابتدأناه
+
+2
+00:00:27,710 --> 00:00:35,620
+سابقا بخصوص موضوع الكوشي sequencesأخر نظرية
+
+3
+00:00:35,620 --> 00:00:41,060
+هناخدها بالنسبة لهذا الموضوع هتكون نظرية التالية
+
+4
+00:00:41,060 --> 00:00:45,580
+لكن في الأول خلينا نراجع بس تعريف الـ Cauchy
+
+5
+00:00:45,580 --> 00:00:48,600
+sequence فطبعا تعريف الـ Cauchy sequence زي ما
+
+6
+00:00:48,600 --> 00:00:52,560
+انتوا شايفين sequence of real numbers is Cauchy if
+
+7
+00:00:52,560 --> 00:00:55,500
+and only if for every epsilon فيه capital N
+
+8
+00:00:55,500 --> 00:01:00,200
+natural number بحيث for every N و M bigger than or
+
+9
+00:01:00,200 --> 00:01:06,260
+equal capital Nالمقارنة بين xn و xm أقل من إبسلون
+
+10
+00:01:06,260 --> 00:01:12,980
+رمارك في ملاحظة هنا it can be easily shown يعني من
+
+11
+00:01:12,980 --> 00:01:17,540
+السهل اثبات أنه sequence of real numbers is Cauchy
+
+12
+00:01:17,540 --> 00:01:23,420
+if and only if limit the distance between xn و xm
+
+13
+00:01:23,420 --> 00:01:29,700
+بالساوي سفر whenever n و m tends to infinityيعني
+
+14
+00:01:29,700 --> 00:01:34,060
+المعنى أخر لما N و M تكون large، و N و M large
+
+15
+00:01:34,060 --> 00:01:39,660
+enough، ال distance between XN و XM بتكون very
+
+16
+00:01:39,660 --> 00:01:44,860
+small، تقول لصفر، فهو نفس المعنى تقريبا، you can
+
+17
+00:01:44,860 --> 00:01:50,260
+write a proof for this remark و it is easy زي ما
+
+18
+00:01:50,260 --> 00:01:54,310
+هو مذكورفي الان definition a sequence of real
+
+19
+00:01:54,310 --> 00:02:02,570
+numbers بنسميها contractive إذا وجد ثابت c عدد بين
+
+20
+00:02:02,570 --> 00:02:08,630
+صفر و واحد عدد موجب بحيث أنه المسافة بين xn plus
+
+21
+00:02:08,630 --> 00:02:14,830
+two و xn plus one أصغر من أو ساوى ثابت c في
+
+22
+00:02:14,830 --> 00:02:20,140
+المسافة بين xn plus one minus xnوهذا الكلام طبعا
+
+23
+00:02:20,140 --> 00:02:24,920
+بيكون متحقق for every natural number in الثابت C
+
+24
+00:02:24,920 --> 00:02:33,760
+هذا بيسمي the constant of the contractive sequence
+
+25
+00:02:33,760 --> 00:02:39,140
+الان النظرية اللي حكيت عنها في المقدمة هي النظرية
+
+26
+00:02:39,140 --> 00:02:46,000
+التالية theorem رقم
+
+27
+00:02:46,000 --> 00:02:47,860
+خمسة و عشرين
+
+28
+00:02:51,670 --> 00:02:59,830
+النظرية هذه بتقول every every contractive every
+
+29
+00:02:59,830 --> 00:03:08,050
+contractive sequence is cauchy كل contractive
+
+30
+00:03:08,050 --> 00:03:15,230
+sequence بتكون cauchy البرهان مش صعب proof for
+
+31
+00:03:15,230 --> 00:03:20,170
+every natural number in apply
+
+32
+00:03:22,820 --> 00:03:28,900
+the defining .. apply
+
+33
+00:03:28,900 --> 00:03:34,980
+the defining condition ال
+
+34
+00:03:34,980 --> 00:03:39,120
+defining condition اللي هو هذا الشرط تبع ال
+
+35
+00:03:39,120 --> 00:03:42,720
+contractive of contractive sequence
+
+36
+00:03:46,850 --> 00:03:52,790
+of contractive sequence لان هنطبق التعريف هذا to
+
+37
+00:03:52,790 --> 00:03:56,830
+get لنحصل
+
+38
+00:03:56,830 --> 00:03:57,450
+على mainly
+
+39
+00:04:01,950 --> 00:04:08,910
+هي عندي absolute xn من التعريف هي absolute xn plus
+
+40
+00:04:08,910 --> 00:04:16,110
+two minus xn plus one less than or equal to c في
+
+41
+00:04:16,110 --> 00:04:25,890
+absolute xn plus one negative xn الآن
+
+42
+00:04:25,890 --> 00:04:35,390
+من نفس التعريف هذا بدل n هناكب N سالب واحد فبصير
+
+43
+00:04:35,390 --> 00:04:40,010
+الطرف الشمال هكذا وهذا بطلع اصغر من ال absolute
+
+44
+00:04:40,010 --> 00:04:45,070
+value هذي اصغر من او ساوي C وفي اندي C تانية فبصير
+
+45
+00:04:45,070 --> 00:04:51,470
+C تربيه في absolute X X
+
+46
+00:04:51,470 --> 00:04:58,190
+N minus X N minus واحد تمام؟
+
+47
+00:04:59,310 --> 00:05:05,730
+والان ممكن من ال defining condition هذا اطبخه على
+
+48
+00:05:05,730 --> 00:05:11,670
+ال absolute value هذه يعني ابدل n هناك ب n سالب
+
+49
+00:05:11,670 --> 00:05:17,410
+اتنين فبصير n بصير الطرف الشمال absolute xn minus
+
+50
+00:05:17,410 --> 00:05:22,540
+xn minus واحدهذا هيطلع أصغر من أوي ساوي C في
+
+51
+00:05:22,540 --> 00:05:27,860
+absolute value تانية وفي عندي C تربية فهيصير عندي
+
+52
+00:05:27,860 --> 00:05:34,960
+C تكايم في absolute xn minus one minus xn minus
+
+53
+00:05:34,960 --> 00:05:42,000
+two وطبعا لو استمرنا على هذا النمط هنصل في الآخر
+
+54
+00:05:42,000 --> 00:05:47,620
+خالص إلىأصغر من أو يساوي c os n في absolute x2
+
+55
+00:05:47,620 --> 00:05:52,980
+negative x1 و بعد هيك بنوقف خلاص لأن هنكرر تطبيق
+
+56
+00:05:52,980 --> 00:05:57,460
+ال defining condition هذا n من المرات و بعد هيك
+
+57
+00:05:57,460 --> 00:06:02,240
+هنوقف لأنه خلاص هنصل ل absolute x2 minus x1 خلصت
+
+58
+00:06:02,240 --> 00:06:04,600
+الحدود أظبط؟
+
+59
+00:06:06,830 --> 00:06:12,490
+تبام اذا انا اندي طلع absolute xn plus two minus
+
+60
+00:06:12,490 --> 00:06:17,730
+xn plus one less than or equal c to n في absolute
+
+61
+00:06:17,730 --> 00:06:23,150
+x two negative x one الكلام هذا صحيح for every n
+
+62
+00:06:23,150 --> 00:06:31,250
+كل الأعداد الطبيعية n الان من ال inequality هذه
+
+63
+00:06:34,380 --> 00:06:39,140
+this inequality and the triangle inequality
+
+64
+00:06:39,140 --> 00:06:50,600
+متباينة المثلث بيؤدوا and geometric progression
+
+65
+00:07:04,750 --> 00:07:11,290
+imply انه .. بيقده انه for .. لو أخدت M أكبر من N
+
+66
+00:07:11,290 --> 00:07:14,870
+فبطلع
+
+67
+00:07:14,870 --> 00:07:23,930
+عندي absolute XM minus XN هذا هيكون باستخدام
+
+68
+00:07:23,930 --> 00:07:36,720
+متبينة المثلث هترح من XMلو طرحت من XM XM-1 ورجعتها
+
+69
+00:07:36,720 --> 00:07:50,860
+ورجعتها و بعدين هطرح XM-2 ورجعها إلى
+
+70
+00:07:50,860 --> 00:08:02,450
+أن أصل إلىxn زاد واحد minus xm شو اللي عملته هنا
+
+71
+00:08:02,450 --> 00:08:09,790
+انا طرحت من xm xm minus one و رجعتها بعدين طرحت xm
+
+72
+00:08:09,790 --> 00:08:14,790
+سالب اتنين و رجعتها و هكذا طرحت xn زاد واحد و
+
+73
+00:08:14,790 --> 00:08:19,550
+رجعتها و طبعا هاي سالب xn هو بوقف لانه بعد كده
+
+74
+00:08:19,550 --> 00:08:30,360
+خلاصالان من ال .. يسمى المتباينة بال star فمن
+
+75
+00:08:30,360 --> 00:08:36,460
+ال star this inequality اللي هي ال star لو
+
+76
+00:08:36,460 --> 00:08:46,240
+أخدت هنا M بدلت N زاد 2 بدلتها ب M معناته N بساوي
+
+77
+00:08:46,240 --> 00:08:56,130
+M سالد 2 صح؟وبالتالي هذا بيصير أصغر
+
+78
+00:08:56,130 --> 00:09:03,190
+من أو ساوي C أُس M اللي هي M سالب اتنين في
+
+79
+00:09:03,190 --> 00:09:09,750
+absolute X2 نيجاتيب X1 إذن هذا باستخدام المتباين
+
+80
+00:09:09,750 --> 00:09:14,290
+أسطار حيث ال M هذه أخدتها بساوي N زي اتنين
+
+81
+00:09:14,290 --> 00:09:19,390
+وبالتالي إذا ال N بساوي M نيجاتيب Twoفال absolute
+
+82
+00:09:19,390 --> 00:09:23,350
+value الأولانية أصغر حسب ال star أصغر من أو ساوي c
+
+83
+00:09:23,350 --> 00:09:27,650
+to m minus 2 في absolute x2 minus 1 Similarly
+
+84
+00:09:27,650 --> 00:09:31,730
+باستخدام ال star المتباين ال star ال absolute
+
+85
+00:09:31,730 --> 00:09:36,490
+value التاني هذه أو second term هذا بيطلع أصغر من
+
+86
+00:09:36,490 --> 00:09:43,530
+أو ساوي c to m minus 3 في absolute x2 minus x1 و
+
+87
+00:09:43,530 --> 00:09:44,130
+هكذا
+
+88
+00:09:46,560 --> 00:09:51,660
+إلى أن نصل آخر حد هيطلع عند you c to n minus واحد
+
+89
+00:09:51,660 --> 00:09:58,380
+في absolute x2 minus x1 طيب
+
+90
+00:09:58,380 --> 00:10:01,880
+ناخد الآن عامل مشترك
+
+91
+00:10:04,360 --> 00:10:11,340
+هذا بيساوي c to n negative two زائد c to n
+
+92
+00:10:11,340 --> 00:10:17,800
+negative three زائد و هكذا to c to n negative one
+
+93
+00:10:17,800 --> 00:10:23,460
+كل هذا مضروب في العامل المشترك absolute x two
+
+94
+00:10:23,460 --> 00:10:28,540
+minus x one المجموع
+
+95
+00:10:28,540 --> 00:10:36,720
+هذا هاخد عامل مشترك c to n minus واحد منهفهيبقى
+
+96
+00:10:36,720 --> 00:10:46,020
+لدي هنا c to m minus n minus 1 وهنا هاخد لو جسمت
+
+97
+00:10:46,020 --> 00:10:52,420
+had على c to n minus 1 هيطلع c to m minus n minus
+
+98
+00:10:52,420 --> 00:10:55,800
+2
+
+99
+00:10:55,800 --> 00:11:06,070
+وهكذا إلى أخر حد هيكون 1و طبعا كل هذا مضروب في
+
+100
+00:11:06,070 --> 00:11:17,890
+absolute x2 minus x1 y ساوي c to n negative one
+
+101
+00:11:17,890 --> 00:11:22,450
+الان هذه عبارة عن geometric progression متوالية
+
+102
+00:11:22,450 --> 00:11:30,230
+هندسية الحد الأول فيها واحد والأساس تبعها cفمجموعة
+
+103
+00:11:30,230 --> 00:11:33,930
+المتوالي الهندسية أو الـ geometric progression
+
+104
+00:11:33,930 --> 00:11:40,210
+مجموعة بساوي الحد الأول واحد سالب الحد الأخير
+
+105
+00:11:40,210 --> 00:11:47,170
+مضروب في الأساس اللي هو C فبطلع C to M negative N
+
+106
+00:11:47,170 --> 00:11:54,070
+على واحد minus الأساسإذن هذا مجموع الـ geometric
+
+107
+00:11:54,070 --> 00:11:58,910
+progression كل هذا مضروب في absolute X2 negative
+
+108
+00:11:58,910 --> 00:12:04,610
+X1 طيب
+
+109
+00:12:04,610 --> 00:12:10,330
+أنا عندي الـC أنا
+
+110
+00:12:10,330 --> 00:12:15,630
+عندي الـC الـC
+
+111
+00:12:15,630 --> 00:12:22,980
+عدد بين 0 و 1وبالتالي c to m minus n بيبقى العدد
+
+112
+00:12:22,980 --> 00:12:32,020
+بين سفر واحد وبالتالي اذا واحد سالب c اكيد هيطلع
+
+113
+00:12:32,020 --> 00:12:37,880
+اكبر من سفر اصغر
+
+114
+00:12:37,880 --> 00:12:38,440
+من واحد
+
+115
+00:12:41,670 --> 00:12:47,450
+أذا هاي أنا عندي هذا هستبدله بأصغر من هاي c to n
+
+116
+00:12:47,450 --> 00:12:53,570
+negative one الآن هذا الكسر ال bus تبعه واحد سالب
+
+117
+00:12:53,570 --> 00:12:59,970
+c to n minus n أصغر من واحد فهذا الكسر أصغر من
+
+118
+00:12:59,970 --> 00:13:06,150
+واحد على واحد سالب c أصغر من واحد على واحد سالب c
+
+119
+00:13:06,150 --> 00:13:10,510
+في absolute x to negative x one
+
+120
+00:13:14,990 --> 00:13:23,230
+طيب انا اندي برضه اخدنا قبل هيك مثال بيقول اذا كان
+
+121
+00:13:23,230 --> 00:13:29,790
+C أكبر من صفر اصغر من واحد هذا بيقدي ان ال limit ل
+
+122
+00:13:29,790 --> 00:13:35,290
+C to N او C to N سالب واحد لما N تقل ل infinity
+
+123
+00:13:35,290 --> 00:13:39,410
+بيساوي صفر فبالاستخدام المثال هذا اللي اثبتناه قبل
+
+124
+00:13:39,410 --> 00:13:47,130
+هيكبنلاحظ ان c to n سالب واحد هنا هذا ثابت وهذا
+
+125
+00:13:47,130 --> 00:13:54,710
+ثابت ف ال c to n سالب واحد تقول للسفر في ثابت تقول
+
+126
+00:13:54,710 --> 00:13:59,730
+لثابت في سفر يعني سفر إذا هذا المقدار كله tends to
+
+127
+00:13:59,730 --> 00:14:03,590
+zero as n tends to infinity
+
+128
+00:14:07,000 --> 00:14:12,180
+وبالتالي اذا هيك احنا اثبتنا ان ال limit ل
+
+129
+00:14:12,180 --> 00:14:19,740
+absolute xm minus xn لما ال M تقول ل infinity
+
+130
+00:14:19,740 --> 00:14:23,680
+بتساوي
+
+131
+00:14:23,680 --> 00:14:32,280
+سفر وطبعا ال M انا ماخدها هنا ال M ماخدها M اكبر
+
+132
+00:14:32,280 --> 00:14:36,860
+من N فلما ال M تقول ل infinity ال M ايضاتقول
+
+133
+00:14:36,860 --> 00:14:42,000
+لإنفينيتي إذا هنا ممكن أحط and M تقول لإنفينيتي
+
+134
+00:14:42,000 --> 00:14:46,220
+إذا هنا أثبتنا إن ال limit ل absolute XM minus XN
+
+135
+00:14:46,220 --> 00:14:51,540
+as N and M both tends to infinity بساوي سفر
+
+136
+00:14:51,540 --> 00:14:56,800
+وبالتالي إذا by above remark حسب ال remark اللي
+
+137
+00:14:56,800 --> 00:15:01,200
+فوق ال sequence XM is Cauchy
+
+138
+00:15:04,210 --> 00:15:12,090
+و هذا بيكمل برهان النظرية تمام؟ واضح؟
+
+139
+00:15:12,090 --> 00:15:16,690
+في أي استفسار؟ في أي سؤال على البرهان؟ في أي قطة
+
+140
+00:15:16,690 --> 00:15:23,570
+مش واضحة؟ is there any question؟
+
+141
+00:15:23,570 --> 00:15:27,790
+okay then this ends
+
+142
+00:15:30,870 --> 00:15:37,390
+section تلاتة خمسة and we are going to start a new
+
+143
+00:15:37,390 --> 00:15:44,490
+section هنبدأ section جديد وهذا ال section بتحدث
+
+144
+00:15:44,490 --> 00:15:55,870
+عن موضوع properly divergent sequences section
+
+145
+00:15:55,870 --> 00:16:00,450
+three point six
+
+146
+00:16:02,990 --> 00:16:11,770
+properly .. properly .. divergent .. divergent ..
+
+147
+00:16:11,770 --> 00:16:12,710
+sequences
+
+148
+00:16:17,180 --> 00:16:21,080
+أي يعني properly .. properly divergence sequence
+
+149
+00:16:21,080 --> 00:16:27,140
+احنا جفنا قبل هيك أمثلة examples about divergence
+
+150
+00:16:27,140 --> 00:16:31,200
+sequences من ال .. ال divergence sequences هذه كان
+
+151
+00:16:31,200 --> 00:16:36,480
+negative one to n بيقولوا أنه هذه has a divergence
+
+152
+00:16:36,480 --> 00:16:40,640
+برضه
+
+153
+00:16:40,640 --> 00:16:48,480
+sequence nDivergent means infinity وبالتالي
+
+154
+00:16:48,480 --> 00:16:51,480
+Divergent ال sequence negative and intense
+
+155
+00:16:51,480 --> 00:16:59,480
+وبالتالي Divergent و هكذا في كتير sequences مر
+
+156
+00:16:59,480 --> 00:17:04,520
+علينا sequences على الأقل هدول إذا ماكانش أكتر
+
+157
+00:17:04,520 --> 00:17:11,930
+وشوفنا ان كل ال sequences هذي are divergentهناك
+
+158
+00:17:11,930 --> 00:17:17,130
+لحد الآن ماتحدثناش عن تفريق التفريق في ال
+
+159
+00:17:17,130 --> 00:17:22,070
+divergence كنا نقول إن إذا كانت ال sequence
+
+160
+00:17:22,070 --> 00:17:26,550
+ماليهاش limit أو تقول ل infinity أو سالب infinity
+
+161
+00:17:26,550 --> 00:17:30,130
+فكنا نقول ال sequence is divergent أو not
+
+162
+00:17:30,130 --> 00:17:34,210
+convergent اليوم ال divergence sequences هنجزقهم
+
+163
+00:17:34,210 --> 00:17:40,450
+إلى نوعين في sequences هنقول عنهم divergentو في
+
+164
+00:17:40,450 --> 00:17:44,850
+نوع معين من الـ divergence sequences هنسميهم
+
+165
+00:17:44,850 --> 00:17:49,810
+properly divergent إذن الـ sequences اللي زي هدول
+
+166
+00:17:49,810 --> 00:17:54,890
+اللي ال limit بتاعتهم إما infinity أو negative
+
+167
+00:17:54,890 --> 00:17:59,110
+infinity طبعا هدول ال sequences are divergent لكن
+
+168
+00:17:59,110 --> 00:18:02,170
+هذا النوع من ال divergence sequences هنسميه
+
+169
+00:18:02,170 --> 00:18:06,410
+properly divergent متباعدة تباعدا صحيحا
+
+170
+00:18:09,530 --> 00:18:13,590
+Okay إذا ال .. ال sequences التلاتي هدول all of
+
+171
+00:18:13,590 --> 00:18:17,690
+them are divergent كلهم ممكن نقول عنهم divergent
+
+172
+00:18:17,690 --> 00:18:23,710
+لكن التنتين هدول الأخرانيين ممكن نقول عنهم أيضا
+
+173
+00:18:23,710 --> 00:18:27,490
+properly divergent أما هذه مقدرش أقول عنها
+
+174
+00:18:27,490 --> 00:18:32,450
+properly divergent مجرد divergent okay تمام إذا
+
+175
+00:18:32,450 --> 00:18:38,270
+نكتب التعريفات هذه definition
+
+176
+00:18:42,170 --> 00:18:49,570
+اتنين ستة وعشرين let
+
+177
+00:18:49,570 --> 00:18:52,590
+x in be sequence of real numbers
+
+178
+00:18:56,340 --> 00:19:05,960
+نقول إن xn tends to infinity أو limit xn as n
+
+179
+00:19:05,960 --> 00:19:14,500
+tends to infinity بساوي infinity اذا تحقق الشرط
+
+180
+00:19:14,500 --> 00:19:15,220
+التالي
+
+181
+00:19:23,170 --> 00:19:31,350
+for every real number alpha there exists capital N
+
+182
+00:19:31,350 --> 00:19:41,150
+depends on alpha natural number such that xn such
+
+183
+00:19:41,150 --> 00:19:47,270
+that لو كان N أكبر
+
+184
+00:19:47,270 --> 00:19:51,150
+من أو ساوي capital N هذا بقدر أن xn أكبر من alpha
+
+185
+00:19:53,550 --> 00:19:56,650
+اللي بتتكلموا لو سمحتوا ماتتكلمش امنع الكلام
+
+186
+00:19:59,910 --> 00:20:04,290
+إذا ما معنى الـ sequence Xn tends to infinity أو
+
+187
+00:20:04,290 --> 00:20:09,410
+limit لها بالساوية infinity معناه لأي number ألفة
+
+188
+00:20:09,410 --> 00:20:12,670
+نقدر نجد ال number الناترال نقدر نجد ال number
+
+189
+00:20:12,670 --> 00:20:13,090
+الناترال نقدر نجد ال number الناترال نقدر نجد ال
+
+190
+00:20:13,090 --> 00:20:14,290
+number الناترال نقدر نجد ال number الناترال نقدر
+
+191
+00:20:14,290 --> 00:20:14,970
+نجد ال number الناترال نقدر نجد ال number الناترال
+
+192
+00:20:14,970 --> 00:20:21,010
+نقدر نجد ال number الناترال نقدر نجد ال number
+
+193
+00:20:21,010 --> 00:20:25,370
+الناترال نقدر نجد ال number الناترال نقدر نجد ال
+
+194
+00:20:25,370 --> 00:20:26,370
+number الناترال نقدر نجد ال number الناترال نقدر
+
+195
+00:20:26,370 --> 00:20:27,990
+نجد ال number الناترال نقدر نجد ال number الناترال
+
+196
+00:20:27,990 --> 00:20:32,450
+نقدر نجد ال numberمن capital N أو كل حدودها for
+
+197
+00:20:32,450 --> 00:20:37,490
+large N أكبر من أي عدد Alpha أي عدد حقيقي Alpha
+
+198
+00:20:37,490 --> 00:20:41,810
+عشوائي okay إذا قدرنا نخلي حدود ال sequence أكبر
+
+199
+00:20:41,810 --> 00:20:46,330
+من أي عدد حقيقي مهما كان فال sequence معناته
+
+200
+00:20:46,330 --> 00:20:54,110
+نهايتها infinity بالمثل ممكن نعرف نقول x n tends
+
+201
+00:20:54,110 --> 00:21:01,710
+to negative infinity أو limitxn as n tends to
+
+202
+00:21:01,710 --> 00:21:07,450
+infinity بساوي negative infinity إذا تحقق الشرط
+
+203
+00:21:07,450 --> 00:21:17,190
+التالي for every beta real numberيوجد capital N
+
+204
+00:21:17,190 --> 00:21:24,390
+يعتمد على ال beta natural number such that لكل N
+
+205
+00:21:24,390 --> 00:21:29,750
+أكبر من أو ساوي capital N بطلع اندي xn أصغر من
+
+206
+00:21:29,750 --> 00:21:34,770
+beta إذا لو قدرت أخلي حدود ال sequence أصغر من أي
+
+207
+00:21:34,770 --> 00:21:40,640
+عدد حقيقي betafor large N فكل .. فالـ sequence هذه
+
+208
+00:21:40,640 --> 00:21:44,780
+بيقول إنها tends to negative infinity أو ال limit
+
+209
+00:21:44,780 --> 00:21:51,020
+بتاعتها is negative infinity هاي أمثلة examples
+
+210
+00:21:51,020 --> 00:22:03,600
+show
+
+211
+00:22:03,600 --> 00:22:06,180
+that limit
+
+212
+00:22:07,910 --> 00:22:13,670
+الـ sequence n بساوي infinity احنا كلنا عارفين
+
+213
+00:22:13,670 --> 00:22:18,790
+that the sequence of natural number ال limit
+
+214
+00:22:18,790 --> 00:22:26,490
+تبعتها infinity لكن ممكن نثبت الان هذا باستخدام
+
+215
+00:22:26,490 --> 00:22:31,590
+التعريف فنشوف
+
+216
+00:22:31,590 --> 00:22:37,560
+هاي البرهان proofحسب التعريف بدنا نبدأ بقول let
+
+217
+00:22:37,560 --> 00:22:45,180
+alpha belonging to R be given ناخد
+
+218
+00:22:45,180 --> 00:22:52,520
+alpha عدد حقيقي عشوائي by
+
+219
+00:22:52,520 --> 00:23:00,260
+Archimedean property حسب خاصية Archimedes
+
+220
+00:23:04,680 --> 00:23:13,220
+يوجد capital N عدد طبيعي بحيث انه capital N أكبر
+
+221
+00:23:13,220 --> 00:23:19,740
+من ال alpha نظبوت؟ هذا حسب ال Archimedean property
+
+222
+00:23:19,740 --> 00:23:22,240
+now
+
+223
+00:23:24,690 --> 00:23:28,990
+لو أخدت N bigger than or equal capital N هذا هيقدي
+
+224
+00:23:28,990 --> 00:23:35,330
+ان XN ال sequence هذه لحد العام XN تبعها ايش
+
+225
+00:23:35,330 --> 00:23:42,730
+بيساوي بيساوي N الان ال N هذه small n أكبر منه
+
+226
+00:23:42,730 --> 00:23:47,010
+يساوي capital N وانا بيختار capital N by
+
+227
+00:23:47,010 --> 00:23:49,350
+Archimedean property أكبر من Alpha
+
+228
+00:23:55,340 --> 00:24:01,740
+وبالتالي إذا ال .. ال .. ال implication هذه تتحقق
+
+229
+00:24:01,740 --> 00:24:08,220
+هنا أثبتنا for any Alpha تنتمي ل R there exists
+
+230
+00:24:08,220 --> 00:24:12,600
+natural number يعتمد على Alpha هيوجه .. يعتمد على
+
+231
+00:24:12,600 --> 00:24:18,160
+Alpha N مرتبطة بAlpha بحيث لكل N أكبر من أوساو
+
+232
+00:24:18,160 --> 00:24:22,740
+capital N طول عندي Xn أكبر من Alpha therefore
+
+233
+00:24:27,120 --> 00:24:31,940
+Alpha capital N definition of limit
+
+234
+00:24:38,380 --> 00:24:43,960
+أو alpha capital N definition بطلع عندي limit xn
+
+235
+00:24:43,960 --> 00:24:50,940
+بساوي infinity طبعا هنا xn مقصود فيها الحد العام
+
+236
+00:24:50,940 --> 00:24:56,440
+لل sequence N يعني xn بساوي N okay تمام واضح
+
+237
+00:24:56,440 --> 00:25:04,360
+البرهان طب هاي مثال تاني show thatإنه limit الـ
+
+238
+00:25:04,360 --> 00:25:06,800
+sequence اللي الحد اللي عام تبقىها negative
+
+239
+00:25:06,800 --> 00:25:16,080
+Interbia equals negative infinity هنطبق
+
+240
+00:25:16,080 --> 00:25:23,660
+التعريف في الجزء التاني ونشوف كيف ممكن نثبت إن ال
+
+241
+00:25:23,660 --> 00:25:28,940
+implication كيف لأي Beta بقدر ألاقي capital N
+
+242
+00:25:28,940 --> 00:25:35,040
+يعتمد على Betaبحيث ان ال implication هذه هي تتحقق
+
+243
+00:25:35,040 --> 00:25:40,320
+هاي
+
+244
+00:25:40,320 --> 00:25:45,800
+البرهان proof بالمناسبة
+
+245
+00:25:45,800 --> 00:25:51,980
+انا ماعنديش يعني عصة سحرية عشان اعرف مسبقا لأي
+
+246
+00:25:51,980 --> 00:25:58,440
+beta كيف اختار ال N ماعيش عصة سحرية فبنعمل
+
+247
+00:25:58,440 --> 00:26:05,060
+analysis تحليلوبنكتشف كيف نختار الـ capital N لأي
+
+248
+00:26:05,060 --> 00:26:12,500
+given Beta إذا هنا حد البرهان بـ let Beta بأي real
+
+249
+00:26:12,500 --> 00:26:16,800
+number بـ given طبعا
+
+250
+00:26:16,800 --> 00:26:21,540
+حسب التعريف عشان أثبت أنه ال sequence الحد العام
+
+251
+00:26:21,540 --> 00:26:22,760
+تبعها XN
+
+252
+00:26:25,210 --> 00:26:29,610
+بساوي سالب n تربية عشان اثبت ان ال limit لل
+
+253
+00:26:29,610 --> 00:26:33,830
+sequence هذه بساوي negative infinity بدي اثبت بدي
+
+254
+00:26:33,830 --> 00:26:37,470
+ارد على ال given beta هذه بcapital N تعتمد عليها
+
+255
+00:26:37,470 --> 00:26:42,510
+بحيث ان ال implication هذه تتحقق فبنيجي بنعمل زي
+
+256
+00:26:42,510 --> 00:26:47,510
+ما عملنا في تعريف epsilon capital N للنهايات انا
+
+257
+00:26:47,510 --> 00:26:53,340
+بقول من الآخر انا عايز ان xnاللي هي سالب enter
+
+258
+00:26:53,340 --> 00:26:58,840
+بيها بدي هذه في النهاية for any given beta real
+
+259
+00:26:58,840 --> 00:27:05,520
+number بدي x in اللي هي negative n squared بديها
+
+260
+00:27:05,520 --> 00:27:12,100
+less than beta تمام؟
+
+261
+00:27:13,650 --> 00:27:19,790
+و طبعا هذا لكل N أكبر من أو ساوي capital N فما هي
+
+262
+00:27:19,790 --> 00:27:27,090
+ال N أنا بدي أجيب ال N اللي بتخلي هذا الكلام صحيح
+
+263
+00:27:27,090 --> 00:27:31,270
+أنا
+
+264
+00:27:31,270 --> 00:27:36,030
+بعرف أنه لازم ال N تكون أكبر من أو ساوي capital N
+
+265
+00:27:36,030 --> 00:27:38,630
+وبالتالي
+
+266
+00:27:40,310 --> 00:27:47,610
+هذا بيقدّي أن N تربية أكبر من أو يساوي N صح؟ أكبر
+
+267
+00:27:47,610 --> 00:27:52,550
+من أو يساوي capital N وهذا
+
+268
+00:27:52,550 --> 00:27:59,530
+بيقدّي أن سالب N تربية أصغر من أو يساوي سالب
+
+269
+00:27:59,530 --> 00:28:00,290
+capital N
+
+270
+00:28:05,790 --> 00:28:13,770
+وانا بدي هذا يطلع أصغر من ال beta عشان يطلع xn
+
+271
+00:28:13,770 --> 00:28:22,870
+أصغر من beta صح؟ إذا بسأل نفسي متى هذا سالب n
+
+272
+00:28:22,870 --> 00:28:27,670
+تربيه اللي هو xn أصغر من beta لإنما negative
+
+273
+00:28:27,670 --> 00:28:33,880
+capital N أصغر من beta إذا for anyfor any beta
+
+274
+00:28:33,880 --> 00:28:40,360
+belonging to R يقول هنا for any beta belonging to
+
+275
+00:28:40,360 --> 00:28:42,640
+R يقول هنا for any beta belonging to R يقول هنا
+
+276
+00:28:42,640 --> 00:28:44,360
+for any beta belonging to R يقول هنا for any beta
+
+277
+00:28:44,360 --> 00:28:44,580
+belonging to R يقول هنا for any beta belonging to
+
+278
+00:28:44,580 --> 00:28:44,580
+R يقول هنا for any beta belonging to R يقول هنا
+
+279
+00:28:44,580 --> 00:28:44,580
+for any beta belonging to R يقول هنا for any beta
+
+280
+00:28:44,580 --> 00:28:44,580
+belonging to R يقول هنا for any beta belonging to
+
+281
+00:28:44,580 --> 00:28:44,580
+R يقول هنا for any beta belonging to R يقول هنا
+
+282
+00:28:44,580 --> 00:28:44,580
+for any beta belonging to R يقول هنا for any beta
+
+283
+00:28:44,580 --> 00:28:44,580
+belonging to R يقول هنا for any beta belonging to
+
+284
+00:28:44,580 --> 00:28:44,580
+R يقول هنا for any beta belonging to R يقول هنا
+
+285
+00:28:44,580 --> 00:28:44,600
+for any beta belonging to R يقول هنا for any beta
+
+286
+00:28:44,600 --> 00:28:44,700
+belonging to R يقول هنا for any beta belonging to
+
+287
+00:28:44,700 --> 00:28:44,720
+R يقول هنا for any beta belonging to R يقول هنا
+
+288
+00:28:44,720 --> 00:28:44,740
+for any beta belonging to R يقول هنا for any beta
+
+289
+00:28:44,740 --> 00:28:44,740
+belonging to R يقول هنا for any beta belonging to
+
+290
+00:28:44,740 --> 00:28:50,520
+R يقول هنا for any beta belonging to R يقول هنا
+
+291
+00:28:50,520 --> 00:28:54,400
+for any beta belonging to R يقول هنا for any beta
+
+292
+00:28:54,400 --> 00:29:04,050
+belonging to R يقولوهذا بقدر اختاره by Archimedean
+
+293
+00:29:04,050 --> 00:29:11,170
+propertyby Archimedean property لأي real number
+
+294
+00:29:11,170 --> 00:29:15,610
+beta سالب beta is real number و بقدر ألاقي capital
+
+295
+00:29:15,610 --> 00:29:20,410
+N أكبر من أي real number by Archimedean property
+
+296
+00:29:20,410 --> 00:29:23,810
+اذا capital N اللي انا عايزها أكبر من ال given
+
+297
+00:29:23,810 --> 00:29:29,270
+بسالب ال given beta أكبر من سالب ال given beta اذا
+
+298
+00:29:29,270 --> 00:29:36,450
+هنا باجي بقول let beta be given it choose using
+
+299
+00:29:38,220 --> 00:29:44,220
+الـ Archimedean property capital
+
+300
+00:29:44,220 --> 00:29:50,640
+N عدد طبيعي natural number such that capital N
+
+301
+00:29:50,640 --> 00:29:56,660
+أكبر من negative beta وهذا مقدر أعمله by
+
+302
+00:29:56,660 --> 00:30:03,340
+Archimedean propertyالان تعالى نشوف اذا انا لأي
+
+303
+00:30:03,340 --> 00:30:08,660
+beta وجدت عدد طبيعي هيه بيعتمد على beta هاي
+
+304
+00:30:08,660 --> 00:30:12,420
+capital M تعتمد على beta مرتبطة فيها بالمتباينة
+
+305
+00:30:12,420 --> 00:30:19,340
+هذهالان فاضل باقي أثبت أنه لو أخدت أي small n أكبر
+
+306
+00:30:19,340 --> 00:30:25,540
+من أو ساوي capital N بدي أثبت أن هذا بيقدي أن xn
+
+307
+00:30:25,540 --> 00:30:30,820
+أصغر من beta طيب هاي n أكبر من أو ساوي capital N
+
+308
+00:30:30,820 --> 00:30:35,400
+شوف أن هذا بيقدي أن n تربية أكبر من أو ساوي small
+
+309
+00:30:35,400 --> 00:30:41,760
+n لأي عداد طبيعي طيب small n أكبر من أو ساوي
+
+310
+00:30:41,760 --> 00:30:42,520
+capital N
+
+311
+00:30:45,500 --> 00:30:51,940
+و capital N أكبر من negative beta حسب اختيارنا اذا
+
+312
+00:30:51,940 --> 00:30:56,480
+هذا بيقدي اضرب في سالب واحد اذا هذا بيقدي ان xn
+
+313
+00:30:56,480 --> 00:31:01,520
+اللي هي سالب او negative n تربيه اضرب في سالب واحد
+
+314
+00:31:01,520 --> 00:31:07,760
+هذا بيصير اصغر من beta وهذه
+
+315
+00:31:07,760 --> 00:31:12,340
+هي ال implication اللي انا عايز احققها صح؟ اذا انا
+
+316
+00:31:12,340 --> 00:31:20,580
+هيندي صارby definition حققت انه for any beta يوجد
+
+317
+00:31:20,580 --> 00:31:26,100
+capital N عدد طبيعي يعتمد على ال beta بحيث لكل N
+
+318
+00:31:26,100 --> 00:31:32,640
+أكبر من أو يساوي capital N هذا بيقدي ان Xn أصغر من
+
+319
+00:31:32,640 --> 00:31:35,060
+beta therefore by definition
+
+320
+00:31:37,910 --> 00:31:44,330
+حسب التعريف التاني بطلع عندي limit xn equals
+
+321
+00:31:44,330 --> 00:31:53,950
+negative infinity وهو المطلوب تمام هيك؟ في أي
+
+322
+00:31:53,950 --> 00:31:59,250
+سؤال؟ في أي استفسار؟
+
+323
+00:31:59,250 --> 00:32:04,730
+طيب ناخد الآن نظرية
+
+324
+00:32:19,760 --> 00:32:32,580
+خلّيني أغير الجلم theorem
+
+325
+00:32:32,580 --> 00:32:40,260
+رقمها تمانية عشرين هذه
+
+326
+00:32:40,260 --> 00:32:49,040
+تعتبر monotone convergence theorem for properly
+
+327
+00:32:55,550 --> 00:33:03,530
+divergent sequences monotone
+
+328
+00:33:03,530 --> 00:33:06,630
+convergence theorem for properly divergent
+
+329
+00:33:06,630 --> 00:33:15,410
+sequences النظرية هذه بتنص على ان a monotone .. a
+
+330
+00:33:15,410 --> 00:33:19,450
+monotone sequence
+
+331
+00:33:21,930 --> 00:33:35,070
+in R is properly is properly divergent if
+
+332
+00:33:35,070 --> 00:33:41,270
+and only if it is unbounded
+
+333
+00:33:41,270 --> 00:33:46,230
+in
+
+334
+00:33:46,230 --> 00:33:50,950
+fact في حقيقة الأمر in fact
+
+335
+00:33:54,620 --> 00:34:00,520
+إذا XN غير
+
+336
+00:34:00,520 --> 00:34:06,580
+مجموعة ومزيد
+
+337
+00:34:06,580 --> 00:34:09,740
+ومزيد
+
+338
+00:34:09,740 --> 00:34:13,280
+ثم
+
+339
+00:34:13,280 --> 00:34:18,400
+XN يتنقل إلى الانفصال
+
+340
+00:34:27,110 --> 00:34:34,510
+وإذا كان Xn غير مجموعة ومتناخصة
+
+341
+00:34:34,510 --> 00:34:38,090
+فإن
+
+342
+00:34:38,090 --> 00:34:43,330
+سيكوان Xn يتنقص إلى نقاط نقاط نقاط نقاط نقاط نقاط
+
+343
+00:34:43,330 --> 00:34:45,210
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+344
+00:34:45,210 --> 00:34:50,090
+نقاط
+
+345
+00:34:50,090 --> 00:34:50,850
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+346
+00:34:50,850 --> 00:34:50,850
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+347
+00:34:50,850 --> 00:34:50,850
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+348
+00:34:50,850 --> 00:34:50,990
+نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط نقاط
+
+349
+00:34:50,990 --> 00:34:51,050
+نقاط نقاط نقاط نقاط نقاط نق
+
+350
+00:35:00,740 --> 00:35:12,340
+الأول ..الأول مصادر يتبع من
+
+351
+00:35:12,340 --> 00:35:16,960
+الـ monotone convergence theorem
+
+352
+00:35:21,340 --> 00:35:24,060
+الـ first statement اللي هو بيقول a monotone
+
+353
+00:35:24,060 --> 00:35:29,100
+sequence of real numbers is properly divergent if
+
+354
+00:35:29,100 --> 00:35:32,300
+and only if it is unbounded هذا نتيجة على الـ
+
+355
+00:35:32,300 --> 00:35:35,740
+monotone convergence theorem لأن الـ monotone
+
+356
+00:35:35,740 --> 00:35:40,340
+convergence theorem اللي أخدناها قبل هيك بتقول a
+
+357
+00:35:40,340 --> 00:35:45,600
+monotone sequence is convergent if and only if it
+
+358
+00:35:45,600 --> 00:35:51,040
+is boundedفمن نفس النظرية ومن نفس النص ممكن نقول
+
+359
+00:35:51,040 --> 00:35:54,960
+الـ monotone sequence is not convergent أو
+
+360
+00:35:54,960 --> 00:36:00,580
+divergent if and only if it is unbounded بغض النظر
+
+361
+00:36:00,580 --> 00:36:04,800
+ال divergence هنا شو نوعه okay إذا ال face
+
+362
+00:36:04,800 --> 00:36:10,000
+statement العبارة الأولى هذه لحد هنانتيجة على او
+
+363
+00:36:10,000 --> 00:36:15,900
+corollary to the monotone convergence theorem الآن
+
+364
+00:36:15,900 --> 00:36:22,400
+بدنا نثبت الأجزاء واحد واثنين الآن خلّينا نثبت
+
+365
+00:36:22,400 --> 00:36:28,260
+الجزء الأول والتاني برهانه بالمثل similar to one
+
+366
+00:36:28,260 --> 00:36:32,660
+إذا هنا نثبت الجزء الأول assume
+
+367
+00:36:35,940 --> 00:36:45,260
+إن XIN is a sequence of real numbers is unbounded
+
+368
+00:36:45,260 --> 00:36:48,540
+and
+
+369
+00:36:48,540 --> 00:36:51,560
+increasing
+
+370
+00:37:02,370 --> 00:37:09,050
+بنثبت ان الـ sequence xn properly
+
+371
+00:37:09,050 --> 00:37:10,890
+divergent to infinity
+
+372
+00:37:28,250 --> 00:37:33,650
+طيب بس نستذكر هنا في هذه المناسبة خلينا نستذكر
+
+373
+00:37:33,650 --> 00:37:39,950
+تعريف ال bounded sequence definition a sequence x
+
+374
+00:37:39,950 --> 00:37:47,230
+in contained in R is bounded is
+
+375
+00:37:47,230 --> 00:37:54,550
+bounded if and only if there exists positive real
+
+376
+00:37:54,550 --> 00:38:04,160
+numberأو حتى عدد حقيقي M بحيث أنه absolute X N
+
+377
+00:38:04,160 --> 00:38:13,300
+أصغر من أو يساوي M for every N ينتمي ل N مش هيك
+
+378
+00:38:13,300 --> 00:38:20,010
+تعريف ال bounded sequence؟وطبعا دايما لأي sequence
+
+379
+00:38:20,010 --> 00:38:24,510
+دايما ال x in بالمناسبة أصغر من أو ساوى القيمة
+
+380
+00:38:24,510 --> 00:38:28,130
+المطلقة تبعته أي real number is less than or equal
+
+381
+00:38:28,130 --> 00:38:34,630
+its absolute value هذا مافيش فيها شكل الان تعالوا
+
+382
+00:38:34,630 --> 00:38:39,390
+نعمل negation لهذا مامعنى ان هنا هقول in a sense
+
+383
+00:38:39,390 --> 00:38:47,030
+ال sequence x in is unboundedأحنا فرضين أن ال
+
+384
+00:38:47,030 --> 00:38:51,770
+sequence تبعتي unbounded و increasing بما أن ال
+
+385
+00:38:51,770 --> 00:38:57,530
+sequence x in is unbounded
+
+386
+00:38:57,530 --> 00:39:03,150
+فلكل
+
+387
+00:39:03,150 --> 00:39:10,730
+in then
+
+388
+00:39:10,730 --> 00:39:24,330
+for every alphafor every alpha عدد حقيقي يوجد
+
+389
+00:39:24,330 --> 00:39:29,210
+by
+
+390
+00:39:29,210 --> 00:39:40,080
+Archimedean propertyلأ يوجد N يعتمد على Alpha عدب
+
+391
+00:39:40,080 --> 00:39:48,060
+طبيعي بحيث ان ال X رقم capital N Alpha هذا بيطلع
+
+392
+00:39:48,060 --> 00:39:55,700
+أكبر من Alpha يعني
+
+393
+00:39:55,700 --> 00:40:00,360
+لو كان هذا ال M أسمنها Alpha
+
+394
+00:40:03,420 --> 00:40:07,900
+معناه أن الـ sequence هذه تكون bounded فما معناه
+
+395
+00:40:07,900 --> 00:40:10,860
+أن الـ sequence هذه تكون unbounded معناه أننا بدنا
+
+396
+00:40:10,860 --> 00:40:15,480
+ننفي الشرط هذا عشان أنفي الشرط هذا هذا معناه أن
+
+397
+00:40:15,480 --> 00:40:20,920
+بدل يوجد alpha موجبة لكل alpha سواء موجبة أو سالبة
+
+398
+00:40:20,920 --> 00:40:25,980
+لكل alpha for any alpha يوجد
+
+399
+00:40:28,130 --> 00:40:34,230
+بدل لكل n عدد طبيعي يوجد عدد طبيعي capital N يوجد
+
+400
+00:40:34,230 --> 00:40:41,740
+واحد capital N عدد طبيعي مافي لكل يوجدبحيث أن هذا
+
+401
+00:40:41,740 --> 00:40:48,720
+الـ xn بدل
+
+402
+00:40:48,720 --> 00:40:52,880
+ما هي أصغر من أو ساوي Alpha نفي xn أصغر من أو ساوي
+
+403
+00:40:52,880 --> 00:41:00,880
+Alpha هو أكبر من Alpha تمام؟ إذن هذا هو نفي الشرط
+
+404
+00:41:00,880 --> 00:41:04,940
+هذا أو نفي boundednessإذا الـ sequence unbounded
+
+405
+00:41:04,940 --> 00:41:09,400
+معناه لأي عدد حقيقي Alpha فهي عدد طبيعي يعتمد على
+
+406
+00:41:09,400 --> 00:41:14,680
+Alpha بحيث ان ال X المؤشر تبعه capital N بيطلع
+
+407
+00:41:14,680 --> 00:41:19,340
+أكبر من Alpha طيب الآن بما أن ال sequence
+
+408
+00:41:19,340 --> 00:41:27,540
+increasing as ال sequence XN is increasing احنا
+
+409
+00:41:27,540 --> 00:41:35,200
+فرضين انها متزايدةWe get نحصل على لو كان n أكبر من
+
+410
+00:41:35,200 --> 00:41:41,620
+أو ساوي n of alpha فهذا بالتأكيد هيقدّي أن x
+
+411
+00:41:41,620 --> 00:41:47,060
+المؤشر تبعها small n أكبر من أو ساوي x المؤشر
+
+412
+00:41:47,060 --> 00:41:52,840
+تبعها n of alpha وهذا من هنا أكبر من alpha
+
+413
+00:41:56,150 --> 00:42:02,150
+أذن خلّيني ألخّص شو عملنا أحنا أثبتنا الآن أن for
+
+414
+00:42:02,150 --> 00:42:06,570
+every alpha real number there exists a natural
+
+415
+00:42:06,570 --> 00:42:14,310
+number depends on alpha هذا هو بحيث أنه لكل N أكبر
+
+416
+00:42:14,310 --> 00:42:20,530
+من أو ساوي capital N طلع ندي xn أكبر من alphaهذا
+
+417
+00:42:20,530 --> 00:42:25,310
+اذا حسب التعريف الأولاني حسب تعريف one by
+
+418
+00:42:25,310 --> 00:42:32,350
+definition one طبعا هذا معناه ان limit x in بساوي
+
+419
+00:42:32,350 --> 00:42:39,170
+infinity وهو المطلوب okay برهان الجزء التاني the
+
+420
+00:42:39,170 --> 00:42:46,290
+proof of part اتنين is similar
+
+421
+00:42:49,840 --> 00:42:55,180
+two one فحاسبكم انتوا تكتبوا البرهان تبعه okay
+
+422
+00:42:55,180 --> 00:43:01,640
+تمام و هيك بنكون كملنا النظرية اذا هنوقف هنا و
+
+423
+00:43:01,640 --> 00:43:10,420
+بنكمل ان شاء الله الموضوع هذا المرة القادمة فشوفكم
+
+424
+00:43:10,420 --> 00:43:11,720
+ان شاء الله المرة الجاية على فيه
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..87b133f431299ba3a41216c55284a1317cdd4cc3
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_postprocess.srt
@@ -0,0 +1,1448 @@
+1
+00:00:00,000 --> 00:00:01,100
+موسيقى
+
+2
+00:00:19,710 --> 00:00:25,810
+السلام عليكم ان شاء الله اليوم هنكمل section أربعة
+
+3
+00:00:25,810 --> 00:00:31,470
+اتنين اللي بدأناها المرة اللي فاتت في المرة
+
+4
+00:00:31,470 --> 00:00:36,810
+السابقة أخدنا نظريات النهايات أو قوانين النهايات
+
+5
+00:00:37,440 --> 00:00:44,420
+للـ functions و كنتائج
+
+6
+00:00:44,420 --> 00:00:50,220
+على قوانين النهايات لدينا النظرية التالية theorem
+
+7
+00:00:50,220 --> 00:00:55,020
+واحد
+
+8
+00:00:56,630 --> 00:01:04,910
+إذا P of X بساوي A N في X to N زائد A N minus one
+
+9
+00:01:04,910 --> 00:01:12,290
+X to N minus one زائد و هكذا زائد A one في X زائد
+
+10
+00:01:12,290 --> 00:01:16,450
+A zero is a polynomial
+
+11
+00:01:25,480 --> 00:01:33,060
+Polynomial function of X then
+
+12
+00:01:33,060 --> 00:01:47,820
+for any C real number limit P of X as X tends to C
+
+13
+00:01:47,820 --> 00:01:53,740
+بساوي قيمة ال polynomial and C then if
+
+14
+00:01:56,280 --> 00:02:06,620
+R of X بساوي P
+
+15
+00:02:06,620 --> 00:02:15,740
+of X over Q of X where
+
+16
+00:02:15,740 --> 00:02:18,860
+P
+
+17
+00:02:18,860 --> 00:02:22,580
+و Q are polynomial
+
+18
+00:02:26,290 --> 00:02:41,850
+الـ X and if Q and C لا تساوي سفر then limit ل R
+
+19
+00:02:41,850 --> 00:02:50,930
+of X as X tends to C بساوي R محسوب عن C البرهين
+
+20
+00:02:50,930 --> 00:02:59,720
+سهلة proveالبرنامج الأول note first
+
+21
+00:02:59,720 --> 00:03:03,500
+that
+
+22
+00:03:03,500 --> 00:03:08,180
+limit
+
+23
+00:03:08,180 --> 00:03:17,480
+ل x as x tends to c بيساوي c بيقدّي أن ال limitلـ
+
+24
+00:03:17,480 --> 00:03:25,240
+x to the power k as x tends to c بساوي c to k for
+
+25
+00:03:25,240 --> 00:03:30,580
+all k بساوي صفر واحد اتنين إلى ما ننهي
+
+26
+00:03:33,860 --> 00:03:37,040
+أثبتنا قبل ذلك أن الـ limit لل identity function
+
+27
+00:03:37,040 --> 00:03:42,500
+لما x تقول ل c بساوة c أثبتنا ذلك باستخدام تعريف
+
+28
+00:03:42,500 --> 00:03:46,480
+epsilon delta قلنا لأي given epsilon choose delta
+
+29
+00:03:46,480 --> 00:03:51,080
+بساوة epsilon الآن
+
+30
+00:03:51,080 --> 00:03:57,010
+حسب نظريات النهاياتlimit x to the power k بيساوي
+
+31
+00:03:57,010 --> 00:04:01,030
+limit x ضرب نفسها مضروبة في نفسها كام المرات
+
+32
+00:04:01,030 --> 00:04:05,250
+واخدنا في المحاضرة الفاترة انه limit f of x الكل
+
+33
+00:04:05,250 --> 00:04:12,470
+أس m بيساوي limit f of x الكل أس m فهذا بيطلع
+
+34
+00:04:12,470 --> 00:04:20,490
+limit x هذا بيطلع c limit تبع ال x c هذا بيطلع c
+
+35
+00:04:20,490 --> 00:04:23,610
+أس k الآن
+
+36
+00:04:28,400 --> 00:04:35,780
+by limit by limit
+
+37
+00:04:35,780 --> 00:04:47,060
+theorem نظرية النهايات بطلع عندي ال limit ل P of X
+
+38
+00:04:47,060 --> 00:04:52,140
+as X tends to C بساوي ال limit هي عندي ال
+
+39
+00:04:52,140 --> 00:04:57,810
+polynomiallimit المجموعة اثبتنا انه limit المجموعة
+
+40
+00:04:57,810 --> 00:05:04,650
+بيستخدم مجموعة limits فlimit x to limit الحد الأول
+
+41
+00:05:04,650 --> 00:05:09,510
+زاد
+
+42
+00:05:09,510 --> 00:05:11,750
+limit الحد التاني
+
+43
+00:05:18,500 --> 00:05:25,660
+وهكذا إذا limit a1 في x as x tends to c زائد limit
+
+44
+00:05:25,660 --> 00:05:35,370
+a0 as x tends to cوالان limit حاصل ثابت فى دالة
+
+45
+00:05:35,370 --> 00:05:42,350
+بساوية ثابت a n المعاملات هذه a n و a n سالب واحد
+
+46
+00:05:42,350 --> 00:05:46,450
+الاخرى دى كلها ثوابت ف limit ثابت فى دالة بساوية
+
+47
+00:05:46,450 --> 00:05:54,710
+ثابت فى limit x أُس n اللى هى c أُس n زايد limit
+
+48
+00:05:54,710 --> 00:06:02,540
+ثابت فى دالة x أُس n سالب واحدبطلع C أُس N سالب
+
+49
+00:06:02,540 --> 00:06:10,780
+واحد وهكذا إلى A واحد في limit X لما X تقوى لـ C
+
+50
+00:06:10,780 --> 00:06:17,180
+بساوي C limit A Zero بطلع A Zero الآن هذه هي نفس
+
+51
+00:06:17,180 --> 00:06:24,820
+ال Polynomial المحسوب عند X بساوي C إذن هذا بثبت
+
+52
+00:06:24,820 --> 00:06:33,430
+الجزء الأول من النظريةلإثبات الجزء التاني Assume
+
+53
+00:06:33,430 --> 00:06:46,170
+أن الـ Q عند الـ C لا يساوي سفر ثم
+
+54
+00:06:46,170 --> 00:06:55,590
+Limit لR of X لما X تقول إلى C بساوي Limit البصر
+
+55
+00:06:58,130 --> 00:07:06,090
+P of X as X tends to C على limit Q of X لما X تقول
+
+56
+00:07:06,090 --> 00:07:12,810
+إلى C الان
+
+57
+00:07:12,810 --> 00:07:16,790
+باستخدام الجزء الأول من النظرية هذه كثيرة عدود
+
+58
+00:07:16,790 --> 00:07:24,430
+وبالتالي limit لها عن C بساوي قيمتها عن C وQ of X
+
+59
+00:07:24,430 --> 00:07:30,350
+برضه كثيرة حدودالقانون اللي عند أي عدد c بيساوي
+
+60
+00:07:30,350 --> 00:07:38,130
+قيمتها عن c ومع أن q of c هنا لا يساوي سفر فبقدر
+
+61
+00:07:38,130 --> 00:07:43,650
+استخدم القانون تبع limit الكسر بساوي أو limit قسم
+
+62
+00:07:43,650 --> 00:07:46,530
+الدالتين بساوي خارج قسم ال limits
+
+63
+00:07:51,050 --> 00:07:55,270
+لو Q of C بيساوي سفر ماقدرش اعمل الكلام هذا لو ال
+
+64
+00:07:55,270 --> 00:07:59,190
+limit هذه بيساوي سفر ماقدرش اوزع ال limit على ال
+
+65
+00:07:59,190 --> 00:08:03,610
+bust و المخان طبما هذا بالظبط هو عبارة عن ال
+
+66
+00:08:03,610 --> 00:08:10,090
+function R محسوب عن C وهذا بكمل برهان الجزء التاني
+
+67
+00:08:10,090 --> 00:08:12,230
+okay هذه أمثلة
+
+68
+00:08:23,910 --> 00:08:29,950
+Find limit لـ
+
+69
+00:08:29,950 --> 00:08:35,650
+x تربية سالب اتنين x زائد واحد لما x تقول لسفر
+
+70
+00:08:35,650 --> 00:08:44,170
+فهذه عبارة عن كثير تحدود P of X فحسب
+
+71
+00:08:44,170 --> 00:08:48,390
+الجزء الأول من النظرية السابقة limit ل P of X يطلع
+
+72
+00:08:48,390 --> 00:08:51,230
+P and سفر صح؟
+
+73
+00:08:53,080 --> 00:08:56,160
+عودة عن x بصفر مباشرة يعني يعني عودة عن x بصفر
+
+74
+00:08:56,160 --> 00:08:58,540
+مباشرة يعني يعني عودة عن x بصفر مباشرة يعني يعني
+
+75
+00:08:58,540 --> 00:08:58,640
+عودة عن x بصفر مباشرة يعني يعني عودة عن x بصفر
+
+76
+00:08:58,640 --> 00:09:03,040
+مباشرة يعني عودة
+
+77
+00:09:03,040 --> 00:09:03,220
+عن x بصفر مباشرة يعني عودة عن x بصفر مباشرة يعني
+
+78
+00:09:03,220 --> 00:09:03,960
+عودة عن x بصفر مباشرة يعني عودة عن x بصفر مباشرة
+
+79
+00:09:03,960 --> 00:09:09,000
+يعني عودة عن x بصفر مباشرة يعني عودة عن x بصفر
+
+80
+00:09:09,000 --> 00:09:14,540
+مباشرة يعني عودة عن x بصفر مباشرة يعني عودة عن x
+
+81
+00:09:14,540 --> 00:09:19,920
+بصفر مباشرة يعني عودة عن x بصفر مباشرة يعني عودة
+
+82
+00:09:24,080 --> 00:09:28,500
+طبعا هذه كثيرة حدود على كثيرة حدود وبنلاحظ انه
+
+83
+00:09:28,500 --> 00:09:32,320
+limit كثيرة الحدود اللي في المقام لما x تقوله سالب
+
+84
+00:09:32,320 --> 00:09:37,320
+واحد بيطلع اتنين لا يساوي سفر اذا حسب ال band
+
+85
+00:09:37,320 --> 00:09:44,960
+التاني بقدر اعوض عن x بيساوي سالب واحد مباشرة يعني
+
+86
+00:09:44,960 --> 00:09:51,040
+لو سميت ال function هذه R of X فالمفروض هذا يطلع R
+
+87
+00:09:51,040 --> 00:09:57,660
+عن سالب واحديعني عوض عن x بساوي سالب واحد فبطلع
+
+88
+00:09:57,660 --> 00:10:02,300
+عندي خمسة في سالب واحد تارديه بطلع خمسة سالب سالب
+
+89
+00:10:02,300 --> 00:10:09,360
+واحد بطلع واحد زائد اتنين على واحد زائد واحد و
+
+90
+00:10:09,360 --> 00:10:14,240
+بطلع تمانية على اتنين بطلع اربعة
+
+91
+00:10:20,930 --> 00:10:26,470
+أو ممكن نستخدم حل تاني نستخدم قوانين نهايات نقول
+
+92
+00:10:26,470 --> 00:10:31,970
+نهاية دالة كسرية زي هذه بساوي limit ال bus علي
+
+93
+00:10:31,970 --> 00:10:35,730
+limit المقام بشرط ان limit المقام ماساوي سفر limit
+
+94
+00:10:35,730 --> 00:10:39,490
+المقام طلعت ماساوي سفر بساوي هي اتنين اذا بقدر
+
+95
+00:10:39,490 --> 00:10:42,150
+استخدم القانون لكن لو كانت limit المقام بساوي سفر
+
+96
+00:10:42,150 --> 00:10:45,130
+اذا مابستطيع استخدم هذا القانون
+
+97
+00:10:50,800 --> 00:10:55,080
+في كثير من النظريات الخاصة بنهايات الـ sequences
+
+98
+00:10:55,080 --> 00:11:00,380
+أيضا في بقبلها نظريات أخرى خاصة بنهايات الـ
+
+99
+00:11:00,380 --> 00:11:04,620
+functions فمن
+
+100
+00:11:04,620 --> 00:11:05,880
+النظريات هذه
+
+101
+00:11:18,220 --> 00:11:25,280
+من هذه النظريات النظرية التالية let f be a
+
+102
+00:11:25,280 --> 00:11:38,240
+function from a to r and c be a cluster point
+
+103
+00:11:38,240 --> 00:11:40,320
+of A
+
+104
+00:11:50,350 --> 00:12:00,110
+suppose أن الـ function f of x قيمها محصورة من
+
+105
+00:12:00,110 --> 00:12:08,770
+العدد a والعدد b لكل x ينتمي إلى a حيث x لا تساوي
+
+106
+00:12:08,770 --> 00:12:18,600
+scإذا كان ال limit ل f of x as x tends to c exist
+
+107
+00:12:18,600 --> 00:12:27,120
+موجودة then ال limit أيضا لل function هتطلع محصورة
+
+108
+00:12:27,120 --> 00:12:34,560
+بين العددين a و b وهي
+
+109
+00:12:34,560 --> 00:12:38,640
+البرهان proof say دعنا
+
+110
+00:12:41,100 --> 00:12:44,140
+نسمي الـ limit لـ f of x
+
+111
+00:12:55,430 --> 00:13:03,430
+العدد حقيقي المطلوب اثبات ال claim التالي المطلوب
+
+112
+00:13:03,430 --> 00:13:08,870
+اثبات ان العدد L أكبر من أو يساوي A أصغر من أو
+
+113
+00:13:08,870 --> 00:13:18,170
+يساوي B لبرهان ذلك to see this we
+
+114
+00:13:18,170 --> 00:13:22,690
+use sequential criterion let
+
+115
+00:13:26,050 --> 00:13:33,090
+x in B sequence in A هدولها مختلفة عن الـ C such
+
+116
+00:13:33,090 --> 00:13:40,430
+that limit x in as N tends to infinity بساوي الـ C
+
+117
+00:13:40,430 --> 00:13:44,870
+طيب،
+
+118
+00:13:44,870 --> 00:13:57,640
+since ال limit ل F of X as X tends to C بساوي Lby
+
+119
+00:13:57,640 --> 00:14:05,680
+sequential .. sequential criterion إذا
+
+120
+00:14:05,680 --> 00:14:09,620
+كانت ال limit لدالة f of x لما x تقول c بساوي L
+
+121
+00:14:09,620 --> 00:14:15,120
+فهذا بيقدّي أنه لأي sequence في ال domain تبعت ال
+
+122
+00:14:15,120 --> 00:14:19,940
+function f مختلفة عن .. حدودها مختلفة عن ال C و
+
+123
+00:14:19,940 --> 00:14:21,280
+نهايتها بساوي C
+
+124
+00:14:24,670 --> 00:14:33,530
+بطلع limit لصورة ال sequence x in under if بتساوي
+
+125
+00:14:33,530 --> 00:14:40,250
+العدد ال .. هذا by sequential criterion تمام؟
+
+126
+00:15:01,600 --> 00:15:07,780
+by hypothesis star هذا
+
+127
+00:15:07,780 --> 00:15:15,580
+الفرض احنا فرضين ان ده قيمها معصورة بين a و b لكل
+
+128
+00:15:15,580 --> 00:15:22,020
+x في a مختلفة عن c فby hypothesis star عندي f of x
+
+129
+00:15:22,020 --> 00:15:28,600
+inبطلع أصغر من أو يساوي B أكبر من أو يساوي ال A
+
+130
+00:15:28,600 --> 00:15:36,200
+وهذا صحيح لكل N ينتمي إلى A لأن كل X in في ال
+
+131
+00:15:36,200 --> 00:15:42,840
+sequence هذه هو عنصر في A ومختلف عن ال C فمن الفرض
+
+132
+00:15:42,840 --> 00:15:48,380
+start بطلع هذا الكلام صحيح now
+
+133
+00:15:48,380 --> 00:15:51,980
+by
+
+134
+00:15:59,090 --> 00:16:03,410
+النظرية الخاصة بالـ sequences بتقول إنه لو في عندي
+
+135
+00:16:03,410 --> 00:16:07,870
+sequence كل حدودها محصورة بين a و b و ال limit لل
+
+136
+00:16:07,870 --> 00:16:12,410
+sequence exist فلازم ال limit تكون محصورة أيضا بين
+
+137
+00:16:12,410 --> 00:16:16,410
+a و b هذه هي ال previous theorem أخناها قبلك و
+
+138
+00:16:16,410 --> 00:16:25,360
+برحلناها by previous theoremلف of xn as n tends to
+
+139
+00:16:25,360 --> 00:16:30,020
+infinity بيطلع أصغر من أو ساوي دي أكبر من أو ساوي
+
+140
+00:16:30,020 --> 00:16:34,500
+لإيه طبما هذا ال limit هذه حسب ال sequential
+
+141
+00:16:34,500 --> 00:16:40,240
+criterion بساوي L إذا بيطلع لدي A أصغر من أو ساوي
+
+142
+00:16:40,240 --> 00:16:47,680
+Lأصغر من أو يساوي بيه وهذا اللي بدنا نثبته ان ال L
+
+143
+00:16:47,680 --> 00:16:52,360
+اللي هي ال limit ده F of X أما X ولا C محصورة بين
+
+144
+00:16:52,360 --> 00:17:02,120
+A وB okay تمام هنا هيك أثبتنا النظرية تمام واضح؟
+
+145
+00:17:02,120 --> 00:17:06,900
+في أي استفسار؟ في أي سؤال؟
+
+146
+00:17:12,010 --> 00:17:17,050
+في كمان نظرية ال sandwich theorem أو ال squeeze
+
+147
+00:17:17,050 --> 00:17:27,130
+theorem لل functions squeeze
+
+148
+00:17:27,130 --> 00:17:32,850
+theorem for
+
+149
+00:17:32,850 --> 00:17:33,710
+functions
+
+150
+00:17:43,800 --> 00:17:54,960
+فا let f و g و h be functions from a to r be
+
+151
+00:17:54,960 --> 00:17:59,340
+functions و
+
+152
+00:17:59,340 --> 00:18:11,160
+c a cluster point of a and
+
+153
+00:18:13,430 --> 00:18:20,070
+f of x أصغر من أو ساوي h of x أصغر من أو ساوي g of
+
+154
+00:18:20,070 --> 00:18:31,510
+x for every x تنتمي إلى a و x different from c إذا
+
+155
+00:18:31,510 --> 00:18:37,650
+كان ال limit لل functions على الأطراف اللي هي f of
+
+156
+00:18:37,650 --> 00:18:46,670
+x as x tends to c بساوي Lوكذلك ال limit لل
+
+157
+00:18:46,670 --> 00:18:51,530
+function g of x as x instances of c بساوي L حيث L
+
+158
+00:18:51,530 --> 00:18:59,660
+عدد حقيقييعني limit ل F و limit ل G exist and C و
+
+159
+00:18:59,660 --> 00:19:04,980
+كلهما بساوي نفس العدد الحقيقي L then limit لدالة
+
+160
+00:19:04,980 --> 00:19:10,720
+المحزورة في الوسط اللي هي H of X as X tends to C
+
+161
+00:19:10,720 --> 00:19:16,520
+أيضا بساوي العدد L و
+
+162
+00:19:16,520 --> 00:19:21,160
+البرهان أزاي برهان
+
+163
+00:19:23,410 --> 00:19:28,750
+النظرية السابقة هنستخدم
+
+164
+00:19:28,750 --> 00:19:37,170
+الـ sequential criterion زائد to use
+
+165
+00:19:37,170 --> 00:19:45,610
+sequential criterion and to squeeze the theorem
+
+166
+00:19:45,610 --> 00:19:50,570
+for
+
+167
+00:19:58,560 --> 00:20:05,040
+sequences فال let
+
+168
+00:20:05,040 --> 00:20:12,340
+xn بي sequence in a حدودها مختلفة عن c such that
+
+169
+00:20:12,340 --> 00:20:16,020
+limit xn بساوي c
+
+170
+00:20:20,090 --> 00:20:26,950
+then عندي بطلع f of xn أصغر من أو ساوي h of xn
+
+171
+00:20:26,950 --> 00:20:34,930
+أصغر من أو ساوي g of xn لكل n وعندي
+
+172
+00:20:34,930 --> 00:20:38,270
+by
+
+173
+00:20:38,270 --> 00:20:49,380
+sequential criterion أنا عندي ال limitF of X لما
+
+174
+00:20:49,380 --> 00:20:57,930
+اكسطر O لـ C بساوي L بيقدي ان ال limitلـ f of x n
+
+175
+00:20:57,930 --> 00:21:06,670
+as n tends to infinity بساوي L and ال limit ل g of
+
+176
+00:21:06,670 --> 00:21:12,010
+x as x tends to c بساوي L احنا فرضين ان ال limit ل
+
+177
+00:21:12,010 --> 00:21:17,030
+g and c بساوي L فهذا بيقدي by sequential criterion
+
+178
+00:21:17,030 --> 00:21:23,830
+ان ال limit ل sequence x n اللي نهايتها c نهاية
+
+179
+00:21:23,830 --> 00:21:24,550
+صورتها
+
+180
+00:21:27,560 --> 00:21:34,660
+هتطلع أيضا بساوي L وبالتالي
+
+181
+00:21:34,660 --> 00:21:46,520
+so by squeeze theorem for sequences يعني
+
+182
+00:21:46,520 --> 00:21:51,800
+عندي تلات متتاليات هذه
+
+183
+00:21:51,800 --> 00:21:57,840
+ال limit بتاعتها بتطلع Lو هذه ال limit تبعتها
+
+184
+00:21:57,840 --> 00:22:05,080
+بيطلع لعدد L لما N تقول infinity إذا
+
+185
+00:22:05,080 --> 00:22:10,960
+ال sequence اللي في وسط ال limit تبعتها limit ال
+
+186
+00:22:10,960 --> 00:22:15,860
+sequence H of X N as N tends to infinity بتطلع
+
+187
+00:22:15,860 --> 00:22:21,060
+بساوي L وبالتالي
+
+188
+00:22:21,060 --> 00:22:22,340
+therefore
+
+189
+00:22:24,080 --> 00:22:31,520
+by sequential criterion مرة أخرى ال sequential
+
+190
+00:22:31,520 --> 00:22:38,660
+criterion بتقول عشان أثبت أن ال limit لل function
+
+191
+00:22:38,660 --> 00:22:46,360
+h of x لما x تقول إلى c بساوي العدد L عشان أثبت
+
+192
+00:22:46,360 --> 00:22:52,380
+limit ال function h لما x تقول إلى c بساوي Lلازم
+
+193
+00:22:52,380 --> 00:22:56,260
+هذا بكافئ حسب الـ sequential criterion هذا بكافئ
+
+194
+00:22:56,260 --> 00:23:00,580
+إننا نثبت لو أخدت أي sequence في مجال الدالة كل
+
+195
+00:23:00,580 --> 00:23:04,000
+حدودها مختلفة عن الـ C و ال sequence نهايتها C
+
+196
+00:23:04,000 --> 00:23:09,400
+فلازم أثبت نهاية صورة ال sequence بالساوية لعدد L
+
+197
+00:23:09,400 --> 00:23:13,640
+وهذا أثبتناه، اذا by sequential criterion بطلع
+
+198
+00:23:13,640 --> 00:23:20,950
+limit H عن C بساوية L وهو المطلوبOkay تمام إذا هذا
+
+199
+00:23:20,950 --> 00:23:25,230
+برهان الـ sequential criteria الـ squeeze theorem
+
+200
+00:23:25,230 --> 00:23:31,070
+for functions تمام و في كتير من نظريات الأخرى اللي
+
+201
+00:23:31,070 --> 00:23:36,370
+أثبتناها بالنسبة لل limits of sequences فيه
+
+202
+00:23:36,370 --> 00:23:38,290
+بيقابلها أو بيوزيها
+
+203
+00:23:40,890 --> 00:23:45,890
+نظريات خاصة بال functions هتشوفوا بعض النظريات هذه
+
+204
+00:23:45,890 --> 00:23:53,470
+في التمرين فهينبرنتلكم بعضهم ناخد الآن بعض الأمثلة
+
+205
+00:23:53,470 --> 00:23:58,010
+على النظريات هذه وخاصة النظرية الأخيرة
+
+206
+00:24:20,680 --> 00:24:30,940
+examples أمثلة واحد show أن ال limit لل function x
+
+207
+00:24:30,940 --> 00:24:41,220
+أس تلاتة على اتنين لما x تقول سفر بساوي سفر و
+
+208
+00:24:41,220 --> 00:24:43,540
+طبعا هنا ال x موجة
+
+209
+00:25:03,540 --> 00:25:07,520
+الفترة المفتوحة من 0 إلى ملانهاية
+
+210
+00:25:12,860 --> 00:25:15,920
+طبعا مافيش ولا قانون من القوانين النهائية اللي
+
+211
+00:25:15,920 --> 00:25:19,680
+اخدناها سابقا بيعطيني ان ال limit هذي بالساوية سفر
+
+212
+00:25:19,680 --> 00:25:24,520
+يعني ماجدرش اقول عوض عن x بساوية سفر بطلع سفر
+
+213
+00:25:24,520 --> 00:25:29,640
+مافيش ماكان لاش قانون زي هذا هذي ليست polynomial x
+
+214
+00:25:29,640 --> 00:25:33,740
+أُس 3 over 2 is not polynomial لو كانت polynomial
+
+215
+00:25:33,740 --> 00:25:39,420
+بنعود على طول مباشرة عن x بالساوية سفر ولكنها ليست
+
+216
+00:25:39,420 --> 00:25:45,270
+polynomialفلبرهان النظرية هذه بناخد الدالة f of x
+
+217
+00:25:45,270 --> 00:25:49,970
+بالساوي x أص تلاتة على اتنين بنلاحظ أولا، لاحظي
+
+218
+00:25:49,970 --> 00:25:56,550
+note ان ال x for
+
+219
+00:25:56,550 --> 00:26:02,610
+x أكبر من سفر أصغر من أو ساوي الواحد بطلع عندي x
+
+220
+00:26:02,610 --> 00:26:07,330
+تقريبا أصغر من أو ساوي x أصغر من أو ساوي الواحد
+
+221
+00:26:11,890 --> 00:26:21,370
+أو X أصغر من أو يساوي X أص نص أصغر من أو يساوي
+
+222
+00:26:21,370 --> 00:26:29,290
+الواحد كلمة
+
+223
+00:26:29,290 --> 00:26:35,050
+X الجدر الترميه ل X بتطلع أكبر من أو يساوي ال X
+
+224
+00:26:35,050 --> 00:26:39,630
+إذا ال X محصولة من سفر واحد وبالتالي
+
+225
+00:26:41,250 --> 00:26:51,130
+لو ضربت .. لو ضربت هنا في .. في x فهذا هيقدّي أن x
+
+226
+00:26:51,130 --> 00:26:57,150
+تربيه أصغر من أو يساوي x أس تلاتة ع اتنين أصغر من
+
+227
+00:26:57,150 --> 00:27:01,430
+أو يساوي x إذن هذا الكلام صحيح
+
+228
+00:27:05,500 --> 00:27:13,040
+هذا كلام صحيح لكل x أكبر من سفر أصغر من أو ساوي
+
+229
+00:27:13,040 --> 00:27:21,180
+الواحد الان لما x تقول إلى سفر x تربيها كثيرة حدود
+
+230
+00:27:21,180 --> 00:27:26,760
+ال limit بتاعتها سفر وال x هذه polynomial لما x
+
+231
+00:27:26,760 --> 00:27:32,820
+تقول لسفر تطلع نهيتها سفر تمام؟ اذا by squeeze
+
+232
+00:27:32,820 --> 00:27:33,400
+theorem
+
+233
+00:27:52,880 --> 00:28:01,340
+المثال التاني show أن limit ال function sin x لما
+
+234
+00:28:01,340 --> 00:28:03,980
+x تقول سفر بسوى سفر
+
+235
+00:28:08,850 --> 00:28:18,290
+لبرهان ذلك فيه متباينة معروفة it is known c
+
+236
+00:28:18,290 --> 00:28:21,870
+chapter
+
+237
+00:28:21,870 --> 00:28:32,370
+8 that sin x دايما أصغر من أو ساوي x أكبر من أو
+
+238
+00:28:32,370 --> 00:28:44,050
+ساوي سالب xلكل X ينتمي إلى R معروف وهذا .. هيتم ..
+
+239
+00:28:44,050 --> 00:28:49,290
+هذا له برهان في chapter 8 في نفس الكتاب تبعنا اللي
+
+240
+00:28:49,290 --> 00:28:52,470
+هياخدوا منكم تحليل حقيقة 2 هياخدوا ال chapter هذا
+
+241
+00:28:52,470 --> 00:28:57,910
+ففيه برهان هناك للمتبايلة هذه او الحقيقة هذه ان
+
+242
+00:28:57,910 --> 00:29:02,350
+sign X دايما اظلم لو ساوي X اكبر من او ساوي
+
+243
+00:29:02,350 --> 00:29:03,270
+negative X
+
+244
+00:29:08,790 --> 00:29:12,770
+هذا الرسم ماتوضح يعني مش حاكته بانا برهان زي ما
+
+245
+00:29:12,770 --> 00:29:18,250
+اقول ان هذا موجود في chapter تمانية فالناس هياخدوا
+
+246
+00:29:18,250 --> 00:29:23,290
+real analysis اتنين هيشوفوه هاي ال .. ال function
+
+247
+00:29:23,290 --> 00:29:34,130
+y بساوي x و هاي ال function و
+
+248
+00:29:34,130 --> 00:29:38,170
+هاد ال function y بساوي negative xو الـ sine
+
+249
+00:29:38,170 --> 00:29:42,170
+function الموجة
+
+250
+00:29:42,170 --> 00:29:52,790
+أو ال wave تبعتها زي هيك إذا
+
+251
+00:29:52,790 --> 00:30:00,090
+هذه ال function y بساوي sin xفلاحظوا ان ال sign
+
+252
+00:30:00,090 --> 00:30:05,330
+function محصورة بين y بساوي x و y بساوي سالب x
+
+253
+00:30:05,330 --> 00:30:11,510
+تمام؟ هذا طبعا من الرسم لكن الرسم ليس برهان أنا
+
+254
+00:30:11,510 --> 00:30:17,230
+مجرد توضيح الأن أنا عندي ال function هذه لما x
+
+255
+00:30:17,230 --> 00:30:22,850
+تقول للصفر لما x تقول للصفر بتقول للصفر وهذه برضه
+
+256
+00:30:22,850 --> 00:30:28,600
+ال function لما x تقول للصفر بتروح للصفرإذا by
+
+257
+00:30:28,600 --> 00:30:36,760
+squeeze theorem .. إذا by squeeze
+
+258
+00:30:36,760 --> 00:30:45,240
+theorem ال limit لل function sin x اللي هي محصورة
+
+259
+00:30:45,240 --> 00:30:51,640
+في النص and السفر مساوي سفر okay تمام؟ لأن هذه
+
+260
+00:30:51,640 --> 00:30:56,760
+حقيقة برهانها بتم هكذا ماذا؟ في أي سؤال؟
+
+261
+00:31:00,750 --> 00:31:05,610
+كمان في مثال
+
+262
+00:31:05,610 --> 00:31:12,310
+تالت ممكن اثبات ان ال cosine ل X لما X تقول ل 0
+
+263
+00:31:12,310 --> 00:31:23,490
+بساوي 1 proof we
+
+264
+00:31:23,490 --> 00:31:31,480
+useالمتباينة في متباينة اللي هي واحد سالب X تربيع
+
+265
+00:31:31,480 --> 00:31:37,240
+اتنين اصغر من او ساوي cosine X اصغر من او ساوي
+
+266
+00:31:37,240 --> 00:31:42,920
+واحد for every X ينتمي الاراضي المتباينة هذه صحيحة
+
+267
+00:31:42,920 --> 00:31:45,680
+لكل الاعداد الحقيقية
+
+268
+00:31:50,520 --> 00:32:01,100
+و هذه برضه ممكن إثباتها في chapter 8
+
+269
+00:32:01,100 --> 00:32:07,460
+في real analysis 2 الان ال function هذه لما x تقول
+
+270
+00:32:07,460 --> 00:32:15,280
+ل 0 بتقول ل 1 tense واحد as x tends to zero صح؟
+
+271
+00:32:15,280 --> 00:32:20,990
+هذه كثيرة حدودعوض عن x بساوية سفر وهذه دالة ثابتة،
+
+272
+00:32:20,990 --> 00:32:27,310
+نهيتها واحد لما x تقول إلى أي حاجة، إذا ممكن نطبق
+
+273
+00:32:27,310 --> 00:32:32,870
+ال squeeze theorem، okay؟ إذا by squeeze theorem،
+
+274
+00:32:32,870 --> 00:32:39,210
+ال limit ل ال function cosine
+
+275
+00:32:39,210 --> 00:32:47,780
+x as x tends to zero تطلع بساوية واحدةتمام؟ okay
+
+276
+00:32:47,780 --> 00:32:53,000
+واضح؟ في كتير من ال limits المعروفة ممكن اثباتها
+
+277
+00:32:53,000 --> 00:33:01,140
+بنفس الطرق هذه خلينا ناخد كمان مثال limit ل
+
+278
+00:33:01,140 --> 00:33:09,120
+function x غرب sin 1 على x as x tends to zero
+
+279
+00:33:09,120 --> 00:33:12,320
+exists و بالساوي سفر
+
+280
+00:33:14,890 --> 00:33:21,610
+فالـ function هنا الـ
+
+281
+00:33:21,610 --> 00:33:25,610
+function اللي بدي أخدلها ال limit هي عبارة عن الـ
+
+282
+00:33:25,610 --> 00:33:33,670
+function f of x بساوي x ضرب sign واحد على x طبعا
+
+283
+00:33:33,670 --> 00:33:38,410
+هنا x لا تساوي سفر ال domain للـ function هذه كل
+
+284
+00:33:38,410 --> 00:33:45,650
+الأعداد الحقيقية معدىالسفر لأن اسمه على سفر هنا مش
+
+285
+00:33:45,650 --> 00:33:54,170
+معرفة تمام؟ بنثبت ان ال limit للدالة هذه عند السفر
+
+286
+00:33:54,170 --> 00:33:59,230
+بساوي سفر بالمناسبة الدالة هذه هي نفس الدالة
+
+287
+00:33:59,230 --> 00:34:07,570
+المرسومة على ال cover تبع الكتاب هذه رسمة الدالة
+
+288
+00:34:09,910 --> 00:34:17,470
+فلس متدالة f of x بالساوي x في sin واحد على x و
+
+289
+00:34:17,470 --> 00:34:19,910
+الدالة هذه زي ما انتوا شايفين المرحلة تبعها كل ما
+
+290
+00:34:19,910 --> 00:34:25,450
+جربته من السفر كل ما جرب من السفر سواء من اليمين
+
+291
+00:34:25,450 --> 00:34:30,810
+او اليسار لكن لإثبات ذلك نستخدم ال squeeze theorem
+
+292
+00:34:30,810 --> 00:34:35,030
+طيب
+
+293
+00:34:39,940 --> 00:34:49,380
+لت x لا يساوي سفر، خلّيني أخد x بساوي سفر، اذا by
+
+294
+00:34:49,380 --> 00:34:53,320
+tricotomy property باستخدام الخاصية الفلاتية، x
+
+295
+00:34:53,320 --> 00:34:59,740
+أكبر من سفر أو x أصغر من سفر، فحناخد في الحالتين
+
+296
+00:35:09,860 --> 00:35:14,980
+طيب، لنثبت أنه
+
+297
+00:35:14,980 --> 00:35:20,940
+لأي x لا تساوي 0 زي هذه سالب absolute x أصغر من أو
+
+298
+00:35:20,940 --> 00:35:28,000
+ساوي x في sin 1 على x أصغر من أو ساوي absolute x،
+
+299
+00:35:28,000 --> 00:35:30,900
+هذا صحيح لكل x لا يساوي 0
+
+300
+00:35:39,610 --> 00:35:44,550
+أنا بدي أثبت أنه لكل x لا يساوي سفر المتباين هذا
+
+301
+00:35:44,550 --> 00:35:54,970
+صحيح ف to see this لإثبات ذلك to see thisfix x لا
+
+302
+00:35:54,970 --> 00:35:59,930
+يساوي سفر خلّيني أخد arbitrary x مختلفة عن السفر و
+
+303
+00:35:59,930 --> 00:36:04,570
+نثبت إن المتباين هي دي صحيحة إلها then إذا أخدت
+
+304
+00:36:04,570 --> 00:36:09,850
+أنا x لا تساوي سفر فإما x أكبر من سفر or x أصغر من
+
+305
+00:36:09,850 --> 00:36:18,760
+سفر فإذا في عندي أنا حالتين نشوف الحالة الأولىلما
+
+306
+00:36:18,760 --> 00:36:24,480
+تكون الـ X أكبر من الزبر هذا بيقدي ان absolute X
+
+307
+00:36:24,480 --> 00:36:37,620
+بالساوي X صح؟ طيب هنا note that sign
+
+308
+00:36:39,330 --> 00:36:45,010
+ثيتا دائما أكبر من أوساو سالب واحد أصغر من أوساو
+
+309
+00:36:45,010 --> 00:36:49,690
+واحد for all ثيتا تنتمي إلى R هذه المتباينة
+
+310
+00:36:49,690 --> 00:36:53,690
+الصحيحة ال absolute maximum value ل سايل واحد وال
+
+311
+00:36:53,690 --> 00:36:59,710
+absolute minimum value سالب واحد خلينا
+
+312
+00:36:59,710 --> 00:37:08,180
+نضرب المتباينة هذه في absolute Xو ال theta هذه
+
+313
+00:37:08,180 --> 00:37:13,580
+تبدلها
+
+314
+00:37:13,580 --> 00:37:20,420
+ب 1 على x ف since sin 1 على x أكبر من أو ساوي سالب
+
+315
+00:37:20,420 --> 00:37:26,020
+واحد أصغر من أو ساوي واحد لكل x لا تساوي سفر هذا
+
+316
+00:37:26,020 --> 00:37:26,660
+بيقدر
+
+317
+00:37:29,420 --> 00:37:33,100
+لو ضربت المتباينة هذه في absolute x اللي هي
+
+318
+00:37:33,100 --> 00:37:40,160
+بالساوي x و absolute x هنا هتطلع موجة بقى فهيطلع
+
+319
+00:37:40,160 --> 00:37:45,300
+عندي سالب absolute x أصغر من أو ساوي x في sin 1
+
+320
+00:37:45,300 --> 00:37:49,760
+على x أصغر من أو ساوي absolute x وبالتالي
+
+321
+00:37:49,760 --> 00:37:52,320
+المتباينة اللي انا عايز اثبتها ده هي صحيحة
+
+322
+00:37:56,180 --> 00:38:00,780
+أضرب هنا في absolute x و absolute x بالساوية x عدد
+
+323
+00:38:00,780 --> 00:38:07,200
+موجب فالمتباينة تبقى أشرتها زي ما هي و هي ضربت في
+
+324
+00:38:07,200 --> 00:38:10,600
+سالب absolute x ال x هنا هي برضه سالب absolute x
+
+325
+00:38:10,600 --> 00:38:14,460
+فهذه المتباينة اللي احنا عايزين نثبتها في الإدعاء
+
+326
+00:38:14,460 --> 00:38:20,200
+تبعنا الحالة التالية لو كانت x سالب
+
+327
+00:38:31,100 --> 00:38:37,440
+كاس اتنين لو كانت x سالبها فهذا بيقدي ان absolute
+
+328
+00:38:37,440 --> 00:38:45,580
+x بساوي سالب x طبعا هذا موجة بقى وبالتالي multiply
+
+329
+00:38:49,120 --> 00:38:54,100
+multiply المتباينة سالب واحد اصغر من او ساوي ساين
+
+330
+00:38:54,100 --> 00:39:00,200
+واحد على اكس اصغر من او ساوي واحد by absolute x
+
+331
+00:39:00,200 --> 00:39:05,240
+بساوي سالب x اكبر من السفر هذا يعني انه
+
+332
+00:39:12,420 --> 00:39:19,180
+فهذا بيقدي انه ايه؟ انه سالب absolute x أصغر من أو
+
+333
+00:39:19,180 --> 00:39:26,300
+يساوي سالب
+
+334
+00:39:26,300 --> 00:39:33,080
+x في sin 1 على x أصغر من أو يساوي absolute ال x
+
+335
+00:39:46,010 --> 00:39:54,230
+نضرب في سالب واحد نضرب في سالب واحد نضرب في سالب
+
+336
+00:39:54,230 --> 00:39:58,910
+واحد نضرب في سالب واحد نضرب في سالب واحد نضرب في
+
+337
+00:39:58,910 --> 00:39:58,970
+سالب واحد نضرب في سالب واحد نضرب في سالب واحد نضرب
+
+338
+00:39:58,970 --> 00:39:59,210
+في سالب واحد نضرب في سالب واحد نضرب في سالب واحد
+
+339
+00:39:59,210 --> 00:40:02,030
+نضرب في سالب واحد نضرب في سالب واحد نضرب في سالب
+
+340
+00:40:02,030 --> 00:40:06,590
+واحد نضرب في سالب واحد نضرب في سالب واحد نضرب في
+
+341
+00:40:06,590 --> 00:40:13,670
+سالب واحدو هذه هي المتباينة اللي احنا عايزين
+
+342
+00:40:13,670 --> 00:40:21,370
+نثبتها صح؟ اذا this completes
+
+343
+00:40:21,370 --> 00:40:24,770
+the
+
+344
+00:40:24,770 --> 00:40:34,330
+proof of the claim اذا هنا برهنة ل claim وهو ان
+
+345
+00:40:34,330 --> 00:40:37,970
+المتباينة هذه صحيحة تمام؟
+
+346
+00:40:40,820 --> 00:40:50,560
+الان من ال claim now
+
+347
+00:40:50,560 --> 00:41:02,560
+by above .. by above the claim and squeeze the
+
+348
+00:41:02,560 --> 00:41:06,560
+theorem since
+
+349
+00:41:06,560 --> 00:41:12,450
+بما أنه ال limitالـ absolute x as x tends to 0
+
+350
+00:41:12,450 --> 00:41:18,310
+بساوي ال limit لسالب absolute x as x tends to 0
+
+351
+00:41:18,310 --> 00:41:26,170
+بساوي 0 we have نحصل على ان ال limit المحصورة في
+
+352
+00:41:26,170 --> 00:41:32,710
+الوسط اللي هي x في sin 1 على x as x tends to 0
+
+353
+00:41:32,710 --> 00:41:39,760
+بساوي أيضا السفر وهو المطلوباذا هنا استخدمنا الـ
+
+354
+00:41:39,760 --> 00:41:45,780
+squeeze theorem تمام؟ لأن هذا برهان أن ال limit لل
+
+355
+00:41:45,780 --> 00:41:51,380
+function x sin 1 على x عندما x تقوى للصفر بيساوي
+
+356
+00:41:51,380 --> 00:41:56,680
+صفر تمام؟ واضح؟ في أي سفسار؟ أي سؤال؟
+
+357
+00:42:03,260 --> 00:42:10,860
+Okay باقي نظرية واحدة في ال section 4-2 فاتحقلوا
+
+358
+00:42:10,860 --> 00:42:17,060
+تقرؤوها تفهموا البرهانة و تبدووا في حال التمرين و
+
+359
+00:42:17,060 --> 00:42:21,360
+ان شاء الله المرة جاية بنكمل نشرح النظرية هذه و
+
+360
+00:42:21,360 --> 00:42:25,820
+بنبدأ section جديد اللي هو section 4-3
+
+361
+00:42:28,450 --> 00:42:33,790
+شكرا لإصداركم وشوفكم ان شاء الله في المحاضرة
+
+362
+00:42:33,790 --> 00:42:34,250
+الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..edb30e2ee911f40ffadfc241f4ee5fa495de1f4c
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/unjPK5-iKR8_raw.srt
@@ -0,0 +1,1452 @@
+1
+00:00:00,000 --> 00:00:01,100
+موسيقى
+
+2
+00:00:19,710 --> 00:00:25,810
+السلام عليكم ان شاء الله اليوم هنكمل section أربعة
+
+3
+00:00:25,810 --> 00:00:31,470
+اتنين اللي بدأناها المرة اللي فاتت في المرة
+
+4
+00:00:31,470 --> 00:00:36,810
+السابقة أخدنا نظريات النهايات أو قوانين النهايات
+
+5
+00:00:37,440 --> 00:00:44,420
+للـ functions و كنتائج
+
+6
+00:00:44,420 --> 00:00:50,220
+على قوانين النهايات لدينا النظرية التالية theorem
+
+7
+00:00:50,220 --> 00:00:55,020
+واحد
+
+8
+00:00:56,630 --> 00:01:04,910
+إذا P of X بساوي A N في X to N زائد A N minus one
+
+9
+00:01:04,910 --> 00:01:12,290
+X to N minus one زائد و هكذا زائد A one في X زائد
+
+10
+00:01:12,290 --> 00:01:16,450
+A zero is a polynomial
+
+11
+00:01:25,480 --> 00:01:33,060
+Polynomial function of X then
+
+12
+00:01:33,060 --> 00:01:47,820
+for any C real number limit P of X as X tends to C
+
+13
+00:01:47,820 --> 00:01:53,740
+بساوي قيمة ال polynomial and C then if
+
+14
+00:01:56,280 --> 00:02:06,620
+R of X بساوي P
+
+15
+00:02:06,620 --> 00:02:15,740
+of X over Q of X where
+
+16
+00:02:15,740 --> 00:02:18,860
+P
+
+17
+00:02:18,860 --> 00:02:22,580
+و Q are polynomial
+
+18
+00:02:26,290 --> 00:02:41,850
+الـ X and if Q and C لا تساوي سفر then limit ل R
+
+19
+00:02:41,850 --> 00:02:50,930
+of X as X tends to C بساوي R محسوب عن C البرهين
+
+20
+00:02:50,930 --> 00:02:59,720
+سهلة proveالبرنامج الأول note first
+
+21
+00:02:59,720 --> 00:03:03,500
+that
+
+22
+00:03:03,500 --> 00:03:08,180
+limit
+
+23
+00:03:08,180 --> 00:03:17,480
+ل x as x tends to c بيساوي c بيقدّي أن ال limitلـ
+
+24
+00:03:17,480 --> 00:03:25,240
+x to the power k as x tends to c بساوي c to k for
+
+25
+00:03:25,240 --> 00:03:30,580
+all k بساوي صفر واحد اتنين إلى ما ننهي
+
+26
+00:03:33,860 --> 00:03:37,040
+أثبتنا قبل ذلك أن الـ limit لل identity function
+
+27
+00:03:37,040 --> 00:03:42,500
+لما x تقول ل c بساوة c أثبتنا ذلك باستخدام تعريف
+
+28
+00:03:42,500 --> 00:03:46,480
+epsilon delta قلنا لأي given epsilon choose delta
+
+29
+00:03:46,480 --> 00:03:51,080
+بساوة epsilon الآن
+
+30
+00:03:51,080 --> 00:03:57,010
+حسب نظريات النهاياتlimit x to the power k بيساوي
+
+31
+00:03:57,010 --> 00:04:01,030
+limit x ضرب نفسها مضروبة في نفسها كام المرات
+
+32
+00:04:01,030 --> 00:04:05,250
+واخدنا في المحاضرة الفاترة انه limit f of x الكل
+
+33
+00:04:05,250 --> 00:04:12,470
+أس m بيساوي limit f of x الكل أس m فهذا بيطلع
+
+34
+00:04:12,470 --> 00:04:20,490
+limit x هذا بيطلع c limit تبع ال x c هذا بيطلع c
+
+35
+00:04:20,490 --> 00:04:23,610
+أس k الآن
+
+36
+00:04:28,400 --> 00:04:35,780
+by limit by limit
+
+37
+00:04:35,780 --> 00:04:47,060
+theorem نظرية النهايات بطلع عندي ال limit ل P of X
+
+38
+00:04:47,060 --> 00:04:52,140
+as X tends to C بساوي ال limit هي عندي ال
+
+39
+00:04:52,140 --> 00:04:57,810
+polynomiallimit المجموعة اثبتنا انه limit المجموعة
+
+40
+00:04:57,810 --> 00:05:04,650
+بيستخدم مجموعة limits فlimit x to limit الحد الأول
+
+41
+00:05:04,650 --> 00:05:09,510
+زاد
+
+42
+00:05:09,510 --> 00:05:11,750
+limit الحد التاني
+
+43
+00:05:18,500 --> 00:05:25,660
+وهكذا إذا limit a1 في x as x tends to c زائد limit
+
+44
+00:05:25,660 --> 00:05:35,370
+a0 as x tends to cوالان limit حاصل ثابت فى دالة
+
+45
+00:05:35,370 --> 00:05:42,350
+بساوية ثابت a n المعاملات هذه a n و a n سالب واحد
+
+46
+00:05:42,350 --> 00:05:46,450
+الاخرى دى كلها ثوابت ف limit ثابت فى دالة بساوية
+
+47
+00:05:46,450 --> 00:05:54,710
+ثابت فى limit x أُس n اللى هى c أُس n زايد limit
+
+48
+00:05:54,710 --> 00:06:02,540
+ثابت فى دالة x أُس n سالب واحدبطلع C أُس N سالب
+
+49
+00:06:02,540 --> 00:06:10,780
+واحد وهكذا إلى A واحد في limit X لما X تقوى لـ C
+
+50
+00:06:10,780 --> 00:06:17,180
+بساوي C limit A Zero بطلع A Zero الآن هذه هي نفس
+
+51
+00:06:17,180 --> 00:06:24,820
+ال Polynomial المحسوب عند X بساوي C إذن هذا بثبت
+
+52
+00:06:24,820 --> 00:06:33,430
+الجزء الأول من النظريةلإثبات الجزء التاني Assume
+
+53
+00:06:33,430 --> 00:06:46,170
+أن الـ Q عند الـ C لا يساوي سفر ثم
+
+54
+00:06:46,170 --> 00:06:55,590
+Limit لR of X لما X تقول إلى C بساوي Limit البصر
+
+55
+00:06:58,130 --> 00:07:06,090
+P of X as X tends to C على limit Q of X لما X تقول
+
+56
+00:07:06,090 --> 00:07:12,810
+إلى C الان
+
+57
+00:07:12,810 --> 00:07:16,790
+باستخدام الجزء الأول من النظرية هذه كثيرة عدود
+
+58
+00:07:16,790 --> 00:07:24,430
+وبالتالي limit لها عن C بساوي قيمتها عن C وQ of X
+
+59
+00:07:24,430 --> 00:07:30,350
+برضه كثيرة حدودالقانون اللي عند أي عدد c بيساوي
+
+60
+00:07:30,350 --> 00:07:38,130
+قيمتها عن c ومع أن q of c هنا لا يساوي سفر فبقدر
+
+61
+00:07:38,130 --> 00:07:43,650
+استخدم القانون تبع limit الكسر بساوي أو limit قسم
+
+62
+00:07:43,650 --> 00:07:46,530
+الدالتين بساوي خارج قسم ال limits
+
+63
+00:07:51,050 --> 00:07:55,270
+لو Q of C بيساوي سفر ماقدرش اعمل الكلام هذا لو ال
+
+64
+00:07:55,270 --> 00:07:59,190
+limit هذه بيساوي سفر ماقدرش اوزع ال limit على ال
+
+65
+00:07:59,190 --> 00:08:03,610
+bust و المخان طبما هذا بالظبط هو عبارة عن ال
+
+66
+00:08:03,610 --> 00:08:10,090
+function R محسوب عن C وهذا بكمل برهان الجزء التاني
+
+67
+00:08:10,090 --> 00:08:12,230
+okay هذه أمثلة
+
+68
+00:08:23,910 --> 00:08:29,950
+Find limit لـ
+
+69
+00:08:29,950 --> 00:08:35,650
+x تربية سالب اتنين x زائد واحد لما x تقول لسفر
+
+70
+00:08:35,650 --> 00:08:44,170
+فهذه عبارة عن كثير تحدود P of X فحسب
+
+71
+00:08:44,170 --> 00:08:48,390
+الجزء الأول من النظرية السابقة limit ل P of X يطلع
+
+72
+00:08:48,390 --> 00:08:51,230
+P and سفر صح؟
+
+73
+00:08:53,080 --> 00:08:56,160
+عودة عن x بصفر مباشرة يعني يعني عودة عن x بصفر
+
+74
+00:08:56,160 --> 00:08:58,540
+مباشرة يعني يعني عودة عن x بصفر مباشرة يعني يعني
+
+75
+00:08:58,540 --> 00:08:58,640
+عودة عن x بصفر مباشرة يعني يعني عودة عن x بصفر
+
+76
+00:08:58,640 --> 00:09:03,040
+مباشرة يعني عودة
+
+77
+00:09:03,040 --> 00:09:03,220
+عن x بصفر مباشرة يعني عودة عن x بصفر مباشرة يعني
+
+78
+00:09:03,220 --> 00:09:03,960
+عودة عن x بصفر مباشرة يعني عودة عن x بصفر مباشرة
+
+79
+00:09:03,960 --> 00:09:09,000
+يعني عودة عن x بصفر مباشرة يعني عودة عن x بصفر
+
+80
+00:09:09,000 --> 00:09:14,540
+مباشرة يعني عودة عن x بصفر مباشرة يعني عودة عن x
+
+81
+00:09:14,540 --> 00:09:19,920
+بصفر مباشرة يعني عودة عن x بصفر مباشرة يعني عودة
+
+82
+00:09:19,920 --> 00:09:19,920
+عن x ب
+
+83
+00:09:24,080 --> 00:09:28,500
+طبعا هذه كثيرة حدود على كثيرة حدود وبنلاحظ انه
+
+84
+00:09:28,500 --> 00:09:32,320
+limit كثيرة الحدود اللي في المقام لما x تقوله سالب
+
+85
+00:09:32,320 --> 00:09:37,320
+واحد بيطلع اتنين لا يساوي سفر اذا حسب ال band
+
+86
+00:09:37,320 --> 00:09:44,960
+التاني بقدر اعوض عن x بيساوي سالب واحد مباشرة يعني
+
+87
+00:09:44,960 --> 00:09:51,040
+لو سميت ال function هذه R of X فالمفروض هذا يطلع R
+
+88
+00:09:51,040 --> 00:09:57,660
+عن سالب واحديعني عوض عن x بساوي سالب واحد فبطلع
+
+89
+00:09:57,660 --> 00:10:02,300
+عندي خمسة في سالب واحد تارديه بطلع خمسة سالب سالب
+
+90
+00:10:02,300 --> 00:10:09,360
+واحد بطلع واحد زائد اتنين على واحد زائد واحد و
+
+91
+00:10:09,360 --> 00:10:14,240
+بطلع تمانية على اتنين بطلع اربعة
+
+92
+00:10:20,930 --> 00:10:26,470
+أو ممكن نستخدم حل تاني نستخدم قوانين نهايات نقول
+
+93
+00:10:26,470 --> 00:10:31,970
+نهاية دالة كسرية زي هذه بساوي limit ال bus علي
+
+94
+00:10:31,970 --> 00:10:35,730
+limit المقام بشرط ان limit المقام ماساوي سفر limit
+
+95
+00:10:35,730 --> 00:10:39,490
+المقام طلعت ماساوي سفر بساوي هي اتنين اذا بقدر
+
+96
+00:10:39,490 --> 00:10:42,150
+استخدم القانون لكن لو كانت limit المقام بساوي سفر
+
+97
+00:10:42,150 --> 00:10:45,130
+اذا مابستطيع استخدم هذا القانون
+
+98
+00:10:50,800 --> 00:10:55,080
+في كثير من النظريات الخاصة بنهايات الـ sequences
+
+99
+00:10:55,080 --> 00:11:00,380
+أيضا في بقبلها نظريات أخرى خاصة بنهايات الـ
+
+100
+00:11:00,380 --> 00:11:04,620
+functions فمن
+
+101
+00:11:04,620 --> 00:11:05,880
+النظريات هذه
+
+102
+00:11:18,220 --> 00:11:25,280
+من هذه النظريات النظرية التالية let f be a
+
+103
+00:11:25,280 --> 00:11:38,240
+function from a to r and c be a cluster point
+
+104
+00:11:38,240 --> 00:11:40,320
+of A
+
+105
+00:11:50,350 --> 00:12:00,110
+suppose أن الـ function f of x قيمها محصورة من
+
+106
+00:12:00,110 --> 00:12:08,770
+العدد a والعدد b لكل x ينتمي إلى a حيث x لا تساوي
+
+107
+00:12:08,770 --> 00:12:18,600
+scإذا كان ال limit ل f of x as x tends to c exist
+
+108
+00:12:18,600 --> 00:12:27,120
+موجودة then ال limit أيضا لل function هتطلع محصورة
+
+109
+00:12:27,120 --> 00:12:34,560
+بين العددين a و b وهي
+
+110
+00:12:34,560 --> 00:12:38,640
+البرهان proof say دعنا
+
+111
+00:12:41,100 --> 00:12:44,140
+نسمي الـ limit لـ f of x
+
+112
+00:12:55,430 --> 00:13:03,430
+العدد حقيقي المطلوب اثبات ال claim التالي المطلوب
+
+113
+00:13:03,430 --> 00:13:08,870
+اثبات ان العدد L أكبر من أو يساوي A أصغر من أو
+
+114
+00:13:08,870 --> 00:13:18,170
+يساوي B لبرهان ذلك to see this we
+
+115
+00:13:18,170 --> 00:13:22,690
+use sequential criterion let
+
+116
+00:13:26,050 --> 00:13:33,090
+x in B sequence in A هدولها مختلفة عن الـ C such
+
+117
+00:13:33,090 --> 00:13:40,430
+that limit x in as N tends to infinity بساوي الـ C
+
+118
+00:13:40,430 --> 00:13:44,870
+طيب،
+
+119
+00:13:44,870 --> 00:13:57,640
+since ال limit ل F of X as X tends to C بساوي Lby
+
+120
+00:13:57,640 --> 00:14:05,680
+sequential .. sequential criterion إذا
+
+121
+00:14:05,680 --> 00:14:09,620
+كانت ال limit لدالة f of x لما x تقول c بساوي L
+
+122
+00:14:09,620 --> 00:14:15,120
+فهذا بيقدّي أنه لأي sequence في ال domain تبعت ال
+
+123
+00:14:15,120 --> 00:14:19,940
+function f مختلفة عن .. حدودها مختلفة عن ال C و
+
+124
+00:14:19,940 --> 00:14:21,280
+نهايتها بساوي C
+
+125
+00:14:24,670 --> 00:14:33,530
+بطلع limit لصورة ال sequence x in under if بتساوي
+
+126
+00:14:33,530 --> 00:14:40,250
+العدد ال .. هذا by sequential criterion تمام؟
+
+127
+00:15:01,600 --> 00:15:07,780
+by hypothesis star هذا
+
+128
+00:15:07,780 --> 00:15:15,580
+الفرض احنا فرضين ان ده قيمها معصورة بين a و b لكل
+
+129
+00:15:15,580 --> 00:15:22,020
+x في a مختلفة عن c فby hypothesis star عندي f of x
+
+130
+00:15:22,020 --> 00:15:28,600
+inبطلع أصغر من أو يساوي B أكبر من أو يساوي ال A
+
+131
+00:15:28,600 --> 00:15:36,200
+وهذا صحيح لكل N ينتمي إلى A لأن كل X in في ال
+
+132
+00:15:36,200 --> 00:15:42,840
+sequence هذه هو عنصر في A ومختلف عن ال C فمن الفرض
+
+133
+00:15:42,840 --> 00:15:48,380
+start بطلع هذا الكلام صحيح now
+
+134
+00:15:48,380 --> 00:15:51,980
+by
+
+135
+00:15:59,090 --> 00:16:03,410
+النظرية الخاصة بالـ sequences بتقول إنه لو في عندي
+
+136
+00:16:03,410 --> 00:16:07,870
+sequence كل حدودها محصورة بين a و b و ال limit لل
+
+137
+00:16:07,870 --> 00:16:12,410
+sequence exist فلازم ال limit تكون محصورة أيضا بين
+
+138
+00:16:12,410 --> 00:16:16,410
+a و b هذه هي ال previous theorem أخناها قبلك و
+
+139
+00:16:16,410 --> 00:16:25,360
+برحلناها by previous theoremلف of xn as n tends to
+
+140
+00:16:25,360 --> 00:16:30,020
+infinity بيطلع أصغر من أو ساوي دي أكبر من أو ساوي
+
+141
+00:16:30,020 --> 00:16:34,500
+لإيه طبما هذا ال limit هذه حسب ال sequential
+
+142
+00:16:34,500 --> 00:16:40,240
+criterion بساوي L إذا بيطلع لدي A أصغر من أو ساوي
+
+143
+00:16:40,240 --> 00:16:47,680
+Lأصغر من أو يساوي بيه وهذا اللي بدنا نثبته ان ال L
+
+144
+00:16:47,680 --> 00:16:52,360
+اللي هي ال limit ده F of X أما X ولا C محصورة بين
+
+145
+00:16:52,360 --> 00:17:02,120
+A وB okay تمام هنا هيك أثبتنا النظرية تمام واضح؟
+
+146
+00:17:02,120 --> 00:17:06,900
+في أي استفسار؟ في أي سؤال؟
+
+147
+00:17:12,010 --> 00:17:17,050
+في كمان نظرية ال sandwich theorem أو ال squeeze
+
+148
+00:17:17,050 --> 00:17:27,130
+theorem لل functions squeeze
+
+149
+00:17:27,130 --> 00:17:32,850
+theorem for
+
+150
+00:17:32,850 --> 00:17:33,710
+functions
+
+151
+00:17:43,800 --> 00:17:54,960
+فا let f و g و h be functions from a to r be
+
+152
+00:17:54,960 --> 00:17:59,340
+functions و
+
+153
+00:17:59,340 --> 00:18:11,160
+c a cluster point of a and
+
+154
+00:18:13,430 --> 00:18:20,070
+f of x أصغر من أو ساوي h of x أصغر من أو ساوي g of
+
+155
+00:18:20,070 --> 00:18:31,510
+x for every x تنتمي إلى a و x different from c إذا
+
+156
+00:18:31,510 --> 00:18:37,650
+كان ال limit لل functions على الأطراف اللي هي f of
+
+157
+00:18:37,650 --> 00:18:46,670
+x as x tends to c بساوي Lوكذلك ال limit لل
+
+158
+00:18:46,670 --> 00:18:51,530
+function g of x as x instances of c بساوي L حيث L
+
+159
+00:18:51,530 --> 00:18:59,660
+عدد حقيقييعني limit ل F و limit ل G exist and C و
+
+160
+00:18:59,660 --> 00:19:04,980
+كلهما بساوي نفس العدد الحقيقي L then limit لدالة
+
+161
+00:19:04,980 --> 00:19:10,720
+المحزورة في الوسط اللي هي H of X as X tends to C
+
+162
+00:19:10,720 --> 00:19:16,520
+أيضا بساوي العدد L و
+
+163
+00:19:16,520 --> 00:19:21,160
+البرهان أزاي برهان
+
+164
+00:19:23,410 --> 00:19:28,750
+النظرية السابقة هنستخدم
+
+165
+00:19:28,750 --> 00:19:37,170
+الـ sequential criterion زائد to use
+
+166
+00:19:37,170 --> 00:19:45,610
+sequential criterion and to squeeze the theorem
+
+167
+00:19:45,610 --> 00:19:50,570
+for
+
+168
+00:19:58,560 --> 00:20:05,040
+sequences فال let
+
+169
+00:20:05,040 --> 00:20:12,340
+xn بي sequence in a حدودها مختلفة عن c such that
+
+170
+00:20:12,340 --> 00:20:16,020
+limit xn بساوي c
+
+171
+00:20:20,090 --> 00:20:26,950
+then عندي بطلع f of xn أصغر من أو ساوي h of xn
+
+172
+00:20:26,950 --> 00:20:34,930
+أصغر من أو ساوي g of xn لكل n وعندي
+
+173
+00:20:34,930 --> 00:20:38,270
+by
+
+174
+00:20:38,270 --> 00:20:49,380
+sequential criterion أنا عندي ال limitF of X لما
+
+175
+00:20:49,380 --> 00:20:57,930
+اكسطر O لـ C بساوي L بيقدي ان ال limitلـ f of x n
+
+176
+00:20:57,930 --> 00:21:06,670
+as n tends to infinity بساوي L and ال limit ل g of
+
+177
+00:21:06,670 --> 00:21:12,010
+x as x tends to c بساوي L احنا فرضين ان ال limit ل
+
+178
+00:21:12,010 --> 00:21:17,030
+g and c بساوي L فهذا بيقدي by sequential criterion
+
+179
+00:21:17,030 --> 00:21:23,830
+ان ال limit ل sequence x n اللي نهايتها c نهاية
+
+180
+00:21:23,830 --> 00:21:24,550
+صورتها
+
+181
+00:21:27,560 --> 00:21:34,660
+هتطلع أيضا بساوي L وبالتالي
+
+182
+00:21:34,660 --> 00:21:46,520
+so by squeeze theorem for sequences يعني
+
+183
+00:21:46,520 --> 00:21:51,800
+عندي تلات متتاليات هذه
+
+184
+00:21:51,800 --> 00:21:57,840
+ال limit بتاعتها بتطلع Lو هذه ال limit تبعتها
+
+185
+00:21:57,840 --> 00:22:05,080
+بيطلع لعدد L لما N تقول infinity إذا
+
+186
+00:22:05,080 --> 00:22:10,960
+ال sequence اللي في وسط ال limit تبعتها limit ال
+
+187
+00:22:10,960 --> 00:22:15,860
+sequence H of X N as N tends to infinity بتطلع
+
+188
+00:22:15,860 --> 00:22:21,060
+بساوي L وبالتالي
+
+189
+00:22:21,060 --> 00:22:22,340
+therefore
+
+190
+00:22:24,080 --> 00:22:31,520
+by sequential criterion مرة أخرى ال sequential
+
+191
+00:22:31,520 --> 00:22:38,660
+criterion بتقول عشان أثبت أن ال limit لل function
+
+192
+00:22:38,660 --> 00:22:46,360
+h of x لما x تقول إلى c بساوي العدد L عشان أثبت
+
+193
+00:22:46,360 --> 00:22:52,380
+limit ال function h لما x تقول إلى c بساوي Lلازم
+
+194
+00:22:52,380 --> 00:22:56,260
+هذا بكافئ حسب الـ sequential criterion هذا بكافئ
+
+195
+00:22:56,260 --> 00:23:00,580
+إننا نثبت لو أخدت أي sequence في مجال الدالة كل
+
+196
+00:23:00,580 --> 00:23:04,000
+حدودها مختلفة عن الـ C و ال sequence نهايتها C
+
+197
+00:23:04,000 --> 00:23:09,400
+فلازم أثبت نهاية صورة ال sequence بالساوية لعدد L
+
+198
+00:23:09,400 --> 00:23:13,640
+وهذا أثبتناه، اذا by sequential criterion بطلع
+
+199
+00:23:13,640 --> 00:23:20,950
+limit H عن C بساوية L وهو المطلوبOkay تمام إذا هذا
+
+200
+00:23:20,950 --> 00:23:25,230
+برهان الـ sequential criteria الـ squeeze theorem
+
+201
+00:23:25,230 --> 00:23:31,070
+for functions تمام و في كتير من نظريات الأخرى اللي
+
+202
+00:23:31,070 --> 00:23:36,370
+أثبتناها بالنسبة لل limits of sequences فيه
+
+203
+00:23:36,370 --> 00:23:38,290
+بيقابلها أو بيوزيها
+
+204
+00:23:40,890 --> 00:23:45,890
+نظريات خاصة بال functions هتشوفوا بعض النظريات هذه
+
+205
+00:23:45,890 --> 00:23:53,470
+في التمرين فهينبرنتلكم بعضهم ناخد الآن بعض الأمثلة
+
+206
+00:23:53,470 --> 00:23:58,010
+على النظريات هذه وخاصة النظرية الأخيرة
+
+207
+00:24:20,680 --> 00:24:30,940
+examples أمثلة واحد show أن ال limit لل function x
+
+208
+00:24:30,940 --> 00:24:41,220
+أس تلاتة على اتنين لما x تقول سفر بساوي سفر و
+
+209
+00:24:41,220 --> 00:24:43,540
+طبعا هنا ال x موجة
+
+210
+00:25:03,540 --> 00:25:07,520
+الفترة المفتوحة من 0 إلى ملانهاية
+
+211
+00:25:12,860 --> 00:25:15,920
+طبعا مافيش ولا قانون من القوانين النهائية اللي
+
+212
+00:25:15,920 --> 00:25:19,680
+اخدناها سابقا بيعطيني ان ال limit هذي بالساوية سفر
+
+213
+00:25:19,680 --> 00:25:24,520
+يعني ماجدرش اقول عوض عن x بساوية سفر بطلع سفر
+
+214
+00:25:24,520 --> 00:25:29,640
+مافيش ماكان لاش قانون زي هذا هذي ليست polynomial x
+
+215
+00:25:29,640 --> 00:25:33,740
+أُس 3 over 2 is not polynomial لو كانت polynomial
+
+216
+00:25:33,740 --> 00:25:39,420
+بنعود على طول مباشرة عن x بالساوية سفر ولكنها ليست
+
+217
+00:25:39,420 --> 00:25:45,270
+polynomialفلبرهان النظرية هذه بناخد الدالة f of x
+
+218
+00:25:45,270 --> 00:25:49,970
+بالساوي x أص تلاتة على اتنين بنلاحظ أولا، لاحظي
+
+219
+00:25:49,970 --> 00:25:56,550
+note ان ال x for
+
+220
+00:25:56,550 --> 00:26:02,610
+x أكبر من سفر أصغر من أو ساوي الواحد بطلع عندي x
+
+221
+00:26:02,610 --> 00:26:07,330
+تقريبا أصغر من أو ساوي x أصغر من أو ساوي الواحد
+
+222
+00:26:11,890 --> 00:26:21,370
+أو X أصغر من أو يساوي X أص نص أصغر من أو يساوي
+
+223
+00:26:21,370 --> 00:26:29,290
+الواحد كلمة
+
+224
+00:26:29,290 --> 00:26:35,050
+X الجدر الترميه ل X بتطلع أكبر من أو يساوي ال X
+
+225
+00:26:35,050 --> 00:26:39,630
+إذا ال X محصولة من سفر واحد وبالتالي
+
+226
+00:26:41,250 --> 00:26:51,130
+لو ضربت .. لو ضربت هنا في .. في x فهذا هيقدّي أن x
+
+227
+00:26:51,130 --> 00:26:57,150
+تربيه أصغر من أو يساوي x أس تلاتة ع اتنين أصغر من
+
+228
+00:26:57,150 --> 00:27:01,430
+أو يساوي x إذن هذا الكلام صحيح
+
+229
+00:27:05,500 --> 00:27:13,040
+هذا كلام صحيح لكل x أكبر من سفر أصغر من أو ساوي
+
+230
+00:27:13,040 --> 00:27:21,180
+الواحد الان لما x تقول إلى سفر x تربيها كثيرة حدود
+
+231
+00:27:21,180 --> 00:27:26,760
+ال limit بتاعتها سفر وال x هذه polynomial لما x
+
+232
+00:27:26,760 --> 00:27:32,820
+تقول لسفر تطلع نهيتها سفر تمام؟ اذا by squeeze
+
+233
+00:27:32,820 --> 00:27:33,400
+theorem
+
+234
+00:27:52,880 --> 00:28:01,340
+المثال التاني show أن limit ال function sin x لما
+
+235
+00:28:01,340 --> 00:28:03,980
+x تقول سفر بسوى سفر
+
+236
+00:28:08,850 --> 00:28:18,290
+لبرهان ذلك فيه متباينة معروفة it is known c
+
+237
+00:28:18,290 --> 00:28:21,870
+chapter
+
+238
+00:28:21,870 --> 00:28:32,370
+8 that sin x دايما أصغر من أو ساوي x أكبر من أو
+
+239
+00:28:32,370 --> 00:28:44,050
+ساوي سالب xلكل X ينتمي إلى R معروف وهذا .. هيتم ..
+
+240
+00:28:44,050 --> 00:28:49,290
+هذا له برهان في chapter 8 في نفس الكتاب تبعنا اللي
+
+241
+00:28:49,290 --> 00:28:52,470
+هياخدوا منكم تحليل حقيقة 2 هياخدوا ال chapter هذا
+
+242
+00:28:52,470 --> 00:28:57,910
+ففيه برهان هناك للمتبايلة هذه او الحقيقة هذه ان
+
+243
+00:28:57,910 --> 00:29:02,350
+sign X دايما اظلم لو ساوي X اكبر من او ساوي
+
+244
+00:29:02,350 --> 00:29:03,270
+negative X
+
+245
+00:29:08,790 --> 00:29:12,770
+هذا الرسم ماتوضح يعني مش حاكته بانا برهان زي ما
+
+246
+00:29:12,770 --> 00:29:18,250
+اقول ان هذا موجود في chapter تمانية فالناس هياخدوا
+
+247
+00:29:18,250 --> 00:29:23,290
+real analysis اتنين هيشوفوه هاي ال .. ال function
+
+248
+00:29:23,290 --> 00:29:34,130
+y بساوي x و هاي ال function و
+
+249
+00:29:34,130 --> 00:29:38,170
+هاد ال function y بساوي negative xو الـ sine
+
+250
+00:29:38,170 --> 00:29:42,170
+function الموجة
+
+251
+00:29:42,170 --> 00:29:52,790
+أو ال wave تبعتها زي هيك إذا
+
+252
+00:29:52,790 --> 00:30:00,090
+هذه ال function y بساوي sin xفلاحظوا ان ال sign
+
+253
+00:30:00,090 --> 00:30:05,330
+function محصورة بين y بساوي x و y بساوي سالب x
+
+254
+00:30:05,330 --> 00:30:11,510
+تمام؟ هذا طبعا من الرسم لكن الرسم ليس برهان أنا
+
+255
+00:30:11,510 --> 00:30:17,230
+مجرد توضيح الأن أنا عندي ال function هذه لما x
+
+256
+00:30:17,230 --> 00:30:22,850
+تقول للصفر لما x تقول للصفر بتقول للصفر وهذه برضه
+
+257
+00:30:22,850 --> 00:30:28,600
+ال function لما x تقول للصفر بتروح للصفرإذا by
+
+258
+00:30:28,600 --> 00:30:36,760
+squeeze theorem .. إذا by squeeze
+
+259
+00:30:36,760 --> 00:30:45,240
+theorem ال limit لل function sin x اللي هي محصورة
+
+260
+00:30:45,240 --> 00:30:51,640
+في النص and السفر مساوي سفر okay تمام؟ لأن هذه
+
+261
+00:30:51,640 --> 00:30:56,760
+حقيقة برهانها بتم هكذا ماذا؟ في أي سؤال؟
+
+262
+00:31:00,750 --> 00:31:05,610
+كمان في مثال
+
+263
+00:31:05,610 --> 00:31:12,310
+تالت ممكن اثبات ان ال cosine ل X لما X تقول ل 0
+
+264
+00:31:12,310 --> 00:31:23,490
+بساوي 1 proof we
+
+265
+00:31:23,490 --> 00:31:31,480
+useالمتباينة في متباينة اللي هي واحد سالب X تربيع
+
+266
+00:31:31,480 --> 00:31:37,240
+اتنين اصغر من او ساوي cosine X اصغر من او ساوي
+
+267
+00:31:37,240 --> 00:31:42,920
+واحد for every X ينتمي الاراضي المتباينة هذه صحيحة
+
+268
+00:31:42,920 --> 00:31:45,680
+لكل الاعداد الحقيقية
+
+269
+00:31:50,520 --> 00:32:01,100
+و هذه برضه ممكن إثباتها في chapter 8
+
+270
+00:32:01,100 --> 00:32:07,460
+في real analysis 2 الان ال function هذه لما x تقول
+
+271
+00:32:07,460 --> 00:32:15,280
+ل 0 بتقول ل 1 tense واحد as x tends to zero صح؟
+
+272
+00:32:15,280 --> 00:32:20,990
+هذه كثيرة حدودعوض عن x بساوية سفر وهذه دالة ثابتة،
+
+273
+00:32:20,990 --> 00:32:27,310
+نهيتها واحد لما x تقول إلى أي حاجة، إذا ممكن نطبق
+
+274
+00:32:27,310 --> 00:32:32,870
+ال squeeze theorem، okay؟ إذا by squeeze theorem،
+
+275
+00:32:32,870 --> 00:32:39,210
+ال limit ل ال function cosine
+
+276
+00:32:39,210 --> 00:32:47,780
+x as x tends to zero تطلع بساوية واحدةتمام؟ okay
+
+277
+00:32:47,780 --> 00:32:53,000
+واضح؟ في كتير من ال limits المعروفة ممكن اثباتها
+
+278
+00:32:53,000 --> 00:33:01,140
+بنفس الطرق هذه خلينا ناخد كمان مثال limit ل
+
+279
+00:33:01,140 --> 00:33:09,120
+function x غرب sin 1 على x as x tends to zero
+
+280
+00:33:09,120 --> 00:33:12,320
+exists و بالساوي سفر
+
+281
+00:33:14,890 --> 00:33:21,610
+فالـ function هنا الـ
+
+282
+00:33:21,610 --> 00:33:25,610
+function اللي بدي أخدلها ال limit هي عبارة عن الـ
+
+283
+00:33:25,610 --> 00:33:33,670
+function f of x بساوي x ضرب sign واحد على x طبعا
+
+284
+00:33:33,670 --> 00:33:38,410
+هنا x لا تساوي سفر ال domain للـ function هذه كل
+
+285
+00:33:38,410 --> 00:33:45,650
+الأعداد الحقيقية معدىالسفر لأن اسمه على سفر هنا مش
+
+286
+00:33:45,650 --> 00:33:54,170
+معرفة تمام؟ بنثبت ان ال limit للدالة هذه عند السفر
+
+287
+00:33:54,170 --> 00:33:59,230
+بساوي سفر بالمناسبة الدالة هذه هي نفس الدالة
+
+288
+00:33:59,230 --> 00:34:07,570
+المرسومة على ال cover تبع الكتاب هذه رسمة الدالة
+
+289
+00:34:09,910 --> 00:34:17,470
+فلس متدالة f of x بالساوي x في sin واحد على x و
+
+290
+00:34:17,470 --> 00:34:19,910
+الدالة هذه زي ما انتوا شايفين المرحلة تبعها كل ما
+
+291
+00:34:19,910 --> 00:34:25,450
+جربته من السفر كل ما جرب من السفر سواء من اليمين
+
+292
+00:34:25,450 --> 00:34:30,810
+او اليسار لكن لإثبات ذلك نستخدم ال squeeze theorem
+
+293
+00:34:30,810 --> 00:34:35,030
+طيب
+
+294
+00:34:39,940 --> 00:34:49,380
+لت x لا يساوي سفر، خلّيني أخد x بساوي سفر، اذا by
+
+295
+00:34:49,380 --> 00:34:53,320
+tricotomy property باستخدام الخاصية الفلاتية، x
+
+296
+00:34:53,320 --> 00:34:59,740
+أكبر من سفر أو x أصغر من سفر، فحناخد في الحالتين
+
+297
+00:35:09,860 --> 00:35:14,980
+طيب، لنثبت أنه
+
+298
+00:35:14,980 --> 00:35:20,940
+لأي x لا تساوي 0 زي هذه سالب absolute x أصغر من أو
+
+299
+00:35:20,940 --> 00:35:28,000
+ساوي x في sin 1 على x أصغر من أو ساوي absolute x،
+
+300
+00:35:28,000 --> 00:35:30,900
+هذا صحيح لكل x لا يساوي 0
+
+301
+00:35:39,610 --> 00:35:44,550
+أنا بدي أثبت أنه لكل x لا يساوي سفر المتباين هذا
+
+302
+00:35:44,550 --> 00:35:54,970
+صحيح ف to see this لإثبات ذلك to see thisfix x لا
+
+303
+00:35:54,970 --> 00:35:59,930
+يساوي سفر خلّيني أخد arbitrary x مختلفة عن السفر و
+
+304
+00:35:59,930 --> 00:36:04,570
+نثبت إن المتباين هي دي صحيحة إلها then إذا أخدت
+
+305
+00:36:04,570 --> 00:36:09,850
+أنا x لا تساوي سفر فإما x أكبر من سفر or x أصغر من
+
+306
+00:36:09,850 --> 00:36:18,760
+سفر فإذا في عندي أنا حالتين نشوف الحالة الأولىلما
+
+307
+00:36:18,760 --> 00:36:24,480
+تكون الـ X أكبر من الزبر هذا بيقدي ان absolute X
+
+308
+00:36:24,480 --> 00:36:37,620
+بالساوي X صح؟ طيب هنا note that sign
+
+309
+00:36:39,330 --> 00:36:45,010
+ثيتا دائما أكبر من أوساو سالب واحد أصغر من أوساو
+
+310
+00:36:45,010 --> 00:36:49,690
+واحد for all ثيتا تنتمي إلى R هذه المتباينة
+
+311
+00:36:49,690 --> 00:36:53,690
+الصحيحة ال absolute maximum value ل سايل واحد وال
+
+312
+00:36:53,690 --> 00:36:59,710
+absolute minimum value سالب واحد خلينا
+
+313
+00:36:59,710 --> 00:37:08,180
+نضرب المتباينة هذه في absolute Xو ال theta هذه
+
+314
+00:37:08,180 --> 00:37:13,580
+تبدلها
+
+315
+00:37:13,580 --> 00:37:20,420
+ب 1 على x ف since sin 1 على x أكبر من أو ساوي سالب
+
+316
+00:37:20,420 --> 00:37:26,020
+واحد أصغر من أو ساوي واحد لكل x لا تساوي سفر هذا
+
+317
+00:37:26,020 --> 00:37:26,660
+بيقدر
+
+318
+00:37:29,420 --> 00:37:33,100
+لو ضربت المتباينة هذه في absolute x اللي هي
+
+319
+00:37:33,100 --> 00:37:40,160
+بالساوي x و absolute x هنا هتطلع موجة بقى فهيطلع
+
+320
+00:37:40,160 --> 00:37:45,300
+عندي سالب absolute x أصغر من أو ساوي x في sin 1
+
+321
+00:37:45,300 --> 00:37:49,760
+على x أصغر من أو ساوي absolute x وبالتالي
+
+322
+00:37:49,760 --> 00:37:52,320
+المتباينة اللي انا عايز اثبتها ده هي صحيحة
+
+323
+00:37:56,180 --> 00:38:00,780
+أضرب هنا في absolute x و absolute x بالساوية x عدد
+
+324
+00:38:00,780 --> 00:38:07,200
+موجب فالمتباينة تبقى أشرتها زي ما هي و هي ضربت في
+
+325
+00:38:07,200 --> 00:38:10,600
+سالب absolute x ال x هنا هي برضه سالب absolute x
+
+326
+00:38:10,600 --> 00:38:14,460
+فهذه المتباينة اللي احنا عايزين نثبتها في الإدعاء
+
+327
+00:38:14,460 --> 00:38:20,200
+تبعنا الحالة التالية لو كانت x سالب
+
+328
+00:38:31,100 --> 00:38:37,440
+كاس اتنين لو كانت x سالبها فهذا بيقدي ان absolute
+
+329
+00:38:37,440 --> 00:38:45,580
+x بساوي سالب x طبعا هذا موجة بقى وبالتالي multiply
+
+330
+00:38:49,120 --> 00:38:54,100
+multiply المتباينة سالب واحد اصغر من او ساوي ساين
+
+331
+00:38:54,100 --> 00:39:00,200
+واحد على اكس اصغر من او ساوي واحد by absolute x
+
+332
+00:39:00,200 --> 00:39:05,240
+بساوي سالب x اكبر من السفر هذا يعني انه
+
+333
+00:39:12,420 --> 00:39:19,180
+فهذا بيقدي انه ايه؟ انه سالب absolute x أصغر من أو
+
+334
+00:39:19,180 --> 00:39:26,300
+يساوي سالب
+
+335
+00:39:26,300 --> 00:39:33,080
+x في sin 1 على x أصغر من أو يساوي absolute ال x
+
+336
+00:39:46,010 --> 00:39:54,230
+نضرب في سالب واحد نضرب في سالب واحد نضرب في سالب
+
+337
+00:39:54,230 --> 00:39:58,910
+واحد نضرب في سالب واحد نضرب في سالب واحد نضرب في
+
+338
+00:39:58,910 --> 00:39:58,970
+سالب واحد نضرب في سالب واحد نضرب في سالب واحد نضرب
+
+339
+00:39:58,970 --> 00:39:59,210
+في سالب واحد نضرب في سالب واحد نضرب في سالب واحد
+
+340
+00:39:59,210 --> 00:40:02,030
+نضرب في سالب واحد نضرب في سالب واحد نضرب في سالب
+
+341
+00:40:02,030 --> 00:40:06,590
+واحد نضرب في سالب واحد نضرب في سالب واحد نضرب في
+
+342
+00:40:06,590 --> 00:40:13,670
+سالب واحدو هذه هي المتباينة اللي احنا عايزين
+
+343
+00:40:13,670 --> 00:40:21,370
+نثبتها صح؟ اذا this completes
+
+344
+00:40:21,370 --> 00:40:24,770
+the
+
+345
+00:40:24,770 --> 00:40:34,330
+proof of the claim اذا هنا برهنة ل claim وهو ان
+
+346
+00:40:34,330 --> 00:40:37,970
+المتباينة هذه صحيحة تمام؟
+
+347
+00:40:40,820 --> 00:40:50,560
+الان من ال claim now
+
+348
+00:40:50,560 --> 00:41:02,560
+by above .. by above the claim and squeeze the
+
+349
+00:41:02,560 --> 00:41:06,560
+theorem since
+
+350
+00:41:06,560 --> 00:41:12,450
+بما أنه ال limitالـ absolute x as x tends to 0
+
+351
+00:41:12,450 --> 00:41:18,310
+بساوي ال limit لسالب absolute x as x tends to 0
+
+352
+00:41:18,310 --> 00:41:26,170
+بساوي 0 we have نحصل على ان ال limit المحصورة في
+
+353
+00:41:26,170 --> 00:41:32,710
+الوسط اللي هي x في sin 1 على x as x tends to 0
+
+354
+00:41:32,710 --> 00:41:39,760
+بساوي أيضا السفر وهو المطلوباذا هنا استخدمنا الـ
+
+355
+00:41:39,760 --> 00:41:45,780
+squeeze theorem تمام؟ لأن هذا برهان أن ال limit لل
+
+356
+00:41:45,780 --> 00:41:51,380
+function x sin 1 على x عندما x تقوى للصفر بيساوي
+
+357
+00:41:51,380 --> 00:41:56,680
+صفر تمام؟ واضح؟ في أي سفسار؟ أي سؤال؟
+
+358
+00:42:03,260 --> 00:42:10,860
+Okay باقي نظرية واحدة في ال section 4-2 فاتحقلوا
+
+359
+00:42:10,860 --> 00:42:17,060
+تقرؤوها تفهموا البرهانة و تبدووا في حال التمرين و
+
+360
+00:42:17,060 --> 00:42:21,360
+ان شاء الله المرة جاية بنكمل نشرح النظرية هذه و
+
+361
+00:42:21,360 --> 00:42:25,820
+بنبدأ section جديد اللي هو section 4-3
+
+362
+00:42:28,450 --> 00:42:33,790
+شكرا لإصداركم وشوفكم ان شاء الله في المحاضرة
+
+363
+00:42:33,790 --> 00:42:34,250
+الجاية
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ff2733ae42bbcc7890cb27a6877d838de06de141
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_postprocess.srt
@@ -0,0 +1,1756 @@
+1
+00:00:20,670 --> 00:00:25,090
+تحدثنا عن ال absolute value و عرفنا ال absolute
+
+2
+00:00:25,090 --> 00:00:29,810
+value لأي real number إما a إذا كان a غير سالب أو
+
+3
+00:00:29,810 --> 00:00:33,310
+سالب العدد نفسه إذا كان العدد سالب نفس التعريف
+
+4
+00:00:33,310 --> 00:00:38,090
+اللي أخدناه في ال calculus و بكل من التعريف
+
+5
+00:00:38,090 --> 00:00:43,740
+باستخدام التعريف و ال logicأو مبادئ الرياضيات ممكن
+
+6
+00:00:43,740 --> 00:00:47,220
+بسهولة نبره ان ال absolute value ل negative a
+
+7
+00:00:47,220 --> 00:00:51,900
+بساوي absolute a ال absolute value لحاصل الضرب
+
+8
+00:00:51,900 --> 00:00:58,720
+بساوي حاصل ضرب ال absolute values لو كان C عدد غير
+
+9
+00:00:58,720 --> 00:01:04,240
+سالب ف ال inequality هذه بتكافئ ال inequality هذه
+
+10
+00:01:06,320 --> 00:01:11,960
+لأي عدد حقيقي ايه دائما أي عدد حقيقي بيكون أصغر من
+
+11
+00:01:11,960 --> 00:01:15,320
+أو يساوي ال absolute value له وأكبر من أو يساوي
+
+12
+00:01:15,320 --> 00:01:19,860
+سالب ال absolute value هذا دائما صحيح في عند ال
+
+13
+00:01:19,860 --> 00:01:24,000
+triangle inequality هذه متباينة كتير مهمة بتقول ان
+
+14
+00:01:24,000 --> 00:01:28,560
+ال absolute value لمجموعة أو الفرق بين عددين
+
+15
+00:01:28,560 --> 00:01:33,000
+الحقيقيين less than or equal لمجموعة ال absolute
+
+16
+00:01:33,000 --> 00:01:38,990
+values للعددينومن المتباينة هذه ممكن نستنتج
+
+17
+00:01:38,990 --> 00:01:44,970
+متباينة أخرى لاتقل عنها أهمية وهي المتباينة
+
+18
+00:01:44,970 --> 00:01:45,730
+التالية
+
+19
+00:01:52,010 --> 00:01:56,450
+اللي هي المتباينة هذه لو أخدت ال absolute value ل
+
+20
+00:01:56,450 --> 00:02:01,010
+A و طرحت منها ال absolute value ل B و أخدت ال
+
+21
+00:02:01,010 --> 00:02:06,830
+absolute value للفرق فهذا دائما أصغر من أو ساوي
+
+22
+00:02:06,830 --> 00:02:14,490
+absolute value للفرق و البرهان تبع ال inequality
+
+23
+00:02:14,490 --> 00:02:19,730
+هذه نتيجة مباشرة على ال triangle inequality
+
+24
+00:02:21,020 --> 00:02:26,900
+فالبراهين سهلة وبسيطة ويعني موجودة عندكم أي حد
+
+25
+00:02:26,900 --> 00:02:31,600
+فيكم بيقدر يقرأها ويفهمها لو مافيش .. في اي مشكلة
+
+26
+00:02:31,600 --> 00:02:36,520
+ممكن يعني اترجعوني خلال الساعات المكتبية او خلال
+
+27
+00:02:36,520 --> 00:02:44,220
+ساعة المناخشة في
+
+28
+00:02:44,220 --> 00:02:55,550
+اللي هو ملاحظة متباينة المثلثممكن تعميمها بدل
+
+29
+00:02:55,550 --> 00:03:02,850
+.. احنا قلنا ان ال absolute value ل a او a1 زائد
+
+30
+00:03:02,850 --> 00:03:12,440
+a2 اصغر من او ساوي absolute a1 زايد absolute a2أو
+
+31
+00:03:12,440 --> 00:03:16,100
+absolute A زي بي less than or equal to absolute A
+
+32
+00:03:16,100 --> 00:03:22,180
+زي absolute B المتباين هذه ممكن تعميمها مش بس ناخد
+
+33
+00:03:22,180 --> 00:03:24,780
+ال absolute value لعددين ممكن ناخد ال absolute
+
+34
+00:03:24,780 --> 00:03:30,320
+value ل N من الأعداد الحقيقية و نجمعهم فهذا أصغر
+
+35
+00:03:30,320 --> 00:03:34,490
+من أو ساوي مجموع ال absolute valuesوطبعا هذا ممكن
+
+36
+00:03:34,490 --> 00:03:39,710
+برهانة .. المتباين هذا ممكن برهانها بسهولة by
+
+37
+00:03:39,710 --> 00:03:46,930
+induction وطبعا ومتباينة في المثلثة okay اعتقد
+
+38
+00:03:46,930 --> 00:03:56,130
+يعني تقريبا هذا اللي شرحنا اليوم
+
+39
+00:03:56,130 --> 00:04:02,320
+بدنا ناخد تعريف ال epsilon neighborhoodفالـ
+
+40
+00:04:02,320 --> 00:04:09,680
+epsilon neverhood أو جوار epsilon لأي عدد حقيقي هي
+
+41
+00:04:09,680 --> 00:04:19,100
+خط العداد الحقيقية وهي العدد a وهي المسافة هذه او
+
+42
+00:04:19,100 --> 00:04:25,800
+النقطة هذه a سالب epsilon والنقطة هذه a موجب
+
+43
+00:04:25,800 --> 00:04:30,580
+epsilon والepsilon هذا عدد موجب
+
+44
+00:04:34,530 --> 00:04:39,430
+أذا الفترة المفتوحة هذه الفترة المفتوحة هذه اللي
+
+45
+00:04:39,430 --> 00:04:44,710
+هي a سالب إبسلون و a موجب إبسلون اللي هي الفترة
+
+46
+00:04:44,710 --> 00:04:49,770
+هذه الفترة
+
+47
+00:04:49,770 --> 00:04:57,230
+المفتوحة هذه بنسميها لحظة هذه الفترة مركزها a و نص
+
+48
+00:04:57,230 --> 00:05:02,490
+قطرها إبسلون عدد موجبفالفترة المفتوحة دي اللي
+
+49
+00:05:02,490 --> 00:05:06,510
+مركزها a و نص قطرة epsilon بنسميها epsilon
+
+50
+00:05:06,510 --> 00:05:12,510
+neighbourhood of a جوار epsilon أو جوار بأمق
+
+51
+00:05:12,510 --> 00:05:18,270
+epsilon للنقطة a وهكذا
+
+52
+00:05:18,270 --> 00:05:22,270
+تعريفه هاي v epsilon ومصدره بالرمز v epsilon ل a
+
+53
+00:05:22,270 --> 00:05:27,920
+جوار epsilon ل aفهو كل الأعداد الحقيقية اللي
+
+54
+00:05:27,920 --> 00:05:33,840
+المسافة بينها وبين ال A أصغر من Epsilon أو هي كل
+
+55
+00:05:33,840 --> 00:05:38,040
+الأعداد الحقيقية اللي بتحقق المتباينة هذه اللي هي
+
+56
+00:05:38,040 --> 00:05:41,780
+بالظبط كل الأعداد الحقيقية X اللي هي كل الأعداد
+
+57
+00:05:41,780 --> 00:05:47,460
+الحقيقية X اللي هي المسافة بينها وبين ال A أصغر من
+
+58
+00:05:47,460 --> 00:05:53,700
+المسافة هذه اللي هي Epsilonواضح تمام فهذا بنسمي
+
+59
+00:05:53,700 --> 00:05:56,800
+ابسلون neighborhood of A في حاجة اسمها
+
+60
+00:05:56,800 --> 00:06:05,340
+neighborhood of A بدون ابسلون فأي
+
+61
+00:06:05,340 --> 00:06:10,280
+.. طبعا اختصار neighborhood اللي بنختصرها NBD
+
+62
+00:06:10,280 --> 00:06:17,000
+neighborhood في تعريف تاني neighborhood of A is
+
+63
+00:06:17,000 --> 00:06:23,380
+any set V of Aبحيث بقدر ألاقي داخلها epsilon
+
+64
+00:06:23,380 --> 00:06:28,340
+neighborhood ل A بقدر ألاقي أنه عبارة عن set V of
+
+65
+00:06:28,340 --> 00:06:32,680
+A بقدر ألاقي داخلها epsilon neighborhood of A كمان
+
+66
+00:06:32,680 --> 00:06:36,720
+مرة يعني ممكن يكون عندي مثلا حاجة زي هذه مثلا set
+
+67
+00:06:36,720 --> 00:06:41,180
+زي هذه وهي
+
+68
+00:06:41,180 --> 00:06:48,860
+النقطة A فترة مثلا نصف مفتوحة زي هذه في هذه الفترة
+
+69
+00:06:50,180 --> 00:06:54,820
+ممكن تكون مثلا a سالب واحد و a موجة باتنين من
+
+70
+00:06:54,820 --> 00:07:01,060
+النقطة هذه ممكن ولا مش ممكن؟ وهذه فترة نص مفتوحة
+
+71
+00:07:01,060 --> 00:07:06,640
+هذه الفترة حسب التعريف b هي عبارة عن epsilon
+
+72
+00:07:06,640 --> 00:07:10,560
+neighborhood ل a لإيه؟ لأشان تكون الفترة هذه
+
+73
+00:07:10,560 --> 00:07:16,460
+epsilon عشان الفترة هذه أو ال set هذه اللي بنسميها
+
+74
+00:07:16,460 --> 00:07:22,600
+v of aعشان تكون neighborhood جوار ل A لازم ألاقي
+
+75
+00:07:22,600 --> 00:07:29,780
+داخلها epsilon neighborhood لل A وهذا موجود هاي
+
+76
+00:07:29,780 --> 00:07:37,080
+باخد المسافة هذه مثلا هاي A سالب نص وهاي A موجب نص
+
+77
+00:07:37,080 --> 00:07:48,080
+مثلا فالفترة هذه الفترة المفتوحة هذههي مركزها a و
+
+78
+00:07:48,080 --> 00:07:53,200
+نص كترها epsilon بساوة نص عدد موجب إذن هذه نجحت في
+
+79
+00:07:53,200 --> 00:08:01,320
+إيجاد v نص ل a و هذه الفترة المفتوحة واقع كليا
+
+80
+00:08:01,320 --> 00:08:07,080
+عبارة عن مجموعة جزئية من الفترة نص المفتوحة هذه
+
+81
+00:08:07,080 --> 00:08:13,880
+اللي هي a سالب واحد و a موجب اتنين okay تمام إذن
+
+82
+00:08:13,880 --> 00:08:20,840
+هذه الفترةعبارة عن neighborhood لـA إذا ال
+
+83
+00:08:20,840 --> 00:08:25,980
+neighborhood هو أي مجموعة تحتوي
+
+84
+00:08:25,980 --> 00:08:30,880
+A بحيث أن أنا أقدر ألاقي داخلها epsilon
+
+85
+00:08:30,880 --> 00:08:36,200
+neighborhood للـA طيب
+
+86
+00:08:36,200 --> 00:08:41,760
+وبالتالي من التعريفين هذول بنلاحظ أنه كل epsilon
+
+87
+00:08:41,760 --> 00:08:46,120
+neighborhoodكل y neighborhood ل a هو neighborhood
+
+88
+00:08:46,120 --> 00:09:02,320
+لكن العكس مش صحيح في نظرية هنا بتقول انه لو
+
+89
+00:09:02,320 --> 00:09:10,500
+كان عندي عدد حقيقي a فالعدد x العدد الحقيقي x بكون
+
+90
+00:09:10,500 --> 00:09:13,140
+موجود في كل neighborhoods ل a
+
+91
+00:09:16,470 --> 00:09:21,330
+لو كان X موجود في كل ال neighborhoods ل A فلازم ال
+
+92
+00:09:21,330 --> 00:09:26,510
+X هذه تساوي A لازم ال X تساوي A كمان مرة النظرية
+
+93
+00:09:26,510 --> 00:09:35,470
+هذه بتقول نعمل رسمة هاي
+
+94
+00:09:35,470 --> 00:09:42,130
+A لو كل Y neighborhood ل X
+
+95
+00:09:48,500 --> 00:09:53,240
+لو .. لو .. لو كل .. لو كل epsilon neighborhood أو
+
+96
+00:09:53,240 --> 00:09:58,800
+كل neighborhood هد عبارة عن neighborhood لإيه؟ لو
+
+97
+00:09:58,800 --> 00:10:03,140
+كل neighborhood لإيه يحتوي ال X؟ لو أي
+
+98
+00:10:03,140 --> 00:10:08,080
+neighborhood لإيه يحتوي ال X؟
+
+99
+00:10:08,080 --> 00:10:12,900
+أو كل الجوارات لل إيه تحتوي ال X؟ فلازم ال X هدي
+
+100
+00:10:12,900 --> 00:10:17,650
+تكون هي نفس الإيهوهذا البرهان بسيط بيعتمد على
+
+101
+00:10:17,650 --> 00:10:23,290
+خاصية أخدناها قبل هيك و قلتلكم عنها مهمة و
+
+102
+00:10:23,290 --> 00:10:28,450
+هنستخدمها لحظة وهي هنشوفها الجيت موجودة في نظرية
+
+103
+00:10:28,450 --> 00:10:35,850
+واحد تمانية البرهان بما
+
+104
+00:10:35,850 --> 00:10:39,970
+أن كل epsilon neighborhood ل a هو عبارة عن
+
+105
+00:10:39,970 --> 00:10:46,180
+neighborhood ل a فمن الفرضكل neighborhood ل A
+
+106
+00:10:46,180 --> 00:10:47,080
+يحتوى X
+
+107
+00:11:03,320 --> 00:11:07,140
+هذا معناه أنه من تعريف الـ epsilon neverhood هذا
+
+108
+00:11:07,140 --> 00:11:11,920
+معناه أن x المسافة بين x و a المسافة بين ال
+
+109
+00:11:11,920 --> 00:11:16,680
+absolute value بين x و a اللي هي المسافة بين x و a
+
+110
+00:11:16,680 --> 00:11:20,760
+أصغر من epsilon وهذا صحيح لكل epsilon أكبر من
+
+111
+00:11:20,760 --> 00:11:27,980
+السفر طب في نظرية واحد تمانية أخدنا المتباينة هذه
+
+112
+00:11:27,980 --> 00:11:34,740
+اللي بتقوللو كان العدد بي أكبر من أو يساوي صفر
+
+113
+00:11:34,740 --> 00:11:41,480
+أصغر من إبسلون لكل إبسلون أكبر من الصفر فهذا بيدّي
+
+114
+00:11:41,480 --> 00:11:47,040
+أن بي بيساوي صفر، بظبط؟ هذه نظرية واحد تمانية أو
+
+115
+00:11:47,040 --> 00:11:53,440
+جزء منها فأنا هي عندي .. هي عندي ال بي هذا عبارة
+
+116
+00:11:53,440 --> 00:11:58,950
+عن عدد غير سالمالقيمة المطلقة لأي عدد هو عدد غير
+
+117
+00:11:58,950 --> 00:12:07,350
+سالب فهذا هو ال B هذا هو العدد B هذا هو هذا
+
+118
+00:12:07,350 --> 00:12:14,070
+ال Bهيعندي بي أكبر من أو ساوي سفر هذا صحيح و أصغر
+
+119
+00:12:14,070 --> 00:12:17,910
+من إبسلون لكل إبسلون أكبر من السفر إذا حسب النظرية
+
+120
+00:12:17,910 --> 00:12:22,550
+هذه الـ b اللي هو absolute x minus a لازم بساوي
+
+121
+00:12:22,550 --> 00:12:27,830
+سفر وبالتالي إذا x سالب a بساوي سفر ومنها بطلع x
+
+122
+00:12:27,830 --> 00:12:35,270
+بساوي a هذا هو برهان النظرية تمام؟ في أي سؤال؟
+
+123
+00:12:39,260 --> 00:12:44,480
+برهان سهل واضح شوف بعض الأمثلة على ال
+
+124
+00:12:44,480 --> 00:12:49,880
+neighborhoods و ال epsilon neighborhoods يتوضح
+
+125
+00:12:49,880 --> 00:12:57,540
+برسمة كل مثال نعمله رسمة لو أخدت المجموعة U لو
+
+126
+00:12:57,540 --> 00:13:03,540
+أخدت U عبارة عن المجموعة أو الفترة المفتوحة من 0
+
+127
+00:13:03,540 --> 00:13:04,340
+إلى 1
+
+128
+00:13:10,710 --> 00:13:19,490
+هذه عبارة عن المجموعة U فالمجموعة
+
+129
+00:13:19,490 --> 00:13:23,610
+U هذه أو الفترة المفتوحة تعتبر neighborhood لأي
+
+130
+00:13:23,610 --> 00:13:32,450
+نقطة داخلها يعني لو أخدت أي نقطة A داخل
+
+131
+00:13:32,450 --> 00:13:39,810
+المجموعة Aفالمجموعة U هي عبارة عن neighborhood لأي
+
+132
+00:13:39,810 --> 00:13:49,690
+نقطة داخلها فلبرهان ذلك .. لبرهان ذلك بدنا ان ..
+
+133
+00:13:49,690 --> 00:13:54,710
+حسب التعريف عشان U تكون neighborhood للنقطة A اللي
+
+134
+00:13:54,710 --> 00:13:59,090
+داخلها لازم نلاقي epsilon neighborhood حوالين الـA
+
+135
+00:13:59,090 --> 00:14:07,420
+ويكون داخل المجموعة U مظبوط، هذا حسب التعريفطيب ال
+
+136
+00:14:07,420 --> 00:14:16,080
+.. بنيجي ناخد epsilon بساوي ال minimum الأصغر بين
+
+137
+00:14:16,080 --> 00:14:22,360
+المسافتين المسافة هذه .. المسافة هذه ايه؟ المسافة
+
+138
+00:14:22,360 --> 00:14:27,660
+هذه ايه؟ طيب و المسافة التانية هذه؟ واحد نقص ايه؟
+
+139
+00:14:27,660 --> 00:14:34,520
+واحد سالب ايه؟فالـ minimum بين العددين a و 1 سالب
+
+140
+00:14:34,520 --> 00:14:41,000
+a بناخده هو ال epsilon تمام؟ في الرسمة .. في
+
+141
+00:14:41,000 --> 00:14:47,360
+الرسمة هذه ال .. الأصغر بين العددين a و 1 سالب a
+
+142
+00:14:47,360 --> 00:14:52,320
+هو ال a فبقى أجي بكون حوالين ال a epsilon
+
+143
+00:14:52,320 --> 00:14:58,940
+neighborhood هي فهذا عبارة عن epsilon neighborhood
+
+144
+00:15:01,920 --> 00:15:08,720
+فالفترة المفتوحة هذه السودا بالتأكيد اذا هنا ال a
+
+145
+00:15:08,720 --> 00:15:14,320
+سالب ابسلون و a موجة بابسلون هذه بالتأكيد واقع
+
+146
+00:15:14,320 --> 00:15:23,590
+داخل ال U حسب الرسم وبالتالي هيني نجحتفي إيجاد V
+
+147
+00:15:23,590 --> 00:15:27,670
+Epsilon أو Epsilon neighborhood ل A و هذا ال
+
+148
+00:15:27,670 --> 00:15:34,310
+Epsilon neighborhood ل A واقع داخل ال U إذن حسب
+
+149
+00:15:34,310 --> 00:15:39,450
+التعريف ال U أبارع ال neighborhood للنقطة A و لما
+
+150
+00:15:39,450 --> 00:15:44,850
+كانت A نقطة عشوائية إذن U أبارع ال neighborhood
+
+151
+00:15:44,850 --> 00:15:49,990
+لأي نقطة داخلها تمام؟
+
+152
+00:15:53,400 --> 00:15:58,160
+Okay مثال تاني لو
+
+153
+00:15:58,160 --> 00:16:01,360
+أخدنا الفترة المغلقة هنا أخدنا U بالساوية الفترة
+
+154
+00:16:01,360 --> 00:16:07,900
+المفتوحة لو أخدنا هذه
+
+155
+00:16:07,900 --> 00:16:14,600
+خط الأعداد وهذه الفترة المغلقة من 0 إلى 1 هذه
+
+156
+00:16:14,600 --> 00:16:16,440
+الفترة نسميها I
+
+157
+00:16:21,110 --> 00:16:28,110
+I بتساوي الفترة المغلفة من 0 إلى 1 طبعا
+
+158
+00:16:28,110 --> 00:16:33,230
+زي ما في المثال السابق الفترة I هذه ممكن اثبات
+
+159
+00:16:33,230 --> 00:16:38,710
+انها neighborhood لأي نقطة داخلها معدى 0 والواحد
+
+160
+00:16:38,710 --> 00:16:46,210
+نفس البرهن لكن الفترة I هذه أو الست I ليست
+
+161
+00:16:46,210 --> 00:16:52,410
+neighborhoodليست I is not neighborhood للعنصر سفر
+
+162
+00:16:52,410 --> 00:16:58,930
+وكذلك ليست neighborhood للعنصر واحد ليه؟ لأنه بقدر
+
+163
+00:16:58,930 --> 00:17:08,670
+ألاقي لأن كل epsilon كل epsilon neighborhood للسفر
+
+164
+00:17:08,670 --> 00:17:13,170
+هاي لو أخدت أي epsilon مهما كانت صغيرة
+
+165
+00:17:15,840 --> 00:17:20,480
+يعني هي epsilon عدد موجب لأن هذا هيصير سفر زاد
+
+166
+00:17:20,480 --> 00:17:25,740
+epsilon وهي كمان المسافة هذه epsilon فهيصير هذه
+
+167
+00:17:25,740 --> 00:17:33,700
+طبعا سالب epsilon فلو أخدت أي فترة مفتوحة بأمق
+
+168
+00:17:33,700 --> 00:17:39,100
+epsilon هيثم epsilon عدد موجبإذا لأي إبسلون عدد
+
+169
+00:17:39,100 --> 00:17:42,900
+موجب لو كونت إبسلون neighborhood للسفر إذا هذه
+
+170
+00:17:42,900 --> 00:17:51,100
+الفترة الحمرة عبارة عن V إبسلون للسفر ف V إبسلون
+
+171
+00:17:51,100 --> 00:17:59,640
+للسفر هذا ليس ليس مجموعة جزئية من I هذا ليس مجموعة
+
+172
+00:17:59,640 --> 00:18:09,360
+جزئية من I لكل إبسلونلكل إبسلون أكبر من السفر لكل
+
+173
+00:18:09,360 --> 00:18:13,560
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+174
+00:18:13,560 --> 00:18:14,260
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+175
+00:18:14,260 --> 00:18:14,560
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+176
+00:18:14,560 --> 00:18:16,940
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+177
+00:18:16,940 --> 00:18:17,100
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+178
+00:18:17,100 --> 00:18:18,560
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+179
+00:18:18,560 --> 00:18:20,920
+إبسلون أكبر من السفر لكل إبسلون أكبر من السفر لكل
+
+180
+00:18:20,920 --> 00:18:26,910
+إبسلون أكبر من السفربالمثل الفترة I ليست نبرهود
+
+181
+00:18:26,910 --> 00:18:33,150
+للواحد لأن أي جوار بعمق إبسلون للواحد لا يقع كله
+
+182
+00:18:33,150 --> 00:18:37,410
+داخل الفترة I الجزء هذا خارج الفترة I، الجزء هذا
+
+183
+00:18:37,410 --> 00:18:40,010
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+184
+00:18:40,010 --> 00:18:40,990
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+185
+00:18:40,990 --> 00:18:41,330
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+186
+00:18:41,330 --> 00:18:42,670
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+187
+00:18:42,670 --> 00:18:44,910
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+188
+00:18:44,910 --> 00:18:46,930
+خارج الفترة I، الجزء هذا خارج الفترة I، الجزء هذا
+
+189
+00:18:46,930 --> 00:18:55,140
+خارج الفترة I، الجزء هذا خارج الفتمثال تالت مثال
+
+190
+00:18:55,140 --> 00:19:01,220
+تالت لو
+
+191
+00:19:01,220 --> 00:19:07,000
+أخدت أي تلت أعداد حقيقية x, y, z لو أخدت أي تلت
+
+192
+00:19:07,000 --> 00:19:18,240
+أعداد حقيقية x و y و z بحيث أنه هاي ال x أصغر من z
+
+193
+00:19:18,240 --> 00:19:28,110
+أصغر من أو يساوي zو X أصغر من أو يساوي Y و Y أصغر
+
+194
+00:19:28,110 --> 00:19:36,110
+من Z فهذا بيقدي لازم يقدي أن المسافة المسافة بين X
+
+195
+00:19:36,110 --> 00:19:43,070
+و Y زاد المسافة بين Y و Z بتساوي المسافة بين X و Z
+
+196
+00:19:43,070 --> 00:19:51,590
+هذا واضح هذا واضح من الرسم لكن طبعا الرسم أمره ما
+
+197
+00:19:51,590 --> 00:19:57,930
+كان برهانأما لكن ممكن يوضح في فكرة البرهان إذا
+
+198
+00:19:57,930 --> 00:20:02,470
+المجموعة القيم المطلقة هذه بساوة القيم المطلقة هذه
+
+199
+00:20:02,470 --> 00:20:07,090
+بس في حالة لما ال X أصغر من Y أو ال Y تكون بين X و
+
+200
+00:20:07,090 --> 00:20:13,290
+Z و X أصغر من أو ساو Z البرهان بكل بساطة كالتالي
+
+201
+00:20:13,290 --> 00:20:19,630
+أنا عندي الفرض تبعي X أصغر من أو ساو Z و Y بين X و
+
+202
+00:20:19,630 --> 00:20:27,690
+Zهذا يجعل القيمة المطلقة لـ x-y .. الآن x أصغر من
+
+203
+00:20:27,690 --> 00:20:31,570
+y، فهذا الفرق سالب، فالقيمة المطلقة لعدد سالب سالب
+
+204
+00:20:31,570 --> 00:20:38,920
+نفسهبالمثل هذا y سالب z و y أصغر من z إذا العدد
+
+205
+00:20:38,920 --> 00:20:41,360
+هذا اللي جوه ال absolute value سالب هذا حسب تعريف
+
+206
+00:20:41,360 --> 00:20:47,840
+القيمة المطلقة هذا بساوي سالبه و كذلك absolute x
+
+207
+00:20:47,840 --> 00:20:52,600
+minus z بساوي سالب العدد نفسه
+
+208
+00:20:57,610 --> 00:21:03,050
+بنجمع هاي absolute x minus y و absolute y minus z
+
+209
+00:21:03,050 --> 00:21:11,070
+فهي عندي بيطلع سالب x موجب y زائد سالب y موجب z و
+
+210
+00:21:11,070 --> 00:21:16,350
+هدوله طبعا بيطلع بساوي المجموع هذا و هذا بيساوي
+
+211
+00:21:16,350 --> 00:21:22,350
+absolute x minus z لأن x أصغر من z okay تمام إذا
+
+212
+00:21:22,350 --> 00:21:30,220
+هذا هو البرهانهذا هو البرامج نعم نعم X أقل من أو
+
+213
+00:21:30,220 --> 00:21:38,740
+يساوي Z نعم شوف هذا تبين ساوي هو يعني سواء X أصغر
+
+214
+00:21:38,740 --> 00:21:43,120
+من أو يساوي Z أو بتساوي في هذه المتباينة صحيحة
+
+215
+00:21:43,120 --> 00:21:52,370
+يعني لو X بتساوي Zيعني في الحالة الخاصة اللي X
+
+216
+00:21:52,370 --> 00:21:57,490
+فيها بتنطبق على Z ف Y هتنطبق على X وعلى Z يعني
+
+217
+00:21:57,490 --> 00:22:00,970
+التلتة هدول هيطلعوا متساويين وبالتالي هذه ال
+
+218
+00:22:00,970 --> 00:22:05,350
+absolute value سفر وهذه سفر سفر زاد سفر بساوي سفر
+
+219
+00:22:05,350 --> 00:22:12,090
+وبالتالي العلاقة أو المعادلة هذه صحيحة okay هذه
+
+220
+00:22:12,090 --> 00:22:16,350
+الحالة الخاصة اللي فيها بيكون X بساوي Z في الحالة
+
+221
+00:22:16,350 --> 00:22:21,690
+هذه لازم يطلع طبعاأي عدد بين X و Z سيكون بيسويهم
+
+222
+00:22:21,690 --> 00:22:28,670
+وبالتالي هذا سفر وهذا سفر وهذا سفر في أي سؤال
+
+223
+00:22:28,670 --> 00:22:37,310
+تاني؟ طيب في كمان مثال رابع
+
+224
+00:22:40,090 --> 00:22:44,170
+برضه كل الأمثلة هذه على التطبيقات على ال absolute
+
+225
+00:22:44,170 --> 00:22:47,490
+value لو في اندي فانكشن يعني افرض انك انت في
+
+226
+00:22:47,490 --> 00:22:51,810
+calculus ولا في course calculus او algebra وطلب
+
+227
+00:22:51,810 --> 00:22:58,150
+منك عرف الدالة f of x علي انها 2x تربيه زي 3x زي 1
+
+228
+00:22:58,150 --> 00:23:03,630
+على 2x زالف 1 والمجال تبع الدالة هذه الفترة
+
+229
+00:23:03,630 --> 00:23:09,210
+المغلقة من 2 ل 3برهن أن القيمة المطلقة لدالة هذا
+
+230
+00:23:09,210 --> 00:23:14,650
+أصغر من أو ساوي 28 على 3 لكل x في المجال تبعها
+
+231
+00:23:14,650 --> 00:23:21,550
+فكيف نبرهن هذا؟ نستخدم خواص القيمة المطلقة بدي أخد
+
+232
+00:23:21,550 --> 00:23:24,850
+.. بدي أثبت absolute f of x أصغر من أو ساوي العدد
+
+233
+00:23:24,850 --> 00:23:32,610
+هذا باخد absolute xبـ absolute f of x فقيمة
+
+234
+00:23:32,610 --> 00:23:36,590
+f of x بساوية قيمة المطلقة للبسط على قيمة المطلقة
+
+235
+00:23:36,590 --> 00:23:40,230
+للمقرنة هذه خاصية من قواصة ال absolute value لأن
+
+236
+00:23:40,230 --> 00:23:44,290
+القسمة هي ضرب وقلنا absolute value للضرب بساوية
+
+237
+00:23:44,290 --> 00:23:48,590
+حاصل ضرب ال absolute valueالان تعالوا نشتغل على
+
+238
+00:23:48,590 --> 00:23:51,790
+الـ BEST هاي الـ BEST هو عبارة عن القيمة المطلقة
+
+239
+00:23:51,790 --> 00:23:56,310
+دي باستخدام ال generalized triangle inequality ال
+
+240
+00:23:56,310 --> 00:23:59,730
+absolute value لمجموعة تلت أعداد أصغر من أو ساوي
+
+241
+00:23:59,730 --> 00:24:06,300
+مجموعة ال absolute values لهمأبسل يوت اتنين X
+
+242
+00:24:06,300 --> 00:24:10,680
+تربية زاد أبسل يوت تلاتة X بيطلع تلاتة أبسل يوت X
+
+243
+00:24:10,680 --> 00:24:12,580
+زاد أبسل يوت واحد بيطلع واحد
+
+244
+00:24:34,520 --> 00:24:39,900
+فنجمع الأرقام هذه بيطلع MD 28 إذا ال absolute
+
+245
+00:24:39,900 --> 00:24:45,520
+value لل bus هذا ال absolute value لل bus هذا طلع
+
+246
+00:24:45,520 --> 00:24:51,080
+أصغر من أو ساوي 28 الآن تعالوا نشوف ال absolute
+
+247
+00:24:51,080 --> 00:24:52,220
+value للمقام
+
+248
+00:24:54,600 --> 00:24:59,580
+absolute 2x سالب واحد نستخدم المرة هذه ال triangle
+
+249
+00:24:59,580 --> 00:25:04,480
+inequality هذه اللي هي absolute absolute a minus
+
+250
+00:25:04,480 --> 00:25:19,040
+absolute b أكبر من أو يساوي absolute a minus b أقل
+
+251
+00:25:19,040 --> 00:25:22,500
+أقل صحيح
+
+252
+00:25:27,790 --> 00:25:34,990
+فانا عندي هنا ال .. انا عندي ال A بساوي اتنين X
+
+253
+00:25:34,990 --> 00:25:42,010
+وال B بساوي واحد لانت
+
+254
+00:25:42,010 --> 00:25:46,590
+هكسلتي سؤال بدون أذن واتكلمتي بدون بسمعليه فمش
+
+255
+00:25:46,590 --> 00:25:53,380
+هجاوب على سؤالكلأ خرجت القاعدة هذه هي المتباينة
+
+256
+00:25:53,380 --> 00:25:58,000
+هذه أخدناها حكينا عنها اليوم صح أظبط لأ بدي أستخدم
+
+257
+00:25:58,000 --> 00:26:02,780
+المتباينة هذه يعني عندي absolute 2x سالب واحد فهي
+
+258
+00:26:02,780 --> 00:26:11,490
+absolute 2x a باخد a بساوي 2x و b بساوي واحدفهذا
+
+259
+00:26:11,490 --> 00:26:17,750
+المفروض يصير أكبر
+
+260
+00:26:17,750 --> 00:26:20,170
+من أو ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من
+
+261
+00:26:20,170 --> 00:26:20,810
+أو ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+262
+00:26:20,810 --> 00:26:29,470
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من
+
+263
+00:26:29,470 --> 00:26:36,150
+أو ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+264
+00:26:36,150 --> 00:26:37,690
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+265
+00:26:37,690 --> 00:26:37,910
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+266
+00:26:37,910 --> 00:26:38,110
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+267
+00:26:38,110 --> 00:26:39,310
+ساوي أكبر من أو ساوي أكبر من أو ساوي أكبر من أو
+
+268
+00:26:39,310 --> 00:26:44,120
+ساوي أكبر من أو ساوي أكبر من أو س-1 وانا عندي x
+
+269
+00:26:44,120 --> 00:26:50,440
+أكبر من أو يساوي ل 2 إذا absolute x أكبر من أو
+
+270
+00:26:50,440 --> 00:26:57,020
+يساوي 2 إذا هذا أكبر من أو يساوي 2 في 2 سالب 1
+
+271
+00:26:57,020 --> 00:27:05,130
+اللي هو 3كمان مرة باستخدام المتباينة هذه absolute
+
+272
+00:27:05,130 --> 00:27:09,470
+a minus b أكبر من أو ساوي absolute a اللي هي 2 في
+
+273
+00:27:09,470 --> 00:27:14,050
+absolute x سالب absolute b اللي هو absolute 1 بطلع
+
+274
+00:27:14,050 --> 00:27:18,630
+1 الآن ال X هذه أكبر من أو ساوي 2 إذا القيمة
+
+275
+00:27:18,630 --> 00:27:24,710
+المطلقة برضه أكبر من أو ساوي ل 2 إذا هذه أكبر من
+
+276
+00:27:24,710 --> 00:27:28,130
+أو ساوي 2 في 2 سالب 1 بطلع 3
+
+277
+00:27:30,990 --> 00:27:38,890
+الان اذا هذا عبارة عن المقام هنا فإذا كان العدد
+
+278
+00:27:38,890 --> 00:27:43,290
+احنا خدنا قبل هيك خاصية ما تنساش الخواص اللي
+
+279
+00:27:43,290 --> 00:27:51,370
+خدناها في بتقول لو كان عندي أنا عدد A غير سالب او
+
+280
+00:27:51,370 --> 00:27:55,290
+اكبر من السفر فمقلوب
+
+281
+00:28:00,600 --> 00:28:10,640
+أو لو كان a أكبر من b أكبر من 0 فهذا بيقدي أن 1
+
+282
+00:28:10,640 --> 00:28:17,520
+على a أصغر من 1 على b، مظبوط؟ فأنا إذا مقلوب ..
+
+283
+00:28:17,520 --> 00:28:22,480
+إذا مقلوب المقدار هذا أصغر من أو ساوي مقلوب
+
+284
+00:28:22,480 --> 00:28:23,140
+التلاتة
+
+285
+00:28:25,760 --> 00:28:30,340
+إذاً هذا بيطلع .. إذاً هذا الـ gate نساوي بيطلع
+
+286
+00:28:30,340 --> 00:28:36,820
+أصغر من أو خلّينا نكتبه على صورة هاي .. هذا عبارة
+
+287
+00:28:36,820 --> 00:28:48,520
+عن absolute 2x تربيع زاد 3x زاد 1 ضرب ضرب 1 على
+
+288
+00:28:48,520 --> 00:28:51,820
+absolute 2x سالب 1
+
+289
+00:28:57,850 --> 00:29:02,550
+يعني انا حولت القسمة لضرب الان الحد الاول اصغر من
+
+290
+00:29:02,550 --> 00:29:06,330
+او ساوي تمانية و عشرين و الحد التاني اللي هو
+
+291
+00:29:06,330 --> 00:29:09,810
+المقلوب absolutely of x سالب واحد اصغر من او ساوي
+
+292
+00:29:09,810 --> 00:29:16,530
+تلاتة عفوا تلت وبالتالي بيطلع اذا absolute f of x
+
+293
+00:29:16,530 --> 00:29:22,950
+اصغر من او ساوي تمانية و عشرين على تلاتة okay تمام
+
+294
+00:29:24,140 --> 00:29:29,260
+في أي سؤال؟ اذا هنا يعني احنا استخدمنا متباينة
+
+295
+00:29:29,260 --> 00:29:34,320
+المثلث العادية ومتباينة المثلث الأخرى اللي هي هذه
+
+296
+00:29:34,320 --> 00:29:47,660
+في اثبات اللي احنا عايزينه في السؤال هذا طيب
+
+297
+00:29:47,660 --> 00:29:53,530
+ننتقل لموضوع تانياللي هو الـ completeness property
+
+298
+00:29:53,530 --> 00:30:02,450
+of R خاصة تمام للأعداد الحقيقية بالمناسبة
+
+299
+00:30:02,450 --> 00:30:07,470
+في مرجع انا يمكن مش مكتوب في ال syllabus عندكم لكن
+
+300
+00:30:07,470 --> 00:30:15,650
+خليني انا احكيلكم عنه المرجع هذا مكتوب باللغة
+
+301
+00:30:15,650 --> 00:30:20,630
+العربية ويمكن يكون سهل بالنسبالكمو مش غلط انكم
+
+302
+00:30:20,630 --> 00:30:29,490
+تقتلوه او تقتلينه اللي هو كتاب كتاب
+
+303
+00:30:29,490 --> 00:30:40,330
+التحليل الحقيقي لجامعة
+
+304
+00:30:40,330 --> 00:30:43,570
+الخدس
+
+305
+00:30:43,570 --> 00:30:46,970
+المفتوحة
+
+306
+00:30:49,530 --> 00:30:57,930
+أنا درست ال course من الكتاب هذا و كتاب رائع جدا و
+
+307
+00:30:57,930 --> 00:31:01,910
+مكتوب من قبل نخبة من المدرسين و أنا رجعت الكتاب و
+
+308
+00:31:01,910 --> 00:31:08,410
+نقحته و عملنا فيه بعض التعديلات فيعني لو اطلعتوا
+
+309
+00:31:08,410 --> 00:31:14,050
+عليه هيكون مفيد جدا بالاضافة لمرجعتنا اللي احنا
+
+310
+00:31:14,050 --> 00:31:15,790
+قررين عليه
+
+311
+00:31:23,230 --> 00:31:29,730
+Tamam؟ Yes sir إذا ال completeness property of R
+
+312
+00:31:29,730 --> 00:31:34,830
+أو خاصية التمام أو الكمال للعداد الحقيقية نشوف شو
+
+313
+00:31:34,830 --> 00:31:39,410
+معناه عشان نصل للخاصية هذه بدنا شوية تعريفات و
+
+314
+00:31:39,410 --> 00:31:45,250
+شوية تمهيل فلو أخدت أي مجموعة جزئية من R
+
+315
+00:31:54,920 --> 00:31:59,360
+هذه خط الأعداد الحقيقية وهذه R وهو محور X ممكن
+
+316
+00:31:59,360 --> 00:32:05,480
+نعتبره R صح؟ ومحور Y برضه R شو الفرق بينهم فنأخد
+
+317
+00:32:05,480 --> 00:32:10,100
+الأعداد الحقيقية هيك رأسية وهذه المجموعة هذه هيك
+
+318
+00:32:10,100 --> 00:32:19,680
+مجموعة الأعداد هذه S هي مجموعة جزئية من R تمام؟
+
+319
+00:32:19,680 --> 00:32:27,650
+هذه المجموعة جزئية من Rفلو أخدت عدد U زي هذا U عدد
+
+320
+00:32:27,650 --> 00:32:33,090
+حقيقي فالعدد هذا بنسميه upper bound للمجموعة
+
+321
+00:32:33,090 --> 00:32:39,910
+الجزئية S إذا كان كل X في
+
+322
+00:32:39,910 --> 00:32:43,530
+المجموعة
+
+323
+00:32:43,530 --> 00:32:48,390
+S أصغر من أو يساوي U إذا مرة تانية العدد U upper
+
+324
+00:32:48,390 --> 00:32:54,220
+bound حد أعلىللمجموعة S إذا كان كل X في S أصغر من
+
+325
+00:32:54,220 --> 00:33:01,840
+أو يساوي الـ U و العدد W بنسميه lower bound أو حد
+
+326
+00:33:01,840 --> 00:33:10,360
+أدنى للمجموعة S إذا كان كل X إذا كان أي عدد X في S
+
+327
+00:33:10,360 --> 00:33:16,020
+أكبر من أو يساوي الـ W أو W أصغر من أو يساوي كل
+
+328
+00:33:16,020 --> 00:33:24,290
+عناصر الـ Sففي الحالة هذه بنسمي w lower bound طيب
+
+329
+00:33:24,290 --> 00:33:30,650
+في شوية ملاحظات عن التعريفين هدول except may or
+
+330
+00:33:30,650 --> 00:33:34,130
+may not have an upper or lower bound على سبيل
+
+331
+00:33:34,130 --> 00:33:37,770
+المثال for example على سبيل المثال الأعداد
+
+332
+00:33:37,770 --> 00:33:45,600
+الحقيقية الأعداد الحقيقية هدا هيالمجموعة ممكن يكون
+
+333
+00:33:45,600 --> 00:33:49,420
+لها upper bound وممكن مايكونش لها upper bound
+
+334
+00:33:49,420 --> 00:33:53,380
+فمثلا
+
+335
+00:33:53,380 --> 00:34:02,320
+الفترة يعني هذه الأعداد هي من صفر إلى واحد أو
+
+336
+00:34:02,320 --> 00:34:09,160
+الفترة المغلقة من صفر إلى واحدهذه الواحد الواحد أو
+
+337
+00:34:09,160 --> 00:34:14,520
+أي U أكبر من أو ساوء الواحد أي عدد حقيقي U أكبر من
+
+338
+00:34:14,520 --> 00:34:19,060
+أو ساوء الواحد بيطلع upper bound لها ليه؟ لأن كل
+
+339
+00:34:19,060 --> 00:34:24,920
+العناصر اللي هنا أصغر من أو ساوء الواحد، صح؟ وكذلك
+
+340
+00:34:24,920 --> 00:34:32,800
+لو أخدت W عدد أصغر من أو ساوء سفر فكل X في الفترة
+
+341
+00:34:32,800 --> 00:34:33,720
+المغلقة هذه
+
+342
+00:34:36,600 --> 00:34:40,120
+كل x في الفترة المغلقة هذه أصغر من أوي ساوي سفر
+
+343
+00:34:40,120 --> 00:34:46,360
+صح؟ وبالتالي أصغر من أوي ساوي w إذا أي عدد حقيقي
+
+344
+00:34:46,360 --> 00:34:50,940
+بساوي سفر w بساوي سفر أو أصغر من سفر عبارة عن
+
+345
+00:34:50,940 --> 00:34:57,120
+lower bound للست S هذهكذلك أي عدد حقيقي U بساوي
+
+346
+00:34:57,120 --> 00:35:01,540
+واحد أو أكبر من واحد بيكون upper bound للمجموعة
+
+347
+00:35:01,540 --> 00:35:06,660
+هذه لأن كل X في المجموعة S هذه أكبر من أو ساوي
+
+348
+00:35:06,660 --> 00:35:13,140
+الواحد، تمام؟ لكن مجموعة الأعداد الحقيقية R أو
+
+349
+00:35:13,140 --> 00:35:18,300
+المجموعة الأعداد الحقيقية R هذه مالهاش upper bound
+
+350
+00:35:18,300 --> 00:35:20,060
+ومالهاش lower bound
+
+351
+00:35:23,110 --> 00:35:28,250
+العداد الحقيقية مافيش عدد حقيقي أكبر من أو ساوي كل
+
+352
+00:35:28,250 --> 00:35:32,390
+العداد الحقيقية ولا فيه نقدر نحط أصبعنا على العدد
+
+353
+00:35:32,390 --> 00:35:38,210
+ومافيش ولا عدد حقيقي أصغر من أو ساوي كل العداد
+
+354
+00:35:38,210 --> 00:35:43,970
+الحقيقية كتور؟ نعم مش الفترة من صفر لواحد مغلقة
+
+355
+00:35:43,970 --> 00:35:48,450
+تكسبها؟ او مفتوحة مش مشكلة طب إذا كانت مغلقة كيف
+
+356
+00:35:48,450 --> 00:35:55,130
+دي يكون ال lower bound أقل من صفر؟لو كانت مغلقة ف
+
+357
+00:35:55,130 --> 00:36:01,850
+.. ف w بساوي سفر عبارة عن lower bound او ال u
+
+358
+00:36:01,850 --> 00:36:06,470
+بتساوي واحد ال upper bound و u بساوي واحد upper
+
+359
+00:36:06,470 --> 00:36:10,670
+bound او حينما كانت مغلقة مش هيكوا بس كمان لو أخدت
+
+360
+00:36:10,670 --> 00:36:16,390
+u بساوي اتنين يعني اكبر من واحد برضه بطلع upper
+
+361
+00:36:16,390 --> 00:36:22,720
+bound صح؟حتى حتى لو مش مغلقة يعني اتنين هذا العدد
+
+362
+00:36:22,720 --> 00:36:26,760
+اتنين اكبر من العناصر اللي هنا كلها ولا لأ حتى لو
+
+363
+00:36:26,760 --> 00:36:34,860
+مش مغلقة سواء
+
+364
+00:36:34,860 --> 00:36:39,580
+الفترة هذه مغلقة او مش مغلقة اي عدد حقيقي اكبر من
+
+365
+00:36:39,580 --> 00:36:44,940
+او يساوي الواحد بيكون upper bound يعني ال upper
+
+366
+00:36:44,940 --> 00:36:47,740
+bound ممكن انه ماينتمش الفترةممكن ينتمي إلى
+
+367
+00:36:47,740 --> 00:36:54,080
+المجموعة وممكن لا ينتمي إليها يعني
+
+368
+00:36:54,080 --> 00:37:01,980
+لو أخدت الفترة مغلقة الفترة
+
+369
+00:37:01,980 --> 00:37:07,140
+مغلقة U بساوة اتنين upper bound لها في شك في ذلك؟
+
+370
+00:37:07,140 --> 00:37:11,060
+لكن ال upper bound هذا مش شرط يكون ينتمي إلى
+
+371
+00:37:11,060 --> 00:37:18,850
+المجموعةالواحد برضه upper bound للمجموعة دي و
+
+372
+00:37:18,850 --> 00:37:21,470
+ينتمي ل .. ممكن ينتمي إذا ال upper bound ممكن
+
+373
+00:37:21,470 --> 00:37:27,410
+ينتمي ل .. للمجموعة ممكن ماينتميش لو أخدت U
+
+374
+00:37:27,410 --> 00:37:32,210
+بالساوي عشرة برضه هذا upper bound لمجموعة S لو
+
+375
+00:37:32,210 --> 00:37:39,560
+أخدت U بالساوي مية برضه هذا upper bound للستو ال u
+
+376
+00:37:39,560 --> 00:37:43,760
+بساوية واحد upper bound لل 6 الان هذا ال u بساوية
+
+377
+00:37:43,760 --> 00:37:48,080
+واحد هو اصغر upper bound هذا ال u بساوية واحد هو
+
+378
+00:37:48,080 --> 00:37:53,420
+اصغر upper bound كذلك لو أخدت w بساوية سفر هذا
+
+379
+00:37:53,420 --> 00:37:58,240
+عبارة عن lower bound لل 6 هذه لأن كل عنصر في ال 6
+
+380
+00:37:58,240 --> 00:38:04,850
+أكبر منها بساوية سفرلو أخدت w بساوي سالب نص، برضه
+
+381
+00:38:04,850 --> 00:38:09,190
+هذا lower bound للست هذه، لو u أخدتها بساوي سالب
+
+382
+00:38:09,190 --> 00:38:16,450
+عشرة، برضه هذا lower bound آخر، وكذا، الان w بساوي
+
+383
+00:38:16,450 --> 00:38:21,710
+سفرهذا عبارة عن أكبر lower bound للمجموعة هذه أكبر
+
+384
+00:38:21,710 --> 00:38:27,990
+حد أدنى، تمام؟ إذن في حد أدنى وفي أكبر حد أدنى، في
+
+385
+00:38:27,990 --> 00:38:33,770
+حد أعلى وفي أصغر حد أعلى، تمام؟ هدولة مهمين، أصغر
+
+386
+00:38:33,770 --> 00:38:38,510
+حد أعلى و أكبر حد أدنى مهمين و هنطلق .. هنعطيهم
+
+387
+00:38:38,510 --> 00:38:45,090
+أسماء كمان شوية، لكن قبلخلّينا نكمل الملاحظات إذا
+
+388
+00:38:45,090 --> 00:38:48,650
+شفنا إحنا أن ال upper bound أو المجموعة ممكن يكون
+
+389
+00:38:48,650 --> 00:38:53,050
+لها upper bound أو مايكونش المجموعة ال upper bound
+
+390
+00:38:53,050 --> 00:38:55,750
+أو ال lower bound ممكن ينتمي للمجموعة و ممكن
+
+391
+00:38:55,750 --> 00:39:01,850
+ماينتميش لها الملاحظة التانية لو كان U upper bound
+
+392
+00:39:01,850 --> 00:39:06,930
+لست S فأي عدد أكبر منه بيطلع upper bound أيضا لنفس
+
+393
+00:39:06,930 --> 00:39:11,820
+الست يعني هي عنديإذا إحنا الآن في الملاحظة التانية
+
+394
+00:39:11,820 --> 00:39:18,140
+لو كان U upper bound لل 6 و أخدت V عدد أكبر من U
+
+395
+00:39:18,140 --> 00:39:24,060
+فال V أيضا هذا بيطلع upper bound لل 6 طيب السؤال
+
+396
+00:39:24,060 --> 00:39:32,760
+اللي بفرح نفسه كام عدد V ممكن ألاقي أكبر من U عدد
+
+397
+00:39:32,760 --> 00:39:39,010
+لانهائيإذا لو كان الست S لها upper bound واحد
+
+398
+00:39:39,010 --> 00:39:42,990
+فممكن ألاقي عدد لانهائي لها من ال upper bounds
+
+399
+00:39:42,990 --> 00:39:48,810
+طبعا؟ إذا قلنا كمان مرة لو كان ال U لها upper
+
+400
+00:39:48,810 --> 00:39:53,450
+bound أو الست لها upper bound U فأي عدد V أكبر من
+
+401
+00:39:53,450 --> 00:39:58,170
+U بيطلع upper bound وبالتالي لو كان الست لها upper
+
+402
+00:39:58,170 --> 00:40:01,450
+bound واحد فبيكون لها infinitely many upper bounds
+
+403
+00:40:02,480 --> 00:40:07,420
+بالمثل لو كان الست لها lower bound فبالتالي بيكون
+
+404
+00:40:07,420 --> 00:40:11,300
+لها infinitely many lower bounds عدد لانها in ال
+
+405
+00:40:11,300 --> 00:40:15,280
+lower bounds طيب المجموعة الخالية المجموعة الخالية
+
+406
+00:40:15,280 --> 00:40:23,000
+بتمتاز بإنه أي عدد حقيقي بيطلع upper bound و lower
+
+407
+00:40:23,000 --> 00:40:29,580
+bound في نفس الوقت هذا بده برهانة هذا مش واضحاللي
+
+408
+00:40:29,580 --> 00:40:32,640
+هو انتبهوا ويرجعوا لمبادئ الرياضيات وقوانين ال
+
+409
+00:40:32,640 --> 00:40:41,060
+logic هاي برهان بالتناقض انا عندي S مجموعة خالية
+
+410
+00:40:41,060 --> 00:40:47,860
+مجموعة محترمة هاي S بساوي فاي مجموعة جزئية من R
+
+411
+00:40:47,860 --> 00:40:54,760
+الفاي مجموعة جزئية محترمة من R هذه المجموعة اي عدد
+
+412
+00:40:54,760 --> 00:40:59,850
+حقيقي مهما كانهو lower bound و upper bound اللي في
+
+413
+00:40:59,850 --> 00:41:06,350
+نفس الوجود خلّينا نثبت أنه لو خدت أي عدد حقيقي R0
+
+414
+00:41:06,350 --> 00:41:16,970
+يعني تاني لـR فبدا أثبت أن هذا is upper bound of
+
+415
+00:41:16,970 --> 00:41:25,050
+Phi بدا أثبت أنه أي عدد حقيقي R0 بيطلع upper bound
+
+416
+00:41:25,050 --> 00:41:34,030
+لـPhiالبرهان ذلك بالتناقض افرض
+
+417
+00:41:34,030 --> 00:41:39,290
+ان ولا عدد حقيقي هو
+
+418
+00:41:39,290 --> 00:41:48,570
+upper bound لفاية يعني
+
+419
+00:41:48,570 --> 00:41:55,030
+يوجد عدد حقيقي R0 ليس upper bound
+
+420
+00:41:58,090 --> 00:42:02,030
+ففي الحالة هذه هذا معناه من تعريف ال upper bound
+
+421
+00:42:02,030 --> 00:42:06,990
+إذا كان R0 ليس upper bound لل set S التي هي Phi
+
+422
+00:42:06,990 --> 00:42:13,110
+معناته بقدر ألاقي أنصر في ال set أكبر من ال R0 ال
+
+423
+00:42:13,110 --> 00:42:20,110
+R0 لا يحد الأنصر من أعلىوهذا بدّيني تناقض، ليش
+
+424
+00:42:20,110 --> 00:42:25,810
+تناقض؟ لأن هذا أدى إلى أن يوجد S أنصر S في capital
+
+425
+00:42:25,810 --> 00:42:31,250
+S اللي هي Phi و Phi خالية، لا يوجد Phi خالية، لا
+
+426
+00:42:31,250 --> 00:42:39,910
+يوجدفيها او داخلها اي عنصر اسمه الـ S اذا الفرض
+
+427
+00:42:39,910 --> 00:42:46,990
+هذا خطأ وبالتالي كل العداد الحقيقية هي upper
+
+428
+00:42:46,990 --> 00:42:53,590
+bounds للمجموعة Phi بالمثل بطريقة مماثلة ممكن نثبت
+
+429
+00:42:53,590 --> 00:42:59,570
+ان نفس العدد او اي عدد R0 هو lower bound للمجموعة
+
+430
+00:42:59,570 --> 00:43:00,050
+Phi
+
+431
+00:43:04,940 --> 00:43:11,500
+طيب نرجع نيجي الأن ل ..
+
+432
+00:43:11,500 --> 00:43:23,360
+شوية تعريفات تانية المجموعة
+
+433
+00:43:23,360 --> 00:43:30,640
+طيب
+
+434
+00:43:30,640 --> 00:43:42,980
+المحاضرة يعنيانتهت تقريبا المرة الجاية هناخد تعريف
+
+435
+00:43:42,980 --> 00:43:50,820
+ال bounded set و تعريف ال supremum اللي هو least
+
+436
+00:43:50,820 --> 00:43:55,360
+upper bound او اصغر حد اعلى و ال optimum اللي هو
+
+437
+00:43:55,360 --> 00:44:01,250
+اكبر حد ادنىوندرس خاصة وبعدين نشوف ما هي ال
+
+438
+00:44:01,250 --> 00:44:05,670
+completeness property أو خاصية التمام اللي كانت
+
+439
+00:44:05,670 --> 00:44:08,590
+عنوان جانبي للجزء هذا من المحاضرة
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_raw.json b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..5a644dac70ca0b9359561f0466aaa1231637b226
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/wn5D8jbScjs_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 3999, "start": 20.67, "end": 39.99, "text": "تحدثنا عن ال absolute value و عرفنا ال absolute value لأي real number إما a إذا كان a غير سالب أو سالب العدد نفسه إذا كان العدد سالب نفس التعريف اللي أخدناه في ال calculus و بكل من التعريف باستخدام التعريف و ال logic", "tokens": [2655, 24401, 12984, 8315, 18871, 2423, 8236, 2158, 4032, 6225, 28480, 8315, 2423, 8236, 2158, 5296, 10721, 1829, 957, 1230, 11933, 15042, 257, 11933, 15730, 25961, 257, 32771, 13546, 8608, 6027, 3555, 34051, 8608, 6027, 3555, 18863, 3215, 3215, 8717, 36178, 3224, 11933, 15730, 25961, 18863, 3215, 3215, 8608, 6027, 3555, 8717, 36178, 16712, 3615, 16572, 5172, 13672, 1829, 5551, 9778, 3215, 8315, 3224, 8978, 2423, 33400, 4032, 4724, 28820, 9154, 16712, 3615, 16572, 5172, 4724, 995, 14851, 9778, 3215, 10943, 16712, 3615, 16572, 5172, 4032, 2423, 9952], "avg_logprob": -0.10103581125816602, "compression_ratio": 1.7883597883597884, "no_speech_prob": 0.0, "words": [{"start": 20.67, "end": 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.. في اي مشكلة ممكن يعني اترجعوني خلال الساعات المكتبية او خلال ساعة المناخشة في اللي هو ملاحظة متباينة المثلث", "tokens": [5172, 6027, 26890, 40294, 9957, 8608, 3224, 37977, 46599, 3794, 1829, 9566, 3660, 4032, 40228, 22653, 3714, 29245, 23328, 3660, 43242, 24793, 36632, 11331, 3215, 8978, 24793, 4724, 1829, 28543, 2288, 7251, 4587, 2288, 10721, 11296, 4032, 33911, 16095, 11296, 45164, 19446, 41185, 8592, 4386, 8978, 1975, 1829, 37893, 28820, 3660, 3714, 43020, 37495, 22653, 1975, 2655, 47341, 3615, 11536, 1829, 16490, 1211, 6027, 21136, 45761, 9307, 9673, 4117, 2655, 3555, 10632, 1975, 2407, 16490, 1211, 6027, 8608, 995, 27884, 9673, 8315, 9778, 8592, 3660, 8978, 13672, 1829, 31439, 3714, 15040, 5016, 19913, 3660, 44650, 3555, 995, 9957, 3660, 9673, 12984, 1211, 12984], "avg_logprob": -0.14520733459637716, "compression_ratio": 1.8368421052631578, "no_speech_prob": 0.0, "words": [{"start": 141.02, "end": 141.98, "word": "فالبراهين", "probability": 0.725830078125}, {"start": 141.98, 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مافيش", "probability": 0.7620442708333334}, {"start": 150.6, "end": 150.74, "word": " ..", "probability": 0.5625}, {"start": 150.74, "end": 150.96, "word": " في", "probability": 0.392333984375}, {"start": 150.96, "end": 151.14, "word": " اي", "probability": 0.6558837890625}, {"start": 151.14, "end": 151.6, "word": " مشكلة", "probability": 0.98291015625}, {"start": 151.6, "end": 152.0, "word": " ممكن", "probability": 0.959716796875}, {"start": 152.0, "end": 152.3, "word": " يعني", "probability": 0.968505859375}, {"start": 152.3, "end": 153.4, "word": " اترجعوني", "probability": 0.815185546875}, {"start": 153.4, "end": 154.54, "word": " خلال", "probability": 0.97314453125}, {"start": 154.54, "end": 154.98, "word": " الساعات", "probability": 0.9803059895833334}, {"start": 154.98, "end": 155.68, "word": " المكتبية", "probability": 0.9876953125}, {"start": 155.68, "end": 155.96, "word": " او", "probability": 0.7626953125}, {"start": 155.96, "end": 156.52, "word": " خلال", "probability": 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{"start": 187.09, "end": 188.03, "word": " a1", "probability": 0.966796875}, {"start": 188.03, "end": 188.53, "word": " زايد", "probability": 0.8414713541666666}, {"start": 188.53, "end": 189.03, "word": " absolute", "probability": 0.97265625}, {"start": 189.03, "end": 189.95, "word": " a2", "probability": 0.99755859375}], "temperature": 1.0}, {"id": 8, "seek": 21258, "start": 192.0, "end": 212.58, "text": "أو absolute A زي بي less than or equal to absolute A زي absolute B المتباين هذه ممكن تعميمها مش بس ناخد ال absolute value لعددين ممكن ناخد ال absolute value ل N من الأعداد الحقيقية و نجمعهم فهذا أصغر من أو ساوي مجموع ال absolute values", "tokens": [10721, 2407, 8236, 316, 30767, 1829, 4724, 1829, 1570, 813, 420, 2681, 281, 8236, 316, 30767, 1829, 8236, 363, 9673, 2655, 3555, 995, 9957, 29538, 3714, 43020, 37279, 2304, 32640, 11296, 37893, 4724, 3794, 8717, 47283, 3215, 2423, 8236, 2158, 5296, 22488, 3215, 9957, 3714, 43020, 8717, 47283, 3215, 2423, 8236, 2158, 5296, 426, 9154, 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1975, 3215, 1863, 7578], "avg_logprob": -0.11061507652676295, "compression_ratio": 1.5133333333333334, "no_speech_prob": 0.0, "words": [{"start": 2614.18, "end": 2615.14, "word": "انتهت", "probability": 0.9300130208333334}, {"start": 2615.14, "end": 2616.1, "word": " تقريبا", "probability": 0.9884765625}, {"start": 2616.1, "end": 2618.88, "word": " المرة", "probability": 0.859130859375}, {"start": 2618.88, "end": 2619.42, "word": " الجاية", "probability": 0.9469401041666666}, {"start": 2619.42, "end": 2622.3, "word": " هناخد", "probability": 0.9329833984375}, {"start": 2622.3, "end": 2622.98, "word": " تعريف", "probability": 0.9856770833333334}, {"start": 2622.98, "end": 2623.22, "word": " ال", "probability": 0.7861328125}, {"start": 2623.22, "end": 2623.64, "word": " bounded", "probability": 0.85546875}, {"start": 2623.64, "end": 2624.14, "word": " set", "probability": 0.96826171875}, {"start": 2624.14, "end": 2626.1, "word": " و", "probability": 0.630859375}, {"start": 2626.1, "end": 2626.74, "word": " تعريف", "probability": 0.9694010416666666}, {"start": 2626.74, "end": 2626.94, "word": " ال", "probability": 0.9169921875}, {"start": 2626.94, "end": 2627.72, "word": " supremum", "probability": 0.8515625}, {"start": 2627.72, "end": 2629.42, "word": " اللي", "probability": 0.901611328125}, {"start": 2629.42, "end": 2629.82, "word": " هو", "probability": 0.9892578125}, {"start": 2629.82, "end": 2630.82, "word": " least", "probability": 0.52685546875}, {"start": 2630.82, "end": 2631.2, "word": " upper", "probability": 0.904296875}, {"start": 2631.2, "end": 2631.66, "word": " bound", "probability": 0.89453125}, {"start": 2631.66, "end": 2631.96, "word": " او", "probability": 0.779541015625}, {"start": 2631.96, "end": 2632.38, "word": " اصغر", "probability": 0.9537353515625}, {"start": 2632.38, "end": 2632.62, "word": " حد", "probability": 0.9892578125}, {"start": 2632.62, "end": 2633.04, "word": " اعلى", "probability": 0.93310546875}, {"start": 2633.04, "end": 2634.08, "word": " و", "probability": 0.85009765625}, {"start": 2634.08, "end": 2634.24, "word": " ال", "probability": 0.97705078125}, {"start": 2634.24, "end": 2634.64, "word": " optimum", "probability": 0.236572265625}, {"start": 2634.64, "end": 2635.1, "word": " اللي", "probability": 0.982421875}, {"start": 2635.1, "end": 2635.36, "word": " هو", "probability": 0.98388671875}, {"start": 2635.36, "end": 2635.86, "word": " اكبر", "probability": 0.9715169270833334}, {"start": 2635.86, "end": 2636.16, "word": " حد", "probability": 0.99609375}, {"start": 2636.16, "end": 2636.66, "word": " ادنى", "probability": 0.878173828125}], "temperature": 1.0}, {"id": 101, "seek": 264859, "start": 2637.51, "end": 2648.59, "text": "وندرس خاصة وبعدين نشوف ما هي ال completeness property أو خاصية التمام اللي كانت عنوان جانبي للجزء هذا من المحاضرة", "tokens": [11536, 3215, 2288, 3794, 16490, 33546, 3660, 46599, 22488, 9957, 8717, 8592, 38688, 19446, 39896, 2423, 1557, 15264, 4707, 34051, 16490, 33546, 10632, 16712, 2304, 10943, 13672, 1829, 25961, 2655, 18871, 2407, 7649, 10874, 7649, 21292, 24976, 7435, 11622, 38207, 23758, 9154, 9673, 5016, 46958, 25720], "avg_logprob": -0.1559175506551215, "compression_ratio": 1.3076923076923077, "no_speech_prob": 0.0, "words": [{"start": 2637.51, "end": 2638.15, "word": "وندرس", "probability": 0.87158203125}, {"start": 2638.15, "end": 2638.77, "word": " خاصة", "probability": 0.7146809895833334}, {"start": 2638.77, "end": 2639.45, "word": " وبعدين", "probability": 0.7626953125}, {"start": 2639.45, "end": 2639.99, "word": " نشوف", "probability": 0.9710286458333334}, {"start": 2639.99, "end": 2640.21, "word": " ما", "probability": 0.8759765625}, {"start": 2640.21, "end": 2640.51, "word": " هي", "probability": 0.88134765625}, {"start": 2640.51, "end": 2641.25, "word": " ال", "probability": 0.8359375}, {"start": 2641.25, "end": 2641.95, "word": " completeness", "probability": 0.639404296875}, {"start": 2641.95, "end": 2642.59, "word": " property", "probability": 0.75048828125}, {"start": 2642.59, "end": 2642.81, "word": " أو", "probability": 0.76904296875}, {"start": 2642.81, "end": 2643.39, "word": " خاصية", "probability": 0.9689127604166666}, {"start": 2643.39, "end": 2644.03, "word": " التمام", "probability": 0.9259440104166666}, {"start": 2644.03, "end": 2645.21, "word": " اللي", "probability": 0.768310546875}, {"start": 2645.21, "end": 2645.67, "word": " كانت", "probability": 0.980712890625}, {"start": 2645.67, "end": 2646.11, "word": " عنوان", "probability": 0.9493815104166666}, {"start": 2646.11, "end": 2646.75, "word": " جانبي", "probability": 0.9905598958333334}, {"start": 2646.75, "end": 2647.45, "word": " للجزء", "probability": 0.86981201171875}, {"start": 2647.45, "end": 2647.79, "word": " هذا", "probability": 0.93408203125}, {"start": 2647.79, "end": 2648.05, "word": " من", "probability": 0.951171875}, {"start": 2648.05, "end": 2648.59, "word": " المحاضرة", "probability": 0.9803466796875}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2649.211, "duration_after_vad": 2471.266249999988}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM_postprocess.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..5bd12bede0f1653ebb45d4cb4bf562eb287b9beb
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM_postprocess.srt
@@ -0,0 +1,1604 @@
+1
+00:00:21,190 --> 00:00:25,470
+في المحاضرة السابقة تعرفنا على أنواع الفترات أو ال
+
+2
+00:00:25,470 --> 00:00:28,950
+intervals وشوفنا
+
+3
+00:00:28,950 --> 00:00:33,290
+أن أول تلت أنواع من الفترات اللي هي ال open و ال
+
+4
+00:00:33,290 --> 00:00:37,110
+closed و ال half open أو half closed intervals كل
+
+5
+00:00:37,110 --> 00:00:43,790
+الفترات هذه كانت bounded intervals أو فترات محدودة
+
+6
+00:00:45,730 --> 00:00:50,070
+الانواع الخامسة الأخرى من النوع الرابع إلى النوع
+
+7
+00:00:50,070 --> 00:00:55,750
+الثامن كلها فترات غير محدودة أو unbounded
+
+8
+00:00:55,750 --> 00:01:02,710
+intervals و الفترات هذه عبارة عن النوع الرابع open
+
+9
+00:01:02,710 --> 00:01:10,410
+left tray شعاع أيصر مفتوح أو open right tray شعاع
+
+10
+00:01:10,410 --> 00:01:16,650
+أيمن مفتوح أو closed left tray شعاع أيمنمغلق
+
+11
+00:01:16,650 --> 00:01:22,890
+closed writer شعاع أيمن مغلق أو شعاع الجبلة شعاع
+
+12
+00:01:22,890 --> 00:01:27,140
+أيصر مغلقو طبعا هذه عبارة عن فترات زي ما احنا
+
+13
+00:01:27,140 --> 00:01:31,660
+شايفين النوع الأخير اللي هو مجموعة كل الأعداد اللي
+
+14
+00:01:31,660 --> 00:01:35,320
+حصلت فيها the set of all real numbers اللي هي
+
+15
+00:01:35,320 --> 00:01:39,160
+الفترة من ثالث من النهاية للملا نهاية وهذه طبعا
+
+16
+00:01:39,160 --> 00:01:45,560
+أيضا unbounded interval فترة غير محصورة أو غير
+
+17
+00:01:45,560 --> 00:01:49,640
+محدودة لو
+
+18
+00:01:49,640 --> 00:01:54,710
+بصينا لأي نوع من أنواع التمانية هذهمن الفترات لو
+
+19
+00:01:54,710 --> 00:02:02,590
+أخذنا أي نوع من الأنواع التمانية فبنلاحظ
+
+20
+00:02:02,590 --> 00:02:10,850
+وسمينا الفترة هذه I لأن لو أخذنا one of the eight
+
+21
+00:02:10,850 --> 00:02:17,250
+types of interval سمناها I فبنلاحظ أن الفترة I هذه
+
+22
+00:02:17,250 --> 00:02:26,690
+بتحقق الخاصية اللي هي أمامنا هذهوهي أنه لو كان في
+
+23
+00:02:26,690 --> 00:02:32,050
+عندي يعني
+
+24
+00:02:32,050 --> 00:02:38,130
+أي فترة زي هذه مثلا
+
+25
+00:02:38,130 --> 00:02:45,290
+الفترة
+
+26
+00:02:45,290 --> 00:02:50,130
+من A إلى مالنهاية لو النقطة هذه A
+
+27
+00:02:57,030 --> 00:03:01,490
+فهذا open right ray لو أخدت أي نقطتين في الفترة
+
+28
+00:03:01,490 --> 00:03:07,810
+هذه x و y واقعت
+
+29
+00:03:07,810 --> 00:03:13,790
+x و y ينتموا للفترة هذه فبنلاحظ أن الفترة المغلقة
+
+30
+00:03:13,790 --> 00:03:19,690
+ما بين x و y the closed interval from x و y is
+
+31
+00:03:19,690 --> 00:03:27,120
+contained الفترة المغلقة هذهبتكون دائما موجودة في
+
+32
+00:03:27,120 --> 00:03:35,620
+الفترة I لأن لو أخدت أي فترة I احنا أخدنا مثال ال
+
+33
+00:03:35,620 --> 00:03:41,100
+open right ray فبنلاحظ أن هذه الفترة أو أي فترة
+
+34
+00:03:41,100 --> 00:03:47,940
+تانية من الأنواع التامية بتحقق الخصية هذه اللي فوق
+
+35
+00:03:47,940 --> 00:03:53,080
+طبعا هذه احنا عطينا رقم أربعة بس عشان كبرنا الخط
+
+36
+00:03:55,050 --> 00:04:01,070
+فرقم أربعة مش دائرة على الشاشة فأي فترة زي هذه
+
+37
+00:04:01,070 --> 00:04:05,310
+بتحقق خاصية أربعة وهو لو أخدت أي نقطتين في الفترة
+
+38
+00:04:05,310 --> 00:04:12,470
+I فالفترة اللي واقع بين X وY أيضا بتكون موجودة
+
+39
+00:04:12,470 --> 00:04:20,290
+داخل الفترة I الآن العكس تبع الكلام هذا صحيح كما
+
+40
+00:04:20,290 --> 00:04:24,510
+هو موضح في نظرية واحد عشرين
+
+41
+00:04:27,170 --> 00:04:34,990
+بمعنى انه لو كان في اندي فترة لو كان في اندي أخدت
+
+42
+00:04:34,990 --> 00:04:39,990
+مجموعة جزئية من R المجموعة هذه عدد عناصرها اكبر من
+
+43
+00:04:39,990 --> 00:04:42,950
+او ساوى اتنين ال cardinal number تبع اكبر من او
+
+44
+00:04:42,950 --> 00:04:46,890
+ساوى اتنين يعني المجموعة هذه فيها على الأقل عنصرين
+
+45
+00:04:46,890 --> 00:04:51,550
+او اكثر ممكن تكون finite set ممكن تكون infinite
+
+46
+00:04:53,020 --> 00:04:55,760
+يكون فيها على الأقل او نصرين فلو كان في عندي
+
+47
+00:04:55,760 --> 00:04:58,980
+مجموعة جزئية من الارض فيها على الأقل او نصرين و
+
+48
+00:04:58,980 --> 00:05:03,600
+بتحقق الخاصية اربعة فلازم المجموعة الجزئية هذه
+
+49
+00:05:03,600 --> 00:05:11,820
+تطلع فترة و البرهان تبع المظرية هذه مش صعب و ممكن
+
+50
+00:05:11,820 --> 00:05:15,840
+يعني احنا كاتبينه بالتفصيل لكم
+
+51
+00:05:18,790 --> 00:05:26,110
+وقلنا أنتم بمكانكم تقرؤوه تفهموه البرهان
+
+52
+00:05:26,110 --> 00:05:32,150
+بنجزه لأربع حالات المجموعة S هذه اللي احنا عايزين
+
+53
+00:05:32,150 --> 00:05:36,190
+نثبت انها interval فيها أربع احتمالات اما بتكون
+
+54
+00:05:36,190 --> 00:05:40,150
+bounded يعني محدودة من الجهتين
+
+55
+00:05:42,590 --> 00:05:48,630
+او بتكون bounded احتمال التاني تكون bounded above
+
+56
+00:05:48,630 --> 00:05:54,090
+but not below او احتمال تالت انها تكون bounded
+
+57
+00:05:54,090 --> 00:05:58,330
+below but not above الاحتمال الرابع انها is
+
+58
+00:05:58,330 --> 00:06:02,250
+neither bounded above nor below يعني لا محدودة من
+
+59
+00:06:02,250 --> 00:06:06,770
+اسفل ولا من اعلى وفي
+
+60
+00:06:06,770 --> 00:06:11,110
+كل حالة من الحلقات الأربعة في كل حالة من الحلقات
+
+61
+00:06:11,110 --> 00:06:18,580
+الأربعةهي الحالة الأولى S bounded في كل حالة من
+
+62
+00:06:18,580 --> 00:06:24,580
+الحالات الأربعة بنحاول نثبت أن ال 6S هي فترة ما
+
+63
+00:06:24,580 --> 00:06:32,320
+فترة معينة فمثلا لنتعلم نشوف الحالة الأولى الحالة
+
+64
+00:06:32,320 --> 00:06:36,260
+الأولى أن S تكون bounded وبالتالي bounded above
+
+65
+00:06:36,260 --> 00:06:39,940
+and bounded belowبما أنها bounded below إذا ال
+
+66
+00:06:39,940 --> 00:06:44,720
+inform تبعها exist سميه a بما أنها bounded above
+
+67
+00:06:44,720 --> 00:06:48,440
+إذا by ال supreme property ال supreme تبعها exist
+
+68
+00:06:48,440 --> 00:06:56,460
+سميه b وبالتالي S بتصير محصورة داخل أو subset من
+
+69
+00:06:56,460 --> 00:07:02,800
+الفترة المغلفة AB طبعا لو قرأته انتوا التفاصيل
+
+70
+00:07:02,800 --> 00:07:07,980
+هتطلع الفترة الست S هذه في نهاية البرهانهتطلع فترة
+
+71
+00:07:07,980 --> 00:07:19,540
+half open زي هذه او half open زي هذه هو
+
+72
+00:07:19,540 --> 00:07:26,700
+بالتالي فترة وهو المطلوب في الحالة التانية اللي
+
+73
+00:07:26,700 --> 00:07:32,300
+هناخد الحالة التانية ان ال set S is bounded above
+
+74
+00:07:32,300 --> 00:07:37,270
+but not bounded belowمحدودة من أعلى لكن مش محدودة
+
+75
+00:07:37,270 --> 00:07:43,390
+من أسفل في الحالة هذه هتطلع ال 6S هذه فترة و هتطلع
+
+76
+00:07:43,390 --> 00:07:48,770
+هتطلع left ray left ray لأنها مش محدودة من أسفل
+
+77
+00:07:48,770 --> 00:07:56,290
+فيعني لو انتوا تابعتوا البرهان هتطلع الفترة او ال
+
+78
+00:07:56,290 --> 00:07:59,250
+6S هي عبارة عن اما
+
+79
+00:08:08,630 --> 00:08:14,990
+هتطلع ال set is عبارة عن closed lift tray زي هذا
+
+80
+00:08:14,990 --> 00:08:20,870
+او open lift tray زي هذا فهنسيبكم
+
+81
+00:08:20,870 --> 00:08:25,130
+تقراوا البرهان مش صعب أكيد كل واحد منكم هيتفهمه
+
+82
+00:08:25,130 --> 00:08:30,450
+الآن الحالات التالتة والرابعة شبيهة بالحالات
+
+83
+00:08:30,450 --> 00:08:36,270
+الأولى والتانية وبالتالي هسيبها اليكم كتمرين
+
+84
+00:08:36,270 --> 00:08:40,940
+exerciseتتدربوا على كتابة او تحاولوا تكتبوا
+
+85
+00:08:40,940 --> 00:08:46,720
+البرهان هيكون شبيه بالحالتين السابقات okay اي
+
+86
+00:08:46,720 --> 00:08:52,900
+طالبة بسيب تمرين زي هذا مطالبة انها تكتب يعني تحل
+
+87
+00:08:52,900 --> 00:08:59,040
+التمرين هذا على الأقل عشان تفهم المادة اذا في عندك
+
+88
+00:08:59,040 --> 00:09:04,340
+اي تساول او استفسار تتصل فيا او ممكن تكتب البرهان
+
+89
+00:09:04,340 --> 00:09:10,150
+على ورقةو تعطيني في أقرب فرصة و أنا بعرجلكيا في
+
+90
+00:09:10,150 --> 00:09:17,510
+المحاضرة اللي بعدها تمام؟ اذا الحالات التالتة و
+
+91
+00:09:17,510 --> 00:09:20,970
+الرابعة شبيهة بالحالات الأولى و التانية و بالتالي
+
+92
+00:09:20,970 --> 00:09:28,910
+هنسيبها .. هنتركها للطالب كتمرين في عندي هنا تعريف
+
+93
+00:09:43,000 --> 00:09:47,580
+لو أخدت sequence of intervals I N هذه عبارة عن
+
+94
+00:09:47,580 --> 00:09:52,920
+فترة الان هذه عبارة عن sequence على سرها فترات I N
+
+95
+00:09:52,920 --> 00:09:58,640
+فال sequence هذه بنسميها nested ال sequence of
+
+96
+00:09:58,640 --> 00:10:04,690
+intervals هذه بنسميها nestedإذا كانت بتحقق الخاصية
+
+97
+00:10:04,690 --> 00:10:08,670
+هذه و هي أن الفترة الأولى تحتوي التانية والتانية
+
+98
+00:10:08,670 --> 00:10:13,690
+تحتوي التالتة و هكذا أو بمعنى آخر الفترة رقم n I n
+
+99
+00:10:13,690 --> 00:10:18,410
+contains الفترة اللي بعدها مباشرة I n زاد واحد لكل
+
+100
+00:10:18,410 --> 00:10:24,420
+n إذا الفترات أو sequence of intervalsthat
+
+101
+00:10:24,420 --> 00:10:29,280
+satisfies this property بنسميها nested it's called
+
+102
+00:10:29,280 --> 00:10:35,600
+nested sequence of intervals هذه مثال مثال على
+
+103
+00:10:35,600 --> 00:10:42,500
+sequence of nested intervals لو أخدت الفترة I N هي
+
+104
+00:10:42,500 --> 00:10:46,400
+الفترة المغلقة من سفر لواحد على N حيث N أي عدد
+
+105
+00:10:46,400 --> 00:10:55,860
+طبيعي فالفترات هذهواضح جدا انها nested هاي خط
+
+106
+00:10:55,860 --> 00:11:10,380
+العداد هاي خط العداد هاي سفر واحد هاي فترة هادي I
+
+107
+00:11:10,380 --> 00:11:16,200
+واحد خد ان بس I واحد I واحد فترة مغلقة من سفر ل
+
+108
+00:11:16,200 --> 00:11:23,640
+واحدI اتنين خد in بساو اتنين معناته هذا نص يعني في
+
+109
+00:11:23,640 --> 00:11:34,520
+عندي فترة الفترة هذه I اتنين I I اتنين I واحد
+
+110
+00:11:34,520 --> 00:11:40,000
+تحتوي I اتنين لو أخدت in تلاتة بيصير في عندي هنا
+
+111
+00:11:40,000 --> 00:11:43,180
+تلت وفي الفترة هذه
+
+112
+00:11:45,730 --> 00:11:53,470
+هذه الفترة I ثلاثة و I ثلاثة واضح هنا contained I
+
+113
+00:11:53,470 --> 00:11:59,530
+اتنين و هكذا اذا واضح هنا ان I one contains I two
+
+114
+00:11:59,530 --> 00:12:07,270
+contains I three and so on و هكذا وبالتالي ال
+
+115
+00:12:07,270 --> 00:12:16,350
+sequence of intervals هذه تطلع nested تمام؟ طيبالـ
+
+116
+00:12:16,350 --> 00:12:21,270
+sequence هذه مش بس listed كمان التقاطع تبعها لا
+
+117
+00:12:21,270 --> 00:12:27,430
+يساوي في يعني لو قطعت الفترات هذه كلها فهنجد أنه
+
+118
+00:12:27,430 --> 00:12:31,230
+التقاطع تبعها لا يساوي في يعني ممكن ألاقي على
+
+119
+00:12:31,230 --> 00:12:36,410
+الأقل أنصر واحد في التقاطع in fact في حقيقة الأمر
+
+120
+00:12:36,410 --> 00:12:42,810
+التقاطع هذا بساوي singleton set zero يعني هنا في
+
+121
+00:12:42,810 --> 00:12:48,530
+الصفر هذا ينتمي للتقاطعهذا الطبعا الكلام مش واضح
+
+122
+00:12:48,530 --> 00:12:54,750
+هذا ليس واضحا فعشان نوضحه او نبرغله to see this
+
+123
+00:12:54,750 --> 00:13:03,170
+لإثبات ان التقاطع هذا الفترات هذه بساوي single
+
+124
+00:13:03,170 --> 00:13:09,790
+consist zero فبنثبت
+
+125
+00:13:09,790 --> 00:13:19,090
+ان مجمعتين بسوا بعضعندى الاحتواء هذا واضح واضح ان
+
+126
+00:13:19,090 --> 00:13:30,410
+السفر ينتمى للفترة السفر واحد على ان اللى هى in
+
+127
+00:13:30,410 --> 00:13:39,690
+لكل n في ان وبالتالي هذا بيقدم السفر ينتمى لتقاطع
+
+128
+00:13:39,690 --> 00:13:45,490
+كل ال inوبالتالي المجموعة اللى فيها العنصر الوحيد
+
+129
+00:13:45,490 --> 00:13:55,690
+سفر subset من تقاطة المجموعات IL تمام هذا واضح لكن
+
+130
+00:13:55,690 --> 00:14:01,970
+مش واضح انه نثبت الآن العكس او ال reverse
+
+131
+00:14:01,970 --> 00:14:06,270
+inclusion عشان نثبت مساواة ان ال set هدى بالساوية
+
+132
+00:14:06,270 --> 00:14:13,480
+هدى باقي نثبت ال inclusion هدىتمام فخلّينا ناخد
+
+133
+00:14:13,480 --> 00:14:22,080
+let x تنتمي للتخاطى لكل ال I N هذا معناه ان X
+
+134
+00:14:22,080 --> 00:14:27,960
+تنتمي ل I N لكل N طب ما ال I N هي عبارة عن سفر
+
+135
+00:14:27,960 --> 00:14:34,380
+فترة مغلقة من سفر ل 1 على N هذا معناه ان X أكبر من
+
+136
+00:14:34,380 --> 00:14:39,500
+أو ساوي سفر أصغر من أو ساوي 1 على N لكل N
+
+137
+00:14:44,000 --> 00:14:50,260
+تمام الان انا بتثبت ان
+
+138
+00:14:50,260 --> 00:14:56,060
+ال X هذا بساوي سفر اش بتثبت انا بتثبت الاحتواء
+
+139
+00:14:56,060 --> 00:15:00,900
+المعاكس اخدت X عنصر في التقاطة بتثبت ان X ينتمي
+
+140
+00:15:00,900 --> 00:15:04,900
+للمجموع عادي يعني X بساوي سفر لان هذه المجموعة
+
+141
+00:15:04,900 --> 00:15:08,640
+عاملة مافيش فيها غير سفر اذا لو اثبتت ان X بساوي
+
+142
+00:15:08,640 --> 00:15:15,780
+سفر بكون اثبتت الاحتواء المعاكسطيب الان نعمل برهار
+
+143
+00:15:15,780 --> 00:15:21,800
+بالتناقض assume نفترض
+
+144
+00:15:21,800 --> 00:15:27,880
+ان x لا يساوي سفر طب
+
+145
+00:15:27,880 --> 00:15:31,780
+انا عندي ال x هنا اكبر من او يساوي سفر والان
+
+146
+00:15:31,780 --> 00:15:34,860
+بيستويش سفر اذا ال x اكبر من سفر
+
+147
+00:15:40,290 --> 00:15:44,590
+طيب by Archimedean property by Archimedean
+
+148
+00:15:44,590 --> 00:15:47,810
+property
+
+149
+00:15:47,810 --> 00:15:53,990
+هذه اللي حاطين عليها علامة النجمة ايش ال
+
+150
+00:15:53,990 --> 00:15:58,830
+Archimedean property بتقول لأي عدد موجب زي X هذا
+
+151
+00:15:58,830 --> 00:16:09,170
+بنقدر نلاقي يوجد عدد طبيعي نسميه N0بحيث انه مقلوب
+
+152
+00:16:09,170 --> 00:16:18,310
+العدد الطبيعي N0 أصغر من ال X من العدد الموجد وهذا
+
+153
+00:16:18,310 --> 00:16:23,710
+بيؤدي لتناقض contradiction هذا بيؤدي لتناقض
+
+154
+00:16:23,710 --> 00:16:28,770
+contradiction ليه؟ لأنه أنا عندي من هنا هذا ال
+
+155
+00:16:28,770 --> 00:16:32,770
+statement الأخير هذا ال statement الأخير هذا
+
+156
+00:16:32,770 --> 00:16:36,960
+بتناقض مع ال statement اللي هنالأن هذا ال
+
+157
+00:16:36,960 --> 00:16:41,780
+statement بيقول بما أن N0 هذا عدد طبيعي إذا لازم X
+
+158
+00:16:41,780 --> 00:16:47,580
+تكون أصغر من أو ساوي واحد على N0 إذا X أصغر من أو
+
+159
+00:16:47,580 --> 00:16:51,360
+ساوي واحد على N0 X أكبر من واحد على N0 هذا تناقض
+
+160
+00:16:51,360 --> 00:17:00,340
+إذا التناقض هذا سببه مين ال assumption تبعنا هذاإن
+
+161
+00:17:00,340 --> 00:17:05,960
+X لا تساوي سفر إذا الصح إن X تساوي سفر كما هو
+
+162
+00:17:05,960 --> 00:17:09,660
+مطلوب وبالتالي هيك منكون أثبتنا إن كل X في التقاطع
+
+163
+00:17:09,660 --> 00:17:15,260
+بيساوي سفر يعني كل أنصر في التقاطع موجود في
+
+164
+00:17:15,260 --> 00:17:20,740
+المجموعة هذه إذا هيك منكون أثبتنا المساواة إن
+
+165
+00:17:20,740 --> 00:17:26,320
+المجموعة هذه فعلا أو التقاطع تبع الفترة I N بيساوي
+
+166
+00:17:26,320 --> 00:17:28,200
+Singleton Zero
+
+167
+00:17:33,320 --> 00:17:51,320
+هذا المثال بيأسس او هو حافظ لنظرية رقم 22 بنلاحظ
+
+168
+00:17:51,320 --> 00:17:56,220
+في المثال هذا ان الفترات هذه طبعا شفنا انها nested
+
+169
+00:17:56,220 --> 00:18:03,170
+متداخلة يعني nested معناها متداخلة بالعربيبعدين
+
+170
+00:18:03,170 --> 00:18:09,630
+واضح ان الفترات هذه bounded محصورة و كذلك الفترات
+
+171
+00:18:09,630 --> 00:18:16,830
+هذه closed فترات مغلقة صح فالنظرية هذه بتعمم
+
+172
+00:18:16,830 --> 00:18:21,510
+المعلومات اللي هنا او النتيجة اللي هنا اللي شفناها
+
+173
+00:18:21,510 --> 00:18:26,350
+في المثال فالخاصية هذه بتسميها nested interval
+
+174
+00:18:26,350 --> 00:18:27,010
+property
+
+175
+00:18:30,610 --> 00:18:34,550
+ماهي النظرية اللي بتقول لو في اندي سيكوانس of
+
+176
+00:18:34,550 --> 00:18:39,890
+closed, bounded و nested intervals nested
+
+177
+00:18:39,890 --> 00:18:45,430
+intervals و closed و bounded تلت ايه؟ تلت صفات
+
+178
+00:18:45,430 --> 00:18:50,890
+فلازم التقاطة تبع الفترات هذه يكون غير خالي non
+
+179
+00:18:50,890 --> 00:18:54,990
+-empty يعني نقدر نلاقي عنصر واحد على الأقل صي
+
+180
+00:18:54,990 --> 00:18:56,970
+ينتمي للتقاطة
+
+181
+00:18:59,530 --> 00:19:06,510
+إضافة لذلك moreover لو ال infimum للأعداد غير
+
+182
+00:19:06,510 --> 00:19:13,690
+السالبة هذه هذه أعداد غير سالبة إذا
+
+183
+00:19:13,690 --> 00:19:17,290
+كان ال infimum لل non-negative real numbers هذه لل
+
+184
+00:19:17,290 --> 00:19:20,450
+sequence of non-negative real numbers بساوي سفر
+
+185
+00:19:21,930 --> 00:19:26,170
+فالعدد ساي هذا بيكون وحيد يعني التقاطع هذا مافيش
+
+186
+00:19:26,170 --> 00:19:31,730
+فيه إلا عنصر واحد بالضبط زي ما شوفنا هنا إذا إن
+
+187
+00:19:31,730 --> 00:19:38,330
+الجزء التاني الجزء التاني من النظرية لو فرضنا إن
+
+188
+00:19:38,330 --> 00:19:42,890
+ال infimum للعداد غير الساري بهذا بالساوي سفر فال
+
+189
+00:19:42,890 --> 00:19:46,110
+ساي بتكون هي نقطة الوحيدة الموجودة في التقاطع تبع
+
+190
+00:19:46,110 --> 00:19:49,770
+الفترات زي ما في المثال بالظبط السفر هي النقطة
+
+191
+00:19:49,770 --> 00:19:57,090
+الوحيدةفي تقاطع الفترات المتداخلة نثبت النظرية هذه
+
+192
+00:20:25,320 --> 00:20:30,080
+فلبرهان النظرية أنا عندي الفترات هذه nested
+
+193
+00:20:30,080 --> 00:20:38,560
+وبالتالي I1 هتكون أكبر فترة تحتوي I2 و I3 و تحتوي
+
+194
+00:20:38,560 --> 00:20:44,100
+كل ال IN لكل الأعداد الطبيعي هذا أكيد يعني
+
+195
+00:20:44,100 --> 00:20:49,920
+وبالتالي هاي عندي هاي الفترة I1
+
+196
+00:20:53,730 --> 00:21:05,150
+هذه الفترة I1 تحتوي الفترة I N وهذه الفترة I N
+
+197
+00:21:05,150 --> 00:21:13,070
+وهذه I 1 فواضح
+
+198
+00:21:13,070 --> 00:21:18,870
+أن A N هتكون أصغر من أو ساوي B 1 لكل N في M
+
+199
+00:21:21,540 --> 00:21:27,040
+وبالتالي بي واحد عبارة عن upper bound لكل العناصر
+
+200
+00:21:27,040 --> 00:21:33,280
+AN إذا ال set هذه AN حيث ان عدد الطبيعي is bounded
+
+201
+00:21:33,280 --> 00:21:39,520
+above by بي واحد وبالتالي حسب ال supremum property
+
+202
+00:21:39,520 --> 00:21:44,500
+ال set هذه بما أنها bounded above إذا ال supremum
+
+203
+00:21:44,500 --> 00:21:50,000
+تبعها exist إذا يوجد عدد حقيقي R هو supremum لل
+
+204
+00:21:50,000 --> 00:21:56,950
+setأو الست هذه لها supremum سميه ساي بما أن هذا
+
+205
+00:21:56,950 --> 00:22:01,530
+الساي supremum لكل
+
+206
+00:22:01,530 --> 00:22:07,570
+عناصر الست AN فهو upper bound لذلك الساي أكبر من
+
+207
+00:22:07,570 --> 00:22:14,930
+أو ساوي كل عناصر الست الان
+
+208
+00:22:28,600 --> 00:22:36,720
+طبعا هذه ال .. لأن أنا في عندي a n أثبتنا أن a n
+
+209
+00:22:36,720 --> 00:22:44,080
+أصغر من أو ساوي ساي لكل n في n هذه
+
+210
+00:22:44,080 --> 00:22:51,700
+رقمها خمسة متبينة طبعا الرقم مش غير الان
+
+211
+00:22:51,700 --> 00:22:52,660
+لو أثبتنا
+
+212
+00:22:58,370 --> 00:23:01,830
+احنا عايزين نثبت انه ال .. احنا عايزين في النهاية
+
+213
+00:23:01,830 --> 00:23:08,270
+نثبت ان الصي هذا ينتمي ل I N اللي هو بالساوية
+
+214
+00:23:08,270 --> 00:23:16,850
+الفترة المغلقة من A N ل B N لكل N في N وبالتالي
+
+215
+00:23:16,850 --> 00:23:21,930
+تقاطع الفترات هذه يحتوي الصي صح؟ هذا اللي احنا
+
+216
+00:23:21,930 --> 00:23:27,050
+عايزين نثبته طيب بنهينا أثبتنا ان الصي أكبر من أو
+
+217
+00:23:27,050 --> 00:23:36,150
+ساوي A Nإذا عايزين نثبت إذا عشان نكمل البرهان في
+
+218
+00:23:36,150 --> 00:23:43,390
+الجزء الأول إذا هنا to prove عشان
+
+219
+00:23:43,390 --> 00:23:56,250
+نثبت الكلام هذا اه we need to show انه
+
+220
+00:23:56,250 --> 00:24:09,930
+الأصغر من أو ساوي BN لكل N عدد طبيعي نسمي هذه 6 لو
+
+221
+00:24:09,930 --> 00:24:16,430
+أثبتنا المتباينة 6 لكل N فمعناته الـ Psi أكبر من
+
+222
+00:24:16,430 --> 00:24:20,270
+أو ساوي AN أصغر من أو ساوي BN يعني تنتمي للفترة
+
+223
+00:24:20,270 --> 00:24:27,300
+المغلقة هذه وبالتالي هذا بقدّيان التقاط على كل آي
+
+224
+00:24:27,300 --> 00:24:36,460
+ان بساوي ساي لأ او يحتوي ساي تنتمي للتقاط على كل
+
+225
+00:24:36,460 --> 00:24:44,120
+آي ان وهذا بيكون برهان هذا بيكمل برهان الجزء الأول
+
+226
+00:24:45,450 --> 00:24:50,850
+إذا احنا عايزين نثبت المتباينة ستة هذه طلعت خمسة
+
+227
+00:24:50,850 --> 00:24:55,590
+وهذه طلعت لوحدها باق نثبت المتباينة هذه ستة
+
+228
+00:24:55,590 --> 00:25:06,630
+فالإثبات المتباينة الستة هذه يكفي لإثبات
+
+229
+00:25:06,630 --> 00:25:08,230
+المتباينة ستة هذه
+
+230
+00:25:13,060 --> 00:25:23,440
+يكفي إثبات المتباينة هذه لكل K لو
+
+231
+00:25:23,440 --> 00:25:32,040
+أنا ثبتت ال N و أثبتت أن BN هذه أكبر من أو ساوي كل
+
+232
+00:25:32,040 --> 00:25:37,760
+العناصر A K لكل K عدد طبيعي معناته BN هذه upper
+
+233
+00:25:37,760 --> 00:25:44,110
+bound Upper bound للمجموع هذهطب احنا قبل شوية قلنا
+
+234
+00:25:44,110 --> 00:25:49,930
+ان ال supremum للمجموعة هذه هو psi انا عندي psi
+
+235
+00:25:49,930 --> 00:25:57,450
+بساوي ال supremum ال supremum للمجموعة هذه بساوي
+
+236
+00:25:57,450 --> 00:26:05,330
+psi فلو اثبتت ان b in upper bound للمجموعة هذهوالـ
+
+237
+00:26:05,330 --> 00:26:10,230
+Psi هو أصغر upper bound، معناته الـ Psi هيطلع أصغر
+
+238
+00:26:10,230 --> 00:26:13,930
+من أو يساوي ال upper bound PN، وبالتالي هيكم نكون
+
+239
+00:26:13,930 --> 00:26:20,160
+أثباتنا ستةواضحة النقطة هذه كمان مرة لإثبات أن
+
+240
+00:26:20,160 --> 00:26:26,180
+المتباينة 6 هذه يكفي أن احنا نثبت أنه لكل fixed
+
+241
+00:26:26,180 --> 00:26:30,840
+لكل fixed in ال b in هذه upper bound لل 6 هذه
+
+242
+00:26:30,840 --> 00:26:35,100
+وبالتالي ال least upper bound لل 6 هذه اللي هو
+
+243
+00:26:35,100 --> 00:26:40,900
+صغير بيطلع أصلا أو يساوي b inطبعا اذا باقى نثبت a
+
+244
+00:26:40,900 --> 00:26:47,600
+المتباينة اللى هنا لثبات المتباينة هذه بنثبت n و
+
+245
+00:26:47,600 --> 00:26:53,740
+بناخد اي arbitrary national number k فبنلاحظ ان ال
+
+246
+00:26:53,740 --> 00:26:58,720
+n اما هتكون اصغر من أو يساوي k او n اكبر من k by
+
+247
+00:26:58,720 --> 00:27:04,040
+trichotomy property ان ك عداد طبيعى اذا اما n اصغر
+
+248
+00:27:04,040 --> 00:27:10,520
+من أو يساوي k او n اكبر من kطيب في الحالة الأولى
+
+249
+00:27:10,520 --> 00:27:16,920
+لو كانت ال N أصغر من أو يساوي K احنا عارفين ان
+
+250
+00:27:16,920 --> 00:27:20,840
+الفترات هذه نستد وبالتالي الفترة اللي ال index
+
+251
+00:27:20,840 --> 00:27:26,380
+تبعها أكبر تطلع هي الأصغر تمام؟
+
+252
+00:27:26,380 --> 00:27:34,690
+إذا أنا بطلع عندي I K subset من I Nوبالتالي هاي
+
+253
+00:27:34,690 --> 00:27:45,030
+نرسم هاي IK هاي الفترة IK نقاط
+
+254
+00:27:45,030 --> 00:27:52,770
+أطرافها AK BK وهاي الفترة وهذه
+
+255
+00:27:52,770 --> 00:28:00,330
+محتوى داخل الفترة IN هاي الفترة IN كبّر
+
+256
+00:28:00,330 --> 00:28:14,950
+شويةهذه الفترة IN هذه نقاط أطرافها فبما
+
+257
+00:28:14,950 --> 00:28:20,520
+أن الفترة IK subset من الفترة INإذا اللي بطلع عندى
+
+258
+00:28:20,520 --> 00:28:26,840
+aK أصغر من أو يساوي bK و bK أصغر من أو يساوي bN
+
+259
+00:28:26,840 --> 00:28:34,060
+وبالتالي هيني أثبتت أن aK أصغر من أو يساوي bN كما
+
+260
+00:28:34,060 --> 00:28:40,540
+هو مطلوب طيب في الحالة التانية افرض أن N هي اللي
+
+261
+00:28:40,540 --> 00:28:46,110
+أكبر من Kففي الحالة هذه بما ان ال sequence in
+
+262
+00:28:46,110 --> 00:29:02,270
+nested إذا in محتوى داخل ik إذا
+
+263
+00:29:02,270 --> 00:29:06,150
+هاي عندي in
+
+264
+00:29:11,080 --> 00:29:20,360
+الفترة IN هذا هي محتوى داخل الفترة IK
+
+265
+00:29:20,360 --> 00:29:23,660
+يعني
+
+266
+00:29:23,660 --> 00:29:28,880
+نقاط أطرافها AK BK وبالتالي في الحالة هذه بيطلع AK
+
+267
+00:29:28,880 --> 00:29:36,400
+أصغر من أو ساوي AN أصغر من أو ساوي BNمظبوط؟ إذا
+
+268
+00:29:36,400 --> 00:29:39,780
+هنا في الحالة التانية برضه أثبتنا أن aK أصغر من أو
+
+269
+00:29:39,780 --> 00:29:45,680
+ساوي bN إذا في الحالتين أنا أثبتت أن aK أصغر من أو
+
+270
+00:29:45,680 --> 00:29:57,480
+ساوي bN و هذا صحيح لكل for any K إذا هذا معناه أنه
+
+271
+00:29:57,480 --> 00:30:02,440
+من هنا بي زي ما قلنا bN is upper bound لل set هذه
+
+272
+00:30:03,020 --> 00:30:07,480
+وبالتالي ال supremum تبع ال 6 اللي هو Psi بيطلع
+
+273
+00:30:07,480 --> 00:30:14,440
+أصغر من أو ساوي P M إذن هذا بثبت اللي هو المتباينة
+
+274
+00:30:14,440 --> 00:30:20,740
+6 الأقل من 6 .. من 5 و 6 زي ما شرحنا بنحصل على إن
+
+275
+00:30:20,740 --> 00:30:21,580
+ال Psi
+
+276
+00:30:29,990 --> 00:30:35,610
+من خمسة و ستة بنحصل على ان الـ Psi هذا هي أكبر من
+
+277
+00:30:35,610 --> 00:30:39,050
+أو الساوي An أصغر من أو الساوي Bn يعني الـ Psi
+
+278
+00:30:39,050 --> 00:30:45,330
+تنتمي لتقاطة الفترات كما هو مطلوب هذا بثبت أو
+
+279
+00:30:45,330 --> 00:30:51,210
+ببرهن الجزء الأول من النظرية لبرهان الجزء التاني
+
+280
+00:31:03,930 --> 00:31:10,910
+الجزء التاني بإن
+
+281
+00:31:10,910 --> 00:31:15,210
+نثبت إن التقاطة هذا الـ psi هذه اللي طلعناها هي
+
+282
+00:31:15,210 --> 00:31:17,470
+النقطة الوحيدة في التقاطة مافيش غيرها
+
+283
+00:31:20,460 --> 00:31:25,380
+طبعا هنا الهدف نحصل عليه تحت الشرط الإضافي ان ال
+
+284
+00:31:25,380 --> 00:31:30,000
+minimum للأعداد الغير سالب هذه لاحظوا ان ان ان ان
+
+285
+00:31:30,000 --> 00:31:33,040
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+286
+00:31:33,040 --> 00:31:34,180
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+287
+00:31:34,180 --> 00:31:39,160
+ان ان ان ان
+
+288
+00:31:39,160 --> 00:31:39,180
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+289
+00:31:39,180 --> 00:31:41,160
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+290
+00:31:41,160 --> 00:31:41,360
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+291
+00:31:41,360 --> 00:31:41,480
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+292
+00:31:49,520 --> 00:31:54,560
+إذا لو كان الانفة من الأعداد السالبة دي بساوي سفر
+
+293
+00:31:54,560 --> 00:32:00,660
+فبنثبت أنه التقاطع في نقطة واحدة هي ساية طيب
+
+294
+00:32:00,660 --> 00:32:06,260
+لبرهان ذلك بنلاحظ أن ال set هذه غير خالية طبعا لإن
+
+295
+00:32:06,260 --> 00:32:10,200
+أنا عندي ال sequence of intervals اللي لها نقاط
+
+296
+00:32:10,200 --> 00:32:14,840
+أطراف فال sequence هذه فيها على الأقل فترة واحدة
+
+297
+00:32:16,200 --> 00:32:23,600
+هذه المجموعة غير خالية وبعدها bounded below by a1
+
+298
+00:32:23,600 --> 00:32:33,700
+يعني احنا عارفين ان a1 هي a1 او
+
+299
+00:32:33,700 --> 00:32:38,460
+الفترة I1 هذه تحتوي كل الفترات
+
+300
+00:32:42,320 --> 00:32:49,760
+تحتوي كل الفترات I N وبالتالي
+
+301
+00:32:49,760 --> 00:32:59,020
+أنا عندي A1 من هنا بتطلع عندي A1 أصغر من أو يساوي
+
+302
+00:32:59,020 --> 00:33:09,420
+B N لكل N في Nوبالتالي a1 is a lower bound أو ال
+
+303
+00:33:09,420 --> 00:33:15,880
+set of all b in is bounded below by a1 وبالتالي by
+
+304
+00:33:15,880 --> 00:33:23,160
+the infimum property يوجد عدد حقيقي eta وهذا هو ال
+
+305
+00:33:23,160 --> 00:33:31,030
+infimum لset of all b in حيث ان عدد طبيعيإن هنا ال
+
+306
+00:33:31,030 --> 00:33:36,110
+set هذي ال set of all b in حيث in natural number
+
+307
+00:33:36,110 --> 00:33:40,270
+bounded below وبالتالي ال infimum تبعها exist
+
+308
+00:33:40,270 --> 00:33:42,790
+خلينا نسميه إيتا
+
+309
+00:33:45,470 --> 00:33:52,890
+الان by an argument by an argument similar to the
+
+310
+00:33:52,890 --> 00:33:57,190
+proof of sex احنا شوفنا قبل شوية متباينة sex
+
+311
+00:33:57,190 --> 00:34:07,030
+أثبتناها فباستخدام برهان شبيه ببرهان المتباينة sex
+
+312
+00:34:07,030 --> 00:34:13,770
+ممكن نصل إلىمتباينة زي المتباينة six اللي هي هذه
+
+313
+00:34:13,770 --> 00:34:20,330
+ان ا ن تطلع اصغر من او ساوي اتا لكل ان لان هذه
+
+314
+00:34:20,330 --> 00:34:27,250
+برهانها زي برهان ستة خلينا نسميها ستة ستة prime
+
+315
+00:34:27,250 --> 00:34:33,690
+برهان هذه نرجع لبرهان ستة ونشوف كيف برهنها ونعمل
+
+316
+00:34:33,690 --> 00:34:34,570
+برهان مشابه
+
+317
+00:34:37,400 --> 00:34:41,240
+Okay، إذا الـ ETA هنا، العدد ETA ده اللي هو الـ
+
+318
+00:34:41,240 --> 00:34:47,440
+infim للست of all BN هذا إيش بيطلع من هنا، من
+
+319
+00:34:47,440 --> 00:34:51,260
+المتبينة الجديدة six prime؟ بيطلع عبارة عن upper
+
+320
+00:34:51,260 --> 00:34:58,300
+bound لكل العناصر A N، هذه ETA upper bound
+
+321
+00:34:58,300 --> 00:35:02,890
+للمجموعة هذهوبالتالي ال supremum للمجموعة هذه
+
+322
+00:35:02,890 --> 00:35:07,190
+بيطلع أصغر من أو يساوي ال upper bound هذا اللي هو
+
+323
+00:35:07,190 --> 00:35:14,130
+Psi إذا بيطلع عندي هنا Psi بيطلع أصغر من أو يساوي
+
+324
+00:35:14,130 --> 00:35:19,010
+Eta إذا
+
+325
+00:35:19,010 --> 00:35:25,870
+طلع عندي أنا هنا Psi أصغر من أو يساوي Eta
+
+326
+00:35:31,240 --> 00:35:39,040
+الان بنلاحظ انه لو اخدت اي x في الفترة رقم in لكل
+
+327
+00:35:39,040 --> 00:35:45,900
+in لو كان x موجود هنا هذا بكافي ان x اكبر من او
+
+328
+00:35:45,900 --> 00:35:54,080
+يساوي psi اصغر من او يساوي eta لتوضيح
+
+329
+00:35:54,080 --> 00:36:04,120
+ذلك او لبرهان ذلك ال ..لو كانت X موجودة هنا فال X
+
+330
+00:36:04,120 --> 00:36:08,480
+تطلع upper bound لل set هذه هنا خلّيني اوضح بالرسم
+
+331
+00:36:08,480 --> 00:36:13,480
+لو
+
+332
+00:36:13,480 --> 00:36:21,520
+كانت ال X تنتبه ل I N الفترة
+
+333
+00:36:21,520 --> 00:36:32,530
+I N وهي الفترة I N وهي ال Xفلو كانت X موجودة في
+
+334
+00:36:32,530 --> 00:36:42,450
+الفترة I N الفترة هذه فهذا معناه أن ال A N أصغر من
+
+335
+00:36:42,450 --> 00:36:49,890
+أو يساوي X لكل N وبالتالي X upper bound للمجموعة
+
+336
+00:36:49,890 --> 00:36:56,770
+هذه وبالتالي ال supremum للمجموعة هذه اللي هو Psi
+
+337
+00:36:56,770 --> 00:37:05,220
+بيطلع أصغر من لو يساوي X، مظبوط؟كذلك من هنا ال X
+
+338
+00:37:05,220 --> 00:37:13,100
+أصغر من أو يساوي BN لكل N وبالتالي ال X أبارع ال
+
+339
+00:37:13,100 --> 00:37:19,340
+lower bound لمجموعة الأعداد BN وبالتالي ال infimum
+
+340
+00:37:19,340 --> 00:37:26,780
+لمجموعة الأعداد BN ال infimum للمجموعة هذه بطلع
+
+341
+00:37:26,780 --> 00:37:30,620
+أكبر من أو يساوي ال X اللي هو lower bound لها صح؟
+
+342
+00:37:31,090 --> 00:37:35,010
+إذا الانفلان المجموعة هذه اللي هو إيتا أكبر من أو
+
+343
+00:37:35,010 --> 00:37:40,750
+ساوي X اللي هو lower bound للمجموعة هذه okay إذا
+
+344
+00:37:40,750 --> 00:37:44,430
+لو هذا الكلام صح لو كانت X موجودة في الفترة I N
+
+345
+00:37:44,430 --> 00:37:51,140
+لكل Nفهذا بيقدي ان ال X أكبر من أو ساوي ساي و أصغر
+
+346
+00:37:51,140 --> 00:37:55,840
+من أو ساوي إيتا يعني X محصورة بين ساي و إيتا و
+
+347
+00:37:55,840 --> 00:38:00,740
+طبعا العكس لو هذا الكلام صحيح بيقدي ان ال X موجودة
+
+348
+00:38:00,740 --> 00:38:07,280
+هنا تمام؟ إذا هذا برهان ال statement اللي هنا الان
+
+349
+00:38:07,280 --> 00:38:11,100
+انا
+
+350
+00:38:11,100 --> 00:38:11,500
+عندي
+
+351
+00:38:29,400 --> 00:38:33,340
+أنا عندي ال infimum للمجموعة هذه احنا فرضين ان ال
+
+352
+00:38:33,340 --> 00:38:40,520
+infimum لمجموعة الأعداد الغير سالبة هذه ال infimum
+
+353
+00:38:40,520 --> 00:38:45,300
+لها بساوة صفر احنا هذا فرضينه عشان نثبت ان التقاطة
+
+354
+00:38:45,300 --> 00:38:52,240
+في نقطة واحدة صح هذا فرض قائم و احنا أخدنا لمّة
+
+355
+00:38:52,240 --> 00:38:58,490
+واحد اتناشرفي بيجي كان بعديها exercise بيجي بعديها
+
+356
+00:38:58,490 --> 00:39:06,370
+exercise بيقول عشان المجموعة S لو كانت المجموعة في
+
+357
+00:39:06,370 --> 00:39:07,490
+إلها lower bound
+
+358
+00:39:10,250 --> 00:39:16,210
+فعشان ال lower bound هذا يكون هو ال infimum هذا
+
+359
+00:39:16,210 --> 00:39:21,290
+بكافي أنه لكل epsilon أكبر من السفر يوجد عنصر S
+
+360
+00:39:21,290 --> 00:39:25,870
+epsilon في المجموعة S بحيث أن العنصر أصغر من ال
+
+361
+00:39:25,870 --> 00:39:27,150
+infimum زائد epsilon
+
+362
+00:39:30,620 --> 00:39:34,020
+طيب إذا أنا بدي أستخدم ال exercise هذا اللي هو
+
+363
+00:39:34,020 --> 00:39:39,760
+شبيه لمّة واحد اتناشر هي عندي المجموعة هذه وهي
+
+364
+00:39:39,760 --> 00:39:44,620
+عندي ال inform تبعها بساوي سفر السفر هذا اللي هو
+
+365
+00:39:44,620 --> 00:39:51,300
+lower bound المجموعة هذه انتوا لاحظوا انه a b n او
+
+366
+00:39:51,300 --> 00:39:53,860
+a n b n
+
+367
+00:39:57,350 --> 00:40:03,870
+بن ماينوس ان أكبر من أو سوى سفر لكل ان فالسفر هذا
+
+368
+00:40:03,870 --> 00:40:08,710
+lower boundللمجموعة الاعداد هذه و هو مش lower
+
+369
+00:40:08,710 --> 00:40:13,030
+bound بس احنا مُعطى .. مُعطى من الفرض انه هو ال
+
+370
+00:40:13,030 --> 00:40:16,510
+infim هو اكبر lower bound اذا حسب التمرين اللي
+
+371
+00:40:16,510 --> 00:40:20,670
+بيجي بعد لما واحد اتناش بما انه السفر هو ال infim
+
+372
+00:40:20,670 --> 00:40:25,670
+للمجموعة هذه اذا لأي epsilon او لكل epsilon اكبر
+
+373
+00:40:25,670 --> 00:40:30,730
+من السفر يوجد عنصر في المجموعة هذه هذا هو
+
+374
+00:40:33,930 --> 00:40:40,010
+ال index تبعه المؤشر تبعه n هسميه ن إبسلون يعتمد
+
+375
+00:40:40,010 --> 00:40:44,550
+على ال epsilon لكل إبسلون فيه عدد طبيعي ن إبسلون
+
+376
+00:40:44,550 --> 00:40:51,110
+وبالتالي فيه يوجد عنصر في ال set هذه أصغر من ال
+
+377
+00:40:51,110 --> 00:40:56,770
+infimum زائد إبسلون هذا حسب ال exercise الآن هذا
+
+378
+00:40:56,770 --> 00:41:06,750
+العدد هذابن إبسلون أكبر من أو ساوي إتا وان
+
+379
+00:41:06,750 --> 00:41:11,230
+إبسلون أصغر من أو ساوي ساي فالفرق بين هدول أصغر من
+
+380
+00:41:11,230 --> 00:41:19,970
+أو ساوي الفرق بين هدول وانا عندي ساي أصغر من أو
+
+381
+00:41:19,970 --> 00:41:25,990
+ساوي إتا فالفرق بين ساي و إتا أكبر من أو ساوي سفر
+
+382
+00:41:26,920 --> 00:41:33,420
+تمام؟ إذا المتباين الأخيرة هذه بحصل منها على إيه؟
+
+383
+00:41:33,420 --> 00:41:37,560
+المتباين الأخير منها هذه بحصل على إنه صفر أصغر من
+
+384
+00:41:37,560 --> 00:41:45,560
+أوي ساوي eta minus epsilon أصغر من epsilon لأ عفوا
+
+385
+00:41:45,560 --> 00:41:48,280
+هذه eta minus psi
+
+386
+00:41:51,930 --> 00:41:58,010
+وهذا صحيح لكل إبسلون أكبر من السفر أي لأي إبسلون
+
+387
+00:41:58,010 --> 00:42:01,830
+أكبر من السفر وصلنا إلى أنه سفر أصغر من أو ساوي
+
+388
+00:42:01,830 --> 00:42:09,810
+eta minus psi و eta minus psi أصغر من إبسلون طيب
+
+389
+00:42:09,810 --> 00:42:13,390
+احنا خلنا قبل هيك لمبة بتقول لو في عندي عدد حقيقي
+
+390
+00:42:13,390 --> 00:42:18,610
+زي هذا غير سالب و أصغر من إبسلون لكل إبسلون أكبر
+
+391
+00:42:18,610 --> 00:42:23,330
+من السفرفهذا بيقدّي ان إيتا العدد هذا بيساوي سفر
+
+392
+00:42:23,330 --> 00:42:32,090
+وبالتالي إذا بيطلع عندى إيتا بيساوي ساي okay تمام
+
+393
+00:42:32,090 --> 00:42:38,410
+إذا هيك بنكون احنا أثبتنا ان إيتا بيساوي ساي
+
+394
+00:42:38,410 --> 00:42:41,610
+وبالتالي ساي هي النقطة الوحيدة
+
+395
+00:42:45,050 --> 00:42:50,830
+إذا ساي هي النقطة الوحيدة اللي موجودة في التقاطة
+
+396
+00:42:50,830 --> 00:42:59,450
+وهذا يكمل برهان النظرية إذا إحنا خلصنا .. يعني
+
+397
+00:42:59,450 --> 00:43:07,010
+انتهينا من برهان النظرية الطويلة هذهو في بعض
+
+398
+00:43:07,010 --> 00:43:10,410
+الملاحظات على النظرية هنشوفها ان شاء الله المرة
+
+399
+00:43:10,410 --> 00:43:17,450
+القادمة فهنوقف لأن الوقت خلص هننهي المحاضرة الآن و
+
+400
+00:43:17,450 --> 00:43:21,990
+نشوف الملاحظات و الحاجات الخاصة بالنظرية في اللقاء
+
+401
+00:43:21,990 --> 00:43:22,990
+القادم ان شاء الله
+
diff --git a/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM_raw.srt b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6915d6fd959d6ff9991ce5b578b6696465c8e319
--- /dev/null
+++ b/PL9fwy3NUQKwZHPs6l8Fr-st-8cVJxmVek/xZkuM2ApdRM_raw.srt
@@ -0,0 +1,1624 @@
+1
+00:00:21,190 --> 00:00:25,470
+في المحاضرة السابقة تعرفنا على أنواع الفترات أو ال
+
+2
+00:00:25,470 --> 00:00:28,950
+intervals وشوفنا
+
+3
+00:00:28,950 --> 00:00:33,290
+أن أول تلت أنواع من الفترات اللي هي ال open و ال
+
+4
+00:00:33,290 --> 00:00:37,110
+closed و ال half open أو half closed intervals كل
+
+5
+00:00:37,110 --> 00:00:43,790
+الفترات هذه كانت bounded intervals أو فترات محدودة
+
+6
+00:00:45,730 --> 00:00:50,070
+الانواع الخامسة الأخرى من النوع الرابع إلى النوع
+
+7
+00:00:50,070 --> 00:00:55,750
+الثامن كلها فترات غير محدودة أو unbounded
+
+8
+00:00:55,750 --> 00:01:02,710
+intervals و الفترات هذه عبارة عن النوع الرابع open
+
+9
+00:01:02,710 --> 00:01:10,410
+left tray شعاع أيصر مفتوح أو open right tray شعاع
+
+10
+00:01:10,410 --> 00:01:16,650
+أيمن مفتوح أو closed left tray شعاع أيمنمغلق
+
+11
+00:01:16,650 --> 00:01:22,890
+closed writer شعاع أيمن مغلق أو شعاع الجبلة شعاع
+
+12
+00:01:22,890 --> 00:01:27,140
+أيصر مغلقو طبعا هذه عبارة عن فترات زي ما احنا
+
+13
+00:01:27,140 --> 00:01:31,660
+شايفين النوع الأخير اللي هو مجموعة كل الأعداد اللي
+
+14
+00:01:31,660 --> 00:01:35,320
+حصلت فيها the set of all real numbers اللي هي
+
+15
+00:01:35,320 --> 00:01:39,160
+الفترة من ثالث من النهاية للملا نهاية وهذه طبعا
+
+16
+00:01:39,160 --> 00:01:45,560
+أيضا unbounded interval فترة غير محصورة أو غير
+
+17
+00:01:45,560 --> 00:01:49,640
+محدودة لو
+
+18
+00:01:49,640 --> 00:01:54,710
+بصينا لأي نوع من أنواع التمانية هذهمن الفترات لو
+
+19
+00:01:54,710 --> 00:02:02,590
+أخذنا أي نوع من الأنواع التمانية فبنلاحظ
+
+20
+00:02:02,590 --> 00:02:10,850
+وسمينا الفترة هذه I لأن لو أخذنا one of the eight
+
+21
+00:02:10,850 --> 00:02:17,250
+types of interval سمناها I فبنلاحظ أن الفترة I هذه
+
+22
+00:02:17,250 --> 00:02:26,690
+بتحقق الخاصية اللي هي أمامنا هذهوهي أنه لو كان في
+
+23
+00:02:26,690 --> 00:02:32,050
+عندي يعني
+
+24
+00:02:32,050 --> 00:02:38,130
+أي فترة زي هذه مثلا
+
+25
+00:02:38,130 --> 00:02:45,290
+الفترة
+
+26
+00:02:45,290 --> 00:02:50,130
+من A إلى مالنهاية لو النقطة هذه A
+
+27
+00:02:57,030 --> 00:03:01,490
+فهذا open right ray لو أخدت أي نقطتين في الفترة
+
+28
+00:03:01,490 --> 00:03:07,810
+هذه x و y واقعت
+
+29
+00:03:07,810 --> 00:03:13,790
+x و y ينتموا للفترة هذه فبنلاحظ أن الفترة المغلقة
+
+30
+00:03:13,790 --> 00:03:19,690
+ما بين x و y the closed interval from x و y is
+
+31
+00:03:19,690 --> 00:03:27,120
+contained الفترة المغلقة هذهبتكون دائما موجودة في
+
+32
+00:03:27,120 --> 00:03:35,620
+الفترة I لأن لو أخدت أي فترة I احنا أخدنا مثال ال
+
+33
+00:03:35,620 --> 00:03:41,100
+open right ray فبنلاحظ أن هذه الفترة أو أي فترة
+
+34
+00:03:41,100 --> 00:03:47,940
+تانية من الأنواع التامية بتحقق الخصية هذه اللي فوق
+
+35
+00:03:47,940 --> 00:03:53,080
+طبعا هذه احنا عطينا رقم أربعة بس عشان كبرنا الخط
+
+36
+00:03:55,050 --> 00:04:01,070
+فرقم أربعة مش دائرة على الشاشة فأي فترة زي هذه
+
+37
+00:04:01,070 --> 00:04:05,310
+بتحقق خاصية أربعة وهو لو أخدت أي نقطتين في الفترة
+
+38
+00:04:05,310 --> 00:04:12,470
+I فالفترة اللي واقع بين X وY أيضا بتكون موجودة
+
+39
+00:04:12,470 --> 00:04:20,290
+داخل الفترة I الآن العكس تبع الكلام هذا صحيح كما
+
+40
+00:04:20,290 --> 00:04:24,510
+هو موضح في نظرية واحد عشرين
+
+41
+00:04:27,170 --> 00:04:34,990
+بمعنى انه لو كان في اندي فترة لو كان في اندي أخدت
+
+42
+00:04:34,990 --> 00:04:39,990
+مجموعة جزئية من R المجموعة هذه عدد عناصرها اكبر من
+
+43
+00:04:39,990 --> 00:04:42,950
+او ساوى اتنين ال cardinal number تبع اكبر من او
+
+44
+00:04:42,950 --> 00:04:46,890
+ساوى اتنين يعني المجموعة هذه فيها على الأقل عنصرين
+
+45
+00:04:46,890 --> 00:04:51,550
+او اكثر ممكن تكون finite set ممكن تكون infinite
+
+46
+00:04:53,020 --> 00:04:55,760
+يكون فيها على الأقل او نصرين فلو كان في عندي
+
+47
+00:04:55,760 --> 00:04:58,980
+مجموعة جزئية من الارض فيها على الأقل او نصرين و
+
+48
+00:04:58,980 --> 00:05:03,600
+بتحقق الخاصية اربعة فلازم المجموعة الجزئية هذه
+
+49
+00:05:03,600 --> 00:05:11,820
+تطلع فترة و البرهان تبع المظرية هذه مش صعب و ممكن
+
+50
+00:05:11,820 --> 00:05:15,840
+يعني احنا كاتبينه بالتفصيل لكم
+
+51
+00:05:18,790 --> 00:05:26,110
+وقلنا أنتم بمكانكم تقرؤوه تفهموه البرهان
+
+52
+00:05:26,110 --> 00:05:32,150
+بنجزه لأربع حالات المجموعة S هذه اللي احنا عايزين
+
+53
+00:05:32,150 --> 00:05:36,190
+نثبت انها interval فيها أربع احتمالات اما بتكون
+
+54
+00:05:36,190 --> 00:05:40,150
+bounded يعني محدودة من الجهتين
+
+55
+00:05:42,590 --> 00:05:48,630
+او بتكون bounded احتمال التاني تكون bounded above
+
+56
+00:05:48,630 --> 00:05:54,090
+but not below او احتمال تالت انها تكون bounded
+
+57
+00:05:54,090 --> 00:05:58,330
+below but not above الاحتمال الرابع انها is
+
+58
+00:05:58,330 --> 00:06:02,250
+neither bounded above nor below يعني لا محدودة من
+
+59
+00:06:02,250 --> 00:06:06,770
+اسفل ولا من اعلى وفي
+
+60
+00:06:06,770 --> 00:06:11,110
+كل حالة من الحلقات الأربعة في كل حالة من الحلقات
+
+61
+00:06:11,110 --> 00:06:18,580
+الأربعةهي الحالة الأولى S bounded في كل حالة من
+
+62
+00:06:18,580 --> 00:06:24,580
+الحالات الأربعة بنحاول نثبت أن ال 6S هي فترة ما
+
+63
+00:06:24,580 --> 00:06:32,320
+فترة معينة فمثلا لنتعلم نشوف الحالة الأولى الحالة
+
+64
+00:06:32,320 --> 00:06:36,260
+الأولى أن S تكون bounded وبالتالي bounded above
+
+65
+00:06:36,260 --> 00:06:39,940
+and bounded belowبما أنها bounded below إذا ال
+
+66
+00:06:39,940 --> 00:06:44,720
+inform تبعها exist سميه a بما أنها bounded above
+
+67
+00:06:44,720 --> 00:06:48,440
+إذا by ال supreme property ال supreme تبعها exist
+
+68
+00:06:48,440 --> 00:06:56,460
+سميه b وبالتالي S بتصير محصورة داخل أو subset من
+
+69
+00:06:56,460 --> 00:07:02,800
+الفترة المغلفة AB طبعا لو قرأته انتوا التفاصيل
+
+70
+00:07:02,800 --> 00:07:07,980
+هتطلع الفترة الست S هذه في نهاية البرهانهتطلع فترة
+
+71
+00:07:07,980 --> 00:07:19,540
+half open زي هذه او half open زي هذه هو
+
+72
+00:07:19,540 --> 00:07:26,700
+بالتالي فترة وهو المطلوب في الحالة التانية اللي
+
+73
+00:07:26,700 --> 00:07:32,300
+هناخد الحالة التانية ان ال set S is bounded above
+
+74
+00:07:32,300 --> 00:07:37,270
+but not bounded belowمحدودة من أعلى لكن مش محدودة
+
+75
+00:07:37,270 --> 00:07:43,390
+من أسفل في الحالة هذه هتطلع ال 6S هذه فترة و هتطلع
+
+76
+00:07:43,390 --> 00:07:48,770
+هتطلع left ray left ray لأنها مش محدودة من أسفل
+
+77
+00:07:48,770 --> 00:07:56,290
+فيعني لو انتوا تابعتوا البرهان هتطلع الفترة او ال
+
+78
+00:07:56,290 --> 00:07:59,250
+6S هي عبارة عن اما
+
+79
+00:08:08,630 --> 00:08:14,990
+هتطلع ال set is عبارة عن closed lift tray زي هذا
+
+80
+00:08:14,990 --> 00:08:20,870
+او open lift tray زي هذا فهنسيبكم
+
+81
+00:08:20,870 --> 00:08:25,130
+تقراوا البرهان مش صعب أكيد كل واحد منكم هيتفهمه
+
+82
+00:08:25,130 --> 00:08:30,450
+الآن الحالات التالتة والرابعة شبيهة بالحالات
+
+83
+00:08:30,450 --> 00:08:36,270
+الأولى والتانية وبالتالي هسيبها اليكم كتمرين
+
+84
+00:08:36,270 --> 00:08:40,940
+exerciseتتدربوا على كتابة او تحاولوا تكتبوا
+
+85
+00:08:40,940 --> 00:08:46,720
+البرهان هيكون شبيه بالحالتين السابقات okay اي
+
+86
+00:08:46,720 --> 00:08:52,900
+طالبة بسيب تمرين زي هذا مطالبة انها تكتب يعني تحل
+
+87
+00:08:52,900 --> 00:08:59,040
+التمرين هذا على الأقل عشان تفهم المادة اذا في عندك
+
+88
+00:08:59,040 --> 00:09:04,340
+اي تساول او استفسار تتصل فيا او ممكن تكتب البرهان
+
+89
+00:09:04,340 --> 00:09:10,150
+على ورقةو تعطيني في أقرب فرصة و أنا بعرجلكيا في
+
+90
+00:09:10,150 --> 00:09:17,510
+المحاضرة اللي بعدها تمام؟ اذا الحالات التالتة و
+
+91
+00:09:17,510 --> 00:09:20,970
+الرابعة شبيهة بالحالات الأولى و التانية و بالتالي
+
+92
+00:09:20,970 --> 00:09:28,910
+هنسيبها .. هنتركها للطالب كتمرين في عندي هنا تعريف
+
+93
+00:09:43,000 --> 00:09:47,580
+لو أخدت sequence of intervals I N هذه عبارة عن
+
+94
+00:09:47,580 --> 00:09:52,920
+فترة الان هذه عبارة عن sequence على سرها فترات I N
+
+95
+00:09:52,920 --> 00:09:58,640
+فال sequence هذه بنسميها nested ال sequence of
+
+96
+00:09:58,640 --> 00:10:04,690
+intervals هذه بنسميها nestedإذا كانت بتحقق الخاصية
+
+97
+00:10:04,690 --> 00:10:08,670
+هذه و هي أن الفترة الأولى تحتوي التانية والتانية
+
+98
+00:10:08,670 --> 00:10:13,690
+تحتوي التالتة و هكذا أو بمعنى آخر الفترة رقم n I n
+
+99
+00:10:13,690 --> 00:10:18,410
+contains الفترة اللي بعدها مباشرة I n زاد واحد لكل
+
+100
+00:10:18,410 --> 00:10:24,420
+n إذا الفترات أو sequence of intervalsthat
+
+101
+00:10:24,420 --> 00:10:29,280
+satisfies this property بنسميها nested it's called
+
+102
+00:10:29,280 --> 00:10:35,600
+nested sequence of intervals هذه مثال مثال على
+
+103
+00:10:35,600 --> 00:10:42,500
+sequence of nested intervals لو أخدت الفترة I N هي
+
+104
+00:10:42,500 --> 00:10:46,400
+الفترة المغلقة من سفر لواحد على N حيث N أي عدد
+
+105
+00:10:46,400 --> 00:10:55,860
+طبيعي فالفترات هذهواضح جدا انها nested هاي خط
+
+106
+00:10:55,860 --> 00:11:10,380
+العداد هاي خط العداد هاي سفر واحد هاي فترة هادي I
+
+107
+00:11:10,380 --> 00:11:16,200
+واحد خد ان بس I واحد I واحد فترة مغلقة من سفر ل
+
+108
+00:11:16,200 --> 00:11:23,640
+واحدI اتنين خد in بساو اتنين معناته هذا نص يعني في
+
+109
+00:11:23,640 --> 00:11:34,520
+عندي فترة الفترة هذه I اتنين I I اتنين I واحد
+
+110
+00:11:34,520 --> 00:11:40,000
+تحتوي I اتنين لو أخدت in تلاتة بيصير في عندي هنا
+
+111
+00:11:40,000 --> 00:11:43,180
+تلت وفي الفترة هذه
+
+112
+00:11:45,730 --> 00:11:53,470
+هذه الفترة I ثلاثة و I ثلاثة واضح هنا contained I
+
+113
+00:11:53,470 --> 00:11:59,530
+اتنين و هكذا اذا واضح هنا ان I one contains I two
+
+114
+00:11:59,530 --> 00:12:07,270
+contains I three and so on و هكذا وبالتالي ال
+
+115
+00:12:07,270 --> 00:12:16,350
+sequence of intervals هذه تطلع nested تمام؟ طيبالـ
+
+116
+00:12:16,350 --> 00:12:21,270
+sequence هذه مش بس listed كمان التقاطع تبعها لا
+
+117
+00:12:21,270 --> 00:12:27,430
+يساوي في يعني لو قطعت الفترات هذه كلها فهنجد أنه
+
+118
+00:12:27,430 --> 00:12:31,230
+التقاطع تبعها لا يساوي في يعني ممكن ألاقي على
+
+119
+00:12:31,230 --> 00:12:36,410
+الأقل أنصر واحد في التقاطع in fact في حقيقة الأمر
+
+120
+00:12:36,410 --> 00:12:42,810
+التقاطع هذا بساوي singleton set zero يعني هنا في
+
+121
+00:12:42,810 --> 00:12:48,530
+الصفر هذا ينتمي للتقاطعهذا الطبعا الكلام مش واضح
+
+122
+00:12:48,530 --> 00:12:54,750
+هذا ليس واضحا فعشان نوضحه او نبرغله to see this
+
+123
+00:12:54,750 --> 00:13:03,170
+لإثبات ان التقاطع هذا الفترات هذه بساوي single
+
+124
+00:13:03,170 --> 00:13:09,790
+consist zero فبنثبت
+
+125
+00:13:09,790 --> 00:13:19,090
+ان مجمعتين بسوا بعضعندى الاحتواء هذا واضح واضح ان
+
+126
+00:13:19,090 --> 00:13:30,410
+السفر ينتمى للفترة السفر واحد على ان اللى هى in
+
+127
+00:13:30,410 --> 00:13:39,690
+لكل n في ان وبالتالي هذا بيقدم السفر ينتمى لتقاطع
+
+128
+00:13:39,690 --> 00:13:45,490
+كل ال inوبالتالي المجموعة اللى فيها العنصر الوحيد
+
+129
+00:13:45,490 --> 00:13:55,690
+سفر subset من تقاطة المجموعات IL تمام هذا واضح لكن
+
+130
+00:13:55,690 --> 00:14:01,970
+مش واضح انه نثبت الآن العكس او ال reverse
+
+131
+00:14:01,970 --> 00:14:06,270
+inclusion عشان نثبت مساواة ان ال set هدى بالساوية
+
+132
+00:14:06,270 --> 00:14:13,480
+هدى باقي نثبت ال inclusion هدىتمام فخلّينا ناخد
+
+133
+00:14:13,480 --> 00:14:22,080
+let x تنتمي للتخاطى لكل ال I N هذا معناه ان X
+
+134
+00:14:22,080 --> 00:14:27,960
+تنتمي ل I N لكل N طب ما ال I N هي عبارة عن سفر
+
+135
+00:14:27,960 --> 00:14:34,380
+فترة مغلقة من سفر ل 1 على N هذا معناه ان X أكبر من
+
+136
+00:14:34,380 --> 00:14:39,500
+أو ساوي سفر أصغر من أو ساوي 1 على N لكل N
+
+137
+00:14:44,000 --> 00:14:50,260
+تمام الان انا بتثبت ان
+
+138
+00:14:50,260 --> 00:14:56,060
+ال X هذا بساوي سفر اش بتثبت انا بتثبت الاحتواء
+
+139
+00:14:56,060 --> 00:15:00,900
+المعاكس اخدت X عنصر في التقاطة بتثبت ان X ينتمي
+
+140
+00:15:00,900 --> 00:15:04,900
+للمجموع عادي يعني X بساوي سفر لان هذه المجموعة
+
+141
+00:15:04,900 --> 00:15:08,640
+عاملة مافيش فيها غير سفر اذا لو اثبتت ان X بساوي
+
+142
+00:15:08,640 --> 00:15:15,780
+سفر بكون اثبتت الاحتواء المعاكسطيب الان نعمل برهار
+
+143
+00:15:15,780 --> 00:15:21,800
+بالتناقض assume نفترض
+
+144
+00:15:21,800 --> 00:15:27,880
+ان x لا يساوي سفر طب
+
+145
+00:15:27,880 --> 00:15:31,780
+انا عندي ال x هنا اكبر من او يساوي سفر والان
+
+146
+00:15:31,780 --> 00:15:34,860
+بيستويش سفر اذا ال x اكبر من سفر
+
+147
+00:15:40,290 --> 00:15:44,590
+طيب by Archimedean property by Archimedean
+
+148
+00:15:44,590 --> 00:15:47,810
+property
+
+149
+00:15:47,810 --> 00:15:53,990
+هذه اللي حاطين عليها علامة النجمة ايش ال
+
+150
+00:15:53,990 --> 00:15:58,830
+Archimedean property بتقول لأي عدد موجب زي X هذا
+
+151
+00:15:58,830 --> 00:16:09,170
+بنقدر نلاقي يوجد عدد طبيعي نسميه N0بحيث انه مقلوب
+
+152
+00:16:09,170 --> 00:16:18,310
+العدد الطبيعي N0 أصغر من ال X من العدد الموجد وهذا
+
+153
+00:16:18,310 --> 00:16:23,710
+بيؤدي لتناقض contradiction هذا بيؤدي لتناقض
+
+154
+00:16:23,710 --> 00:16:28,770
+contradiction ليه؟ لأنه أنا عندي من هنا هذا ال
+
+155
+00:16:28,770 --> 00:16:32,770
+statement الأخير هذا ال statement الأخير هذا
+
+156
+00:16:32,770 --> 00:16:36,960
+بتناقض مع ال statement اللي هنالأن هذا ال
+
+157
+00:16:36,960 --> 00:16:41,780
+statement بيقول بما أن N0 هذا عدد طبيعي إذا لازم X
+
+158
+00:16:41,780 --> 00:16:47,580
+تكون أصغر من أو ساوي واحد على N0 إذا X أصغر من أو
+
+159
+00:16:47,580 --> 00:16:51,360
+ساوي واحد على N0 X أكبر من واحد على N0 هذا تناقض
+
+160
+00:16:51,360 --> 00:17:00,340
+إذا التناقض هذا سببه مين ال assumption تبعنا هذاإن
+
+161
+00:17:00,340 --> 00:17:05,960
+X لا تساوي سفر إذا الصح إن X تساوي سفر كما هو
+
+162
+00:17:05,960 --> 00:17:09,660
+مطلوب وبالتالي هيك منكون أثبتنا إن كل X في التقاطع
+
+163
+00:17:09,660 --> 00:17:15,260
+بيساوي سفر يعني كل أنصر في التقاطع موجود في
+
+164
+00:17:15,260 --> 00:17:20,740
+المجموعة هذه إذا هيك منكون أثبتنا المساواة إن
+
+165
+00:17:20,740 --> 00:17:26,320
+المجموعة هذه فعلا أو التقاطع تبع الفترة I N بيساوي
+
+166
+00:17:26,320 --> 00:17:28,200
+Singleton Zero
+
+167
+00:17:33,320 --> 00:17:51,320
+هذا المثال بيأسس او هو حافظ لنظرية رقم 22 بنلاحظ
+
+168
+00:17:51,320 --> 00:17:56,220
+في المثال هذا ان الفترات هذه طبعا شفنا انها nested
+
+169
+00:17:56,220 --> 00:18:03,170
+متداخلة يعني nested معناها متداخلة بالعربيبعدين
+
+170
+00:18:03,170 --> 00:18:09,630
+واضح ان الفترات هذه bounded محصورة و كذلك الفترات
+
+171
+00:18:09,630 --> 00:18:16,830
+هذه closed فترات مغلقة صح فالنظرية هذه بتعمم
+
+172
+00:18:16,830 --> 00:18:21,510
+المعلومات اللي هنا او النتيجة اللي هنا اللي شفناها
+
+173
+00:18:21,510 --> 00:18:26,350
+في المثال فالخاصية هذه بتسميها nested interval
+
+174
+00:18:26,350 --> 00:18:27,010
+property
+
+175
+00:18:30,610 --> 00:18:34,550
+ماهي النظرية اللي بتقول لو في اندي سيكوانس of
+
+176
+00:18:34,550 --> 00:18:39,890
+closed, bounded و nested intervals nested
+
+177
+00:18:39,890 --> 00:18:45,430
+intervals و closed و bounded تلت ايه؟ تلت صفات
+
+178
+00:18:45,430 --> 00:18:50,890
+فلازم التقاطة تبع الفترات هذه يكون غير خالي non
+
+179
+00:18:50,890 --> 00:18:54,990
+-empty يعني نقدر نلاقي عنصر واحد على الأقل صي
+
+180
+00:18:54,990 --> 00:18:56,970
+ينتمي للتقاطة
+
+181
+00:18:59,530 --> 00:19:06,510
+إضافة لذلك moreover لو ال infimum للأعداد غير
+
+182
+00:19:06,510 --> 00:19:13,690
+السالبة هذه هذه أعداد غير سالبة إذا
+
+183
+00:19:13,690 --> 00:19:17,290
+كان ال infimum لل non-negative real numbers هذه لل
+
+184
+00:19:17,290 --> 00:19:20,450
+sequence of non-negative real numbers بساوي سفر
+
+185
+00:19:21,930 --> 00:19:26,170
+فالعدد ساي هذا بيكون وحيد يعني التقاطع هذا مافيش
+
+186
+00:19:26,170 --> 00:19:31,730
+فيه إلا عنصر واحد بالضبط زي ما شوفنا هنا إذا إن
+
+187
+00:19:31,730 --> 00:19:38,330
+الجزء التاني الجزء التاني من النظرية لو فرضنا إن
+
+188
+00:19:38,330 --> 00:19:42,890
+ال infimum للعداد غير الساري بهذا بالساوي سفر فال
+
+189
+00:19:42,890 --> 00:19:46,110
+ساي بتكون هي نقطة الوحيدة الموجودة في التقاطع تبع
+
+190
+00:19:46,110 --> 00:19:49,770
+الفترات زي ما في المثال بالظبط السفر هي النقطة
+
+191
+00:19:49,770 --> 00:19:57,090
+الوحيدةفي تقاطع الفترات المتداخلة نثبت النظرية هذه
+
+192
+00:20:25,320 --> 00:20:30,080
+فلبرهان النظرية أنا عندي الفترات هذه nested
+
+193
+00:20:30,080 --> 00:20:38,560
+وبالتالي I1 هتكون أكبر فترة تحتوي I2 و I3 و تحتوي
+
+194
+00:20:38,560 --> 00:20:44,100
+كل ال IN لكل الأعداد الطبيعي هذا أكيد يعني
+
+195
+00:20:44,100 --> 00:20:49,920
+وبالتالي هاي عندي هاي الفترة I1
+
+196
+00:20:53,730 --> 00:21:05,150
+هذه الفترة I1 تحتوي الفترة I N وهذه الفترة I N
+
+197
+00:21:05,150 --> 00:21:13,070
+وهذه I 1 فواضح
+
+198
+00:21:13,070 --> 00:21:18,870
+أن A N هتكون أصغر من أو ساوي B 1 لكل N في M
+
+199
+00:21:21,540 --> 00:21:27,040
+وبالتالي بي واحد عبارة عن upper bound لكل العناصر
+
+200
+00:21:27,040 --> 00:21:33,280
+AN إذا ال set هذه AN حيث ان عدد الطبيعي is bounded
+
+201
+00:21:33,280 --> 00:21:39,520
+above by بي واحد وبالتالي حسب ال supremum property
+
+202
+00:21:39,520 --> 00:21:44,500
+ال set هذه بما أنها bounded above إذا ال supremum
+
+203
+00:21:44,500 --> 00:21:50,000
+تبعها exist إذا يوجد عدد حقيقي R هو supremum لل
+
+204
+00:21:50,000 --> 00:21:56,950
+setأو الست هذه لها supremum سميه ساي بما أن هذا
+
+205
+00:21:56,950 --> 00:22:01,530
+الساي supremum لكل
+
+206
+00:22:01,530 --> 00:22:07,570
+عناصر الست AN فهو upper bound لذلك الساي أكبر من
+
+207
+00:22:07,570 --> 00:22:14,930
+أو ساوي كل عناصر الست الان
+
+208
+00:22:28,600 --> 00:22:36,720
+طبعا هذه ال .. لأن أنا في عندي a n أثبتنا أن a n
+
+209
+00:22:36,720 --> 00:22:44,080
+أصغر من أو ساوي ساي لكل n في n هذه
+
+210
+00:22:44,080 --> 00:22:51,700
+رقمها خمسة متبينة طبعا الرقم مش غير الان
+
+211
+00:22:51,700 --> 00:22:52,660
+لو أثبتنا
+
+212
+00:22:58,370 --> 00:23:01,830
+احنا عايزين نثبت انه ال .. احنا عايزين في النهاية
+
+213
+00:23:01,830 --> 00:23:08,270
+نثبت ان الصي هذا ينتمي ل I N اللي هو بالساوية
+
+214
+00:23:08,270 --> 00:23:16,850
+الفترة المغلقة من A N ل B N لكل N في N وبالتالي
+
+215
+00:23:16,850 --> 00:23:21,930
+تقاطع الفترات هذه يحتوي الصي صح؟ هذا اللي احنا
+
+216
+00:23:21,930 --> 00:23:27,050
+عايزين نثبته طيب بنهينا أثبتنا ان الصي أكبر من أو
+
+217
+00:23:27,050 --> 00:23:36,150
+ساوي A Nإذا عايزين نثبت إذا عشان نكمل البرهان في
+
+218
+00:23:36,150 --> 00:23:43,390
+الجزء الأول إذا هنا to prove عشان
+
+219
+00:23:43,390 --> 00:23:56,250
+نثبت الكلام هذا اه we need to show انه
+
+220
+00:23:56,250 --> 00:24:09,930
+الأصغر من أو ساوي BN لكل N عدد طبيعي نسمي هذه 6 لو
+
+221
+00:24:09,930 --> 00:24:16,430
+أثبتنا المتباينة 6 لكل N فمعناته الـ Psi أكبر من
+
+222
+00:24:16,430 --> 00:24:20,270
+أو ساوي AN أصغر من أو ساوي BN يعني تنتمي للفترة
+
+223
+00:24:20,270 --> 00:24:27,300
+المغلقة هذه وبالتالي هذا بقدّيان التقاط على كل آي
+
+224
+00:24:27,300 --> 00:24:36,460
+ان بساوي ساي لأ او يحتوي ساي تنتمي للتقاط على كل
+
+225
+00:24:36,460 --> 00:24:44,120
+آي ان وهذا بيكون برهان هذا بيكمل برهان الجزء الأول
+
+226
+00:24:45,450 --> 00:24:50,850
+إذا احنا عايزين نثبت المتباينة ستة هذه طلعت خمسة
+
+227
+00:24:50,850 --> 00:24:55,590
+وهذه طلعت لوحدها باق نثبت المتباينة هذه ستة
+
+228
+00:24:55,590 --> 00:25:06,630
+فالإثبات المتباينة الستة هذه يكفي لإثبات
+
+229
+00:25:06,630 --> 00:25:08,230
+المتباينة ستة هذه
+
+230
+00:25:13,060 --> 00:25:23,440
+يكفي إثبات المتباينة هذه لكل K لو
+
+231
+00:25:23,440 --> 00:25:32,040
+أنا ثبتت ال N و أثبتت أن BN هذه أكبر من أو ساوي كل
+
+232
+00:25:32,040 --> 00:25:37,760
+العناصر A K لكل K عدد طبيعي معناته BN هذه upper
+
+233
+00:25:37,760 --> 00:25:44,110
+bound Upper bound للمجموع هذهطب احنا قبل شوية قلنا
+
+234
+00:25:44,110 --> 00:25:49,930
+ان ال supremum للمجموعة هذه هو psi انا عندي psi
+
+235
+00:25:49,930 --> 00:25:57,450
+بساوي ال supremum ال supremum للمجموعة هذه بساوي
+
+236
+00:25:57,450 --> 00:26:05,330
+psi فلو اثبتت ان b in upper bound للمجموعة هذهوالـ
+
+237
+00:26:05,330 --> 00:26:10,230
+Psi هو أصغر upper bound، معناته الـ Psi هيطلع أصغر
+
+238
+00:26:10,230 --> 00:26:13,930
+من أو يساوي ال upper bound PN، وبالتالي هيكم نكون
+
+239
+00:26:13,930 --> 00:26:20,160
+أثباتنا ستةواضحة النقطة هذه كمان مرة لإثبات أن
+
+240
+00:26:20,160 --> 00:26:26,180
+المتباينة 6 هذه يكفي أن احنا نثبت أنه لكل fixed
+
+241
+00:26:26,180 --> 00:26:30,840
+لكل fixed in ال b in هذه upper bound لل 6 هذه
+
+242
+00:26:30,840 --> 00:26:35,100
+وبالتالي ال least upper bound لل 6 هذه اللي هو
+
+243
+00:26:35,100 --> 00:26:40,900
+صغير بيطلع أصلا أو يساوي b inطبعا اذا باقى نثبت a
+
+244
+00:26:40,900 --> 00:26:47,600
+المتباينة اللى هنا لثبات المتباينة هذه بنثبت n و
+
+245
+00:26:47,600 --> 00:26:53,740
+بناخد اي arbitrary national number k فبنلاحظ ان ال
+
+246
+00:26:53,740 --> 00:26:58,720
+n اما هتكون اصغر من أو يساوي k او n اكبر من k by
+
+247
+00:26:58,720 --> 00:27:04,040
+trichotomy property ان ك عداد طبيعى اذا اما n اصغر
+
+248
+00:27:04,040 --> 00:27:10,520
+من أو يساوي k او n اكبر من kطيب في الحالة الأولى
+
+249
+00:27:10,520 --> 00:27:16,920
+لو كانت ال N أصغر من أو يساوي K احنا عارفين ان
+
+250
+00:27:16,920 --> 00:27:20,840
+الفترات هذه نستد وبالتالي الفترة اللي ال index
+
+251
+00:27:20,840 --> 00:27:26,380
+تبعها أكبر تطلع هي الأصغر تمام؟
+
+252
+00:27:26,380 --> 00:27:34,690
+إذا أنا بطلع عندي I K subset من I Nوبالتالي هاي
+
+253
+00:27:34,690 --> 00:27:45,030
+نرسم هاي IK هاي الفترة IK نقاط
+
+254
+00:27:45,030 --> 00:27:52,770
+أطرافها AK BK وهاي الفترة وهذه
+
+255
+00:27:52,770 --> 00:28:00,330
+محتوى داخل الفترة IN هاي الفترة IN كبّر
+
+256
+00:28:00,330 --> 00:28:14,950
+شويةهذه الفترة IN هذه نقاط أطرافها فبما
+
+257
+00:28:14,950 --> 00:28:20,520
+أن الفترة IK subset من الفترة INإذا اللي بطلع عندى
+
+258
+00:28:20,520 --> 00:28:26,840
+aK أصغر من أو يساوي bK و bK أصغر من أو يساوي bN
+
+259
+00:28:26,840 --> 00:28:34,060
+وبالتالي هيني أثبتت أن aK أصغر من أو يساوي bN كما
+
+260
+00:28:34,060 --> 00:28:40,540
+هو مطلوب طيب في الحالة التانية افرض أن N هي اللي
+
+261
+00:28:40,540 --> 00:28:46,110
+أكبر من Kففي الحالة هذه بما ان ال sequence in
+
+262
+00:28:46,110 --> 00:29:02,270
+nested إذا in محتوى داخل ik إذا
+
+263
+00:29:02,270 --> 00:29:06,150
+هاي عندي in
+
+264
+00:29:11,080 --> 00:29:20,360
+الفترة IN هذا هي محتوى داخل الفترة IK
+
+265
+00:29:20,360 --> 00:29:23,660
+يعني
+
+266
+00:29:23,660 --> 00:29:28,880
+نقاط أطرافها AK BK وبالتالي في الحالة هذه بيطلع AK
+
+267
+00:29:28,880 --> 00:29:36,400
+أصغر من أو ساوي AN أصغر من أو ساوي BNمظبوط؟ إذا
+
+268
+00:29:36,400 --> 00:29:39,780
+هنا في الحالة التانية برضه أثبتنا أن aK أصغر من أو
+
+269
+00:29:39,780 --> 00:29:45,680
+ساوي bN إذا في الحالتين أنا أثبتت أن aK أصغر من أو
+
+270
+00:29:45,680 --> 00:29:57,480
+ساوي bN و هذا صحيح لكل for any K إذا هذا معناه أنه
+
+271
+00:29:57,480 --> 00:30:02,440
+من هنا بي زي ما قلنا bN is upper bound لل set هذه
+
+272
+00:30:03,020 --> 00:30:07,480
+وبالتالي ال supremum تبع ال 6 اللي هو Psi بيطلع
+
+273
+00:30:07,480 --> 00:30:14,440
+أصغر من أو ساوي P M إذن هذا بثبت اللي هو المتباينة
+
+274
+00:30:14,440 --> 00:30:20,740
+6 الأقل من 6 .. من 5 و 6 زي ما شرحنا بنحصل على إن
+
+275
+00:30:20,740 --> 00:30:21,580
+ال Psi
+
+276
+00:30:29,990 --> 00:30:35,610
+من خمسة و ستة بنحصل على ان الـ Psi هذا هي أكبر من
+
+277
+00:30:35,610 --> 00:30:39,050
+أو الساوي An أصغر من أو الساوي Bn يعني الـ Psi
+
+278
+00:30:39,050 --> 00:30:45,330
+تنتمي لتقاطة الفترات كما هو مطلوب هذا بثبت أو
+
+279
+00:30:45,330 --> 00:30:51,210
+ببرهن الجزء الأول من النظرية لبرهان الجزء التاني
+
+280
+00:31:03,930 --> 00:31:10,910
+الجزء التاني بإن
+
+281
+00:31:10,910 --> 00:31:15,210
+نثبت إن التقاطة هذا الـ psi هذه اللي طلعناها هي
+
+282
+00:31:15,210 --> 00:31:17,470
+النقطة الوحيدة في التقاطة مافيش غيرها
+
+283
+00:31:20,460 --> 00:31:25,380
+طبعا هنا الهدف نحصل عليه تحت الشرط الإضافي ان ال
+
+284
+00:31:25,380 --> 00:31:30,000
+minimum للأعداد الغير سالب هذه لاحظوا ان ان ان ان
+
+285
+00:31:30,000 --> 00:31:33,040
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+286
+00:31:33,040 --> 00:31:34,180
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+287
+00:31:34,180 --> 00:31:39,160
+ان ان ان ان
+
+288
+00:31:39,160 --> 00:31:39,160
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+289
+00:31:39,160 --> 00:31:39,180
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+290
+00:31:39,180 --> 00:31:41,160
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+291
+00:31:41,160 --> 00:31:41,360
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+292
+00:31:41,360 --> 00:31:41,360
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+293
+00:31:41,360 --> 00:31:41,480
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+294
+00:31:41,480 --> 00:31:41,480
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+295
+00:31:41,480 --> 00:31:41,480
+ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان ان
+
+296
+00:31:41,480 --> 00:31:41,480
+ان ان ان ان ان ان
+
+297
+00:31:49,520 --> 00:31:54,560
+إذا لو كان الانفة من الأعداد السالبة دي بساوي سفر
+
+298
+00:31:54,560 --> 00:32:00,660
+فبنثبت أنه التقاطع في نقطة واحدة هي ساية طيب
+
+299
+00:32:00,660 --> 00:32:06,260
+لبرهان ذلك بنلاحظ أن ال set هذه غير خالية طبعا لإن
+
+300
+00:32:06,260 --> 00:32:10,200
+أنا عندي ال sequence of intervals اللي لها نقاط
+
+301
+00:32:10,200 --> 00:32:14,840
+أطراف فال sequence هذه فيها على الأقل فترة واحدة
+
+302
+00:32:16,200 --> 00:32:23,600
+هذه المجموعة غير خالية وبعدها bounded below by a1
+
+303
+00:32:23,600 --> 00:32:33,700
+يعني احنا عارفين ان a1 هي a1 او
+
+304
+00:32:33,700 --> 00:32:38,460
+الفترة I1 هذه تحتوي كل الفترات
+
+305
+00:32:42,320 --> 00:32:49,760
+تحتوي كل الفترات I N وبالتالي
+
+306
+00:32:49,760 --> 00:32:59,020
+أنا عندي A1 من هنا بتطلع عندي A1 أصغر من أو يساوي
+
+307
+00:32:59,020 --> 00:33:09,420
+B N لكل N في Nوبالتالي a1 is a lower bound أو ال
+
+308
+00:33:09,420 --> 00:33:15,880
+set of all b in is bounded below by a1 وبالتالي by
+
+309
+00:33:15,880 --> 00:33:23,160
+the infimum property يوجد عدد حقيقي eta وهذا هو ال
+
+310
+00:33:23,160 --> 00:33:31,030
+infimum لset of all b in حيث ان عدد طبيعيإن هنا ال
+
+311
+00:33:31,030 --> 00:33:36,110
+set هذي ال set of all b in حيث in natural number
+
+312
+00:33:36,110 --> 00:33:40,270
+bounded below وبالتالي ال infimum تبعها exist
+
+313
+00:33:40,270 --> 00:33:42,790
+خلينا نسميه إيتا
+
+314
+00:33:45,470 --> 00:33:52,890
+الان by an argument by an argument similar to the
+
+315
+00:33:52,890 --> 00:33:57,190
+proof of sex احنا شوفنا قبل شوية متباينة sex
+
+316
+00:33:57,190 --> 00:34:07,030
+أثبتناها فباستخدام برهان شبيه ببرهان المتباينة sex
+
+317
+00:34:07,030 --> 00:34:13,770
+ممكن نصل إلىمتباينة زي المتباينة six اللي هي هذه
+
+318
+00:34:13,770 --> 00:34:20,330
+ان ا ن تطلع اصغر من او ساوي اتا لكل ان لان هذه
+
+319
+00:34:20,330 --> 00:34:27,250
+برهانها زي برهان ستة خلينا نسميها ستة ستة prime
+
+320
+00:34:27,250 --> 00:34:33,690
+برهان هذه نرجع لبرهان ستة ونشوف كيف برهنها ونعمل
+
+321
+00:34:33,690 --> 00:34:34,570
+برهان مشابه
+
+322
+00:34:37,400 --> 00:34:41,240
+Okay، إذا الـ ETA هنا، العدد ETA ده اللي هو الـ
+
+323
+00:34:41,240 --> 00:34:47,440
+infim للست of all BN هذا إيش بيطلع من هنا، من
+
+324
+00:34:47,440 --> 00:34:51,260
+المتبينة الجديدة six prime؟ بيطلع عبارة عن upper
+
+325
+00:34:51,260 --> 00:34:58,300
+bound لكل العناصر A N، هذه ETA upper bound
+
+326
+00:34:58,300 --> 00:35:02,890
+للمجموعة هذهوبالتالي ال supremum للمجموعة هذه
+
+327
+00:35:02,890 --> 00:35:07,190
+بيطلع أصغر من أو يساوي ال upper bound هذا اللي هو
+
+328
+00:35:07,190 --> 00:35:14,130
+Psi إذا بيطلع عندي هنا Psi بيطلع أصغر من أو يساوي
+
+329
+00:35:14,130 --> 00:35:19,010
+Eta إذا
+
+330
+00:35:19,010 --> 00:35:25,870
+طلع عندي أنا هنا Psi أصغر من أو يساوي Eta
+
+331
+00:35:31,240 --> 00:35:39,040
+الان بنلاحظ انه لو اخدت اي x في الفترة رقم in لكل
+
+332
+00:35:39,040 --> 00:35:45,900
+in لو كان x موجود هنا هذا بكافي ان x اكبر من او
+
+333
+00:35:45,900 --> 00:35:54,080
+يساوي psi اصغر من او يساوي eta لتوضيح
+
+334
+00:35:54,080 --> 00:36:04,120
+ذلك او لبرهان ذلك ال ..لو كانت X موجودة هنا فال X
+
+335
+00:36:04,120 --> 00:36:08,480
+تطلع upper bound لل set هذه هنا خلّيني اوضح بالرسم
+
+336
+00:36:08,480 --> 00:36:13,480
+لو
+
+337
+00:36:13,480 --> 00:36:21,520
+كانت ال X تنتبه ل I N الفترة
+
+338
+00:36:21,520 --> 00:36:32,530
+I N وهي الفترة I N وهي ال Xفلو كانت X موجودة في
+
+339
+00:36:32,530 --> 00:36:42,450
+الفترة I N الفترة هذه فهذا معناه أن ال A N أصغر من
+
+340
+00:36:42,450 --> 00:36:49,890
+أو يساوي X لكل N وبالتالي X upper bound للمجموعة
+
+341
+00:36:49,890 --> 00:36:56,770
+هذه وبالتالي ال supremum للمجموعة هذه اللي هو Psi
+
+342
+00:36:56,770 --> 00:37:05,220
+بيطلع أصغر من لو يساوي X، مظبوط؟كذلك من هنا ال X
+
+343
+00:37:05,220 --> 00:37:13,100
+أصغر من أو يساوي BN لكل N وبالتالي ال X أبارع ال
+
+344
+00:37:13,100 --> 00:37:19,340
+lower bound لمجموعة الأعداد BN وبالتالي ال infimum
+
+345
+00:37:19,340 --> 00:37:26,780
+لمجموعة الأعداد BN ال infimum للمجموعة هذه بطلع
+
+346
+00:37:26,780 --> 00:37:30,620
+أكبر من أو يساوي ال X اللي هو lower bound لها صح؟
+
+347
+00:37:31,090 --> 00:37:35,010
+إذا الانفلان المجموعة هذه اللي هو إيتا أكبر من أو
+
+348
+00:37:35,010 --> 00:37:40,750
+ساوي X اللي هو lower bound للمجموعة هذه okay إذا
+
+349
+00:37:40,750 --> 00:37:44,430
+لو هذا الكلام صح لو كانت X موجودة في الفترة I N
+
+350
+00:37:44,430 --> 00:37:51,140
+لكل Nفهذا بيقدي ان ال X أكبر من أو ساوي ساي و أصغر
+
+351
+00:37:51,140 --> 00:37:55,840
+من أو ساوي إيتا يعني X محصورة بين ساي و إيتا و
+
+352
+00:37:55,840 --> 00:38:00,740
+طبعا العكس لو هذا الكلام صحيح بيقدي ان ال X موجودة
+
+353
+00:38:00,740 --> 00:38:07,280
+هنا تمام؟ إذا هذا برهان ال statement اللي هنا الان
+
+354
+00:38:07,280 --> 00:38:11,100
+انا
+
+355
+00:38:11,100 --> 00:38:11,500
+عندي
+
+356
+00:38:29,400 --> 00:38:33,340
+أنا عندي ال infimum للمجموعة هذه احنا فرضين ان ال
+
+357
+00:38:33,340 --> 00:38:40,520
+infimum لمجموعة الأعداد الغير سالبة هذه ال infimum
+
+358
+00:38:40,520 --> 00:38:45,300
+لها بساوة صفر احنا هذا فرضينه عشان نثبت ان التقاطة
+
+359
+00:38:45,300 --> 00:38:52,240
+في نقطة واحدة صح هذا فرض قائم و احنا أخدنا لمّة
+
+360
+00:38:52,240 --> 00:38:58,490
+واحد اتناشرفي بيجي كان بعديها exercise بيجي بعديها
+
+361
+00:38:58,490 --> 00:39:06,370
+exercise بيقول عشان المجموعة S لو كانت المجموعة في
+
+362
+00:39:06,370 --> 00:39:07,490
+إلها lower bound
+
+363
+00:39:10,250 --> 00:39:16,210
+فعشان ال lower bound هذا يكون هو ال infimum هذا
+
+364
+00:39:16,210 --> 00:39:21,290
+بكافي أنه لكل epsilon أكبر من السفر يوجد عنصر S
+
+365
+00:39:21,290 --> 00:39:25,870
+epsilon في المجموعة S بحيث أن العنصر أصغر من ال
+
+366
+00:39:25,870 --> 00:39:27,150
+infimum زائد epsilon
+
+367
+00:39:30,620 --> 00:39:34,020
+طيب إذا أنا بدي أستخدم ال exercise هذا اللي هو
+
+368
+00:39:34,020 --> 00:39:39,760
+شبيه لمّة واحد اتناشر هي عندي المجموعة هذه وهي
+
+369
+00:39:39,760 --> 00:39:44,620
+عندي ال inform تبعها بساوي سفر السفر هذا اللي هو
+
+370
+00:39:44,620 --> 00:39:51,300
+lower bound المجموعة هذه انتوا لاحظوا انه a b n او
+
+371
+00:39:51,300 --> 00:39:53,860
+a n b n
+
+372
+00:39:57,350 --> 00:40:03,870
+بن ماينوس ان أكبر من أو سوى سفر لكل ان فالسفر هذا
+
+373
+00:40:03,870 --> 00:40:08,710
+lower boundللمجموعة الاعداد هذه و هو مش lower
+
+374
+00:40:08,710 --> 00:40:13,030
+bound بس احنا مُعطى .. مُعطى من الفرض انه هو ال
+
+375
+00:40:13,030 --> 00:40:16,510
+infim هو اكبر lower bound اذا حسب التمرين اللي
+
+376
+00:40:16,510 --> 00:40:20,670
+بيجي بعد لما واحد اتناش بما انه السفر هو ال infim
+
+377
+00:40:20,670 --> 00:40:25,670
+للمجموعة هذه اذا لأي epsilon او لكل epsilon اكبر
+
+378
+00:40:25,670 --> 00:40:30,730
+من السفر يوجد عنصر في المجموعة هذه هذا هو
+
+379
+00:40:33,930 --> 00:40:40,010
+ال index تبعه المؤشر تبعه n هسميه ن إبسلون يعتمد
+
+380
+00:40:40,010 --> 00:40:44,550
+على ال epsilon لكل إبسلون فيه عدد طبيعي ن إبسلون
+
+381
+00:40:44,550 --> 00:40:51,110
+وبالتالي فيه يوجد عنصر في ال set هذه أصغر من ال
+
+382
+00:40:51,110 --> 00:40:56,770
+infimum زائد إبسلون هذا حسب ال exercise الآن هذا
+
+383
+00:40:56,770 --> 00:41:06,750
+العدد هذابن إبسلون أكبر من أو ساوي إتا وان
+
+384
+00:41:06,750 --> 00:41:11,230
+إبسلون أصغر من أو ساوي ساي فالفرق بين هدول أصغر من
+
+385
+00:41:11,230 --> 00:41:19,970
+أو ساوي الفرق بين هدول وانا عندي ساي أصغر من أو
+
+386
+00:41:19,970 --> 00:41:25,990
+ساوي إتا فالفرق بين ساي و إتا أكبر من أو ساوي سفر
+
+387
+00:41:26,920 --> 00:41:33,420
+تمام؟ إذا المتباين الأخيرة هذه بحصل منها على إيه؟
+
+388
+00:41:33,420 --> 00:41:37,560
+المتباين الأخير منها هذه بحصل على إنه صفر أصغر من
+
+389
+00:41:37,560 --> 00:41:45,560
+أوي ساوي eta minus epsilon أصغر من epsilon لأ عفوا
+
+390
+00:41:45,560 --> 00:41:48,280
+هذه eta minus psi
+
+391
+00:41:51,930 --> 00:41:58,010
+وهذا صحيح لكل إبسلون أكبر من السفر أي لأي إبسلون
+
+392
+00:41:58,010 --> 00:42:01,830
+أكبر من السفر وصلنا إلى أنه سفر أصغر من أو ساوي
+
+393
+00:42:01,830 --> 00:42:09,810
+eta minus psi و eta minus psi أصغر من إبسلون طيب
+
+394
+00:42:09,810 --> 00:42:13,390
+احنا خلنا قبل هيك لمبة بتقول لو في عندي عدد حقيقي
+
+395
+00:42:13,390 --> 00:42:18,610
+زي هذا غير سالب و أصغر من إبسلون لكل إبسلون أكبر
+
+396
+00:42:18,610 --> 00:42:23,330
+من السفرفهذا بيقدّي ان إيتا العدد هذا بيساوي سفر
+
+397
+00:42:23,330 --> 00:42:32,090
+وبالتالي إذا بيطلع عندى إيتا بيساوي ساي okay تمام
+
+398
+00:42:32,090 --> 00:42:38,410
+إذا هيك بنكون احنا أثبتنا ان إيتا بيساوي ساي
+
+399
+00:42:38,410 --> 00:42:41,610
+وبالتالي ساي هي النقطة الوحيدة
+
+400
+00:42:45,050 --> 00:42:50,830
+إذا ساي هي النقطة الوحيدة اللي موجودة في التقاطة
+
+401
+00:42:50,830 --> 00:42:59,450
+وهذا يكمل برهان النظرية إذا إحنا خلصنا .. يعني
+
+402
+00:42:59,450 --> 00:43:07,010
+انتهينا من برهان النظرية الطويلة هذهو في بعض
+
+403
+00:43:07,010 --> 00:43:10,410
+الملاحظات على النظرية هنشوفها ان شاء الله المرة
+
+404
+00:43:10,410 --> 00:43:17,450
+القادمة فهنوقف لأن الوقت خلص هننهي المحاضرة الآن و
+
+405
+00:43:17,450 --> 00:43:21,990
+نشوف الملاحظات و الحاجات الخاصة بالنظرية في اللقاء
+
+406
+00:43:21,990 --> 00:43:22,990
+القادم ان شاء الله
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/0MG_WyJdBaA.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/0MG_WyJdBaA.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1e7e460173a95e8a14fd65315c28406564475736
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/0MG_WyJdBaA.srt
@@ -0,0 +1,1599 @@
+1
+00:00:01,430 --> 00:00:05,370
+بسم الله الرحمن الرحيم مرة ثانية إن شاء الله هنبدأ
+
+2
+00:00:05,370 --> 00:00:10,610
+الآن نكمل محاضرة الـ Hypertension/high blood 
+
+3
+00:00:10,610 --> 00:00:17,210
+pressure part 2 الجزء الثاني كنا تحدثنا وحكينا عن
+
+4
+00:00:17,210 --> 00:00:21,470
+الـ statistics وقلنا عن أهم definition of
+
+5
+00:00:21,470 --> 00:00:25,250
+hypertension sustained elevation of the systolic
+
+6
+00:00:25,250 --> 00:00:30,850
+blood pressure أكثر من 139 سيستوليك ودايستوليك أكثر 
+
+7
+00:00:30,850 --> 00:00:35,210
+من 89 وطبعا ما ينفعش أجيء مرة واحدة وأقول الناس أنت 
+
+8
+00:00:35,210 --> 00:00:41,720
+في عندك ضغط وما ينفعش أجيء اتحاد الظروف وأن المريض
+
+9
+00:00:41,720 --> 00:00:47,320
+الإنسان اللي بده يجيس ماخد كافيين وعمل exercise و
+
+10
+00:00:47,320 --> 00:00:51,420
+smoking في آخر ثلاثين دقيقة وأن الـ cuff تبع الـ
+
+11
+00:00:51,420 --> 00:00:57,360
+قياس الضغط لابد أن يكون مناسب وأن أجيء أكثر من مرة
+
+12
+00:00:57,360 --> 00:01:01,580
+وكنا عملنا classification إذا فاكرين للـ blood
+
+13
+00:01:01,580 --> 00:01:04,980
+pressure normal pre hypertension stage one
+
+14
+00:01:04,980 --> 00:01:08,480
+hypertension و stage two hypertension قلنا أيش الـ
+
+15
+00:01:08,480 --> 00:01:13,460
+pre hypertension وأنه ما يحتاجش علاج إلا في بعض
+
+16
+00:01:13,460 --> 00:01:23,760
+الحالات معينة وقلنا لـ Stage 1 و Stage 2 ماذا
+
+17
+00:01:23,760 --> 00:01:29,820
+تعني؟ وقلنا إن الـ hypertension من أهم الأمراض
+
+18
+00:01:29,820 --> 00:01:33,920
+اللي بتمثل risk factors إلى جانب risk factors أخرى
+
+19
+00:01:33,920 --> 00:01:40,790
+عند مرضى القلب والجهاز الوعائي وكنا حددنا كمان
+
+20
+00:01:40,790 --> 00:01:44,390
+وقلنا في نوبات بتصير اسمها hypertensive crisis
+
+21
+00:01:44,390 --> 00:01:47,510
+وفرقنا الـ hypertensive crisis قلنا في منها
+
+22
+00:01:47,510 --> 00:01:52,270
+emergency وفي urgency قلنا الـ hypertensive urgency
+
+23
+00:01:52,270 --> 00:01:56,510
+هو الـ no progressive target organ dysfunction
+
+24
+00:01:56,510 --> 00:01:59,450
+بنقول عنه accelerated hypertension وفي الـ
+
+25
+00:01:59,450 --> 00:02:02,670
+hypertensive emergency بيصير عندنا progressive end
+
+26
+00:02:02,670 --> 00:02:07,920
+organ dysfunction زي headache زي فتق وزي حاجات زي
+
+27
+00:02:07,920 --> 00:02:12,980
+هيك وبنسميه كمان Malignant Hypertension قلنا الـ
+
+28
+00:02:12,980 --> 00:02:18,380
+Hypertensive Emergency غالباً
+
+29
+00:02:18,380 --> 00:02:22,360
+سبب تبعه Under-controlled Hypertension والـ
+
+30
+00:02:22,360 --> 00:02:29,220
+Hypertensive Emergency في الغالب بيبقى نتيجة مشكلة
+
+31
+00:02:29,220 --> 00:02:34,040
+أكبر وقلنا إنه ممكن يسبب لنا مشاكل ثانية زي
+
+32
+00:02:34,040 --> 00:02:38,020
+Encephalopathy، Left Ventricle Failure، Acute MI
+
+33
+00:02:38,020 --> 00:02:43,340
+و Descending Aortic Aneurysm وطلبنا من حضراتكوا أيش
+
+34
+00:02:43,340 --> 00:02:46,600
+معنى Descending Aortic Aneurysm وأيش فيه له
+
+35
+00:02:46,600 --> 00:02:52,460
+تفاصيل وقلنا إن الـ Hypertensive Crisis ممكن يجيء
+
+36
+00:02:52,460 --> 00:02:56,300
+كصورة Hypertensive Encephalopathy، Confusion مثلا،
+
+37
+00:02:56,300 --> 00:02:59,870
+Renal Insufficiency، أورينا الـ Failure، Heart
+
+38
+00:02:59,870 --> 00:03:03,310
+Failure، قلة مونة رئة ديما، Diaphoresis كنا طلبنا
+
+39
+00:03:03,310 --> 00:03:06,410
+يا جماعة حاولوا تعرفونا أيش معنى الـ Diaphoresis
+
+40
+00:03:06,410 --> 00:03:14,260
+ومعظمكم بعت لي التصوّر عن الـ Diaphoresis وتحدثنا في 
+
+41
+00:03:14,260 --> 00:03:16,920
+الآخر في الآخر عن اللي هي الـ Types of
+
+42
+00:03:16,920 --> 00:03:20,120
+Hypertension قلنا فيه Hypertension Primary
+
+43
+00:03:20,120 --> 00:03:22,960
+Hypertension اللي هو الـ Disease الـ Hypertension
+
+44
+00:03:22,960 --> 00:03:27,700
+itself is the disease وده بيمثل 95% من حالات الـ
+
+45
+00:03:27,700 --> 00:03:31,260
+Hypertension وفيه secondary هو less common حوالي 5
+
+46
+00:03:31,260 --> 00:03:36,230
+% وعادة بيبقى فيه سبب ثاني هو اللي أدى
+
+47
+00:03:36,230 --> 00:03:40,110
+للـ hypertension بيقول عنه سكندري هايبرتنشن بيقول لنا
+
+48
+00:03:40,110 --> 00:03:44,270
+فيه causes وكل واحد انشغلته شوية الأسبوع اللي في
+
+49
+00:03:44,270 --> 00:03:49,570
+الأيام اللي فاتته حاولت أفصله في Sleep apnea أيش
+
+50
+00:03:49,570 --> 00:03:53,450
+يعني Sleep apnea أو Obstructive Apnea، drug-induced
+
+51
+00:03:53,450 --> 00:03:55,710
+causes، chronic kidney disease
+
+52
+00:03:58,540 --> 00:04:05,120
+مع Cushing syndrome، Coarctation of the aorta and hyper
+
+53
+00:04:05,120 --> 00:04:09,260
+and hypothyroidism هذه من الأسباب إلى جانب أسباب 
+
+54
+00:04:09,260 --> 00:04:14,520
+أخرى زي الـ hyper-aldosteronism على سبيل المثال
+
+55
+00:04:14,520 --> 00:04:21,710
+ويقول لنا الأسباب الـ secondary على رأسها الـ Renal
+
+56
+00:04:21,710 --> 00:04:24,690
+Parenchymal Disease أيش يعني Parenchymal Disease؟
+
+57
+00:04:24,690 --> 00:04:29,870
+يعني من الـ Kidney itself، مش من الـ Vascularity
+
+58
+00:04:29,870 --> 00:04:33,550
+تبعها، من الـ Renal، من الـ Kidney نفسها و Common
+
+59
+00:04:33,550 --> 00:04:38,130
+Cause يُعتبر بيمثل حوالي 5% من أسباب secondary
+
+60
+00:04:40,580 --> 00:04:45,220
+هايبرتنشن وممكن يكون هو ناتج عن مشكلة يعني واحد 
+
+61
+00:04:45,220 --> 00:04:48,040
+عنده hypertension تعمله مشاكل في الكلية لكن مشاكل
+
+62
+00:04:48,040 --> 00:04:54,400
+الكلية ممكن تعمل برضه hypertension وطبعا الأسباب 
+
+63
+00:04:54,400 --> 00:04:59,360
+الكتيرة في الموضوع هذا وأسباب جديدة الـ Renal
+
+64
+00:04:59,360 --> 00:05:02,540
+Disease إلها إتيولوجيا كتير زي الـ Glomerulonephritis 
+
+65
+00:05:02,540 --> 00:05:06,180
+زي الـ Nephrotic Syndrome زي الـ Pyelonephritis،
+
+66
+00:05:06,180 --> 00:05:10,080
+الالتهابات كلها زي الـ Obstructive Stones كتير
+
+67
+00:05:10,080 --> 00:05:13,740
+حاجات ممكن تعمل لنا Renal Disease وهذه ممكن تعمل لنا
+
+68
+00:05:13,740 --> 00:05:16,940
+بالتالي Hypertension وفيه إن الـ Renovascular
+
+69
+00:05:16,940 --> 00:05:24,330
+Hypertension وهذا يعني رينو فاسكولر رين يعني كلية 
+
+70
+00:05:24,330 --> 00:05:28,790
+فاسكولر أو عيط الكلية الشرايين والـ vein اللي 
+
+71
+00:05:28,790 --> 00:05:33,990
+طالعين ودخلين على الكلية ولما يصير فيه تضيّق سبب
+
+72
+00:05:33,990 --> 00:05:37,470
+من الأسباب، هنحكي أيش أسباب التضيّق، هذا ممكن يعملي
+
+73
+00:05:37,470 --> 00:05:40,790
+hypertension زي أيش التضيّق الـ atherosclerosis،
+
+74
+00:05:40,790 --> 00:05:44,070
+كلكم عارفين الـ atherosclerosis اللي هو تيبس
+
+75
+00:05:44,070 --> 00:05:49,010
+الشرايين في 95% لـ 90% زي ما بنشوفه عند الـ Old
+
+76
+00:05:49,010 --> 00:05:52,470
+Ages، فينا الـ fibromuscular dysplasia وهذا بيصير
+
+77
+00:05:52,470 --> 00:05:55,910
+أكثر في الـ young patient، خاصة female، 10% لـ 25
+
+78
+00:05:55,910 --> 00:06:00,950
+%، جال يعني وممكن طبعا الـ Aortic Dissection أو
+
+79
+00:06:00,950 --> 00:06:04,190
+الـ Renal Dissection أو الـ Thromboembolic Emboli
+
+80
+00:06:04,190 --> 00:06:08,510
+أو الـ Post Transplantation Stenosis أو الـ Post
+
+81
+00:06:08,510 --> 00:06:13,170
+Radiation أو الـ CVD هذه
+
+82
+00:06:13,170 --> 00:06:16,810
+الأمراض كلها ممكن تديني وتعملي Renovascular
+
+83
+00:06:16,810 --> 00:06:19,970
+Hypertension أو تعملي مشكلة في الـ Renovascular
+
+84
+00:06:19,970 --> 00:06:25,150
+وبالتالي Hypertension طب وإحنا ليش خايفين من الـ
+
+85
+00:06:25,150 --> 00:06:29,240
+Hypertension؟ يا سيدي ضغطه عالي ضغطه عالي هو يعني
+
+86
+00:06:29,240 --> 00:06:35,760
+أيَش هيعمل مثلا سؤال كثير مهم ومحق ليش أنا بخاف من
+
+87
+00:06:35,760 --> 00:06:41,140
+الـ hypertension يا جماعة الـ hypertension هو لو لو 
+
+88
+00:06:41,140 --> 00:06:45,340
+لو مرة واحدة ارتفعت ضغط طبعا ما أقصدش عن الـ
+
+89
+00:06:45,340 --> 00:06:48,700
+emergency والـ urgency لكن لمرة مش هيعمل لي مشكلة
+
+90
+00:06:48,700 --> 00:06:55,740
+لكن بيكونوا تتصوروا أنه الضغط هو عبارة عن تأثير
+
+91
+00:06:55,740 --> 00:07:05,640
+مباشر شديد على الشرايين طول الوقت هذا أيش بيقدّم؟
+
+92
+00:07:05,640 --> 00:07:08,940
+بيقدّم إنه يصير عندنا changes في الـ vessels
+
+93
+00:07:08,940 --> 00:07:13,740
+وطبعا،
+
+94
+00:07:13,740 --> 00:07:17,480
+وهذا بيقدّم في الآخر لـ failure للـ organs هذا
+
+95
+00:07:17,480 --> 00:07:21,910
+المشكلة تبقى أتنا كلها على سبيل المثال لو جينا 
+
+96
+00:07:21,910 --> 00:07:27,050
+قلنا في الـ Retina في العين على سبيل المثال ممكن 
+
+97
+00:07:27,050 --> 00:07:30,030
+تعمل لي مشكلة وهذا بإمكاننا نشوفه في حاجة اسمها الـ
+
+98
+00:07:30,030 --> 00:07:34,070
+Funduscope الـ blood vessels ممكن تتأثر اللي في 
+
+99
+00:07:34,070 --> 00:07:39,110
+العين فأنا ممكن من خلال فحص Funduscope للعين، لقاع
+
+100
+00:07:39,110 --> 00:07:42,530
+العين أعرف إن فلان عنده hypertension ليش؟ لأن 
+
+101
+00:07:42,530 --> 00:07:46,610
+الشرايين اللي في قاع العين زي ما إحنا هنشوفها 
+
+102
+00:07:46,610 --> 00:07:48,710
+الـ gate أتصور إنه هي الصورة
+
+103
+00:07:52,140 --> 00:07:59,840
+الرتينة أو الشبكية والشرايين اللي فيها تظهر تغيرات
+
+104
+00:07:59,840 --> 00:08:03,400
+ناشئة أو ناجمة عن الـ hypertension فبتدلني أنه في
+
+105
+00:08:03,400 --> 00:08:06,400
+hypertension تغيرات هذه إنه ممكن أنا ألاقي 
+
+106
+00:08:06,400 --> 00:08:09,620
+hemorrhages على سبيل المثال النقطة البيضة هي اللي 
+
+107
+00:08:09,620 --> 00:08:14,580
+موجودة على الـ Retina ممكن ألاقي Exudate وهذه 
+
+108
+00:08:14,580 --> 00:08:17,980
+عبارة عن Fatted Deposits وممكن ألاقي حاجة اسمها
+
+109
+00:08:17,980 --> 00:08:23,300
+Cotton Wool يعني زي القطن زي كوام القطن Spots هذه 
+
+110
+00:08:23,300 --> 00:08:28,000
+نتيجة Micro Strokes بتصير فين؟ الصورة C عندنا
+
+111
+00:08:28,000 --> 00:08:30,100
+هنا بتصير في الـ Retina
+
+112
+00:08:33,060 --> 00:08:39,100
+الشرايين في كل مكان ممكن تتأثر بارتفاع ضغط الدم من 
+
+113
+00:08:39,100 --> 00:08:42,860
+الأمثلة الأخرى غير جذية الـ I أو الـ Retinopathy
+
+114
+00:08:42,860 --> 00:08:48,240
+اللي حكينا عنها اللي هو nervous system الـ nervous
+
+115
+00:08:48,240 --> 00:08:52,200
+system أنا مش هتوسع الآن لأن المفروض أنه أنتم
+
+116
+00:08:52,200 --> 00:08:55,320
+أخذتوه في موضوع الـ stroke أو هتاخذوه في موضوع الـ
+
+117
+00:08:55,320 --> 00:08:58,600
+stroke اللي هو الـ hemorrhagic والـ ischemic stroke
+
+118
+00:08:58,600 --> 00:09:04,040
+فممكن نتيجة ارتفاع ضغط الدم يصير عندنا stroke و
+
+119
+00:09:04,040 --> 00:09:07,060
+يصير عندنا بعد هيك cerebral atrophy and dementia
+
+120
+00:09:09,370 --> 00:09:12,530
+وذكرنا قبل شوية إن الـ brain stroke في منه 
+
+121
+00:09:12,530 --> 00:09:15,770
+hemorrhagic يعني بيصير عندنا انفجار لشريان من 
+
+122
+00:09:15,770 --> 00:09:20,870
+الشرايين أو ischemic بيصير عندنا thrombus أو 
+
+123
+00:09:20,870 --> 00:09:23,670
+embolus وهذا بيسدّ للشريان اللي رايح على المنطقة
+
+124
+00:09:23,670 --> 00:09:28,650
+اللي فلانية في الدماغ فبيعمل لي أيش؟ بيعمل لي 
+
+125
+00:09:28,650 --> 00:09:34,370
+ischemic stroke طبعا من الحاجات اللي بتنعمل كمان 
+
+126
+00:09:34,370 --> 00:09:39,180
+إن الـ effects on the kidneys بمعنى، زي ما إحنا 
+
+127
+00:09:39,180 --> 00:09:41,520
+شايفين أو الـ «effects on the cardiovascular
+
+128
+00:09:41,520 --> 00:09:45,840
+system» بمعنى إن الـ hypertension بيأثر لي على 
+
+129
+00:09:45,840 --> 00:09:49,160
+القلب، بيأثر لي على الـ kidney، بيأثر لي على الـ 
+
+130
+00:09:49,160 --> 00:09:53,340
+nervous system، بيأثر لي على الـ eyes، بيأثر لي على
+
+131
+00:09:53,340 --> 00:09:55,980
+حاجات كثيرة، إحنا بعض الأحيان كذلك، مش كلها كذلك،
+
+132
+00:09:55,980 --> 00:10:02,140
+بس إحنا كذا كلنا، أهم أربع أعضاء بتتأثر بارتفاع
+
+133
+00:10:02,140 --> 00:10:08,180
+ضغط الدم، heart، brain، kidney، الـ Arteries and 
+
+134
+00:10:08,180 --> 00:10:13,540
+Retinopathy أو الـ Retina أو الشبكية كلها بتتأثر
+
+135
+00:10:13,540 --> 00:10:19,020
+هذه هي الـ Target Organs وطبعا أي حد بتبع دكتور 
+
+136
+00:10:19,020 --> 00:10:22,380
+الـ Hypertension لابد أن يعمل Consultations 
+
+137
+00:10:22,380 --> 00:10:26,480
+للمرضى هؤلاء بشأن الحاجات هذه والانتباه اللي لها
+
+138
+00:10:26,480 --> 00:10:29,480
+طيب إيه بيعمل في الـ cardiovascular؟ بتديني
+
+139
+00:10:29,480 --> 00:10:33,160
+Ventricular Hypertrophy؟ مع dysfunction؟ مع 
+
+140
+00:10:33,160 --> 00:10:39,280
+failure؟ كيف يعني؟ تصوّروا أن القلب بضخ؟ وبضغط ضد
+
+141
+00:10:39,280 --> 00:10:46,480
+ضغط عالي فبضطر القلب يعمل عشان يتغلب على الضغط هذا
+
+142
+00:10:46,480 --> 00:10:50,180
+يعمل dilation في الأول وبعدين hypertrophy فبيصير
+
+143
+00:10:50,180 --> 00:10:54,520
+تضخم وبعد التضخم طبعاً بيصير failure بيوقف يعني
+
+144
+00:10:54,520 --> 00:10:58,620
+ممكن يمشي شهر شهرين لكن بعدها بيقدرش القلب يكمل
+
+145
+00:10:58,980 --> 00:11:01,780
+طبعاً نتيجته التضخم بيصير عندنا حاجة اسمها
+
+146
+00:11:01,780 --> 00:11:04,540
+Arrhythmias إيش يعني Arrhythmias؟ Arrhythmias
+
+147
+00:11:04,540 --> 00:11:09,280
+معناته تغير
+
+148
+00:11:09,280 --> 00:11:15,070
+في شكل النبضات وعدد النبضات القلب ممكن يصير
+
+149
+00:11:15,070 --> 00:11:18,610
+عندي MI هنحكي عنها الـ Ischemic heart disease بعدين
+
+150
+00:11:18,610 --> 00:11:22,950
+هيك وطبعاً ممكن يعمل لي arterial aneurysm مع
+
+151
+00:11:22,950 --> 00:11:28,010
+dissection مع rupture وحكينا عن الكلى وحكينا عن الـ
+
+152
+00:11:28,010 --> 00:11:32,670
+nervous system وحكينا عن الـ eye واللي ممكن يعملها
+
+153
+00:11:32,670 --> 00:11:36,930
+مش بس احنا كنا حكينا تو إنه بيصير عندنا زي النزيف
+
+154
+00:11:36,930 --> 00:11:41,210
+في الـ retina بعض الأحيان وبيعمل لي impairment للـ
+
+155
+00:11:41,210 --> 00:11:45,590
+vision لكن كمان الـ vitreous العين فيها مية
+
+156
+00:11:45,590 --> 00:11:48,730
+من جوا، هذا بيقول أنا vitreous body بيعمل لي
+
+157
+00:11:48,730 --> 00:11:52,270
+vitreous hemorrhage وممكن يعمل لي انفصال شبكي
+
+158
+00:11:52,270 --> 00:11:56,870
+retinal detachment الموضوع مش بيظل لحدها ممكن
+
+159
+00:11:56,870 --> 00:12:00,690
+يصير عندي neuropathy of the nerves leading to
+
+160
+00:12:00,690 --> 00:12:04,710
+extraocular muscle paralysis and dysfunction انتشرت
+
+161
+00:12:04,710 --> 00:12:05,330
+الموضوع
+
+162
+00:12:08,020 --> 00:12:13,080
+طب احنا هو أنا يعني الآن كويس ومنيح لكن أنا إيش
+
+163
+00:12:13,080 --> 00:12:18,360
+بدي من المريض
+
+164
+00:12:18,360 --> 00:12:25,740
+أجاني عنده ضغط إيش اللي بديه منه أول حاجة بدي هذا
+
+165
+00:12:25,740 --> 00:12:29,220
+اللي بنقول عنه الـ objective of the evaluation إيش
+
+166
+00:12:29,220 --> 00:12:33,490
+بدي منه؟ to assess lifestyle and identify other
+
+167
+00:12:33,490 --> 00:12:36,210
+cardiovascular risk factors أدور إذا في أي مرض 
+
+168
+00:12:36,210 --> 00:12:41,910
+ثاني عنده وأشوف الـ lifestyle تبعه that may
+
+169
+00:12:41,910 --> 00:12:45,150
+affect prognosis and guide treatment واللي ممكن
+
+170
+00:12:45,150 --> 00:12:49,950
+تأثر على الـ prognosis والعلاج الحاجة الثانية،
+
+171
+00:12:49,950 --> 00:12:54,850
+بدي أحاول أعرف والله إذا كان السبب secondary أو
+
+172
+00:12:54,850 --> 00:12:57,210
+إذا كان الـ hypertension secondary، بدي أعرف السبب
+
+173
+00:12:57,210 --> 00:13:05,370
+مهم الأمر الثالث بدي أشوف إذا كان الأعضاء تأثرت
+
+174
+00:13:05,370 --> 00:13:08,350
+يعني أنا ما يكفي إن أنا فلان عرفت إنه في عنده ضغط
+
+175
+00:13:08,350 --> 00:13:12,070
+بدي أشوف هل عينه فيها مشكلة بدي أبعته على دكتور
+
+176
+00:13:12,070 --> 00:13:16,310
+العيون يعمل له Funduscopy بدي أبعته على دكتور
+
+177
+00:13:16,310 --> 00:13:21,110
+القلب يعمل له تخطيط قلب ويشوف إذا فيه Arrhythmias
+
+178
+00:13:21,110 --> 00:13:25,030
+يعمل له echocardiography ويشوف إذا في عنده أي
+
+179
+00:13:25,030 --> 00:13:30,040
+hypertrophy أود أن أرسله إلى دكتور الكلى لأرى
+
+180
+00:13:30,040 --> 00:13:34,420
+إذا كان هناك مشاكل في الكلى وهل هو مجرم؟ هذه
+
+181
+00:13:34,420 --> 00:13:41,600
+أساسياته لماذا أريد أن أصل مع المريض هذا؟ لازم
+
+182
+00:13:41,600 --> 00:13:45,200
+أبحث عن سبب إذا كان هناك سبب أرى هل أثرت على أي
+
+183
+00:13:45,200 --> 00:13:48,780
+أعضاء عنده وأرى إذا كان عنده مشاكل أخرى، فهي تصعب
+
+184
+00:13:48,780 --> 00:13:54,080
+الأمر الـ Goals of Treatment طبعاً أنا أريد أن
+
+185
+00:13:54,080 --> 00:14:01,460
+أنزل ضغطه في الآخر ننزل ضغطه السيستوليك والدياستوليك
+
+186
+00:14:01,460 --> 00:14:08,200
+لأجل من 140 لـ 90 يعني من 139 لـ 89 هذه الأساس تبع
+
+187
+00:14:08,200 --> 00:14:11,880
+الموضوع لكن في ناس ثانية بنبدأ ننزلها لأجل من هيك
+
+188
+00:14:17,330 --> 00:14:20,870
+زي على سبيل المثال، الناس اللي عندها Diabetes سكري
+
+189
+00:14:20,870 --> 00:14:24,730
+أو عندها Renal Disease الـ Blood Pressure Goal أقل
+
+190
+00:14:24,730 --> 00:14:31,470
+من 130 لـ 80، هننزلها وطبعاً
+
+191
+00:14:31,470 --> 00:14:36,810
+أحاول قدر الإمكان أنظم أمور المريض مع مشاكل
+
+192
+00:14:36,810 --> 00:14:42,490
+الـ Killer ومع مشاكل القلب طيب، ونزلنا الضغط، يعني
+
+193
+00:14:42,490 --> 00:14:48,300
+وإحنا هل بتنزلنا الضغط ممكن إنه يتحسن وضع المريض
+
+194
+00:14:48,300 --> 00:14:54,700
+ممكن؟ إيش رأيكم أنتم؟ آه، ما هو هذا الهدف في
+
+195
+00:14:54,700 --> 00:14:58,480
+الموضوع وبتحسنوا، لاجوا من خلال الإحصائيات والـ
+
+196
+00:14:58,480 --> 00:15:03,780
+Statistics Studies يعني، لاجوا إنه إذا أنا أديت
+
+197
+00:15:03,780 --> 00:15:08,620
+علاج وتحسن المريض Reduction in stroke incidence
+
+198
+00:15:08,620 --> 00:15:15,060
+تقريباً من 35% لـ 40%، هذا رقم مهول، رقم ممتاز مش رقم
+
+199
+00:15:15,060 --> 00:15:20,040
+بسيط، أنا تقريباً فوق الـ Tilt نزلنا الـ stroke
+
+200
+00:15:20,040 --> 00:15:24,300
+incidence الإمكانية إن يحصل فيه stroke incidence
+
+201
+00:15:24,300 --> 00:15:29,340
+مش بس الـ MI الـ Myocardial Infection الجلطة
+
+202
+00:15:29,340 --> 00:15:35,740
+القلبية تنزل إمكانية إنها تحدث عند المريض طبع الضغط
+
+203
+00:15:35,740 --> 00:15:40,540
+لو عالجته بطريقة سليمة من 20% لـ 25% رقم رهيب
+
+204
+00:15:40,540 --> 00:15:43,480
+وطبعاً هذه الأرقام تتحسن مع الوقت يا جماعة يعني
+
+205
+00:15:43,480 --> 00:15:49,180
+بمعنى كل ما تطور الطب وكل ما المريض أو المريض أو
+
+206
+00:15:49,180 --> 00:15:54,550
+الإنسان بشكل عام عاوز معايا أكتر كل ما تحسنت
+
+207
+00:15:54,550 --> 00:15:59,210
+الأمور أكتر مش بس حتى الـ heart failure heart
+
+208
+00:15:59,210 --> 00:16:05,590
+failure فشل القلب بيقل ليش؟ لأنه مش مضطر يضخ ضد ضغط
+
+209
+00:16:05,590 --> 00:16:09,910
+عالي أكتر من 50% هذه الـ benefits of treatment يعني
+
+210
+00:16:09,910 --> 00:16:14,550
+لأ احنا مش بس يعني مش بس يعني وهذا طبعاً مش لازم
+
+211
+00:16:14,550 --> 00:16:17,110
+يكون بس بالدواء يا جماعة لأنه أنا في عندي عدة طرق
+
+212
+00:16:17,110 --> 00:16:21,880
+to manage a hypertension هنحكي عنها بعدين بس
+
+213
+00:16:21,880 --> 00:16:27,140
+يعني جزماً إذا أدينا علاج بيتحسن المريض بنسب معينة
+
+214
+00:16:27,140 --> 00:16:31,600
+ومش بس بخفف من إمكانية الـ Complications الـ
+
+215
+00:16:31,600 --> 00:16:35,480
+Complications اللي هي Stroke Myocardial Infection
+
+216
+00:16:35,480 --> 00:16:38,660
+Heart Failure ويمكن لو نطلع على باقي الأعضاء
+
+217
+00:16:38,660 --> 00:16:45,600
+هنلاقيها أنها بتتحسن كمان بنسب أو بأخرى طيب يمكن
+
+218
+00:16:45,600 --> 00:16:50,240
+كتير هتسمعوا مني خلال الفصل موضوع الـ lifestyle
+
+219
+00:16:50,240 --> 00:16:58,620
+modification لسبب إنه هو جزء في العلاج الطبيعي من
+
+220
+00:17:00,620 --> 00:17:05,840
+الـ Medical System بمعنى، ما أقدرش واحد يجي يقول
+
+221
+00:17:05,840 --> 00:17:10,600
+أنا بس علاج طبيعي، أنا باجي وبعمله وبدلكه و
+
+222
+00:17:10,600 --> 00:17:14,500
+بمشيه وبظبط له العضلات وبس، لأ هو جزء من الـ
+
+223
+00:17:14,500 --> 00:17:17,700
+Education للناس، بيكون فيها أخصائي العلاج الطبيعي،
+
+224
+00:17:17,700 --> 00:17:20,000
+بيكون فيها الممرض، بيكون فيها الدكتور، يعني زي ما
+
+225
+00:17:20,000 --> 00:17:23,560
+يقولوها Teamwork، وفي الآخر المريض يا جماعة بقعد
+
+226
+00:17:23,560 --> 00:17:28,520
+عنده فترة أكثر من أكثر المهن اللي بقعد المريض فيها
+
+227
+00:17:28,520 --> 00:17:34,420
+مع المعالج العلاج الطبيعي أكثر المهن الطبية أقصد
+
+228
+00:17:34,420 --> 00:17:40,800
+صحياً يعني الطبيب خمس دقائق عشر دقائق الممرض يعني
+
+229
+00:17:40,800 --> 00:17:43,540
+زي ما بيقولوها زي النحلة بروح بعمل الشغل وبطلع
+
+230
+00:17:43,540 --> 00:17:46,120
+ما بقعدش عند المريض ولا المريض موجود عنده لكن العلاج
+
+231
+00:17:46,120 --> 00:17:51,810
+الطبيعي لأ فيه مما فيه فيه فيه فترة فالفترة هذه بتكون
+
+232
+00:17:51,810 --> 00:17:54,770
+غنية في الغالب بيظلوا هم مش ساكتين هما الاثنين تبع
+
+233
+00:17:54,770 --> 00:17:58,390
+العلاج الطبيعي والمريض بيصير المريض يبدأ يسأل
+
+234
+00:17:58,390 --> 00:18:01,530
+ما ينفعش أخصائي العلاج الطبيعي يقول والله لأ مش
+
+235
+00:18:01,530 --> 00:18:05,610
+عارف ما مرش عليا ما ينفعش وأنا مطلوب منه يجاوب صح
+
+236
+00:18:05,610 --> 00:18:11,730
+عشان ما تفهمونيش غلط بس مطلوب إنه يجاوبه صح ويدله
+
+237
+00:18:11,730 --> 00:18:16,510
+صح فرصة إنه أنا أستاذ الجهاز الهضمي أحاول أفهم الناس و
+
+238
+00:18:16,510 --> 00:18:22,450
+أشرح لهم شرح صحيح طبعاً زي ما قلنا بنضطر نشرح، فمهم
+
+239
+00:18:22,450 --> 00:18:26,250
+تعرفوا الحاجات هذه أنتم، إنه لازم أشرح للمريض عن
+
+240
+00:18:26,250 --> 00:18:29,790
+الـ Life Style Modification، هذا في كل الأمراض
+
+241
+00:18:29,790 --> 00:18:34,650
+غالباً حالياً، على رأسها الـ Weight Reduction، طبعاً
+
+242
+00:18:34,650 --> 00:18:37,590
+الناس بتكره الكلمة هذه، بدك تنزل وزنك، ما بتحبوش
+
+243
+00:18:37,590 --> 00:18:38,210
+ينزلوا وزنهم
+
+244
+00:18:41,190 --> 00:18:47,090
+أو العمر كبير، 60 أو 65 سنة ولم تكن قادرة على
+
+245
+00:18:47,090 --> 00:18:50,410
+المشي ولم تكن قادرة على الذهاب للجيم وليس هناك
+
+246
+00:18:50,410 --> 00:18:55,470
+جيم، يعني أمور كثيرة تمنع الموضوع هذا، لكن يا
+
+247
+00:18:55,470 --> 00:18:59,900
+جماعة إذا وصلت لـ Normal Body Weight Body Mass
+
+248
+00:18:59,900 --> 00:19:07,640
+Index من 18.5 إلى 24.9 كيلو جرام لكل متر مربع ممكن
+
+249
+00:19:07,640 --> 00:19:13,340
+ينزل الضغط من 5 إلى 20 ملم زئبق
+
+250
+00:19:13,340 --> 00:19:22,420
+لكل 10 كيلو جرام هذا
+
+251
+00:19:22,420 --> 00:19:29,610
+رقم كبير يعني أنا لو كان الضغط 145 على 90 ونزلنا
+
+252
+00:19:29,610 --> 00:19:34,130
+لحد زي ما بنقول معناته احنا في نعمة كبيرة حينزل
+
+253
+00:19:34,130 --> 00:19:38,450
+حينزل الضغط على الطبيعي عملياً بس بالـ body weight
+
+254
+00:19:38,450 --> 00:19:43,610
+reduction توي حكينا عن واحدة اسمها body mass index
+
+255
+00:19:43,610 --> 00:19:48,870
+هذه طبعاً هي الـ homework الوحيد اللي هتعمله what is
+
+256
+00:19:48,870 --> 00:19:59,500
+body mass index إيش بيمثل كيف من حسب والدرجات تبعته
+
+257
+00:19:59,500 --> 00:20:08,000
+درجات تبعته فيه درجات وهذا بدي إياه بس إيش بدي copy
+
+258
+00:20:08,000 --> 00:20:13,940
+paste بدي إياه مكتوب بخط اليد ومصور وتبعثوا لي إياه لما
+
+259
+00:20:13,940 --> 00:20:18,500
+أبعث لكم الفيديو هحدد لكم على الـ model لو أكتشف
+
+260
+00:20:18,500 --> 00:20:25,500
+الإجابة يبقى بخط اليد جدول body mass index الشرح
+
+261
+00:20:25,500 --> 00:20:32,540
+ماذا يعني كيف من حسب والدرجات تبعته okay الحاجة
+
+262
+00:20:32,540 --> 00:20:35,460
+الثانية اللي أنا عشان من الـ lifestyle modification
+
+263
+00:20:35,460 --> 00:20:41,740
+بعملها هي حاجة اسمها dash eating plan هنحكي عنها
+
+264
+00:20:41,740 --> 00:20:47,090
+بعد شوية وهذه لو أنا مشيت فيها بالمناسبة هي الـ
+
+265
+00:20:47,090 --> 00:20:52,110
+«Dash High» تتناسب جداً جداً مع حياة الإنسان الصحي
+
+266
+00:20:52,110 --> 00:20:58,190
+هو اللي بنعمله الـ «Accidentally Lifestyle» اللي
+
+267
+00:20:58,190 --> 00:21:02,290
+بنشتغله اللابتوب والجوال وطول النهار عن الـ «نت»
+
+268
+00:21:02,290 --> 00:21:08,490
+والكلونة جانان كبيرة ومجات قهوة وما أعرفش الشاورما
+
+269
+00:21:08,490 --> 00:21:13,190
+هذه كلها أصلاً مش الوضع الطبيعي هذه مش هي الوضع
+
+270
+00:21:13,190 --> 00:21:17,850
+الطبيعي الوضع الطبيعي إنه أنا بأكل Fruit كفاية،
+
+271
+00:21:17,850 --> 00:21:22,270
+Vegetable كفاية اللي هي Low-fat في الغالب، بأخذ
+
+272
+00:21:22,270 --> 00:21:28,070
+منتجات لبنية Low-fat كفاية، الأكل تبعي بيبقى مش
+
+273
+00:21:28,070 --> 00:21:37,990
+Full of saturated fat، هذا الأصل وبيتزا واندومي
+
+274
+00:21:37,990 --> 00:21:40,790
+وما أعرفش إيش، لأ مش هو هذا الأصل، الأصل إنه أنا
+
+275
+00:21:40,790 --> 00:21:43,890
+بأكل طبيعي زي بني آدمين عادي، fruits
+
+276
+00:21:43,890 --> 00:21:48,450
+وvegetables، قطعة لحمة وفواكه وخضار وهي الأكل
+
+277
+00:21:48,450 --> 00:21:52,050
+تبعنا، هذا الأصل، هذا الموضوع لو أنا مشيت عليه
+
+278
+00:21:52,050 --> 00:21:57,610
+اللي هو الـ «Dash Eating Plan» بنزل الضغط من 8 إلى
+
+279
+00:21:57,610 --> 00:21:59,590
+14 systolic
+
+280
+00:22:01,730 --> 00:22:09,310
+طبعاً كثير بنسمع عن خفف الملح في أكلك Reduce 
+
+281
+00:22:09,310 --> 00:22:16,310
+Dietary Sodium Intake To No More Than حوالي إحنا 
+
+282
+00:22:16,310 --> 00:22:23,930
+اللي بنقوله 2.5 جرام في النهار وهذا بننزّل 2 إلى 8
+
+283
+00:22:23,930 --> 00:22:30,850
+في الضغط السيستولي، الحاجة الأخيرة اللي بنقول عنها
+
+284
+00:22:30,850 --> 00:22:33,270
+لموضوع الـ Lifestyle Modification اللي هي الـ
+
+285
+00:22:33,270 --> 00:22:38,810
+Physical Activity بنلاقي إنه كل ما مشينا بِرِسْك 
+
+286
+00:22:38,810 --> 00:22:44,370
+ووكينج يعني بهمة لتلاتين دقيقة Most days of the
+
+287
+00:22:44,370 --> 00:22:49,520
+week هذه ممكن تنزّل من أربعة لتسعة في الضغط، طبعاً
+
+288
+00:22:49,520 --> 00:22:52,120
+لما أنا بعمل كل الـ activities هايم أنا ضغطي هينزل
+
+289
+00:22:52,120 --> 00:22:55,880
+بصورة واضحة، وأنا لسه في Lifestyle Modification أنا
+
+290
+00:22:55,880 --> 00:23:00,780
+ما دخلتش لسه على Drugs في 
+
+291
+00:23:00,780 --> 00:23:03,620
+ناس ميّشي بيقولك طب وأنت قاعد بتقولي أجلّل أكلّك كيف
+
+292
+00:23:03,620 --> 00:23:06,280
+بدأ أجلّل الأكل في هذا الوضع يا جماعة بدنا نحاول 
+
+293
+00:23:06,280 --> 00:23:09,940
+قدر الإمكان نعمل Dietary changes نقول عنها كيف
+
+294
+00:23:09,940 --> 00:23:15,460
+يعني؟ يعني أجلّل الـ portion size تبع الأكل مش بس طب 
+
+295
+00:23:15,460 --> 00:23:18,900
+كيف بدأ أجلّله؟ دايماً أنا بنصحه لما تيجي تأكل يا
+
+296
+00:23:18,900 --> 00:23:24,180
+جماعة أكل في صحن صغير، صحن صغير حطّه في صحن صغير كلّ
+
+297
+00:23:24,180 --> 00:23:27,340
+ما كبرت الصحن عين... تعرفوا عين البني آدم فارغة
+
+298
+00:23:27,340 --> 00:23:31,160
+بدّه يعبيّه أكتر لأ، حطّ في صحن صغير جد ما تحطّ فيه 
+
+299
+00:23:31,160 --> 00:23:35,600
+بيبيّن كتير، يعني اضحك على حالك هيك بمعنى آخر الأمر
+
+300
+00:23:35,600 --> 00:23:38,620
+التاني لما تستعمله يا بتستعمل معلقة صغيرة، المعلقة 
+
+301
+00:23:38,620 --> 00:23:44,020
+بتاعة الشاي في الأكل عشان تطوّره وأنت بتأكله إمّا 
+
+302
+00:23:44,020 --> 00:23:48,140
+بتستخدم الشوكة، الشوكة بيضلّش عليها رزّ وحاجات تأكل
+
+303
+00:23:48,140 --> 00:23:51,600
+كتير، خاصة لما يكون مقارنة ورزّ وكذا حاول الشوكة
+
+304
+00:23:51,600 --> 00:23:57,980
+الشوكة بتأخدش كتير فبتطول في الأكل هذه من الحاجات 
+
+305
+00:23:57,980 --> 00:24:03,380
+اللي إحنا ممكن نستخدمها عشان أجلّل مثلاً الخبز على 
+
+306
+00:24:03,380 --> 00:24:07,260
+سبيل المثال، دايماً ما تأكلش الطازة على الآخر الطريق 
+
+307
+00:24:07,260 --> 00:24:10,780
+لأن هذا سهل المضغ وسهل البلع وسهل كل حاجة، خذ
+
+308
+00:24:10,780 --> 00:24:16,230
+دايماً الناشف أو المحمّرس عشان أقل قيمة غذائية من
+
+309
+00:24:16,230 --> 00:24:19,210
+ناحية الـ Calories لأ لأنه بطوّل وأنا بمضغ فيه و
+
+310
+00:24:19,210 --> 00:24:22,750
+أنا بأكل فيه فهذه بتخلّي الأمر الثالث والمهم
+
+311
+00:24:22,750 --> 00:24:27,110
+حاولوا قدر الإمكان تبعدوا عن السفرة عن جوع السفرة
+
+312
+00:24:27,110 --> 00:24:32,610
+هذه لللمّة اللي فيها كل ما لذّ وطاب، حاولوا تبعدوا
+
+313
+00:24:32,610 --> 00:24:40,640
+عنها، هذه بتضمنلك أنّ أنت تأخذ كميات بسيطة من الأكل، 
+
+314
+00:24:40,640 --> 00:24:43,260
+ساندوشات أنت عارف إيش اللي بدّك إيّاه أو بتأخذ على
+
+315
+00:24:43,260 --> 00:24:46,460
+الطريقة الأمريكية كله لنفسه صحن صغير على جدّك 
+
+316
+00:24:46,460 --> 00:24:50,140
+وبتقعد فيه وبتاكل وخلصنا، أما السفرة هات إيدك وهات
+
+317
+00:24:50,140 --> 00:24:53,660
+رجلك وعبيّ وعبيّ وعبيّ وعبيّ، هذه مشكلة كبيرة، صحيح؟
+
+318
+00:24:53,660 --> 00:24:57,140
+وبتخلّي الواحد يظلّ يأكل بلا حدود، بلا حدود، خاصة لو
+
+319
+00:24:57,140 --> 00:25:01,160
+في تسلية، جباله، في منافسة وهذا بيأكل وهذا بتأكل
+
+320
+00:25:01,160 --> 00:25:06,510
+وهذا بيأكل وهذا بتأكل هذه مشكلة، طيّب الأمر الثاني
+
+321
+00:25:06,510 --> 00:25:12,170
+نحاول نجلّل... نجلّل، نحن لا نحرم حالنا يا جماعة
+
+322
+00:25:12,170 --> 00:25:15,750
+أنا مش طالب من حدّ يحرم حاله، لكن Reduce Portion
+
+323
+00:25:15,750 --> 00:25:19,950
+Size or Frequency of Consumption of Calorie 
+
+324
+00:25:19,950 --> 00:25:24,550
+Containing Beverages مش ممنوع ولا حرام ولا عيب 
+
+325
+00:25:24,550 --> 00:25:29,170
+أشرب كوكا كولا بس ممنوع أشرب جنيتين وتلاتة أو جنية
+
+326
+00:25:29,170 --> 00:25:33,920
+كبيرة أو علبتين وتلاتة أو أربعة في النهار أو جنية
+
+327
+00:25:33,920 --> 00:25:37,620
+لتر أو جنيتيه لتر في النهار، لكن مسموح لي أشرب شوية 
+
+328
+00:25:37,620 --> 00:25:42,480
+كوكا كولا مش قضية كبيرة، الأمر الثاني اللي هو Decrease
+
+329
+00:25:42,480 --> 00:25:45,400
+Time in Sedentary Behavior طول النهار قاعد على
+
+330
+00:25:45,400 --> 00:25:50,460
+التلفزيون وعلى الـ Video Games وشغال Online وطبعاً 
+
+331
+00:25:50,460 --> 00:25:56,320
+أكل أكل أكل أكل وخاصة الأمر الآخر اللي هي Regular 
+
+332
+00:25:56,320 --> 00:26:01,240
+Physical Activity Avoidance of Tobacco Use طبعاً و
+
+333
+00:26:01,240 --> 00:26:06,920
+Stress Management يبقى إحنا هنا مش طالبين معجزات
+
+334
+00:26:06,920 --> 00:26:15,080
+عملياً المفروض أنّه الأكل الصحي هيك زي ما قلنا كمان 
+
+335
+00:26:15,080 --> 00:26:20,940
+مرة أجلّل الـ Food اشتغل Exercise Manage Weight Stop
+
+336
+00:26:20,940 --> 00:26:26,620
+Smoking Cut Salt Limit Alcohol Manage Stress كلام 
+
+337
+00:26:26,620 --> 00:26:31,630
+معقول ما فيش عليه كلام كتير طب إيش الـ Dash-Diet؟
+
+338
+00:26:31,630 --> 00:26:34,930
+الـ Dash-Diet عبارة اختصار طبعاً كلمة الـ Dash
+
+339
+00:26:34,930 --> 00:26:41,430
+الـ D عن Dietary Approach أو Approaches الـ A Stop
+
+340
+00:26:41,430 --> 00:26:47,690
+S Hypertension الـ H Diet Nutritional Therapy
+
+341
+00:26:47,690 --> 00:26:51,930
+اسمها وطبعاً أجراء عنها كتير مكتوب عليها الأساس
+
+342
+00:26:51,930 --> 00:26:59,690
+تبعها Sodium Restriction مش فسيخ، Sodium Restriction
+
+343
+00:26:59,690 --> 00:27:04,970
+حاول نجلّل الملح قدر الإمكان، من الأساسيات هي Diet
+
+344
+00:27:04,970 --> 00:27:09,310
+Rich in Vegetable Fruit and Non-Fat Dietary
+
+345
+00:27:09,310 --> 00:27:15,850
+Products دايماً حتى لو أخذت منتجات لبنية منتجات
+
+346
+00:27:15,850 --> 00:27:20,490
+حليب لابد أن يكون كمية الفات وهنا سنتحدث بعد ذلك
+
+347
+00:27:20,490 --> 00:27:24,990
+في محاضرة كاملة ممكن أنتم تتحدثوا فيها عن أنواع
+
+348
+00:27:24,990 --> 00:27:33,030
+الدهون أو Calorie Restriction if Overweight وقدر
+
+349
+00:27:33,030 --> 00:27:38,030
+الإمكان One Medication يعني إذا اشتغلت عليها هاي زي ما
+
+350
+00:27:38,030 --> 00:27:41,750
+حكينا على الحاجات اللي قبل هيك تقريباً بتأثر تأثير
+
+351
+00:27:42,730 --> 00:27:47,950
+الدواء، تلقى واحد One Medication هذا بالنسبة للـ
+
+352
+00:27:47,950 --> 00:27:51,190
+Dash ياريت يكون فيه قراءة حرة عن الـ Dash يعني كلّ 
+
+353
+00:27:51,190 --> 00:27:54,610
+واحد يحاول يشوف له حاجة يجراها عن الـ Dash لأنّه 
+
+354
+00:27:54,610 --> 00:27:57,370
+لابد أنّه نعرف عنها، أنا مش هحكي عنها تفاصيل زي ما 
+
+355
+00:27:57,370 --> 00:28:04,050
+قلت لكم لكن هي أساسياتها، طبعاً فينا الآن موضوع بعد
+
+356
+00:28:04,050 --> 00:28:06,670
+ما خلّصنا اللي هو العلاج
+
+357
+00:28:09,880 --> 00:28:17,540
+زي عندما خلّصنا العلاج بالـ Lifestyle Modification
+
+358
+00:28:17,540 --> 00:28:23,260
+نحكي عن العلاج بالأدوية واللي دا داير طبعاً الآن هو
+
+359
+00:28:23,260 --> 00:28:26,480
+الشغال زي ما إحنا عارفين أن الـ Blood Pressure هو
+
+360
+00:28:26,480 --> 00:28:31,080
+عبارة عن ناتج الـ Cardiac Output، ما جدّش الجلب بدخ، 
+
+361
+00:28:31,080 --> 00:28:36,340
+ما جدّش فيه مقاومة في الشرايين مقابل الدخ اللي بيصير
+
+362
+00:28:36,340 --> 00:28:40,260
+فيه، اللي هو Blood Pressure يساوي Cardiac Output ضرب
+
+363
+00:28:40,260 --> 00:28:46,520
+Systemic Vascular Resistance وبنستخدم أدوية كلها
+
+364
+00:28:46,520 --> 00:28:50,840
+تخفّف إمّا من الـ Cardiac Output أو بتخفّف من الـ
+
+365
+00:28:50,840 --> 00:28:54,660
+Systemic Vascular Resistance بتجلّل منه وبتزيد
+
+366
+00:28:54,660 --> 00:28:59,120
+Cardiac Output فبتجلّل
+
+367
+00:28:59,120 --> 00:29:06,470
+Blood Pressure من الأدوية اللي بنستخدمها الـ 
+
+368
+00:29:06,470 --> 00:29:13,770
+Diuretics هي الأدوية التي تزيد كمية البول لأنها
+
+369
+00:29:13,770 --> 00:29:18,850
+بتخفّف من الـ Systemic Vascular Resistance وبتخفّف
+
+370
+00:29:18,850 --> 00:29:23,610
+من كمية الـ Cardiac Output الـ Beta Blockers الـ 
+
+371
+00:29:23,610 --> 00:29:28,170
+Direct Vasodilators لأنها بتوسّع الشرايين فالضغط
+
+372
+00:29:28,170 --> 00:29:28,670
+تخفّ
+
+373
+00:29:31,760 --> 00:29:36,420
+ونفس الفكرة وفيها الـ Calcium Channel Blockers 
+
+374
+00:29:38,420 --> 00:29:42,580
+أيضاً من الأدوية أو المجموعات الأدوية اللي
+
+375
+00:29:42,580 --> 00:29:48,680
+بيستخدمها كـ Blood Pressure Reducer طبعاً الآن أنا
+
+376
+00:29:48,680 --> 00:29:51,540
+مش هشرح عنهم هدول ولا إحنا لازمنا حقيقة بس ده يعني
+
+377
+00:29:51,540 --> 00:29:55,180
+كويس لو نعرف المجموعات كفاية الـ Drugs أنتو مش 
+
+378
+00:29:55,180 --> 00:29:58,980
+مطلوبة منكم لكن يعرف إحنا وين اتجاهنا في الأدوية
+
+379
+00:29:58,980 --> 00:30:05,780
+إيه رايحين، هذا بالنسبة لمحاضرة الـ Hypertension وزي
+
+380
+00:30:05,780 --> 00:30:08,580
+ما تحدثنا يا جماعة موضوع الـ Hypertension موضوع
+
+381
+00:30:08,580 --> 00:30:14,040
+أضخم من هيك بكتير في عليه كتب أصلاً، لكن إحنا أخدنا
+
+382
+00:30:14,040 --> 00:30:18,540
+من زاوية خاصة بينا في العلاج الطبيعي بحيث أنّه يكفي
+
+383
+00:30:18,540 --> 00:30:22,800
+عملياً
+
+384
+00:30:22,800 --> 00:30:27,160
+لإلنا ويكون عندنا فكرة عن الموضوع بس بدأ أذكركم في
+
+385
+00:30:27,160 --> 00:30:33,100
+سؤال أوجهناه، الـ Body Mass Index ما هو؟ كيف يتمّ 
+
+386
+00:30:33,100 --> 00:30:39,810
+حسابه؟ وجدول عن الدرجات تبعته وهذا كلّه مكتوب بخطّ
+
+387
+00:30:39,810 --> 00:30:43,850
+اليد وتبعتوليها على الـ Model، تحديد الوجبة تبع الـ
+
+388
+00:30:43,850 --> 00:30:49,030
+Model على الـ Model هتلاقي الكلام هذا 
+
+389
+00:30:52,220 --> 00:30:55,960
+يعني إن شاء الله تكونوا، أنا عارف إنّه التعليم
+
+390
+00:30:55,960 --> 00:31:00,580
+بالطريقة هذه مش هو، لكن أنا هكون سعيد بكلّ ملاحظة
+
+391
+00:31:00,580 --> 00:31:05,760
+ممكن ملاحظة إيجابية طبعاً إنّه أقدر أحسّن في وصول 
+
+392
+00:31:05,760 --> 00:31:13,120
+المعلومة إليكم بأي صورة لأن الفرصة هذه مش موجودة
+
+393
+00:31:13,120 --> 00:31:19,410
+في تعليم الإلكتروني، لكن لو كان وجاهي بقدر واحد 
+
+394
+00:31:19,410 --> 00:31:24,570
+يشرح ويزيد ويعيد وكذا لكن الآن حقيقة محصور، لأيّ 
+
+395
+00:31:24,570 --> 00:31:28,190
+حدّ بدّه أي حاجة على الرقم اللي عندكم على الواتساب
+
+396
+00:31:28,190 --> 00:31:32,670
+يرسل لي مباشرة ما عندي مشكلة، حتى يعني بدون... بدون
+
+397
+00:31:32,670 --> 00:31:37,310
+ما تتبايلوا على فريزة ابعتوا لي لو في أي حاجة إيش
+
+398
+00:31:37,310 --> 00:31:42,450
+بقدر أعمل حاول... حاول نخفّف بدنا نعدّي هذه الفترة
+
+399
+00:31:42,450 --> 00:31:45,570
+على خير ونشوف وشكم على خير إن شاء الله السلام 
+
+400
+00:31:45,570 --> 00:31:45,870
+عليكم
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/FR-vBag9siY_postprocess.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/FR-vBag9siY_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..4df3310cca3cbcb2c41377a5255b51164cea4970
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/FR-vBag9siY_postprocess.srt
@@ -0,0 +1,832 @@
+1
+00:00:01,350 --> 00:00:03,470
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:03,470 --> 00:00:07,530
+والصلاة والسلام علي سيدنا محمد سيد المرسلين أجمعين
+
+3
+00:00:07,530 --> 00:00:13,950
+يعني اليوم ان شاء الله نحاول نتحدث عن الاخر محاضرة
+
+4
+00:00:13,950 --> 00:00:19,470
+هي في الصدرية عملياطبعا ممكن في الآخر يطلع معنا
+
+5
+00:00:19,470 --> 00:00:23,910
+كمان محاضرة اسمها الـ Core Pulmonale لكن الآن COPD
+
+6
+00:00:23,910 --> 00:00:27,490
+Chronic Obstructive Pulmonary Disease حسب كيف
+
+7
+00:00:27,490 --> 00:00:30,030
+بيكفل وجد بالنسبة لموضوع الـ Core Pulmonale ممكن
+
+8
+00:00:30,030 --> 00:00:34,030
+نعملها محاضرة ممكن ننسيبها حسب إيش بيصير معنا لكن
+
+9
+00:00:34,030 --> 00:00:41,470
+الـCOPDهي اختصار COPD هذه الكلمة هتسمعوها كتير في
+
+10
+00:00:41,470 --> 00:00:46,510
+أجسام الباطنة أو في المستشفيات COPD فيش هتقول
+
+11
+00:00:46,510 --> 00:00:51,590
+chronic obstructive pulmonary disease COPD اختصار
+
+12
+00:00:51,590 --> 00:00:55,130
+ل chronic obstructive pulmonary disease في طبعا
+
+13
+00:00:55,130 --> 00:01:03,360
+اختصارات تانية للـ COPDزي chronic obstructive lung
+
+14
+00:01:03,360 --> 00:01:07,500
+disease نفس الفكرة أو chronic lower respiratory
+
+15
+00:01:07,500 --> 00:01:12,680
+disease CLRD ماعرفش إذا في حد بيستخدمها لكن هي من
+
+16
+00:01:12,680 --> 00:01:16,560
+الإختصارات المتعارف عليها لكن أشهر اختصار اللي هو
+
+17
+00:01:16,560 --> 00:01:23,320
+CBD عالمياوهو COPD مش مرض عمليًا هو umbrella term
+
+18
+00:01:23,320 --> 00:01:27,080
+used to describe progressive lung disease
+
+19
+00:01:27,080 --> 00:01:30,560
+progressive lung obstructive disease من الإسم
+
+20
+00:01:30,560 --> 00:01:34,920
+chronic obstructive pulmonary disease الحاجات اللي
+
+21
+00:01:34,920 --> 00:01:38,300
+بتيجي تحت اللي درسناها ذهبت نذاكر مع بعض إيش
+
+22
+00:01:38,300 --> 00:01:43,900
+الحاجات اللي كانت chronic obstructive diseases
+
+23
+00:01:43,900 --> 00:01:49,610
+lung diseases صح انفيزيمة صحيح ميالميةChronic
+
+24
+00:01:49,610 --> 00:01:55,260
+Bronchitisأزمة، بس مش الأزمة العادية طبعا، اللي هي
+
+25
+00:01:55,260 --> 00:01:58,340
+الـirreversible أو الـrefractory، اللي هي لا
+
+26
+00:01:58,340 --> 00:02:05,040
+تتجاوب ولا تستجيب للأدوية المعهودة، وsevere
+
+27
+00:02:05,040 --> 00:02:08,820
+bronchiectasis كمان، يبقى هي عبارة عن umbrella
+
+28
+00:02:08,820 --> 00:02:13,080
+term used to describe progressive lung disease
+
+29
+00:02:13,080 --> 00:02:16,900
+which include emphysema 1, chronic bronchitis 2,
+
+30
+00:02:17,440 --> 00:02:20,500
+refractory irreversible asthma 3, severe
+
+31
+00:02:20,500 --> 00:02:28,350
+bronchiectasisرقم أربع طيب جداش يعني الـ COPD
+
+32
+00:02:28,350 --> 00:02:32,850
+عاملينها محاضرة لحالها وكذا و هي عبارة عن زي ما
+
+33
+00:02:32,850 --> 00:02:37,670
+قلنا umbrella term جداش حجمها في العالم كتير حجمها
+
+34
+00:02:37,670 --> 00:02:40,110
+في العالم في ال national heart and lung blood
+
+35
+00:02:40,110 --> 00:02:43,970
+institute في أمريكا قال إنه في عنده في أمريكا
+
+36
+00:02:43,970 --> 00:02:47,690
+اتناشر مليون adult عندهم COPD اتناشر مليون adult
+
+37
+00:02:47,690 --> 00:02:52,120
+طب هو كام جداش عدد سكان أمريكا تقريبا 300 مليونهذه
+
+38
+00:02:52,120 --> 00:02:56,320
+نسبة كويسة وفي 12 مليون تانيات Undiagnosed أو
+
+39
+00:02:56,320 --> 00:02:59,020
+Developing COPD جاعدين بيعملوا Developing COPD
+
+40
+00:02:59,020 --> 00:03:03,680
+يعني أنتوا أنا بحكي عن 25 مليون 25 مليون لو كنا
+
+41
+00:03:03,680 --> 00:03:10,510
+250 مليون إحنا بنحكي عن 10%يبقى طيب 10% وهذا بتشكل
+
+42
+00:03:10,510 --> 00:03:15,610
+ثقة المهول على الاقتصاد في أي بلد لما يكون في عندي
+
+43
+00:03:15,610 --> 00:03:20,150
+10% من الناس في عندهم مشكلة في الـ Lung في الـ
+
+44
+00:03:20,150 --> 00:03:26,130
+Economy في الـ USA Economy بيكلف الـ COPD تقريبا
+
+45
+00:03:26,130 --> 00:03:32,650
+32 مليون دولار بليون دولار يعني في direct أو
+
+46
+00:03:32,650 --> 00:03:37,220
+indirect cost في سنة 2002 كان هذا الموضوعأحنا
+
+47
+00:03:37,220 --> 00:03:40,460
+مابنحكيش قاعدين عن مرض بسيط عشرة نفار أو عشرين
+
+48
+00:03:40,460 --> 00:03:43,560
+نفار أو خمسين نفار، لا، احنا بحكي عن اقتصاد بلد
+
+49
+00:03:45,330 --> 00:03:51,550
+مصانع تعمل لهذا الموضوع، مصانع الأسباب أو المسببات
+
+50
+00:03:51,550 --> 00:03:56,430
+للموضوع هذا، جداش بتكسب و جداش بدخل جيوبها من هذا
+
+51
+00:03:56,430 --> 00:04:00,550
+الموضوع، اللي هي مصانع الدخان، a person with سؤب
+
+52
+00:04:00,550 --> 00:04:05,590
+دي dies every four minutes in the U.S. في أمريكا
+
+53
+00:04:05,590 --> 00:04:20,370
+واحد من سؤب دي بيموت كل أربع دقايقSOPD will be
+
+54
+00:04:20,370 --> 00:04:26,010
+the third leading cause of death in the US by 2020
+
+55
+00:04:32,610 --> 00:04:37,390
+that over حوالي 210 مليون people هذا طبعا إذا
+
+56
+00:04:37,390 --> 00:04:42,650
+حسبناها worldwide have COPD الأرقام مش ضرورية كتير
+
+57
+00:04:42,650 --> 00:04:46,170
+يا جماعة لكن أنا بال statistics بتاع الـCOPD بحاول
+
+58
+00:04:46,170 --> 00:04:52,290
+أقولكوا أنه قداش الـCOPD موجود في العالم وإيه له
+
+59
+00:04:52,290 --> 00:04:57,490
+مشاكله مش موضوع بسيط وسهل لأ وطبعا هذا ما فعلته اي
+
+60
+00:04:57,490 --> 00:05:06,020
+دينا مرة تانية يعنيالـ COPD هو الـ lung is a lung
+
+61
+00:05:06,020 --> 00:05:09,100
+alignment that is characterized by a persistent
+
+62
+00:05:09,100 --> 00:05:17,940
+blockage of airflow from the lungs طبعاً اسم المرض
+
+63
+00:05:17,940 --> 00:05:21,220
+Obstructive فالـ COPD Obstructive فتصبح عندي
+
+64
+00:05:21,220 --> 00:05:25,140
+Obstruction وهي اللي بتعمل ليه المشكلة it is under
+
+65
+00:05:25,140 --> 00:05:28,690
+diagnosedlife-threatening lung disease that
+
+66
+00:05:28,690 --> 00:05:32,230
+interferes with normal breathing and is not fully
+
+67
+00:05:32,230 --> 00:05:35,850
+reversible يعني ممكن واحد تلاقيه متنفس عادي لكن هو
+
+68
+00:05:35,850 --> 00:05:40,470
+عنده COPD ممكن يكون وضعه كويس وهو عنده COPD فهو
+
+69
+00:05:40,470 --> 00:05:44,230
+عشان هيك بنقول عنه under diagnosed يعني مش كل واحد
+
+70
+00:05:44,230 --> 00:05:51,630
+بنقدر نعرف أو هو وما بشكيش بنقدر نقول عنه COPD ال
+
+71
+00:05:51,630 --> 00:05:57,900
+most cases of COPD بتيجي كخدوا بالكوا result of
+
+72
+00:05:57,900 --> 00:06:05,380
+long term exposure to lung irritants أي lung
+
+73
+00:06:05,380 --> 00:06:08,140
+irritants، بغض النظر الشيية الآن، هحنا هنحكي عن
+
+74
+00:06:08,140 --> 00:06:11,840
+التدخين وعن كذا، لكن لو واحد بيشتغل في مصنع بترول
+
+75
+00:06:11,840 --> 00:06:17,480
+أو واحد بيشتغل في مصانع الفحم أو واحد بيشتغل في
+
+76
+00:06:17,480 --> 00:06:23,170
+منجرة، في مناجر الخشباللي بتدخل على أو واحد بيشتغل
+
+77
+00:06:23,170 --> 00:06:26,810
+فيه مصانع ما نعرف الـ Parfum، الرواي حي الـ
+
+78
+00:06:26,810 --> 00:06:31,150
+Irritants اللي فيها اللي بتدخل في الصناعات، بتعملي
+
+79
+00:06:31,150 --> 00:06:35,250
+Irritation للـ Lung، بتعملي Damaging للـ Lung،
+
+80
+00:06:35,250 --> 00:06:39,210
+ممكن تعملي Soapy D، ده الـ Most Common، لكن هي الـ
+
+81
+00:06:39,210 --> 00:06:42,530
+Most Common Irritants that causes Soapy D،
+
+82
+00:06:42,530 --> 00:06:45,570
+cigarette smoke، طبعاً هنا cigarette smoke لأن هو
+
+83
+00:06:45,570 --> 00:06:48,390
+هذا الغالبية، لكن احنا عندنا الأن أنواع أخرى من ال
+
+84
+00:06:48,390 --> 00:06:55,710
+smokingايوة ايوة
+
+85
+00:06:55,710 --> 00:07:01,270
+ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوة
+
+86
+00:07:01,270 --> 00:07:06,570
+ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوة
+
+87
+00:07:06,570 --> 00:07:09,390
+ايوة ايوة ايوة ايوة ايوة ايوة ايوة ايوةPassive
+
+88
+00:07:09,390 --> 00:07:12,590
+smoking او secondhand smoking ايش هي secondhand
+
+89
+00:07:12,590 --> 00:07:16,170
+smoking؟ طبعا في حاجات تانية كتير لـ smoking لكن
+
+90
+00:07:16,170 --> 00:07:19,470
+ايش ال secondhand smoking او ال passive smoking؟
+
+91
+00:07:19,470 --> 00:07:24,270
+اه انه .. انه .. انه الطرف التاني او الآخر اللي مع
+
+92
+00:07:24,270 --> 00:07:29,140
+المدخنعمليًا هو بيقول عنه second أو passive smoker
+
+93
+00:07:29,140 --> 00:07:33,060
+هو مابدخنش في الحقيقة لكن بياخد موضوع التدخين
+
+94
+00:07:33,060 --> 00:07:38,660
+يتضرر من موضوع التدخين من المدخن الحقيقي في rare
+
+95
+00:07:38,660 --> 00:07:41,720
+cases طبعًا COPD حكينا عنها قبل هيك جون a genetic
+
+96
+00:07:41,720 --> 00:07:45,200
+condition called Alpha-1-antitrypsin deficiency
+
+97
+00:07:45,200 --> 00:07:48,720
+ممكن يعمل إيه؟ يقول لها بيعمل emphysema إذا فاكرين
+
+98
+00:07:48,720 --> 00:07:52,840
+فهو بيبقى إن ال emphysema واحد من ال من ال من ال
+
+99
+00:07:52,840 --> 00:08:01,100
+COPD فهو causing COPDمين هم أكتر ناس مين هم أكتر
+
+100
+00:08:01,100 --> 00:08:05,640
+ناس أكتر
+
+101
+00:08:05,640 --> 00:08:11,480
+ناس at risk هم الناس people who smoke or are
+
+102
+00:08:11,480 --> 00:08:16,020
+exposed to smoke name people who have a family
+
+103
+00:08:16,020 --> 00:08:19,800
+history of COPD are more likely to develop the
+
+104
+00:08:19,800 --> 00:08:24,390
+disease if they smokeوالتخلص من تجارب طفولة طويلة
+
+105
+00:08:24,390 --> 00:08:30,920
+للغاية على الرغم من أنها فاكتورة خطر لـ COPDالناس
+
+106
+00:08:30,920 --> 00:08:34,900
+اللي بتشتغل في مصانع زي ما قلنا القار والأسفل مثلا
+
+107
+00:08:34,900 --> 00:08:42,260
+مصانع الفحم، مصانع الصناعات اللي فيها long
+
+108
+00:08:42,260 --> 00:08:47,340
+irritants هدول كلهم ناس liable المناجر، liable أن
+
+109
+00:08:47,340 --> 00:08:51,300
+يصير عندهم سوق طبعا مش يوم يشتغل أو لا يومين، لازم
+
+110
+00:08:51,300 --> 00:08:56,310
+يكون long term exposureحوالي 90% of COPD deaths
+
+111
+00:08:56,310 --> 00:09:00,910
+occur in low and middle income countries where
+
+112
+00:09:00,910 --> 00:09:04,870
+effective strategies for prevention and control
+
+113
+00:09:04,870 --> 00:09:09,930
+are not always implemented or accessible طبعا هو
+
+114
+00:09:09,930 --> 00:09:13,550
+ممكن واحد يقولك يعني نوقف صناعة الفحم، نوقف صناعة
+
+115
+00:09:13,550 --> 00:09:17,130
+الأسفر تلقار، لأ احنا مانوقفش، لكن لابد ان يكون
+
+116
+00:09:17,130 --> 00:09:23,160
+هناكقوانين تحمي العامل من الاستنشاق المستمر أو
+
+117
+00:09:23,160 --> 00:09:27,080
+أجهزة تحمي العامل من الاستنشاق المستمر بما فيها
+
+118
+00:09:27,080 --> 00:09:32,880
+الكمامات، الطرق المعهودة لحماية العامل من تنفس هذه
+
+119
+00:09:32,880 --> 00:09:38,460
+المواداللي هي المضرة للجهاز التنفسي لكن إنك توقف
+
+120
+00:09:38,460 --> 00:09:42,060
+الصناعات طبعاً مش واري توقف الصناعات لكن حماية
+
+121
+00:09:42,060 --> 00:09:47,880
+العمال سواء في الوقت أو في الأدوات للتخفيف من أثر
+
+122
+00:09:47,880 --> 00:09:53,910
+هذه الـIrritants على الجسمالـ Symptoms اللي
+
+123
+00:09:53,910 --> 00:09:56,090
+بيلاقيها نفس اللي شفناها في الـ Obstructive
+
+124
+00:09:56,090 --> 00:10:03,150
+Bronchitis وفي الانفيزيمة وفي الأزمة اللي هو
+
+125
+00:10:03,150 --> 00:10:08,190
+Breathlessnessيعني بينقطع نفسه abnormal sputum
+
+126
+00:10:08,190 --> 00:10:12,730
+بطلع بالغم كتير يعني mix of saliva and mucus in
+
+127
+00:10:12,730 --> 00:10:18,970
+the airway كحة مستمرة كحة زي ما بقولوها مزمنة
+
+128
+00:10:18,970 --> 00:10:23,220
+daily activities can become very difficultas the
+
+129
+00:10:23,220 --> 00:10:28,480
+condition gradually worsens بصير طبعا صعب بعد هيك
+
+130
+00:10:28,480 --> 00:10:33,180
+على المريض اللي عنده سؤوبة دي يقدر يجوم بالأعمال
+
+131
+00:10:33,180 --> 00:10:37,580
+اليومية even انه يطلع بس للجامع أو كذا بيصير يضايق
+
+132
+00:10:37,580 --> 00:10:41,820
+ومايقدرش يتنفس من الحاجات اللي
+
+133
+00:10:45,050 --> 00:10:48,890
+الـ COPD قولنا هو Chronic Air Flow Limitation
+
+134
+00:10:48,890 --> 00:10:53,370
+انفيزيمة وChronic Bronchitis على سبيل المثال ويكون
+
+135
+00:10:53,370 --> 00:10:58,150
+فيها المريض easily fatigued frequent respiratory
+
+136
+00:10:58,150 --> 00:11:03,910
+infection use of accessory muscles to breathe
+
+137
+00:11:03,910 --> 00:11:08,110
+أورتبنك كورب المنالي ممكن يصير
+
+138
+00:11:12,100 --> 00:11:17,680
+ممكن طبعا يضعف، ممكن يصير عندنا wheezes، ممكن يصير
+
+139
+00:11:17,680 --> 00:11:21,160
+عندنا chronic cough، ممكن يكون عندنا parallel
+
+140
+00:11:21,160 --> 00:11:26,660
+chest dyspnea، ممكن يكون عندنا الـexpiratory time
+
+141
+00:11:26,660 --> 00:11:33,740
+prolong، نتيجة الـobstruction، بيصير عملية إخراج
+
+142
+00:11:33,740 --> 00:11:39,160
+النفس أصعب كمانوممكن طبعاً حاجة أخدناها قبل هيك
+
+143
+00:11:39,160 --> 00:11:43,120
+يدخل المريض في نتيجة الهيبوكسيا في Digital
+
+144
+00:11:43,120 --> 00:11:46,140
+Clapping، إيش يعني Digital Clapping؟ اللي هو
+
+145
+00:11:46,140 --> 00:11:51,840
+Clapping of the fingers، nails of the fingers كيف
+
+146
+00:11:51,840 --> 00:11:56,000
+بنشخصه؟ ال simple diagnostic test بنقول عنه
+
+147
+00:11:56,000 --> 00:12:00,360
+Spirometry حكينا عنه measures how much airالشخص
+
+148
+00:12:00,360 --> 00:12:06,540
+يستطيع انهاء وانهاء ويستطيع انهاء وانهاء ويستطيع
+
+149
+00:12:06,540 --> 00:12:06,560
+انهاء ويستطيع انهاء ويستطيع انهاء ويستطيع انهاء
+
+150
+00:12:06,560 --> 00:12:07,660
+ويستطيع انهاء ويستطيع انهاء ويستطيع انهاء ويستطيع
+
+151
+00:12:07,660 --> 00:12:08,440
+انهاء ويستطيع انهاء ويستطيع انهاء ويستطيع انهاء
+
+152
+00:12:08,440 --> 00:12:08,640
+انهاء ويستطيع انهاء ويستطيع انهاء ويستطيع انهاء
+
+153
+00:12:08,640 --> 00:12:13,120
+ويستطيع انهاء ويستطيع انهاء ويستطيع انهاء ويستطيع
+
+154
+00:12:13,120 --> 00:12:16,440
+انهاء ويستطيع
+
+155
+00:12:16,440 --> 00:12:28,400
+انهاء ويستطيع انهاء ويستطيع
+
+156
+00:12:29,080 --> 00:12:33,740
+no cure الحل
+
+157
+00:12:33,740 --> 00:12:37,600
+اول اش الباب اللي بيجيك منه الريح سد و استريح
+
+158
+00:12:37,600 --> 00:12:44,540
+quitting smoking وقف التدخين مش تخفي في التدخين
+
+159
+00:12:44,540 --> 00:12:49,240
+وقف التدخين التدخين the most important stepan
+
+160
+00:12:49,240 --> 00:12:52,660
+individual can take to treats OBD هذا رقم واحد إذا
+
+161
+00:12:52,660 --> 00:12:55,860
+في تدخين يجب أن يقف التدخين ماوفيش تدخين معناته
+
+162
+00:12:55,860 --> 00:13:00,880
+إحنا قاعدين بنروح اتجاه الهوية الأمور التانية طبعا
+
+163
+00:13:00,880 --> 00:13:05,120
+كلهيتها include vaccination طبعا عشان أحميه من أنه
+
+164
+00:13:05,120 --> 00:13:07,900
+يصير عنده pneumonia ويصير عنده مشاكل زي ال
+
+165
+00:13:07,900 --> 00:13:11,380
+pulmonary rehabilitation للصدر oxygen therapy
+
+166
+00:13:11,380 --> 00:13:16,370
+surgery ممكن يحتاج أنهيصير عندى في منطقة معينة الـ
+
+167
+00:13:16,370 --> 00:13:20,030
+lung emphysema أو كذا، يصير بدها removal أو حاجة،
+
+168
+00:13:20,030 --> 00:13:25,170
+هذه العلاجات بعض الأحيان زى على سبيل المثال الـ
+
+169
+00:13:25,170 --> 00:13:29,490
+antibiotic في بعض الأحيان للناس اللى بيصير عندهم
+
+170
+00:13:29,490 --> 00:13:34,430
+infection و هيك، سوء بديه symptoms usually slowly
+
+171
+00:13:34,430 --> 00:13:38,170
+worsens over time، بسوقه، انا جيت لو صار عنده رش
+
+172
+00:13:38,170 --> 00:13:44,710
+حم في الونزاأو infection بيديني symptoms أشد وأكتر
+
+173
+00:13:44,710 --> 00:13:52,500
+نحاول جدر الإمكان avoid lung irritantsget ongoing
+
+174
+00:13:52,500 --> 00:13:55,840
+care manage the disease and its symptoms prepare
+
+175
+00:13:55,840 --> 00:13:59,700
+for emergencies وتكون جاهز ومعروف للمستشفى اللي
+
+176
+00:13:59,700 --> 00:14:03,420
+حاول الواحد أنه هذا دايما بيجي يسوقه بالدين وبندي
+
+177
+00:14:03,420 --> 00:14:07,100
+بعض الأحيان مضادات حيوية بزيادة ليش؟ لأن هدول
+
+178
+00:14:07,100 --> 00:14:14,880
+الناس liable يدخلوا في chest infectionوبندى كمان
+
+179
+00:14:14,880 --> 00:14:21,140
+مواصعات للشعب الهوائية وبعضها أحياناً كورتازونات
+
+180
+00:14:21,140 --> 00:14:24,600
+على أساس تخفف من الـ inflammation اللى موجود وهذا
+
+181
+00:14:24,600 --> 00:14:32,220
+بحسن وضع الـ breathing يبقى احنا هان بنحكى عن
+
+182
+00:14:32,220 --> 00:14:37,800
+الـCOPD اللى هو umbrella
+
+183
+00:14:38,530 --> 00:14:42,710
+التي تسمى كـ progressive lung disease زي
+
+184
+00:14:42,710 --> 00:14:45,130
+الانفزيامة، chronic bronchitis، refractory أو
+
+185
+00:14:45,130 --> 00:14:50,030
+irreversible أزمة وsevere bronchectasis وقلنا إن
+
+186
+00:14:50,030 --> 00:14:54,770
+الـCOPD هو عبارة عن obstructive pulmonary disease
+
+187
+00:14:54,770 --> 00:14:58,390
+if the lung alignment that is characterized by a
+
+188
+00:14:58,390 --> 00:15:03,550
+persistent blockage of airway from the lung وكتير
+
+189
+00:15:03,550 --> 00:15:06,980
+من الناس يبقى في عنده symptomsبيبقى ماعندوش
+
+190
+00:15:06,980 --> 00:15:11,620
+symptoms لكن هو عنده COPD. The most common
+
+191
+00:15:11,620 --> 00:15:14,820
+irritants that cause COPD، سيجارة smoking، لكن في
+
+192
+00:15:14,820 --> 00:15:19,760
+حاجات تانية كمان مهم نعرفها، الناس اللي أترسك أكتر
+
+193
+00:15:19,760 --> 00:15:25,300
+هم الناس الـ Smoker و الـ exposed to smoke، الناس
+
+194
+00:15:25,300 --> 00:15:29,760
+اللي عند family hustle و في COPD و بدخنه، الناس
+
+195
+00:15:29,760 --> 00:15:34,250
+الـ long term exposureto other lung irritants زي
+
+196
+00:15:34,250 --> 00:15:40,470
+مثلًا ما قلنا مصانع الفحم وتسعين في المية من
+
+197
+00:15:40,470 --> 00:15:44,850
+الـdeaths بتصير في الناس اللي هي مستوى الاقتصادي
+
+198
+00:15:44,850 --> 00:15:49,950
+تبعها سيء واللي لا تحافظ على صحتها بصورة عامة
+
+199
+00:15:49,950 --> 00:15:54,510
+الـdiagnosis بالـSpirometer والتريتمنت COPD has no
+
+200
+00:15:54,510 --> 00:15:58,850
+cureأفضل حاجة، الناس اللي بتدخن، توقف التدخين،
+
+201
+00:15:58,850 --> 00:16:01,550
+وبدها بعد هيك Oxygen Therapy، Pulmonary
+
+202
+00:16:01,550 --> 00:16:06,490
+Rehabilitation، Vaccine، وزي ما قلنا، بعض الأحيان
+
+203
+00:16:06,490 --> 00:16:11,110
+بندّي مضادات حيوية لحماية المريض من التهيابات،
+
+204
+00:16:11,110 --> 00:16:17,570
+ممكن إنها تسويق حالته أكتر وأكترويعطيكم العافية
+
+205
+00:16:17,570 --> 00:16:24,330
+ويعني هي محاضرة قصيرة لشك لكن مبدأيًا هي بتضم كتير
+
+206
+00:16:24,330 --> 00:16:28,570
+من المواضيع اللي حكيناها بهيك إحنا بنبقى خلصنا
+
+207
+00:16:28,570 --> 00:16:33,370
+اللي هو موضوع ال respiratory system وحننجل بعد هيك
+
+208
+00:16:33,370 --> 00:16:35,510
+لموضوع آخر إن شاء الله تعالى
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/KitY0z_MKw8_postprocess.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/KitY0z_MKw8_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3f31769cc5afd1fdb247ed477c8b05949398965c
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/KitY0z_MKw8_postprocess.srt
@@ -0,0 +1,1540 @@
+1
+00:00:04,150 --> 00:00:07,790
+طيب بسم الله الرحمن الرحيم اليوم يا جماعة بنكمل
+
+2
+00:00:07,790 --> 00:00:12,990
+محاضرة ال diabetes mellitus اللي هي نوعا ما طويلة
+
+3
+00:00:12,990 --> 00:00:19,270
+صبح عليكم شر الله تعالى و كنا اتحدثنا طبعا حاجات
+
+4
+00:00:19,270 --> 00:00:23,870
+كتير عن ال diabetes mellitus type 1 to 1 to
+
+5
+00:00:23,870 --> 00:00:28,230
+gestational diabetes و secondary diabetes و ال
+
+6
+00:00:28,230 --> 00:00:33,670
+LADA و المودي كل هذه الحاجات اتحدثنا عنهاوبدأنا
+
+7
+00:00:33,670 --> 00:00:37,990
+نحكي عن treatment أو management of diabetes
+
+8
+00:00:37,990 --> 00:00:43,400
+mellitus ووصلنا في لحظة من اللحظاتاللي هي ال
+
+9
+00:00:43,400 --> 00:00:49,700
+subcutaneous treatment اللي هو الإنسولين، فاكرين؟
+
+10
+00:00:49,700 --> 00:00:53,840
+وكنا في المرة اللي فاتت حكينا بدنا على ال WhatsApp
+
+11
+00:00:53,840 --> 00:00:59,380
+للناس اللي لسه ما عملت تبعتلي،
+
+12
+00:00:59,380 --> 00:01:03,740
+مديش أقول تفصيل، لكن سطرين، تلاتة عن بعض الحاجات،
+
+13
+00:01:03,740 --> 00:01:08,790
+بس بذكركوا فيهمالناس اللي معاملة ال share rate على
+
+14
+00:01:08,790 --> 00:01:12,830
+ال whatsapp مش على ال model اتحدثنا عن ال insulin
+
+15
+00:01:12,830 --> 00:01:18,770
+او طرق اعطاء ال insulin وقولنا ان ال insulin احنا
+
+16
+00:01:18,770 --> 00:01:25,370
+عادة بنعطي للديابة S1 كل ديابة S1 بياخد type 1
+
+17
+00:01:25,760 --> 00:01:30,040
+بياخد انسولين الـ diabetes too نبدأ معاه ب
+
+18
+00:01:30,040 --> 00:01:33,760
+lifestyle modification
+
+19
+00:01:33,760 --> 00:01:43,170
+ممكن بعد هيك نديله oral antidiabeticوإذا ما زبطش
+
+20
+00:01:43,170 --> 00:01:47,530
+ندّي combination انسولين مع oral antidiabetic
+
+21
+00:01:47,530 --> 00:01:53,650
+treatment طبعا مع exercise بالنسبة
+
+22
+00:01:53,650 --> 00:02:00,630
+لل .. وقولنا طبعا من مشاكل الانسولينهيبوغليسيميا
+
+23
+00:02:00,630 --> 00:02:09,530
+الهيبوغليسيميا
+
+24
+00:02:09,530 --> 00:02:12,050
+الهيبوغليسيميا الهيبوغليسيميا الهيبوغليسيميا
+
+25
+00:02:12,050 --> 00:02:12,730
+الهيبوغليسيميا الهيبوغليسيميا الهيبوغليسيميا
+
+26
+00:02:12,730 --> 00:02:12,930
+الهيبوغليسيميا الهيبوغليسيميا الهيبوغليسيميا
+
+27
+00:02:12,930 --> 00:02:13,070
+الهيبوغليسيميا الهيبوغليسيميا الهيبوغليسيميا
+
+28
+00:02:13,070 --> 00:02:13,810
+الهيبوغليسيميا الهيبوغليسيميا الهيبوغليسيميا
+
+29
+00:02:13,810 --> 00:02:18,290
+الهيبوغليسيميا الهيبوغليسيميا الهيبووأحنا كنا
+
+30
+00:02:18,290 --> 00:02:22,990
+نتوردكم على صورة كيف أن البني آدم ممكن بطريقة
+
+31
+00:02:22,990 --> 00:02:30,730
+دورانية أنه نعمل ال injection في أماكن مختلفة خلال
+
+32
+00:02:30,730 --> 00:02:37,070
+الأوقات، حتى لا يحصل أي مشاكل، هذه الصور كانت
+
+33
+00:02:37,070 --> 00:02:41,850
+لـLibodystrophyوبدأنا نتحدث عن ال drug therapy
+
+34
+00:02:41,850 --> 00:02:46,090
+اللي هي ال oral agents، diabetes mellitus oral
+
+35
+00:02:46,090 --> 00:02:49,990
+agents وهو طبعا مش انسلين على كل حالة، طبعا لم
+
+36
+00:02:49,990 --> 00:02:55,320
+يستطيع الإنسان الحاليأو الطب الحالي أنه يصنع
+
+37
+00:02:55,320 --> 00:03:01,320
+Insulin عن طريق الفم، ليش؟ لأنه هو أصلا يعني عملية
+
+38
+00:03:01,320 --> 00:03:08,200
+الهدم تبعه لا تتم أو تتم في الماعدة
+
+39
+00:03:08,200 --> 00:03:11,880
+بصورة كبيرة ولذلك لم يستطيعوا، بحاول الآن يعملوا
+
+40
+00:03:11,880 --> 00:03:16,640
+منه بخاخ، لسه ما زالت الأمور في الطريق للحل، أنا
+
+41
+00:03:16,640 --> 00:03:22,380
+واثقلأن في يوم من الأيام سيحدث أنسلين غير أورالي
+
+42
+00:03:22,380 --> 00:03:28,120
+سبكوتينياس ممكن يكون أورالي ممكن يكون نازل على
+
+43
+00:03:28,120 --> 00:03:34,300
+سبيل المثال بخاخ أو حاجة كل شيء ممكن يصير ال drug
+
+44
+00:03:34,300 --> 00:03:38,940
+therapyوهو بيحس الـ mechanism in which insulin and
+
+45
+00:03:38,940 --> 00:03:42,300
+glucose are produced and used by the body يعني هو
+
+46
+00:03:42,300 --> 00:03:48,420
+بيعمل عملية تحفيز لإفراز الإنسلين هي هذه الحاجة
+
+47
+00:03:48,420 --> 00:03:56,420
+الأساسية عنده إنه بيحاول يحفز إفراز الإنسلين، هذه
+
+48
+00:03:56,420 --> 00:03:57,440
+شغلته الأساسية
+
+49
+00:04:03,690 --> 00:04:08,530
+وزي ما قلنا، في الـ pancreas، وبعمل reduction، الـ
+
+50
+00:04:08,530 --> 00:04:12,530
+glucose reduction، by liver، وبيزيد، زي ما قلنا،
+
+51
+00:04:12,530 --> 00:04:16,890
+حساسية الإنسولين أو الجسم أو خلايا الجسم
+
+52
+00:04:16,890 --> 00:04:22,030
+للإنسولين، تقريبا، هذه أهم الحاجات اللي بيعملها
+
+53
+00:04:22,030 --> 00:04:25,890
+الـ oral anti-diabetic طبعا، ليس بعيدا، اللي هو
+
+54
+00:04:25,890 --> 00:04:29,130
+الـ nutritional therapy، nutritional therapy
+
+55
+00:04:32,130 --> 00:04:38,950
+هو حاجة نطلبها يعني هي نفس ال healthy eating plan
+
+56
+00:04:38,950 --> 00:04:44,890
+مش مش بس مريض أو عيان ال diabetes إنما حتى الإنسان
+
+57
+00:04:44,890 --> 00:04:52,090
+الطبيعي لابد إنه ياكل أكل ياكل أكل healthy أصلا
+
+58
+00:04:55,020 --> 00:04:59,060
+وطب احنا ايش بنوم؟ ايش بدنا منها؟ ايش ال goal تبع
+
+59
+00:04:59,060 --> 00:05:04,900
+ال nutritional therapy؟ assist
+
+60
+00:05:04,900 --> 00:05:08,320
+people to make changes in nutrition and exercise
+
+61
+00:05:08,320 --> 00:05:11,860
+لأنه زي ما قلنا كتير من الناس دول أصلا كانت
+
+62
+00:05:11,860 --> 00:05:16,500
+ممارساتهم التخذوية سيئة كتير على وعلى ال genetic
+
+63
+00:05:16,500 --> 00:05:23,380
+factors فهو الممارسة التخذوية كانت سيئةفاحنا
+
+64
+00:05:23,380 --> 00:05:28,080
+بنحاول نغير ال habits تبعتهم ونغير ال exercise
+
+65
+00:05:28,080 --> 00:05:31,140
+habits that will lead to improved metabolic
+
+66
+00:05:31,140 --> 00:05:36,240
+control من
+
+67
+00:05:36,240 --> 00:05:40,100
+الإجراءات اللي بعملها ناخد compositionاللي هو
+
+68
+00:05:40,100 --> 00:05:43,900
+المحتويات، بدل ما أنا معظم أكل سكر أو معظم أكل
+
+69
+00:05:43,900 --> 00:05:47,900
+دهون أو معظم أكل كذا، بحاول أعمل اللي هي الـmix
+
+70
+00:05:47,900 --> 00:05:52,540
+الطبيعي للـfood، meal plan developed with
+
+71
+00:05:52,540 --> 00:05:55,500
+dietician، يفضل حقيقة، خاصة لما يكون السكر من
+
+72
+00:05:55,500 --> 00:05:59,220
+النوع الـuncontrolled، ويكون طبعا زي ما قلنا
+
+73
+00:05:59,220 --> 00:06:00,460
+balanced،
+
+74
+00:06:02,490 --> 00:06:07,990
+ومش ممنوع، هذا القناة اللي بنكررها دايما، مش ممنوع
+
+75
+00:06:07,990 --> 00:06:15,930
+أي أكل عن أي مريض سكر، لكن خير الأمور الوسط ودايما
+
+76
+00:06:15,930 --> 00:06:20,470
+بالاتفاق مع الـ«دياتيشين» وممكن واحد يأكل بعض
+
+77
+00:06:20,470 --> 00:06:24,750
+الحاجات الـExercise هو essential part of diabetes
+
+78
+00:06:24,750 --> 00:06:28,890
+managementالـ exercise مش مقصود فيه يا جماعة ألبس
+
+79
+00:06:28,890 --> 00:06:34,750
+ال joking suit و ال boat و أروح أجري على البحر،
+
+80
+00:06:34,750 --> 00:06:41,950
+انما ال exercise جزء منه المشي
+
+81
+00:06:43,330 --> 00:06:48,150
+خطوات كثيرة خلال النهار هذا المقصود في الموضوع
+
+82
+00:06:48,150 --> 00:06:53,230
+طبعا لو صحيح في بعض نوعيات الرياضات لمانع من ذلك
+
+83
+00:06:53,230 --> 00:06:57,770
+لكن حتى لا نثقل على الناس و لازم كل يوم تنزل تجري
+
+84
+00:06:57,770 --> 00:07:01,990
+لإنما الحركة العادية ولكن بصورة أكبر و بصورة تكون
+
+85
+00:07:01,990 --> 00:07:06,850
+فعالة يبدو كويسة وإيش بيعمل ال exercise increase
+
+86
+00:07:06,850 --> 00:07:12,480
+insulin sensitivityبنزل السكر وبنزل اللي هو الـ
+
+87
+00:07:12,480 --> 00:07:17,620
+Insulin Resistance توكسمول Carbohydrate Snakes كل
+
+88
+00:07:17,620 --> 00:07:21,240
+30 Minutes During Exercise To Prevent Hypoglycemia
+
+89
+00:07:21,240 --> 00:07:24,700
+لما احنا عارفين ان رياضة تستهلك الطاقة، تستهلك
+
+90
+00:07:24,700 --> 00:07:30,270
+السكر، فانا ماقدش ادخل العيان في Hypoglycemiaفهو
+
+91
+00:07:30,270 --> 00:07:34,650
+ممكن ياكله أي حاجة خفيفة و هو ماشي طبعا مش واجبة،
+
+92
+00:07:34,650 --> 00:07:40,310
+snakes، يعني حاجة سريعة بحيث أن السكر يظل محافظ
+
+93
+00:07:42,090 --> 00:07:46,430
+ممكن نعمل أكسرسايز بعد الميال أكسرسايز should be
+
+94
+00:07:46,430 --> 00:07:54,010
+individualized كل بني آدم حسب وضعه مريض السكر
+
+95
+00:07:54,010 --> 00:07:58,650
+وعنده مشاكل مفاصل تختلف أكسرسايزه عن مريض السكر
+
+96
+00:07:58,650 --> 00:08:02,390
+واللي عنده ضغط وهكذا monitor blood glucose level
+
+97
+00:08:02,390 --> 00:08:06,090
+before during and after أكسرسايز ضروري طبعا من
+
+98
+00:08:06,090 --> 00:08:11,660
+الأكسرسايز المحبوبة جدا جدا غير المشياللي هي موضوع
+
+99
+00:08:11,660 --> 00:08:18,320
+السباحة swimming من
+
+100
+00:08:18,320 --> 00:08:23,060
+الأساسيات بتاعة الـ diabetes mellitus اللي هو الـ
+
+101
+00:08:23,060 --> 00:08:25,900
+monitoring blood glucose اللي احنا بيقول عنها الـ
+
+102
+00:08:25,900 --> 00:08:30,120
+self monitoring blood glucose وهذا من الأساسيات ال
+
+103
+00:08:30,120 --> 00:08:36,000
+schooling أو التعليمهؤلاء المرضى لازم أعلمه من بدر
+
+104
+00:08:36,000 --> 00:08:40,200
+يعمل «self-monitoring» يفحص لنفسه، ليش؟ لأن هو
+
+105
+00:08:40,200 --> 00:08:44,460
+بيصير بعد ذلك مش معقول كل شويه بدي أخد decisionمن
+
+106
+00:08:44,460 --> 00:08:47,240
+اللي بتصل في الطبيب أو في الـDietician لأ إنما هو
+
+107
+00:08:47,240 --> 00:08:52,660
+بتعلم بعلمه عن طريق بعين القوائم لما يكون سكرك كذا
+
+108
+00:08:52,660 --> 00:08:57,640
+أخد كذا و أعطي كذا و هيك يعني فـself monitoring of
+
+109
+00:08:57,640 --> 00:09:02,260
+blood glucose SMBG allows self management decision
+
+110
+00:09:02,260 --> 00:09:06,060
+regarding diet exercise and medication important
+
+111
+00:09:06,060 --> 00:09:11,210
+for detecting episodic hyperglycemiaوهيبوغلايسيميا
+
+112
+00:09:11,210 --> 00:09:15,050
+وpatient education زي ما قلنا is crucial مافيش
+
+113
+00:09:15,050 --> 00:09:19,410
+مجال لازم يتعلم لأنه في أحضن اللحظات لازم يتمد على
+
+114
+00:09:19,410 --> 00:09:23,830
+نفسه حتى لو كان طفل صغير احنا أنا بعرف أطفال سبعة
+
+115
+00:09:23,830 --> 00:09:29,130
+و تمان سنين اتعلم كيف يعمل self-monitoring أمر سهل
+
+116
+00:09:29,130 --> 00:09:35,700
+وبسيط من الحاجات اللي تقريبا الآنبحاول العلم أو
+
+117
+00:09:35,700 --> 00:09:39,660
+الطب يوجدها اللي هي بما أنه أنا في type one على
+
+118
+00:09:39,660 --> 00:09:44,440
+سبيل المثال عندي بانكرياس Ilet cells أو beta cells
+
+119
+00:09:44,440 --> 00:09:51,020
+destructed اللي هو موضوع بانكرياس transplantation
+
+120
+00:09:53,750 --> 00:09:57,130
+وطبعاً هذا بنعمله حقيقة في الـDiabetes I ويكون
+
+121
+00:09:57,130 --> 00:10:02,510
+معاه اللي هو End-stage Renal Disease ويكون لديه أو
+
+122
+00:10:02,510 --> 00:10:07,170
+مخطط لديه ترانسبلانت كيدنيا تخلص من الحاجة من
+
+123
+00:10:07,170 --> 00:10:10,110
+الإنسولين الإكسيجيناسي بعد هيك، محتاج إنسولين
+
+124
+00:10:10,110 --> 00:10:13,750
+براني، ويمكن أيضا تخلص من الـhypoglycemia
+
+125
+00:10:13,750 --> 00:10:17,160
+والـhyperglycemiaبالمناسبة بس لأنه أجاني سؤال
+
+126
+00:10:17,160 --> 00:10:23,080
+الـ«Gestational Diabetes» أحنا أعتبرناه نوع من
+
+127
+00:10:23,080 --> 00:10:27,990
+أنواع الـType 2والـ Type 2 بنقدر ندّي علاج أي oral
+
+128
+00:10:27,990 --> 00:10:32,990
+مثلا ديابتكا لكن لأ ال gestational بندّي انسولين
+
+129
+00:10:32,990 --> 00:10:37,690
+فكان السؤال ليش طيب ندّي انسولين وندّيهوش oral انت
+
+130
+00:10:37,690 --> 00:10:41,790
+ديابتكا هو عشان ال side effects تبعت اللي علاجه أن
+
+131
+00:10:41,790 --> 00:10:46,710
+الانسولين ليس side effects على الجنين لكن ال oral
+
+132
+00:10:46,710 --> 00:10:53,770
+antidiabetic فيلها side effects ففضلنا موضوع ال
+
+133
+00:10:55,000 --> 00:11:05,980
+الإنسلين من الـ Complication بتاعة الـ Diabetes من
+
+134
+00:11:05,980 --> 00:11:11,080
+الـ Complication فينا طبعا Acute Complications حنا
+
+135
+00:11:11,080 --> 00:11:18,360
+دلوقت دخلنا على موضوع غير في الأهمية ليش؟
+
+136
+00:11:18,360 --> 00:11:23,690
+لأنه خلاص الواحد ممكن يعيشويتعايش طبعا يتعايش الآن
+
+137
+00:11:23,690 --> 00:11:30,250
+هذه كلمة صرعت العصر الحالي التعايش ممكن واحد
+
+138
+00:11:30,250 --> 00:11:35,290
+يتعايش مع الـ diabetes لكن يريد أن ياخد باله من
+
+139
+00:11:35,290 --> 00:11:42,250
+الـ acute and chronic complications إيش ال acute
+
+140
+00:11:42,250 --> 00:11:47,150
+complications؟ على رأسها الـ hypoglycemia وحاجة
+
+141
+00:11:47,150 --> 00:11:52,220
+تانية الـ hyperglycemiaالـ hypoglycemia إنه أنا
+
+142
+00:11:52,220 --> 00:11:57,740
+أدي too much insulin أو oral agents يعني، بغض
+
+143
+00:11:57,740 --> 00:12:05,100
+النظر، إنما أدي بزيادة، وماكنش باخد سكر، يعني
+
+144
+00:12:05,100 --> 00:12:09,840
+بمعنى مثلا، أنا باخد قبل 1.25 ساعة الإنسولين
+
+145
+00:12:12,400 --> 00:12:16,220
+على أساس إنه شوي يلا بدأ أفطر أو بدأ أتغدى أو كذا
+
+146
+00:12:16,220 --> 00:12:21,840
+وبيجي بعد ما أخدت الإنسولين حاجة بتخليني أنسى أكل
+
+147
+00:12:21,840 --> 00:12:25,840
+أو أطلع مشوار أو استدعوني هان أو كذا، بشتغل
+
+148
+00:12:25,840 --> 00:12:29,680
+الإنسولين، بنزل السكر، بدخل في hypoglycemia، سلامة
+
+149
+00:12:29,680 --> 00:12:33,540
+سلمك وتعيش مشكلة كبيرة، hypoglycemia ممكن تأثرلي
+
+150
+00:12:33,540 --> 00:12:38,700
+وتعمللي brain damageيبقى لازم يكون in relationship
+
+151
+00:12:38,700 --> 00:12:43,460
+to glucose availability لازم أدي الإنسولين، بمعنى
+
+152
+00:12:43,460 --> 00:12:52,440
+لازم أعطي الإنسولين وأكل على طول، مش أخد يعني كمية
+
+153
+00:12:52,440 --> 00:12:57,950
+الأكل اللي أنا محتاجهاكيف تظهر الـ Hypoglycemia؟
+
+154
+00:12:57,950 --> 00:13:03,270
+Symptoms and Signs Confusion Irritability Anxiety
+
+155
+00:13:03,270 --> 00:13:10,520
+Tachycardia Diaphoresis تعرق Tremor رجةبصير واحد
+
+156
+00:13:10,520 --> 00:13:15,240
+زي الجعان، weakness تعب، visual disturbance، مش
+
+157
+00:13:15,240 --> 00:13:19,720
+شايف زي الناس، if untreated، بيصير انه loss of
+
+158
+00:13:19,720 --> 00:13:23,320
+consciousness، lock، loss of consciousness،
+
+159
+00:13:23,320 --> 00:13:28,660
+seizure، كومة، وممكن death طب العلاقة الآن يا
+
+160
+00:13:28,660 --> 00:13:34,900
+جماعة، لو انا، لو انا في حد، بعرف انه سكري، لكن
+
+161
+00:13:34,900 --> 00:13:40,660
+اغيب، وانا مش عارفهل اللي صار معاه hypoglycemia
+
+162
+00:13:40,660 --> 00:13:45,500
+ولا hyperglycemia؟ أكل بزيادة ولا ماكلش؟ كيف
+
+163
+00:13:45,500 --> 00:13:54,020
+أتعامل معاه؟ أه، إيش رايك؟ إيش رايك؟ أتعامل معاه
+
+164
+00:13:54,020 --> 00:14:04,180
+hypoglycemia ولا hyperglycemia؟ صح، برضه صح أه،
+
+165
+00:14:04,180 --> 00:14:07,540
+بتعامل معاه أنه hypoglycemia، ليش؟
+
+166
+00:14:10,450 --> 00:14:15,530
+لأنه هي الأخطر فانا باقد بمبدأ الحيطة أكتر طب إيش
+
+167
+00:14:15,530 --> 00:14:23,070
+بعمل؟ بدي سكر طب افرض هو مافد هو hyperglycemia ودت
+
+168
+00:14:23,070 --> 00:14:27,850
+السكر ممكن مش هيضره كتير لكن لو مادتوش سكر هينضر
+
+169
+00:14:27,850 --> 00:14:32,510
+جدا يبقى دائما بتعامل في حالات ال coma في مرضى
+
+170
+00:14:33,160 --> 00:14:37,460
+الديابتس على أساس إنه hypoglycemia لما يكون عنده
+
+171
+00:14:37,460 --> 00:14:40,260
+acute loss of consciousness على أساس إنه
+
+172
+00:14:40,260 --> 00:14:44,900
+hypoglycemia مش hyperglycemia لأن الـ hypoglycemia
+
+173
+00:14:44,900 --> 00:14:50,780
+أخطر من الـ hyperglycemia في النتائج تبعتها يبقى
+
+174
+00:14:50,780 --> 00:14:54,460
+من أهم ال acute complications of diabetes mellitus
+
+175
+00:14:54,460 --> 00:15:00,260
+اللي هي الـ hypoglycemia طب إيش يعمل لو صار عنده
+
+176
+00:15:00,260 --> 00:15:06,280
+hypoglycemiaهو طبعا إذا كان ولّا لسه by
+
+177
+00:15:06,280 --> 00:15:12,880
+consciousness صح صح بس الله مقن tremor و hunger و
+
+178
+00:15:12,880 --> 00:15:16,160
+weakness و visual disturbance تاكد كارديا confused
+
+179
+00:15:16,160 --> 00:15:19,480
+بدي حاجة fruit choice
+
+180
+00:15:22,280 --> 00:15:29,680
+عشان حاجة simple carbohydrate أو إذا موجود برا فيه
+
+181
+00:15:29,680 --> 00:15:37,920
+أقراس سكر موجودة برا ممكن يعطيها الواحد، طبعا مش
+
+182
+00:15:37,920 --> 00:15:44,500
+أروح أدي جطعة بيتزا، مينفعش، ليش؟ لأن هذا complex
+
+183
+00:15:44,500 --> 00:15:48,460
+carbohydrate، بدي أنا حاجة simple على طول، بمجرد
+
+184
+00:15:48,460 --> 00:15:54,640
+ما اشربهاتكون متوسطة وغيرت وضع الـhypoglycemia
+
+185
+00:15:54,640 --> 00:15:58,860
+avoid sweets with fat لأن عملية الabsorption تبقى
+
+186
+00:15:58,860 --> 00:16:04,500
+أصعب وممكن كل 15 دقيقة أعملها لحد ما يصير السكر
+
+187
+00:16:04,500 --> 00:16:09,780
+تبعه فوق السبعين تقريبا و بعدين يأكل طبعا هذه حاجة
+
+188
+00:16:09,780 --> 00:16:13,640
+بتصير يوميا أن يصير عنده symptoms and signs of
+
+189
+00:16:13,640 --> 00:16:18,190
+hypoglycemiaلسة مش انا مابحكيش كلها الـ loss of
+
+190
+00:16:18,190 --> 00:16:20,550
+consciousness و seizure و coma و death لأ مش هذا
+
+191
+00:16:20,550 --> 00:16:25,270
+الموضوع لأ إنما هذه عوارض طبيعية فممكن هو ياخدلها
+
+192
+00:16:25,270 --> 00:16:31,610
+ياخدلها سكر سمبل سكر الآن في لكن حاجة تانية بتصير
+
+193
+00:16:31,610 --> 00:16:35,310
+لو أنا عندي واحد coma مثلا أو seizure بدي حاجة
+
+194
+00:16:35,310 --> 00:16:36,330
+اسمها glucagon
+
+195
+00:16:40,210 --> 00:16:45,890
+هو بدي إما IM أو بدي subcutaneous عبارة عن إبرة
+
+196
+00:16:45,890 --> 00:16:52,630
+pen، زي الجلم، بغزها في جسمه، وهذا إيش بيعمل؟ هو
+
+197
+00:16:52,630 --> 00:16:57,890
+عبارة عن الـ counter player تبع من؟ تبع الإنسولين،
+
+198
+00:16:57,890 --> 00:17:02,770
+هو دائما على تضاد من الإنسولين، الإنسولين بنزل
+
+199
+00:17:02,770 --> 00:17:04,890
+السكر، وهذا برفع السكر
+
+200
+00:17:07,440 --> 00:17:12,580
+وكمان ممكن ينعطق بنفس الصورة من المشاكل والـ acute
+
+201
+00:17:12,580 --> 00:17:15,460
+complications اللي بتسير في الـ diabetes طبعا في
+
+202
+00:17:15,460 --> 00:17:20,020
+حاجات أكتر بس أنا هنا حاكي عنها اللي هي الـ
+
+203
+00:17:20,020 --> 00:17:23,780
+diabetic ketoacetosis اللي احنا بيقول عنها ايش؟
+
+204
+00:17:23,780 --> 00:17:24,300
+DKA
+
+205
+00:17:28,020 --> 00:17:33,780
+Diabetic ketoacidosis يعني سكري كيتو من كيتونز لأن
+
+206
+00:17:33,780 --> 00:17:38,400
+مشكلتي هي مشكلة كيتونز و Acidosis حموضة حموضة
+
+207
+00:17:38,400 --> 00:17:42,920
+الكيتونز اللي بتصير وهذه عبارة عن life-threatening
+
+208
+00:17:42,920 --> 00:17:48,860
+complication of type 1ممكن تجي في type 2 لكن هي
+
+209
+00:17:48,860 --> 00:17:51,820
+mainly في type 1 في كتير والأحيان تبقى ال fast
+
+210
+00:17:51,820 --> 00:17:56,740
+presentation عند الأطفال أو عند الأصغار في ال type
+
+211
+00:17:56,740 --> 00:18:02,180
+1 اللي هي بيبقى عنده سكري هو لكن بيجي infection
+
+212
+00:18:02,180 --> 00:18:07,620
+بتيجي أي stressors زي psychological stressors أو
+
+213
+00:18:07,620 --> 00:18:12,820
+واحد بياخد انسلين وما خدش الانسلين تبعه نسي، ضاع،
+
+214
+00:18:12,820 --> 00:18:17,600
+ماعرفش إيش بدوشيأخذ الإنسولين in compliance، عدم
+
+215
+00:18:17,600 --> 00:18:21,960
+تعاون، وما أخدش الإنسولين بأي صورة من الصور، ممكن
+
+216
+00:18:21,960 --> 00:18:26,560
+يقلل السكر إلى درجات 400، 500، 600، 700، ويدخل في
+
+217
+00:18:26,560 --> 00:18:31,400
+DKA، يمجد الـDKA هو عبارة عن hyperglycemia
+
+218
+00:18:31,400 --> 00:18:36,780
+وtypical symptoms تبعتهأو طبعا في الـ Undiagnosed
+
+219
+00:18:36,780 --> 00:18:39,740
+Diabetes، واحد مش معروف انه Diabetes، اتعرض ل
+
+220
+00:18:39,740 --> 00:18:43,280
+-infection، لجينا عنده إيش، صار decay، من
+
+221
+00:18:43,280 --> 00:18:45,840
+الـTypical Symptoms، vomiting, dehydration,
+
+222
+00:18:47,280 --> 00:18:50,220
+gasping, breathing, confusion and occasionally
+
+223
+00:18:50,220 --> 00:18:58,230
+comaحقيقة صورة عنيفة من الـ Hyperglycemia والعلاج
+
+224
+00:18:58,230 --> 00:19:02,330
+تبعها بسيط جدا، بس أهم شيء يكون الـ index of
+
+225
+00:19:02,330 --> 00:19:08,510
+suspicion عند الواحد عالي وكبير، IV fluid، ليش؟
+
+226
+00:19:08,510 --> 00:19:12,390
+لأن هدول المرضى بدخلوا في dehydration، لازم أدي
+
+227
+00:19:12,390 --> 00:19:16,490
+إنسولين، ليش؟ عشان suppress the production of
+
+228
+00:19:16,490 --> 00:19:22,100
+ketones in the bodyوأشوف إيش ال underlying cause،
+
+229
+00:19:22,100 --> 00:19:26,660
+إيش اللي عملني المشكلة، والله إذا كان زي ما
+
+230
+00:19:26,660 --> 00:19:30,600
+بقولوها infection، هعالج ال infection، إذا مزعلة
+
+231
+00:19:30,600 --> 00:19:36,360
+مع جوزها، مرضيها، إذا بدأت طلج،غير الوضع المهم
+
+232
+00:19:36,360 --> 00:19:40,080
+يعني بدنا نشوف حل في الموضوع لازم أحل الموضوع
+
+233
+00:19:40,080 --> 00:19:43,320
+close observation to prevent and identify
+
+234
+00:19:43,320 --> 00:19:46,100
+complications و بالمناسبة الـ diabetic
+
+235
+00:19:46,100 --> 00:19:51,080
+ketoacidoses غالبيتهم 90% بدخلوا عناية مركزة ليش؟
+
+236
+00:19:51,080 --> 00:19:54,080
+لأنه زي ما قلنا is a vigorous condition of
+
+237
+00:19:54,080 --> 00:19:58,920
+hyperglycemia ممكن تعملي مشاكل كبيرة و هيك احنا
+
+238
+00:19:58,920 --> 00:20:03,600
+بننجل لمين؟ بننجل لل chronic complication of
+
+239
+00:20:03,600 --> 00:20:08,200
+diabetesالـ Chronic Complication كرونك يعني يعني
+
+240
+00:20:08,200 --> 00:20:10,880
+يعني بدي سنوات عشان تصير الـ Complication هاي ممكن
+
+241
+00:20:10,880 --> 00:20:16,980
+توصل عشر سنوات خمسة عشر سنة بعد ما اكتشف اللي هو
+
+242
+00:20:16,980 --> 00:20:25,400
+ال ال diabetes من ال chronic ال chronic
+
+243
+00:20:25,400 --> 00:20:28,780
+complications
+
+244
+00:20:28,780 --> 00:20:33,510
+of diabetes mellitusعنا حاجات بنقول عنها
+
+245
+00:20:33,510 --> 00:20:38,570
+macrovascular وهذا طبعا أنا بحكي عنها معظمها بتكون
+
+246
+00:20:38,570 --> 00:20:48,610
+تأثيرها على الشراين أو الشعيرات الدموية وهي
+
+247
+00:20:48,610 --> 00:20:53,270
+بالتالي بتأثرلي على العضو المقصود يعني شغلها
+
+248
+00:20:53,270 --> 00:20:59,650
+مشكلتها في الشراينأو طبعاً بتأثرلي على الأعصاب
+
+249
+00:20:59,650 --> 00:21:04,010
+الطرفية مباشرة أو الشراين اللي بتغذيها هيك تقريبا
+
+250
+00:21:04,010 --> 00:21:11,590
+الفكرة يعني بمعنى بتعملي حاجة اسمها Angiopathy مرض
+
+251
+00:21:11,590 --> 00:21:17,230
+أنجيو اللي هو الواعي الدموي blood vessel disease
+
+252
+00:21:17,230 --> 00:21:22,970
+من خلال تأثيرها على blood vessel بيصير عملية
+
+253
+00:21:22,970 --> 00:21:28,950
+المشاكل في العضو المكسورماكروفاسكولر معناته بتصيب
+
+254
+00:21:28,950 --> 00:21:34,490
+للـ large and mid-sized vessel وطبعاً هذا إليه
+
+255
+00:21:34,490 --> 00:21:39,830
+علاقة حقيقة بالـ lipid و metabolism of diabetes to
+
+256
+00:21:39,830 --> 00:21:42,550
+altered lipid يعني علاقة بين ال lipid metabolism
+
+257
+00:21:42,550 --> 00:21:48,290
+مع ال diabetes طبعا بنتصور الأماكن اللي الشرايين
+
+258
+00:21:48,290 --> 00:21:55,490
+فيها بتتأثر بارتفاع السكر شراين الدماغ شراين القلب
+
+259
+00:21:57,600 --> 00:22:03,180
+الشرايين الطرفية طب هو ضايل حاجة مثلا مش ضايل حاجة
+
+260
+00:22:03,180 --> 00:22:07,460
+صحيح كلامه هو هدوله هو أي البني آدم زي ما يقولوها
+
+261
+00:22:07,460 --> 00:22:13,700
+مخه قلبه وشرايين طرفية و micro vascular وهدى
+
+262
+00:22:13,700 --> 00:22:17,920
+بنشوفها due to thickening of a small vessel
+
+263
+00:22:17,920 --> 00:22:23,160
+membrane هنجي لايلها بعد شوية طيب هوالـ Epidemic
+
+264
+00:22:23,160 --> 00:22:30,800
+of Chronic Diabetic Complications بتصيب كتير يعني
+
+265
+00:22:30,800 --> 00:22:36,100
+كتير كتير؟ أه يعني متوقع إنه من الأرقام اللي
+
+266
+00:22:36,100 --> 00:22:45,680
+حكيناها إن حوالي في 2025 كان 380 مليون إنسان
+
+267
+00:22:47,110 --> 00:22:53,170
+انصاب بمشاكل في الـ Chronic Complications وزي ما
+
+268
+00:22:53,170 --> 00:22:59,990
+احنا شايفين على في الصورة انه Retinopathy اللي هي
+
+269
+00:22:59,990 --> 00:23:07,510
+شبكية العين وبالتالي بتعمل Blindness in people of
+
+270
+00:23:07,510 --> 00:23:11,770
+working age يعني مش بس Blindness لو واحد عمره 70
+
+271
+00:23:11,770 --> 00:23:17,000
+ولا 80 سنة لأ لو واحد بشتغل يعني لسه منتجريتينو
+
+272
+00:23:17,000 --> 00:23:21,840
+باثي ريتينو
+
+273
+00:23:21,840 --> 00:23:29,720
+باثي ريتينو شبكية باثي مرض شبكية نفرو كيلة نفرو
+
+274
+00:23:29,720 --> 00:23:33,620
+باثي في الكيلة بيصير عندك مشاكل 16% of new
+
+275
+00:23:33,620 --> 00:23:39,060
+patients needing renal replacement therapy ناس
+
+276
+00:23:39,060 --> 00:23:46,420
+بتحتاج زرع كيلة بسبب السكريأيش كمان؟ في عندنا كمان
+
+277
+00:23:46,420 --> 00:23:50,540
+macrovascular and cerebrovascular disease اتنين
+
+278
+00:23:50,540 --> 00:23:55,840
+لتلات مرات إمكانية الإصابة أو ال risk of coronary
+
+279
+00:23:55,840 --> 00:24:03,020
+heart disease and stroke food problems 15% of
+
+280
+00:24:03,020 --> 00:24:09,800
+people with diabetes developFood Ulcers 5-15% of
+
+281
+00:24:09,800 --> 00:24:13,680
+people with diabetic food ulcers need amputation
+
+282
+00:24:13,680 --> 00:24:21,580
+Ergtyral dysfunction 50% من الرجال ممكن يؤثر علي
+
+283
+00:24:21,610 --> 00:24:25,990
+with long-standing diabetes يكون عندهم erectile
+
+284
+00:24:25,990 --> 00:24:31,270
+dysfunction هو طبعا ماعرفش إذا الكل فاهمها لكن على
+
+285
+00:24:31,270 --> 00:24:37,430
+أي حال هو عبارة عن ضعف في الانقصاب عند الرجالالـ
+
+286
+00:24:37,430 --> 00:24:42,410
+50% اللي عنده سكري يصاب بمشكلة زي هيك يعني أنا
+
+287
+00:24:42,410 --> 00:24:48,890
+بحكي جاعد عن مشاكل بتأثر على الحياة مباشرة سواء في
+
+288
+00:24:48,890 --> 00:24:52,710
+الـ Retina، سواء في الكلة، سواء في القلب، سواء في
+
+289
+00:24:52,710 --> 00:24:57,590
+الـ Sexual Life، هذه كلها بتتأثر بصورة أو بأخرى
+
+290
+00:24:57,590 --> 00:25:03,930
+بموضوع اللي هو Diabetes Militas
+
+291
+00:25:06,260 --> 00:25:10,060
+الـ Microvascular كمان مرة نعود نعيد Microvascular
+
+292
+00:25:10,060 --> 00:25:14,360
+زي الـ Retinopathy، leading cause of new
+
+293
+00:25:14,360 --> 00:25:18,940
+blindness، واحد ما عندهوش برض في العينولكن صار
+
+294
+00:25:18,940 --> 00:25:23,440
+عنده سكري من الأسباب الأساسية للـblindness فيها
+
+295
+00:25:23,440 --> 00:25:26,580
+وفي مرضى السكر اللي بيصير عنده hemorrhage بيصير
+
+296
+00:25:26,580 --> 00:25:31,060
+retinal detachment انفصال شبكي، nephropathy بيؤدي
+
+297
+00:25:31,060 --> 00:25:36,160
+لإنه يكون في عندي damage للشرايين اللي رايحة على
+
+298
+00:25:36,160 --> 00:25:41,380
+الوحدات التصفية الـglomeruli، وحدات التصفية في
+
+299
+00:25:41,380 --> 00:25:50,750
+الكلةو فبتدى الى ESRD فكروا
+
+300
+00:25:50,750 --> 00:25:59,710
+معاى لأ مش هنعملها assignment ولا حاجة المرة هى
+
+301
+00:25:59,710 --> 00:26:05,640
+بدون assignment ان شاء الله تعالىإيش الـ ESRD؟
+
+302
+00:26:05,640 --> 00:26:10,940
+اللي هو End Stage Renal Disease أو Failure End
+
+303
+00:26:10,940 --> 00:26:14,500
+Stage Renal Disease End Stage Renal Disease يعني
+
+304
+00:26:14,500 --> 00:26:20,040
+مرض كلا في نهايته End Stage Renal Disease من
+
+305
+00:26:20,040 --> 00:26:24,740
+الأسباب الأساسية السكري وطبعا حكينا نسبة 18% بسيطة
+
+306
+00:26:24,740 --> 00:26:28,080
+الـ Neuropathy آه، هنا، هنا، هنا عندنا حاجة
+
+307
+00:26:28,080 --> 00:26:35,750
+بنشوفها باستمرارإنه أن الـ neuropathy مرض في العصب
+
+308
+00:26:35,750 --> 00:26:39,710
+sensory neuropathy الـ sensation يصير لـ loss
+
+309
+00:26:39,710 --> 00:26:44,070
+abnormal sensation مش بس loss إنما كمان abnormal
+
+310
+00:26:44,070 --> 00:26:49,050
+pain of hands or foot can progress to partial أو
+
+311
+00:26:49,050 --> 00:26:51,490
+complete loss of sensitivity to touch أو
+
+312
+00:26:51,490 --> 00:26:57,250
+temperature high risk of injury without pain في إن
+
+313
+00:26:57,250 --> 00:27:01,120
+الـ autonomic neuropathyالـ Sensory يعني فيها
+
+314
+00:27:01,120 --> 00:27:04,580
+sensation الـ Autonomic اللي هي الأعضاء اللي
+
+315
+00:27:04,580 --> 00:27:12,360
+بتشتغل بدون الإحساس لكن بتشتغل من نفسها زي بيصير
+
+316
+00:27:12,360 --> 00:27:17,900
+إن حاجة اسمها «hypoglycemic unawareness» تصوروا إن
+
+317
+00:27:17,900 --> 00:27:26,230
+هدول المرضىفي الآخر نتيجة الـ Autonomic Neuropathy
+
+318
+00:27:26,230 --> 00:27:30,090
+بيبطل يحس إنه صار عنده Hypoglycemia شوفت الـ
+
+319
+00:27:30,090 --> 00:27:34,670
+Symptoms اللي اتحدثنا عنها قبل شوية إنه يحس إنه
+
+320
+00:27:34,670 --> 00:27:41,150
+عنده confusion، إنه عنده جعان، إنه tremor، هذا كله
+
+321
+00:27:41,150 --> 00:27:44,990
+بيبطل يحصل Hypoglycemic Unawareness
+
+322
+00:27:47,240 --> 00:27:53,340
+بيقدرش يستوعب جسمه إنه دخل في hypoglycemia إيش
+
+323
+00:27:53,340 --> 00:27:58,820
+كمان؟ silent MI إيش يعني؟ myocardial infection
+
+324
+00:27:58,820 --> 00:28:05,260
+silent بيصير تجيله MI وما
+
+325
+00:28:05,260 --> 00:28:11,780
+يحسش فيها يعني بمعنى أخر بموت بهدوءبدون ما يحس
+
+326
+00:28:11,780 --> 00:28:18,340
+بوجع، عادة هو الوجع اللي بيصير نتيجة autonomic
+
+327
+00:28:18,340 --> 00:28:24,660
+nerves و nerves طبعا بيحس الواحد بالpain، بدي
+
+328
+00:28:24,660 --> 00:28:30,880
+إشارة لدماغ، الدماغ، و هيك، لكن في حالة السكر
+
+329
+00:28:30,880 --> 00:28:35,260
+بيصير عندي silent MI، rectile dysfunction و
+
+330
+00:28:35,260 --> 00:28:39,110
+decreased libidoوالـ Neurogenic Bladder ممكن يصير
+
+331
+00:28:39,110 --> 00:28:42,030
+عندنا مشكلة في الـ what؟ في الـ urine retention
+
+332
+00:28:42,030 --> 00:28:47,610
+هذه من الحاجات اللي بتصير في Neuropathy لكن الـ
+
+333
+00:28:47,610 --> 00:28:49,830
+Neuropathy بتلعب دور مش بس في الـ Sensory and
+
+334
+00:28:49,830 --> 00:28:54,430
+Autonomic Neuropathy إنما بتلعب دور كمان في الـ
+
+335
+00:28:54,430 --> 00:29:01,020
+Diabetic Food لأن احنا حكينا قبل شويةالمشكلة هي أن
+
+336
+00:29:01,020 --> 00:29:06,860
+المشكلة المكروفسكولارية بيعملني peripheral vein
+
+337
+00:29:06,860 --> 00:29:11,640
+disease بيعملني
+
+338
+00:29:11,640 --> 00:29:16,920
+peripheral artery disease وهذا بيقللني supply of
+
+339
+00:29:16,920 --> 00:29:21,140
+oxygen و ال white disease و ال nutrition فبيعملني
+
+340
+00:29:21,140 --> 00:29:28,070
+مشكلة في الفي الـ food بيجي زيادة عليها sensory
+
+341
+00:29:28,070 --> 00:29:31,750
+neuropathy بنجرح طلعوا لا في تغذية زي الناس لل
+
+342
+00:29:31,750 --> 00:29:37,770
+food و بيجي جرح عليها ففي الآخر بيصير عنده
+
+343
+00:29:37,770 --> 00:29:44,290
+diabetic foodمهم جدا جدا مريض السكري يتعلم كيف
+
+344
+00:29:44,290 --> 00:29:49,330
+يجسج أضافره ينتبه من الصدمات للرجل لأنه أي جرح
+
+345
+00:29:49,330 --> 00:29:54,070
+ممكن لأنه شريينه اللي نازلة عن الرجل اللي هو ال
+
+346
+00:29:54,070 --> 00:29:58,870
+peripheral vascular disease اللي عنده أصلا مش واصل
+
+347
+00:29:58,870 --> 00:30:05,540
+دم كفاية و ال sensoryنيرو باتي شغالة على بودنه
+
+348
+00:30:05,540 --> 00:30:10,540
+فانا عندي عاملين اتنين بيمنعوا انه اي جرح يطيب
+
+349
+00:30:10,540 --> 00:30:15,120
+بسهولة او ببساطة فانا لازم انتبه للقضية هذه لازم
+
+350
+00:30:15,120 --> 00:30:19,200
+اعلمه teach prevention of ulceration and injury
+
+351
+00:30:19,200 --> 00:30:25,040
+وهذا الديابتك فوت من الحاجات الأساسية اللي ممكن
+
+352
+00:30:25,040 --> 00:30:32,280
+تؤدي ل non-traumatic non-traumatic amputation
+
+353
+00:30:33,320 --> 00:30:35,620
+«نانتراوماتيك» يعني مش بحاجة السيارة أو لحاجة أنم
+
+354
+00:30:35,620 --> 00:30:38,420
+السكري، يبقى أهم سبب في الـ«نانتراوماتيك
+
+355
+00:30:38,420 --> 00:30:42,460
+أمبيوتاشن» هو الـDiabetes mellitus، من الحاجات
+
+356
+00:30:42,460 --> 00:30:47,360
+اللي كمان بتأثر هو بيصير هدول الناس ليبقى اللي
+
+357
+00:30:47,360 --> 00:30:51,460
+يصير عندهم infection، نتيجة immune deficiency،
+
+358
+00:30:51,460 --> 00:30:57,600
+وأصلا أي واحد بمرض، نتيجة الـsensory neuropathy
+
+359
+00:30:57,600 --> 00:31:01,540
+اللي حكينا عنها قبل شوية، بتصير صعوبة جدا أنك
+
+360
+00:31:01,540 --> 00:31:06,480
+تنتبهلهاوزي ما قلنا الـ decreased circulation
+
+361
+00:31:06,480 --> 00:31:11,160
+delay or prevent immune response هذا ال infection
+
+362
+00:31:11,160 --> 00:31:16,820
+بخصوص اللي هو الـ diabetic foot هذه عوامل الـ
+
+363
+00:31:16,820 --> 00:31:19,960
+immune deficiency delay detection due to sensory
+
+364
+00:31:19,960 --> 00:31:23,100
+neuropathy decreased circulation هذه كلها بتقدر
+
+365
+00:31:23,100 --> 00:31:28,020
+لإيش الـ diabetic foot يعني زي ما قلتلكوا موضوعنا
+
+366
+00:31:28,020 --> 00:31:35,030
+موضوعمش المفروض أنه بده خلينا نقول ممكن محاضرتين،
+
+367
+00:31:35,030 --> 00:31:38,610
+لأ، تلت محاضرات، ساعة و نص، ساعة و نص، ساعة و نص،
+
+368
+00:31:38,610 --> 00:31:42,630
+ساعة و نص، ساعة و نص، ساعة و نص، ساعة و نص، ساعة و
+
+369
+00:31:42,630 --> 00:31:42,790
+نص، ساعة و نص، ساعة و نص، ساعة و نص، ساعة و نص،
+
+370
+00:31:42,790 --> 00:31:42,850
+ساعة و نص، ساعة و نص، ساعة و نص، ساعة و نص، ساعة و
+
+371
+00:31:42,850 --> 00:31:44,490
+ساعة و نص، ساعة و نص، ساعة و نص، ساعة و نص، ساعة و
+
+372
+00:31:44,490 --> 00:31:52,560
+نص، ساعة و نص،لكن هذا اللي طالع معانا الآن، إن شاء
+
+373
+00:31:52,560 --> 00:31:55,980
+الله، حتى نعدي هذه الفترة على خير، بالنسبة لموضوع
+
+374
+00:31:55,980 --> 00:31:58,760
+الـDiabetes، من أهم الموضوعات على الإطلاق، وانا
+
+375
+00:31:58,760 --> 00:32:02,880
+الآن بقولكوا جاعد، إن الموضوع هذا هو موضوع
+
+376
+00:32:02,880 --> 00:32:07,140
+الـHypertension، على الأجل من أي امتحان هتدخلوا
+
+377
+00:32:07,140 --> 00:32:12,540
+معايا في الباطنة، هيكونوا بمثل من 10% لـ20% من
+
+378
+00:32:12,540 --> 00:32:16,520
+الامتحان، في الغالب، أي امتحان النهائي أو حتى
+
+379
+00:32:16,520 --> 00:32:19,930
+الـ…ماشي، إن شاء الله تعالى،موضوعين مهمين غير
+
+380
+00:32:19,930 --> 00:32:23,210
+الامتحان إنهم هما موضوعين، موضوعين من مواضيع
+
+381
+00:32:23,210 --> 00:32:28,690
+الحياة المهمة اللي لازم أي حد بيشتغل في الحياة
+
+382
+00:32:28,690 --> 00:32:33,210
+العامة تبع الصحية الطبية سواء دكتور، سواء ممرض،
+
+383
+00:32:33,210 --> 00:32:36,450
+سواء علاج طبيعي، سواء مختبر، لازم يكون نوعا ما
+
+384
+00:32:36,450 --> 00:32:40,410
+بيفهم في الموضوعين هدول، الله يصبحكم بالخير
+
+385
+00:32:40,410 --> 00:32:41,830
+ويعطيكم العافية إن شاء الله السلامة
+
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/Pm6glBUJbFE.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/Pm6glBUJbFE.srt
new file mode 100644
index 0000000000000000000000000000000000000000..256da9c01591d0a7d8669cb69a0f0f389c5308b9
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/Pm6glBUJbFE.srt
@@ -0,0 +1,1711 @@
+1
+00:00:00,000 --> 00:00:03,320
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:03,320 --> 00:00:05,900
+الصلاة و السلام علي سيدنا محمد و سيد المرسلين
+
+3
+00:00:05,900 --> 00:00:12,860
+أجمعين اليوم حقيقة هنحكي في موضوع غاية في الأهمية
+
+4
+00:00:12,860 --> 00:00:19,100
+كثير منتشر عند بعضكم كمان، عند أهليكم، عند الكل
+
+5
+00:00:19,100 --> 00:00:26,260
+عملي جدا الموضوع هذا، مين عمره ما جا ه أو مين عمره 
+
+6
+00:00:26,260 --> 00:00:32,680
+أجاه اللي بنقول عنها الحزاز أو الحموضة يعني حاجة
+
+7
+00:00:32,680 --> 00:00:36,400
+حراقة شديدة أو نقول في ناس شعور عن تم المعدة
+
+8
+00:00:36,400 --> 00:00:42,700
+لفوق زي ما احنا شايفين في الصورة نار مشتعلة في
+
+9
+00:00:42,700 --> 00:00:46,540
+منطقة تم المعدة وطبعا تم المعدة موجودة في منطقة
+
+10
+00:00:46,540 --> 00:00:50,700
+الصدر يعني قريب من القلب قريب من الرئتين قريب من
+
+11
+00:00:50,700 --> 00:00:57,030
+حاجات كثيرة هذا إيش بيقولوا عنه؟ Gastro-Oesophageal
+
+12
+00:00:57,030 --> 00:01:03,130
+Reflux، مش Reflex، Reflux Disease الكل ممكن يكون
+
+13
+00:01:03,130 --> 00:01:09,630
+معاه نتيجة
+
+14
+00:01:09,630 --> 00:01:14,610
+أكل معين، طريقة معينة في الأكل، يصير عنده Gastro
+
+15
+00:01:14,610 --> 00:01:15,310
+يعني معدّى،
+
+16
+00:01:19,830 --> 00:01:25,230
+الحامض أو الأكل من المعدة إلى المريء وهو عبارة عن
+
+17
+00:01:25,230 --> 00:01:29,030
+Disease عالميا gastroesophageal reflux disease
+
+18
+00:01:29,030 --> 00:01:34,910
+اللي بنقول عنه إيش؟ GERD زي GERD يعني GERD اختصار
+
+19
+00:01:34,910 --> 00:01:39,250
+ل gastroesophageal reflux disease زي ما قلنا كثيرا
+
+20
+00:01:39,250 --> 00:01:43,170
+من الناس بيصير معاهم مين بيصير معها أو بيصير
+
+21
+00:01:43,170 --> 00:01:48,840
+معها زي هيك؟ حموضة أو شعور أو حزاز زي ما نقول عنها
+
+22
+00:01:48,840 --> 00:01:53,980
+أو ارتجاع حتى في بعض الناس في نصف الليل لما بتعشى 
+
+23
+00:01:53,980 --> 00:01:57,920
+متأخر ممكن يكون زي المخنوق ليش؟ لأنه صار عنده
+
+24
+00:01:57,920 --> 00:02:03,940
+ارتجاع للمادة الحامضة هذه وين طلعت من المعدة
+
+25
+00:02:03,940 --> 00:02:08,240
+للمريء ومش بس وراحت دخلت على الحنجرة راحت دخلت على
+
+26
+00:02:08,240 --> 00:02:11,320
+الـ larynx على الحنجرة وعلى الحنجرة وهذه من الحاجات
+
+27
+00:02:11,320 --> 00:02:16,690
+الصعبة اللي بيحس الواحد بعض الأحيان بالليل يتحدث
+
+28
+00:02:16,690 --> 00:02:20,290
+الآن عن الـ GERD gastroesophageal reflux disease
+
+29
+00:02:20,290 --> 00:02:26,520
+وبنقول عنه كمان acid reflux وهو عبارة عن a chronic
+
+30
+00:02:26,520 --> 00:02:30,380
+illness that affects خمسة لسبعة في المئة of the
+
+31
+00:02:30,380 --> 00:02:35,320
+world population وأنا بأكد هذا خمسة لسبعة لكل الـ
+
+32
+00:02:35,320 --> 00:02:39,320
+world لكن في منطقة البحر الأبيض المتوسط في بلادنا
+
+33
+00:02:39,320 --> 00:02:42,700
+الناس اللي بتستخدم الشطة والفلفل في المكسيك وكذا
+
+34
+00:02:42,700 --> 00:02:46,020
+هتلاقيها أعلى بكثير and it's associated with a
+
+35
+00:02:46,020 --> 00:02:49,880
+serious medical complication if untreated قضية
+
+36
+00:02:49,880 --> 00:02:55,190
+الآن أنا مش بس بحكي عن بس حموضة ويلا وبأخذ حاجة
+
+37
+00:02:55,190 --> 00:02:58,270
+وبمشي لحالة يعني بعض الأحيان بتعمل مشاكل هنتعرف
+
+38
+00:02:58,270 --> 00:03:01,090
+على الموضوع هذا من خلال المحاضرة it's a condition
+
+39
+00:03:01,090 --> 00:03:05,570
+in which the liquid content السائل الموجود في
+
+40
+00:03:05,570 --> 00:03:10,310
+المعدة of the stomach regurgitates بيصير له عملية
+
+41
+00:03:10,310 --> 00:03:16,830
+إيش؟ ارتجاع back up or reflux into the esophagus
+
+42
+00:03:16,830 --> 00:03:21,370
+the liquid can inflame هذا طبعا الـ liquid عبارة
+
+43
+00:03:21,370 --> 00:03:30,350
+عمليا عن ماء نار حمض شديد الحموضة حمض
+
+44
+00:03:30,350 --> 00:03:41,530
+الـ HCL وهذا الحمض المفروض أنه لا يهضم المعدة لأن
+
+45
+00:03:41,530 --> 00:03:46,470
+المعدة مهيأة بجدار يحميها من الحمض لكن إذا الحمض 
+
+46
+00:03:46,470 --> 00:03:51,410
+هذا راح على منطقة ما فيهاش هذا الجدار زي المريء
+
+47
+00:03:51,410 --> 00:03:56,400
+المريئي This occurs in a minority of patients
+
+48
+00:03:56,400 --> 00:04:01,920
+ساعتها بيصير عندي مشكلة محترمة شايفين الصورة 
+
+49
+00:04:01,920 --> 00:04:06,500
+الصغيرة هنا اللي فيها احمرار هذه احتقان واحمرار
+
+50
+00:04:06,500 --> 00:04:12,320
+والتهاب تم المعدة الصورة هذه شكلها تصوروا الواحد
+
+51
+00:04:12,320 --> 00:04:15,640
+لما بيكون على جلده في حاجة محمرة وكذا بيعرفش يقعد
+
+52
+00:04:15,640 --> 00:04:19,900
+منها فبالتالي لما تكون على تم المعدة مش بس بتعمل زي
+
+53
+00:04:19,900 --> 00:04:23,100
+ما قلنا حموضة وارتجاع لكن هتعمل حاجات ثانية بعد
+
+54
+00:04:23,100 --> 00:04:30,300
+شوية The regurgitated liquid usually contains حمض
+
+55
+00:04:30,300 --> 00:04:36,610
+ومع البيبسين بتفرزه مين؟ بتفرزه المعدة بيبسين is
+
+56
+00:04:36,610 --> 00:04:41,230
+an enzyme that begins the digestion of protein in
+
+57
+00:04:41,230 --> 00:04:45,560
+the stomach هتجد الـ Reflux liquid ممكن زي ما قلنا
+
+58
+00:04:45,560 --> 00:04:48,780
+يكون فيه أيضًا إيش؟ بايل إيش بايل؟ العصارة الصفراوية
+
+59
+00:04:48,780 --> 00:04:52,860
+هذه اللي بتبقى جاية من وين؟ من الدودنوم الـ 12 جاية
+
+60
+00:04:52,860 --> 00:04:57,960
+على المعدة لكن حقيقة هذا المحتوى تبع الحمض أو تبع 
+
+61
+00:04:57,960 --> 00:05:01,880
+الـ Fluid اللي بيسير في عملية ارتجاع عبارة عن حمض
+
+62
+00:05:01,880 --> 00:05:07,420
+بيبسين وكمان فيه بايل اللي هو المادة الصفراء اللي 
+
+63
+00:05:07,420 --> 00:05:13,660
+بترجع من من الدودنوم بتيجي على بتيجي على المعدة لكن
+
+64
+00:05:13,660 --> 00:05:18,440
+أسوأهم أو أكثرهم serious هو اللي بيعمل لي المشكلة
+
+65
+00:05:18,440 --> 00:05:22,240
+كلها هو الحمض عشان يكونوا برضه واضحين الـ Pepsin
+
+66
+00:05:22,240 --> 00:05:28,060
+والـ Bile قد يؤذي الـ Esophagus ولكن دوره في بناء
+
+67
+00:05:28,060 --> 00:05:34,880
+Esophagus التهاب ودمج ليس واضحا كرم أسد يعني
+
+68
+00:05:34,880 --> 00:05:39,500
+صريح بلعب دور لكن مش بالوضوح اللي بيعملوا الحمض
+
+69
+00:05:39,500 --> 00:05:42,640
+طيب إيش الـ Symptoms اللي بتختلف عن اللي تبوّت حدثنا
+
+70
+00:05:42,640 --> 00:05:49,120
+عنه؟ طبعًا الـ Disease اللي هو الـ GERD ممكن يكون
+
+71
+00:05:49,120 --> 00:05:54,660
+موجود بدون أعراض asymptomatic وممكن يكون
+
+72
+00:05:54,660 --> 00:06:00,500
+symptomatic إذا الـ acid only backs up إذا الحمض
+
+73
+00:06:00,500 --> 00:06:05,680
+هذا بيرجع لحد مسافة بس الـ maria الـ esophagus الـ 
+
+74
+00:06:05,680 --> 00:06:09,460
+geared symptoms بتلاقيها دائما بس heart burn heart
+
+75
+00:06:09,460 --> 00:06:16,280
+burn يعني حموضة heart قلب burn حرقة الحرقة القلب
+
+76
+00:06:16,280 --> 00:06:20,200
+هم زمان لما سموها heartburn كانوا يفكروا أنه حاجة
+
+77
+00:06:20,200 --> 00:06:24,980
+لها علاقة بالقلب وقريبة من القلب فجاء يقولوا عنه
+
+78
+00:06:24,980 --> 00:06:29,140
+heartburn ما زال الواحد بيقول عنه heartburn حتى الآن
+
+79
+00:06:29,140 --> 00:06:34,940
+لكنها ما لهاش علاقة بالقلب if the acid إذا الحمض
+
+80
+00:06:34,940 --> 00:06:40,340
+رجع لحد الـ larynx يعني رجع في الـ esophagus لفوق
+
+81
+00:06:40,340 --> 00:06:47,100
+لفوق لفوق ونزل لحد منطقة الحنجرة إن هم هناك فيه
+
+82
+00:06:47,100 --> 00:06:50,280
+نقطة التقاء الـSleeper النائم و الـ wake up
+
+83
+00:06:50,280 --> 00:06:54,220
+coughing and choking زي ما حكينا coughing and
+
+84
+00:06:54,220 --> 00:07:01,220
+choking sometimes GERD can cause serious
+
+85
+00:07:01,220 --> 00:07:04,000
+complications including inflammation of the
+
+86
+00:07:04,000 --> 00:07:08,300
+esophagus from stomach acid that causes bleeding
+
+87
+00:07:08,300 --> 00:07:09,520
+or ulcer
+
+88
+00:07:16,470 --> 00:07:21,170
+طيب بيبقى كمان مرة أنه sometimes GERD can cause
+
+89
+00:07:21,170 --> 00:07:25,290
+serious complications including inflammation بما
+
+90
+00:07:25,290 --> 00:07:30,810
+فيها الالتهابات للمريء لكن ممكن بعض الأحيان توصل
+
+91
+00:07:30,810 --> 00:07:37,800
+لأن الالتهابات هذه يحصل فيها bleeding أو ulcers يعني
+
+92
+00:07:37,800 --> 00:07:41,320
+جروح نتيجة الموضوع هذا وهذا اللي بنقول عنه احنا 
+
+93
+00:07:41,320 --> 00:07:45,580
+بعد شوية اللي هو apoptic ulcer مش بس في بعض
+
+94
+00:07:45,580 --> 00:07:50,840
+الحالات في بعض الحالات بينشأ إن حاجة اسمها بارت
+
+95
+00:07:50,840 --> 00:07:54,420
+إيزوفيقيا إيزوفيقيا إيزوفيقيا إيزوفيقيا إيزوفيقيا
+
+96
+00:07:56,330 --> 00:08:00,270
+هذا عبارة عن تغيير في الخلايا للحمض طول الوقت نازل
+
+97
+00:08:00,270 --> 00:08:05,470
+يغيرها بيصير عملية تغيير داخلي للخلايا تاعت الـ
+
+98
+00:08:05,470 --> 00:08:09,750
+Esophagus نتيجة زي ما قلنا ضغط الحمض عليها طول
+
+99
+00:08:09,750 --> 00:08:13,490
+الوقت وهذا ممكن يكون brain cancer أو ممكن lead to
+
+100
+00:08:13,490 --> 00:08:17,430
+brain cancer يبقى أو to cancer بارت Esophagus هو
+
+101
+00:08:17,430 --> 00:08:21,250
+عبارة عن brain cancer condition ناتجة عن طريق عن 
+
+102
+00:08:21,250 --> 00:08:25,550
+موضوع الـ GERD Most patients with GERD experience
+
+103
+00:08:32,810 --> 00:08:35,850
+أو محاولة النوم أو أول النوم، يعني بتلاقي إنهم
+
+104
+00:08:35,850 --> 00:08:40,870
+بتعبوا منها أكثر و الـ severity بتزيد أكثر، خاصة
+
+105
+00:08:40,870 --> 00:08:47,030
+لو كانوا من النوع اللي بتعشى متأخر باستمرار يبقى
+
+106
+00:08:47,030 --> 00:08:50,390
+الـ Symptoms persistent heartburn يعني مش من مرة
+
+107
+00:08:50,390 --> 00:08:53,530
+واحدة حموضة بقدر أقول أنا GERD أو مرتين، لا، 
+
+108
+00:08:53,530 --> 00:08:57,130
+احنا بنحكي عن Chronic Condition اسمها GERD اللي
+
+109
+00:08:57,130 --> 00:09:01,310
+هو Gastro-Oesophageal Reflux الـ GERD وهذه persistent
+
+110
+00:09:01,310 --> 00:09:08,170
+heartburn حموضة مستمرة وبتيجي كمان مع حاجات ثانية
+
+111
+00:09:08,170 --> 00:09:11,770
+زي الـ Nausea بيصير نفسه زي المايع، Coughing،
+
+112
+00:09:11,770 --> 00:09:16,410
+بيصير يكح، يعني الحمض بتقول لا للرئتين فبيعمل لي
+
+113
+00:09:16,410 --> 00:09:21,570
+coughing sore taste in the mouth طبعا لأنه 
+
+114
+00:09:21,570 --> 00:09:24,470
+هيكيت بصل حمض كمان بعض الأحيان للطم الـ
+
+115
+00:09:24,470 --> 00:09:27,590
+respiratory conditions زي الأسباب pneumonia و
+
+116
+00:09:27,590 --> 00:09:30,930
+chronic bronchitis كثيرا من الأحيان يا جماعة خليني
+
+117
+00:09:30,930 --> 00:09:36,870
+أدخل على طب الأطفال شوية أطفال بيجينا بـ recurrent
+
+118
+00:09:36,870 --> 00:09:42,230
+chest infection التهابات الصدر المتكررة من أهم 
+
+119
+00:09:42,230 --> 00:09:46,170
+الأسباب أو الـ differential diagnosis تبعها الأسباب
+
+120
+00:09:46,170 --> 00:09:49,670
+اللي بتعمل chest infection متكرر اللي هي
+
+121
+00:09:49,670 --> 00:09:55,330
+gastroesophageal reflux disease إلى جانب أمراض
+
+122
+00:09:55,330 --> 00:10:00,830
+القلب على سبيل المثال إلى جانب ضعف الـ immunity إلى 
+
+123
+00:10:00,830 --> 00:10:06,820
+جانب وجود جسم غريب بلع جسم غريب على سبيل المثال هذه
+
+124
+00:10:06,820 --> 00:10:10,080
+أربعة conditions حكيتها الآن أنا مش موجودة في
+
+125
+00:10:10,080 --> 00:10:14,020
+السلايدات أربعة conditions بتعمل إيش؟ recurrent
+
+126
+00:10:14,020 --> 00:10:20,220
+chest infection منها كمان الخامس cystic fibrosis
+
+127
+00:10:20,220 --> 00:10:26,600
+يبقى خمس حاجات المفروض أنه أنا لما أشوف حالة دائما
+
+128
+00:10:26,600 --> 00:10:31,020
+بيصير عندها chronic أو بيشوف حالة بيصير عندها
+
+129
+00:10:31,020 --> 00:10:35,750
+pneumonia Recurrent Pneumonia أو Recurrent Chest
+
+130
+00:10:35,750 --> 00:10:40,250
+Infection المفروض أنه بفكر فيهم heart disease
+
+131
+00:10:40,250 --> 00:10:45,790
+كنجينة الـ heart disease foreign body inhalation الـ
+
+132
+00:10:45,790 --> 00:10:51,910
+immunity system defect اللي بنحكي عنه
+
+133
+00:10:51,910 --> 00:10:56,930
+gastroesophageal reflux واتحدتنا
+
+134
+00:10:56,930 --> 00:11:01,290
+عن اللي هو الـ foreign cystic fibrosis أو foreign body
+
+135
+00:11:02,570 --> 00:11:10,960
+الـ Causes تبعت الـ GERD طبعا حسبها أسباب وهي
+
+136
+00:11:10,960 --> 00:11:14,660
+نفسها محفزات الـ Age كل ما كبر الواحد كل ما صارت
+
+137
+00:11:14,660 --> 00:11:19,620
+أكثر نوعية الأكل الناس اللي بتاكل Spicy دائما اللي
+
+138
+00:11:19,620 --> 00:11:23,860
+دائما أكلها خذنا نقول بندورة معججة على سبيل المثال
+
+139
+00:11:23,860 --> 00:11:29,140
+اللي بيأكلوا دائما البيتزا الناس اللي بتاكل فلفل
+
+140
+00:11:29,140 --> 00:11:32,860
+كثيرا على سبيل المثال وأملاح كثيرة الناس اللي بتاكل 
+
+141
+00:11:32,860 --> 00:11:37,910
+متأخر بتاكل متأخر و أكل كتير يعني بتضرب المعدة
+
+142
+00:11:37,910 --> 00:11:42,130
+الناس اللي بتاخد alcohol الناس اللي بتدخل ال
+
+143
+00:11:42,130 --> 00:11:46,760
+pregnancy ليش الـ pregnancy؟ والـ Obesity لأنه
+
+144
+00:11:46,760 --> 00:11:51,500
+بنتيجة الـ Pregnancy المعدة مضغوطة لفوق فالحامض 
+
+145
+00:11:51,500 --> 00:11:54,960
+بتجمع بصورة أكبر و بيصل للـ Esophagus بصورة أكبر
+
+146
+00:11:54,960 --> 00:11:59,040
+و الـ Obesity نفس الكلام في هناك certain foods
+
+147
+00:11:59,040 --> 00:12:03,680
+associated with reflux events طبعا مش معناته بس هم
+
+148
+00:12:03,680 --> 00:12:06,580
+هدول جماعة كل واحد فيه إلو حاجة بتعمله
+
+149
+00:12:06,580 --> 00:12:12,360
+Gastroesophageal refluxاللي ممكن يعمل لي ممكن
+
+150
+00:12:12,360 --> 00:12:17,600
+مايعملش للآخر و هيك لكن فينا تجارب عصابية مثال يمكن
+
+151
+00:12:17,600 --> 00:12:21,820
+كلكم بتستحضروا اللي بيأكلوا الجرشة بالشاي عصابي
+
+152
+00:12:21,820 --> 00:12:25,600
+مثال عصابي مثال اللي بيأكلوا الفلافل بالشاي بيصير
+
+153
+00:12:25,600 --> 00:12:32,360
+معاهم حموضة عصابي مثال فالـ Spicy Foods الـ Garlic
+
+154
+00:12:32,360 --> 00:12:39,920
+and Onion الشطة التومة والبصلCitrus fruits طبعاً
+
+155
+00:12:39,920 --> 00:12:43,900
+لأنه احنا قاعدين نحكي عن حمضيات حوامض من اسمها
+
+156
+00:12:43,900 --> 00:12:49,120
+طبعا ممكن ما بتتعاملش عند كتير ناس لكن لو الواحد
+
+157
+00:12:49,120 --> 00:12:52,340
+liable وهو معرض أو في عنده مشاكل لو أكلها ممكن
+
+158
+00:12:52,340 --> 00:12:55,670
+تتعامل معاه مشاكله كل المشروبات أو المشروبات اللي
+
+159
+00:12:55,670 --> 00:12:59,650
+فيها كافيين نقصدش بس القهوة هان ولا بنقصد بس
+
+160
+00:12:59,650 --> 00:13:03,390
+الكوكاكولا السوداء لكن أي مشروب فيه كافيين ممكن
+
+161
+00:13:03,390 --> 00:13:08,510
+يعملنا هيك شوكليت شوكولاتة Fatty and Fried Foods mint
+
+162
+00:13:08,510 --> 00:13:14,850
+flavorings اللي هي الحاجات اللي عليها الحلو اللي
+
+163
+00:13:14,850 --> 00:13:20,430
+عليه نعنع نعنع بصورة عامة وفي حاجات الـ Tomato paste
+
+164
+00:13:20,430 --> 00:13:23,590
+Foods زي الـ Spaghetti والشيلي والبيتزا والحاجات
+
+165
+00:13:23,590 --> 00:13:26,810
+هذه الـ risk factors اتحدتنا قبل شويه وقولنا ال
+
+166
+00:13:26,810 --> 00:13:31,530
+Obesity Pregnancy, Hepatic Ulcer الناس اللي في
+
+167
+00:13:31,530 --> 00:13:36,350
+عندها Delaying of Stomach Emptying طبعا ليش؟ لأنه
+
+168
+00:13:36,350 --> 00:13:40,290
+الـ Stomach بتصير تحبس والحمض بزيد فهذا كله يتوقّع
+
+169
+00:13:40,290 --> 00:13:47,010
+يعمل ايه؟ يعمل ايه ارتفاع الحمض في المريض إلى جانب
+
+170
+00:13:47,010 --> 00:13:51,190
+زي مثلا الـ Diabetes أو الـ Ulcer أو Abnormal Nerve
+
+171
+00:13:51,190 --> 00:13:54,690
+أو Muscle Function can delay emptying of the
+
+172
+00:13:54,690 --> 00:13:58,370
+Stomach وهذا بيخلي الحامض Back Up into the
+
+173
+00:13:58,370 --> 00:14:05,150
+Esophagus الأزمة ليش؟ لأنه في عندنا some Asthma
+
+174
+00:14:05,150 --> 00:14:12,540
+Medications that widen أو Dilate Airways ممكن أيضًا
+
+175
+00:14:12,540 --> 00:14:20,400
+أن يتراجع مخرج الإيزوفاجيا السفلي Sphincter اللي مع
+
+176
+00:14:20,400 --> 00:14:30,320
+المعدة ويجعل الحامض يرجع إلى الإيزوفاجيا السفلي لأن
+
+177
+00:14:30,320 --> 00:14:33,740
+One of the many complications of Diabetes is
+
+178
+00:14:33,740 --> 00:14:38,340
+Gastroparesis يعني زي ما بيقولوها ضعف حركة المعدة
+
+179
+00:14:38,340 --> 00:14:42,900
+وهذا الموضوع يجعل الأكل والحامض يظل فترة طويلة في
+
+180
+00:14:42,900 --> 00:14:46,920
+المعدة، لماذا؟ لأن الـ Emptying بطيء، فهذا بدي فرصة
+
+181
+00:14:46,920 --> 00:14:51,440
+للحامض أنه 
+
+182
+00:14:51,440 --> 00:14:55,620
+يرجع على المريض طب كيف بنشخص الـ GERD،
+
+183
+00:14:55,620 --> 00:15:00,860
+Gastroesophageal Reflux؟ تشخيصه طبعاً فيه Different
+
+184
+00:15:00,860 --> 00:15:05,460
+Modalities بما فيها Upper Gastrointestinal
+
+185
+00:15:05,460 --> 00:15:11,370
+Endoscopy من المريء لمنطقة المعدة والمريء من فوق ممكن
+
+186
+00:15:11,370 --> 00:15:16,130
+كمان بما إنه احنا قاعدين نقول عن Acid Activity في
+
+187
+00:15:16,130 --> 00:15:19,430
+المنطقة هذه ممكن نعمله Esophageal pH Monitoring
+
+188
+00:15:19,430 --> 00:15:25,190
+بدخل واحد زي حاجة هيك Sensor عند ثم المعدة وبيقيس
+
+189
+00:15:25,190 --> 00:15:30,250
+حموضة الحموضة
+
+190
+00:15:30,250 --> 00:15:35,920
+في المنطقة حوالين بين المعدة و بين المريء هذا على مدى
+
+191
+00:15:35,920 --> 00:15:41,080
+ساعات معينة 24 ساعة وفيه Software على الكمبيوتر
+
+192
+00:15:41,080 --> 00:15:46,040
+السنسور بتفرج على الكمبيوتر وبنشوف جداش النسب
+
+193
+00:15:46,040 --> 00:15:47,900
+وجداش الحموضة
+
+194
+00:15:59,550 --> 00:16:04,590
+ولا طريقة منهم تدّيك مية في المية الحل وكل واحدة
+
+195
+00:16:04,590 --> 00:16:15,270
+لها مشاكل كل واحدة لها مشاكل من الطرق الاندوسكوبي
+
+196
+00:16:15,270 --> 00:16:19,670
+فقط لا يحتاج لشعور بشكل كامل الـ pH Monitoring ليس
+
+197
+00:16:19,670 --> 00:16:20,630
+دائما متوفر
+
+198
+00:16:23,320 --> 00:16:27,960
+ولكن بقولكوا إيش اللي بيصير الآن، فيش حد من حد ما
+
+199
+00:16:27,960 --> 00:16:30,480
+بيشكي من Gastro-Oesophageal Reflux Disease بروح
+
+200
+00:16:30,480 --> 00:16:33,960
+بعمل فحوصات بالصورة اللي مكتوبة هنا، اللي هي قضية
+
+201
+00:16:33,960 --> 00:16:37,820
+إنه منظار أو pH Monitoring أو Radiological
+
+202
+00:16:37,820 --> 00:16:42,800
+Studies، ها إيش بنعمل؟ بنعمل حاجة اسمها GERD
+
+203
+00:16:42,800 --> 00:16:46,000
+Treatment-Based Approach، يعني على الـ Clinical
+
+204
+00:16:46,000 --> 00:16:50,060
+اللي شكّيها المريض، وبما إنه منتشر كتير، فبندّي
+
+205
+00:16:50,060 --> 00:16:51,740
+علاج لهذا الموضوع
+
+206
+00:16:55,460 --> 00:17:00,180
+وبندّي أدوية وزى ما قلنا بنخفف عنه اللى هو الفحوصات
+
+207
+00:17:00,180 --> 00:17:05,460
+هاي وبترايه الآن فرضا ما تحسنش مع العلاج طبعا مع
+
+208
+00:17:05,460 --> 00:17:08,340
+الـ Diet يعني هو فيه قول له Diet ونقوله إيش يعمل في
+
+209
+00:17:08,340 --> 00:17:10,900
+قضية الأكل لإنه مابنفعش أنا أقول والله عندي
+
+210
+00:17:10,900 --> 00:17:16,730
+Gastroesophageal Reflux Disease وبدأ أظل أشرب
+
+211
+00:17:16,730 --> 00:17:21,070
+Alcohol أو بدأ أظل أدخل أو بدأ أظل مع الـ Pregnancy
+
+212
+00:17:21,070 --> 00:17:25,130
+بنتهي الموضوع أو بدأ أظل أكل حاجات Spicy أو Tomato
+
+213
+00:17:25,130 --> 00:17:29,610
+أو Mint أو كذا، ماينفعش المفروض أنه أنا مع العلاج
+
+214
+00:17:29,610 --> 00:17:34,580
+بأعمل تحمية اللي هو Lifestyle Modification وإذا
+
+215
+00:17:34,580 --> 00:17:38,300
+ظلّ، طبعاً ساعتها بدك، بده الواحد يضطر يروح يعمل
+
+216
+00:17:38,300 --> 00:17:43,540
+فحوصات ليشوف Differential Diagnoses لأنه ممكن يكون
+
+217
+00:17:43,540 --> 00:17:49,060
+عنده Peptic Ulcer، ممكن يكون عندنا Cancer على سبيل
+
+218
+00:17:49,060 --> 00:17:52,220
+المثال و الـ Presentation تبعه بالصورة هذه،
+
+219
+00:17:52,220 --> 00:17:55,440
+فالمفروض أنه الواحد بياخد باله من الموضوع هذا لو
+
+220
+00:17:55,440 --> 00:17:59,880
+ما اتحسنش الـ Treatment Method to effectively treat
+
+221
+00:17:59,880 --> 00:18:04,940
+GERD range from Lifestyle Measures to the use of
+
+222
+00:18:04,940 --> 00:18:09,640
+Geared Medication أو إذا في الآخر في الآخر في
+
+223
+00:18:09,640 --> 00:18:14,740
+الآخر في مشاكل بنعمل Surgical Procedures لهدّ الدرجة
+
+224
+00:18:14,740 --> 00:18:16,900
+أه ما هو في ناس عندها مشكلة في الموضوع
+
+225
+00:18:27,770 --> 00:18:31,890
+زي ما قلنا كمان مرة مهم جدا جدا للناس اللي عندهم
+
+226
+00:18:31,890 --> 00:18:35,050
+Persistent Heartburn أو Other Chronic and
+
+227
+00:18:35,050 --> 00:18:38,190
+Recurrent Symptoms of GERD to seek an Accurate
+
+228
+00:18:38,190 --> 00:18:41,930
+Diagnosis ماكان فيش أنه أنا أظن أبلع في حاجة في
+
+229
+00:18:41,930 --> 00:18:45,930
+السوق موجودة اسمها راني تيد راني تيد عبارة حبوب
+
+230
+00:18:45,930 --> 00:18:49,690
+بيقولوا عنها جرش وهذه بتخفف الحموظة لأن الراني تيد
+
+231
+00:18:49,690 --> 00:18:53,710
+هذه عبارة عن حاجة بتعمل Neutralization تتفاعل مع
+
+232
+00:18:53,710 --> 00:18:59,400
+الحمض بتعمل Neutralization للحمض وخلاص وبرتاح الشخص
+
+233
+00:18:59,400 --> 00:19:02,660
+أو في ناس بتاخد مالوكس أو في ناس بتاكل عدس أو في
+
+234
+00:19:02,660 --> 00:19:06,020
+ناس بتشرب حليب طبعا الحليب إلو موضوع تاني لإن هو
+
+235
+00:19:06,020 --> 00:19:09,540
+يعتبر غلط لإن هو ساعتها بيعمل الـ Release ساعتها
+
+236
+00:19:09,540 --> 00:19:13,360
+ساعتها لكن بعد شوية الحموضة بتزيد أو في ناس بتاكل
+
+237
+00:19:13,360 --> 00:19:18,020
+موز أو في ناس تشرب مية المهم إنه كل واحد إلو
+
+238
+00:19:18,020 --> 00:19:20,480
+طريقته في إنه يعمل Release للـ Heart Pain بس مش هي
+
+239
+00:19:20,480 --> 00:19:25,180
+هذه الجضية الجضية إنه إنه ما يصيرش هذا Chronic، مش
+
+240
+00:19:25,180 --> 00:19:28,060
+كل ما أكل، كل ما أفطر، كل ما أتغدى، كل ما أتعشى،
+
+241
+00:19:28,060 --> 00:19:31,120
+بيصير معايا، هذا الموضوع بيصير كبير في الأحوال
+
+242
+00:19:31,120 --> 00:19:34,060
+اللي زي هي الـ Medications اللي إحنا بنديها في
+
+243
+00:19:34,060 --> 00:19:36,680
+الحقيقة للـ Gastroesophageal Reflux Disease، توّ
+
+244
+00:19:36,680 --> 00:19:40,880
+حكيت أنا عن واحد منهم الـ Anti-acid، إيش يعني Anti
+
+245
+00:19:40,880 --> 00:19:46,250
+-acid؟ إشي مضاد للحموضة زي الـ Maalox أو زي ما يكون
+
+246
+00:19:46,250 --> 00:19:51,490
+فيه تمس في عصابية المثال الـ Ranitidine عصابية المثال وهذا
+
+247
+00:19:51,490 --> 00:19:58,350
+إيش بيعمل ايه؟ Neutralize Stomach Acid and can
+
+248
+00:19:58,350 --> 00:20:03,210
+provide quick relief بتحسن على السريع الحامض خلاص
+
+249
+00:20:03,210 --> 00:20:08,190
+متعدل راحة المشكلة Antacid Alone won't heal an
+
+250
+00:20:08,190 --> 00:20:12,450
+inflamed Esophagus damaged by Stomach Acid يعني ما
+
+251
+00:20:12,450 --> 00:20:15,070
+تتصورش أنه لو في أنا في عندي Inflammation أنه
+
+252
+00:20:15,070 --> 00:20:20,580
+هتروح بالحالة هو بس الحامض خف لكن الـ الـ الـ الـ
+
+253
+00:20:20,580 --> 00:20:21,060
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ 
+
+254
+00:20:21,060 --> 00:20:22,940
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+255
+00:20:22,940 --> 00:20:29,340
+الـ الـ الـ الـ الـ الـ الـ الـ الـ 
+
+256
+00:20:29,340 --> 00:20:33,020
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+257
+00:20:33,020 --> 00:20:34,880
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+258
+00:20:34,880 --> 00:20:37,280
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+259
+00:20:37,280 --> 00:20:40,340
+الـ الـ 
+
+260
+00:20:41,610 --> 00:20:44,950
+اللي هي شغالة على طريقة الباب اللي بيجيك منه الريح
+
+261
+00:20:44,950 --> 00:20:50,010
+سده واستريح اللي هي الـ PPI بيقولوا عنها PPI أو
+
+262
+00:20:50,010 --> 00:20:56,150
+Proton Pump Inhibitors زي الـ Omeprazole مشهور الـ ..
+
+263
+00:20:56,150 --> 00:20:59,190
+الـ .. These GERD Medications إيش بتعملوا؟ بتعملوا
+
+264
+00:20:59,190 --> 00:21:04,830
+Blockage of Acid Production تماما، بيبطل يطلع حامض
+
+265
+00:21:04,830 --> 00:21:12,130
+وهذه طبعا بتدي وقت فرصة لإن الـ Damaged أو Inflamed
+
+266
+00:21:12,130 --> 00:21:17,810
+Esophageal Tissue تتهيئ زمان
+
+267
+00:21:17,810 --> 00:21:22,810
+كانت لازم تنكتب روشتة، الآن صارت تنكتب صارت عادة
+
+268
+00:21:22,810 --> 00:21:26,790
+تشتريها من الصيدلية، بنقول عنها OTC، إيش يعني OTC؟
+
+269
+00:21:26,790 --> 00:21:28,890
+Over The Counter، يعني بتقدر تروح الصيدلية تقول له
+
+270
+00:21:28,890 --> 00:21:31,190
+والله ديلي أوميبرازول، بتديك أوميبرازول، طبعا مش في
+
+271
+00:21:31,190 --> 00:21:34,310
+كل البلاد أنا بحكي عن برا، هنا برا بتشتري الصيدلية و
+
+272
+00:21:34,310 --> 00:21:37,350
+الصيدلي والعيلة وكل البلد، مافيش مشكلة، إيش اللي
+
+273
+00:21:37,350 --> 00:21:37,910
+بدك اياها؟
+
+274
+00:21:41,130 --> 00:21:44,890
+الآن يعني خلاص صار مش لازم ينكتب بروشيطة ممكن
+
+275
+00:21:44,890 --> 00:21:47,730
+الواحد يجيبه لحاله طبعا بس هذا خدوا بالكوا بيغش
+
+276
+00:21:47,730 --> 00:21:52,190
+الموضوع هذا لأنه لأنه لأنه القضية ممكن مايكونش
+
+277
+00:21:52,190 --> 00:21:53,570
+موضوع موضوع بس
+
+278
+00:21:56,130 --> 00:21:59,890
+ممكن يكون Peptic Ulcer، ممكن يكون Cancer، ممكن يكون
+
+279
+00:21:59,890 --> 00:22:02,990
+Barrett's Esophagus، ممكن يكون حاجات كتير، فيبقى ياخد
+
+280
+00:22:02,990 --> 00:22:06,550
+باله الواحد إذا دلّ كرونيك الموضوع، لابد من التوجه 
+
+281
+00:22:06,550 --> 00:22:10,190
+للطبيب لمعرفة السبب اللي عملنا الموضوع هذا الـH2
+
+282
+00:22:10,190 --> 00:22:14,290
+-Receptors اللي هو الهستامين Two Receptors
+
+283
+00:22:14,290 --> 00:22:21,870
+Blockers، هذه نوعية أخرى من الـMedication زي الـ
+
+284
+00:22:21,870 --> 00:22:24,550
+Ranitidine كتير منكم بيعرفوا عنه يقولوا سنة تقريبا
+
+285
+00:22:24,550 --> 00:22:29,370
+موقف الـ Ranitidine لأنه هذا إيش بعملنا؟ بعملنا
+
+286
+00:22:29,370 --> 00:22:33,770
+reduction of the production of acid reduction بس
+
+287
+00:22:33,770 --> 00:22:42,170
+الـ PPI عملنا blockage للـ production هذا عملنا بس
+
+288
+00:22:42,170 --> 00:22:46,230
+reduction للـ production they don't act as quickly
+
+289
+00:22:46,230 --> 00:22:50,250
+as antacid ولكنها تقدر تقدر تقدر تقدر تقدر تقدر
+
+290
+00:22:50,250 --> 00:22:57,670
+تقدر تقدر تقدر تقدر تقدر تقدر تقدر
+
+291
+00:22:57,830 --> 00:23:02,550
+مين اللي بيريحنا بسرعة؟ اللي هو الـ Antacid لكن
+
+292
+00:23:02,550 --> 00:23:07,150
+هاي الـ Proton Pump Inhibitor أو الـ PPI والـ H2
+
+293
+00:23:07,150 --> 00:23:11,470
+Receptor Blockers هذه بتديناش Release سريع لكن
+
+294
+00:23:11,470 --> 00:23:17,530
+مفعولها أكبر، ليش؟ لأنها عمليا بتخفف أو بتوقف
+
+295
+00:23:17,530 --> 00:23:21,510
+إفراز الحامض اللي أهم من هيك يا جماعة وهذا لاحظناه
+
+296
+00:23:21,510 --> 00:23:27,790
+عند ناس كتير اللي هي الـ  Guard Diet الـ Gear Diet 
+
+297
+00:23:27,790 --> 00:23:34,390
+إنه أنا أحاول أمتنع عن الباكلات أو الأدوية اللي
+
+298
+00:23:34,390 --> 00:23:41,470
+بتسبب للموضوع هذا Avoid foods such as citrus،
+
+299
+00:23:41,470 --> 00:23:47,130
+tomato، coffee، هذي بتعملي directly irritation
+
+300
+00:23:47,130 --> 00:23:52,270
+للـ mucosa، onion، شوكليت، peppermint، طبعا الكمية
+
+301
+00:23:52,270 --> 00:23:55,530
+كمان بتلعب دوري يا بابا، إحنا الآن في عصر المجات
+
+302
+00:23:55,530 --> 00:23:59,110
+للأسف كيف يعني عصر المجات؟ صارت الناس لما بدأت تشرب
+
+303
+00:23:59,110 --> 00:24:02,150
+زمان كنا نشرب الجهوة السادة والمرة في فنجان صغير
+
+304
+00:24:02,150 --> 00:24:05,730
+نصّه هيك أو فنجان على أجال الفنجان الجهوة التركي
+
+305
+00:24:05,730 --> 00:24:09,710
+العادي، هذا الآن صار كله بيشرب في مجات والكمية
+
+306
+00:24:09,710 --> 00:24:13,750
+الكبيرة هيك كمان بتأثرنا على الموضوع اللي
+
+307
+00:24:13,750 --> 00:24:17,410
+الـ peppermint and any food with high fat content
+
+308
+00:24:17,410 --> 00:24:21,830
+that affects pressure in the stomach Limit or cut
+
+309
+00:24:21,830 --> 00:24:24,910
+the use of non-steroidal ناس اللي بتاخد دايكلوفين
+
+310
+00:24:24,910 --> 00:24:29,850
+و Ibuprofen that may irritate the esophagus and
+
+311
+00:24:29,850 --> 00:24:34,110
+stomach lining Individuals with GERD should avoid
+
+312
+00:24:34,110 --> 00:24:40,770
+whole milk Try to maintain a low-fat diet by
+
+313
+00:24:40,770 --> 00:24:44,670
+staying away from fatty meats and processed meats
+
+314
+00:24:44,670 --> 00:24:48,030
+called cuts اللي هي المرتدلة والحاجات هذه If you
+
+315
+00:24:48,030 --> 00:24:52,450
+must have coffee أو tea حاول تشوف الكافي اللي بدون
+
+316
+00:24:52,450 --> 00:24:55,350
+كافيين أو herbal tea، شاي، عشاب
+
+317
+00:25:00,960 --> 00:25:06,320
+Eat smaller meals Control your weight Don't smoke
+
+318
+00:25:06,320 --> 00:25:10,740
+Eliminate hot pain trigger Don't lie down after a
+
+319
+00:25:10,740 --> 00:25:16,900
+meal Raise the head of the bed وهذا بتدخلنا بعد
+
+320
+00:25:16,900 --> 00:25:21,640
+شوية في موضوع الـ Peptic Ulcer
+
+321
+00:25:23,880 --> 00:25:31,020
+والـ Peptic Ulcer Disease بيو دي بيو دي فيه منه
+
+322
+00:25:31,020 --> 00:25:33,240
+gastric ulcer زي ما احنا شايفين في الصورة وفيه
+
+323
+00:25:33,240 --> 00:25:38,340
+duodenal ulcer يعني ulcer اللي هو المعدة و ulcer أو
+
+324
+00:25:38,340 --> 00:25:42,420
+قرحة المعدة و قرحة الاثني عشر الـ Peptic Ulcer
+
+325
+00:25:42,420 --> 00:25:46,640
+Disease known as Peptic Ulcer or Stomach Ulcer is
+
+326
+00:25:46,640 --> 00:25:50,770
+a prick أو breakage in the lining of the stomach،
+
+327
+00:25:50,770 --> 00:25:55,410
+first part of the small intestine أو الدودنوم زي
+
+328
+00:25:55,410 --> 00:26:00,750
+ما بنقول و occasionally the lower esophagus الـ
+
+329
+00:26:00,750 --> 00:26:04,790
+ulcer in the stomach is known as a gastric ulcer
+
+330
+00:26:04,790 --> 00:26:08,670
+في حين اللي هو في الـ small أو first part of the
+
+331
+00:26:08,670 --> 00:26:15,050
+intestine بنقول عنه duodenal ulcerالخطر العملي
+
+332
+00:26:15,050 --> 00:26:20,530
+لتطور من الـ Peptic Ulcer هو حوالي 10% تقريبا،
+
+333
+00:26:20,530 --> 00:26:24,450
+طبعا أنا أعتقد أن الموضوع أكبر من هيك وأكثر من
+
+334
+00:26:24,450 --> 00:26:28,170
+هيك، لكن ليش؟ لأن كتير من الناس بتعالج نفسها
+
+335
+00:26:28,170 --> 00:26:32,230
+بالطرق الـ Herbal العادية بدون متوجه للإحصائيات،
+
+336
+00:26:32,230 --> 00:26:36,670
+في البلدان الغربية الـ Bad scent of people with
+
+337
+00:26:36,670 --> 00:26:40,410
+Helicobacter pylori، الـ Helicobacter pylori هي
+
+338
+00:26:40,410 --> 00:26:47,020
+عبارة عن عبارة عن اللي هو البكتيريا اللي الآن اتهمت
+
+339
+00:26:47,020 --> 00:26:51,080
+اتهام مباشر بإنها سبب الـ Peptic و Duodenal Ulcer
+
+340
+00:26:51,080 --> 00:26:55,160
+اللي هي البكتيريا
+
+341
+00:26:55,160 --> 00:27:00,060
+العصوية أو حاجة زي هي Helicobacter pylori
+
+342
+00:27:00,060 --> 00:27:04,900
+Infection Roughly Match Age 20% من الناس عمرهم
+
+343
+00:27:04,900 --> 00:27:10,460
+عشرين سنة عندهم Helicobacter ثلاثين في المية من
+
+344
+00:27:10,460 --> 00:27:12,720
+عمرهم ثلاثين ثمانين في المية من عمرهم ثمانين عندهم
+
+345
+00:27:12,720 --> 00:27:14,900
+Helicobacter infection
+
+346
+00:27:17,650 --> 00:27:20,410
+وزي ما قلنا الـ ECO-FACTOR من أهم الأسباب اللي
+
+347
+00:27:20,410 --> 00:27:23,950
+بتعملنا Gastroesophageal، عفوا، Peptic Ulcer
+
+348
+00:27:23,950 --> 00:27:28,910
+Transmission is by food contaminated groundwater و
+
+349
+00:27:28,910 --> 00:27:33,190
+through human saliva زي kissing مثلا أو sharing
+
+350
+00:27:33,190 --> 00:27:38,830
+food utensils استعمال الأدوات والمعالق والشوك ورا
+
+351
+00:27:38,830 --> 00:27:42,870
+بعض من الـ Symptoms والـ Signs تبقى الـ Peptic
+
+352
+00:27:42,870 --> 00:27:47,680
+Ulcer الـ Abdominal Pain بالذات Epigastric المنطقة
+
+353
+00:27:47,680 --> 00:27:53,640
+اللي هي تحت الجلَب زي ما احنا شايفين في الصورة في
+
+354
+00:27:53,640 --> 00:27:59,280
+حالة duodenal ulcer، الشعور يظهر خلال ثلاث ساعات بعد
+
+355
+00:27:59,280 --> 00:28:03,260
+أخذ طعام، يعني في الـ Gastric Ulcers، على السريع
+
+356
+00:28:03,260 --> 00:28:06,840
+الوجع، بتاكل و بتتوجع، الـ duodenal ulcer بعد ثلاث
+
+357
+00:28:06,840 --> 00:28:10,260
+ساعات، من ساعته لثلاث ساعات بعد الأكل، Bloating،
+
+358
+00:28:10,260 --> 00:28:13,600
+بيصير إنه نفخة، Abdominal Fullness، حاسس بطن مليان،
+
+359
+00:28:13,600 --> 00:28:17,760
+Nausea and vomiting، ممكن يكون كمان، من الـ Signs
+
+360
+00:28:17,760 --> 00:28:23,050
+and Symptoms، Loss of Appetite، Weight Loss،
+
+361
+00:28:23,050 --> 00:28:30,690
+Hematemesis، Hematemesis
+
+362
+00:28:30,690 --> 00:28:34,730
+ميلينا، ميلينا
+
+363
+00:28:34,730 --> 00:28:38,570
+هو الدم اللي بنزل من المعدة لكن بدخل على الجهاز
+
+364
+00:28:38,570 --> 00:28:43,950
+الهضمي وبتفاعل مع الحامض بالـ DNA بيصيروا عملية
+
+365
+00:28:43,950 --> 00:28:48,110
+Oxidation للـ Iron اللي فيه من اللي في الـ
+
+366
+00:28:48,110 --> 00:28:53,810
+Hemoglobin وهذا عبارة عن دم بس بشكله زي الزفتة
+
+367
+00:28:53,810 --> 00:29:01,190
+الطريقة فاول Smelling Faces البراز زي الزفتة هيك
+
+368
+00:29:01,190 --> 00:29:05,650
+ريحته سيئة و بنلاقيها نزلت مع البراز هذا عبارة عن
+
+369
+00:29:05,650 --> 00:29:15,030
+دم لكن صار له عملية Oxidation الأسباب هي أهم متهم
+
+370
+00:29:15,030 --> 00:29:18,290
+major causative factor is chronic inflammation
+
+371
+00:29:18,290 --> 00:29:23,810
+بالـ Helicobacter pylori that colonize the mucosa
+
+372
+00:29:23,810 --> 00:29:30,250
+60% of gastric ulcer هي السبب 50 إلى 75% of Duodenal
+
+373
+00:29:30,250 --> 00:29:31,410
+ulcer
+
+374
+00:29:33,500 --> 00:29:37,420
+المتهم الثاني اللي هي الـ Non-steroidal أدوية Non
+
+375
+00:29:37,420 --> 00:29:41,740
+-steroidal Anti-inflammatory drugs وزي ما قلنا هي
+
+376
+00:29:41,740 --> 00:29:47,220
+المتهم الآخر، المتهم الثاني يعني
+
+377
+00:29:47,220 --> 00:29:54,820
+معلومة مشهورة عند الكل ونتيجة كثرة وانتشار استعمال
+
+378
+00:29:56,270 --> 00:29:58,790
+اللي هو الـ Non-Steroidal Anti-Inflammatory Drugs
+
+379
+00:29:58,790 --> 00:30:04,530
+الـ ميكانيزم هي موجودة عندكم مكتوبة أدت إلى إنه
+
+380
+00:30:04,530 --> 00:30:07,570
+يصير عندنا Peptic Ulcer أو Duodenal Ulcer والحاجة
+
+381
+00:30:07,570 --> 00:30:10,670
+الثالثة واللي برضه مشهورة بصورة كبيرة لـ Stress
+
+382
+00:30:10,670 --> 00:30:14,890
+due to serious health problems واحنا بنقول غالبا
+
+383
+00:30:14,890 --> 00:30:20,830
+الناس اللي بتـ damp في العناية المركزة لسبب آخر
+
+384
+00:30:20,830 --> 00:30:23,730
+نتيجة الـ stress اللي صار معهم لازم تكون حاجة
+
+385
+00:30:23,730 --> 00:30:27,410
+كبيرة تعملنا الموضوع والحاجة الرابعة من الـ causes
+
+386
+00:30:27,410 --> 00:30:29,890
+اللي هو الـ diet الشيء الوحيد هو الواقعيات البيوترية
+
+387
+00:30:29,890 --> 00:30:35,290
+مثل المشروبات والتناول الكافيين والكوفي والكحول
+
+388
+00:30:35,290 --> 00:30:40,330
+أيضا مفهوما لأسباب أو Exacerbation العوامل وطبعا
+
+389
+00:30:40,330 --> 00:30:46,450
+النار كمان لاجوا يعني بعض الـ studies لاجت فيه
+
+390
+00:30:46,450 --> 00:30:50,530
+علاقة بالموضوع الـ diagnosis How to diagnose
+
+391
+00:30:50,530 --> 00:30:56,830
+mainly based on the characteristic symptoms
+
+392
+00:31:02,120 --> 00:31:15,880
+وهنا بيصير عندنا
+
+393
+00:31:15,880 --> 00:31:22,180
+في assignment How to diagnose غير اللي أنا كاتبه
+
+394
+00:31:22,180 --> 00:31:22,380
+هنا
+
+395
+00:31:26,110 --> 00:31:31,270
+الـ Helicobacter Infestation
+
+396
+00:31:31,270 --> 00:31:37,510
+عند المريض كيف الـ diagnosis تبع الـ Helicobacter؟
+
+397
+00:31:37,510 --> 00:31:40,470
+إيش الفحوصات اللي بتنعمل؟ في ثلاث أربع فحوصات، شيء
+
+398
+00:31:40,470 --> 00:31:47,070
+من الدم، شيء من الـ stool، شيء من الـ antibody، شيء
+
+399
+00:31:47,070 --> 00:31:54,290
+من النفس، بدي الطرق، إيش أفضلها ممكن نعمل
+
+400
+00:31:54,290 --> 00:31:59,190
+comparison بين الطرق بس جدول هيك الطرق اللي بعمل
+
+401
+00:31:59,190 --> 00:32:03,230
+فيها فحص وإيش أفضلها هذا الـ assignment التبعنا
+
+402
+00:32:05,270 --> 00:32:07,450
+Gastrointestinal bleeding طبعا بدل ما أنا عندي
+
+403
+00:32:07,450 --> 00:32:10,210
+جرح ممكن يصير عندي نزيف The most common
+
+404
+00:32:10,210 --> 00:32:13,050
+complication لهذا الدرجة آه طبعا sudden large
+
+405
+00:32:13,050 --> 00:32:15,790
+bleeding can be life threatening وطبعا الـ bleeding
+
+406
+00:32:15,790 --> 00:32:21,330
+ممكن يدخلنا بـ hypovolemic shock زي ما الكل عارف إيش
+
+407
+00:32:21,330 --> 00:32:25,310
+كمان perforation آه ممكن يصير عندي مش بس bleeding
+
+408
+00:32:25,310 --> 00:32:28,730
+تفتح المعدة لبرا وهذه طبعا catastrophic
+
+409
+00:32:28,730 --> 00:32:35,360
+complication ممكن نتيجة أنه بعد فترة لو كان عندي
+
+410
+00:32:35,360 --> 00:32:37,840
+على الـ gastric outlet على مخرج المعدة فيه عندي
+
+411
+00:32:37,840 --> 00:32:47,140
+هناك قرحة وصارت طابت أو راجت عملت الـ Scarring و
+
+412
+00:32:47,140 --> 00:32:51,600
+بعد الـ Scarring عملت له obstruction of the outlet
+
+413
+00:32:51,600 --> 00:32:55,860
+وطبعا اللي حكينا عنه في الـ differential يعني لازم
+
+414
+00:32:55,860 --> 00:32:57,900
+أحطه في الـ differential diagnosis ما هوش بس
+
+415
+00:32:57,900 --> 00:33:03,360
+complication هو الـ cancer وطبعا لابد من عينة
+
+416
+00:33:04,010 --> 00:33:12,070
+العلاج الحقيقة بيتمثل في أكثر من حاجة الحاجة
+
+417
+00:33:12,070 --> 00:33:15,790
+الأولى acid reducing medication الحاجة الثانية لل
+
+418
+00:33:15,790 --> 00:33:19,390
+infection لازم أدي علاج two antibiotics يبقى بدي
+
+419
+00:33:19,390 --> 00:33:23,030
+شيء مضاد للحاجة اسمها triple therapy triple يعني
+
+420
+00:33:23,030 --> 00:33:28,230
+ثلاثي therapy اثنين مضادات حيوية واحد antacid أو 
+
+421
+00:33:28,230 --> 00:33:33,530
+acid reducing medication زي الـ PPI معاه لما يكون
+
+422
+00:33:33,530 --> 00:33:43,790
+عندنا Peripheric Ulcer «العوامل 
+
+423
+00:33:43,790 --> 00:33:51,710
+التي تؤدي» أو «التجارب» لـ Gastric Ulcer اللي هو
+
+424
+00:33:51,710 --> 00:33:56,730
+Reduction of Drug Intake حاول أنتبه للـ
+
+425
+00:33:56,730 --> 00:34:02,050
+Helicobacter، الـ Diet، stress، smoking and alcohol
+
+426
+00:34:02,050 --> 00:34:06,730
+اللي هي تحت اسم lifestyle risk factors دائماً إذا 
+
+427
+00:34:06,730 --> 00:34:13,750
+إذا اتغلبت على المسببات بتغلب على المرض بإذن الله
+
+428
+00:34:13,750 --> 00:34:19,090
+تعالى وربنا هو الشافي وبتمنّى لك يوم سعيد 
diff --git a/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/wKbEPHOGwtg_postprocess.srt b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/wKbEPHOGwtg_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e7e89c3a1a7de9d29014fc1140d6cfad2230c11e
--- /dev/null
+++ b/PL9fwy3NUQKwaf4aPlmtIQXiASpDxVLEnO/wKbEPHOGwtg_postprocess.srt
@@ -0,0 +1,1892 @@
+1
+00:00:01,250 --> 00:00:03,270
+بسم الله الرحمن الرحيم والحمد لله رب العالمين
+
+2
+00:00:03,270 --> 00:00:06,770
+والصلاة والسلام علي سيدنا محمد وسيدنا أجمعين اليوم
+
+3
+00:00:06,770 --> 00:00:09,530
+ان شاء الله تعالى هنتحدث عن ال disorders of
+
+4
+00:00:09,530 --> 00:00:17,010
+respiratory system لكن قبل ما نبدأ هنحاول هذه
+
+5
+00:00:17,010 --> 00:00:23,830
+المحاضرة هتكون على two partsهنحكي في الأول عن شوية
+
+6
+00:00:23,830 --> 00:00:29,290
+general symptoms of respiratory disease زي على
+
+7
+00:00:29,290 --> 00:00:32,750
+سبيل المثال، إيش الـhypoxia؟ تعريف الـhypoxia
+
+8
+00:00:32,750 --> 00:00:39,050
+تعريف الـhypoxia هي decrease level of oxygen in
+
+9
+00:00:39,050 --> 00:00:44,510
+the tissue، مش في blood الـhypoxemia هو decrease
+
+10
+00:00:44,510 --> 00:00:49,560
+level of oxygen in arterial bloodالـ hypercapnia
+
+11
+00:00:49,560 --> 00:00:53,560
+increased level of CO2 in the blood والتسنية
+
+12
+00:00:53,560 --> 00:00:59,140
+difficult breathingالتكيبنية rapid rate of
+
+13
+00:00:59,140 --> 00:01:03,740
+breathing سيانوزز حكينا عنها قبل هيك بلوش
+
+14
+00:01:03,740 --> 00:01:07,860
+discoloration of the skin and mucous membranes due
+
+15
+00:01:07,860 --> 00:01:11,860
+to poor oxygenation of the blood و hemoptysis كنا
+
+16
+00:01:11,860 --> 00:01:17,900
+حكينا الفرق بين hemoptysis و ash و hematemesis و
+
+17
+00:01:17,900 --> 00:01:21,200
+hematuria hemoptysis هو إبانة عن blood in the
+
+18
+00:01:21,200 --> 00:01:24,990
+sputumهذه التعريفات بسيطة، يجب أن نكون عارفينها
+
+19
+00:01:24,990 --> 00:01:28,910
+ونحكي عن الـ Disorders أو Diseases of the
+
+20
+00:01:28,910 --> 00:01:32,890
+Respiratory System هنتحدث في الأول عن تجسيمة
+
+21
+00:01:32,890 --> 00:01:37,880
+Respiratory Infectionالإنفكشن في الـ Respiratory
+
+22
+00:01:37,880 --> 00:01:41,020
+Tract يمكن أن يحدث في الـ Upper Respiratory Tract
+
+23
+00:01:41,020 --> 00:01:48,020
+Infection اختصارها URT
+
+24
+00:01:48,020 --> 00:01:51,080
+أو Upper Respiratory Tract Infection وفيه Lower
+
+25
+00:01:51,080 --> 00:01:54,200
+Respiratory Tract Infection وممكن يكونوا مخصصين مع
+
+26
+00:01:54,200 --> 00:02:01,340
+بعضالـ الـ الـ organisms اللي بتعملنا infections
+
+27
+00:02:01,340 --> 00:02:06,480
+في ال respiratory system طبعا ال ال ال ال ال
+
+28
+00:02:06,480 --> 00:02:11,540
+bacteria ال viruses و ال fungi لابد من الانتباه أن
+
+29
+00:02:11,540 --> 00:02:14,820
+the majority of upper respiratory tract infections
+
+30
+00:02:14,820 --> 00:02:16,380
+are caused by virus
+
+31
+00:02:18,980 --> 00:02:21,840
+الرينوفيروس والبرانفلونزافيروس والبرانفلونزافيروس
+
+32
+00:02:21,840 --> 00:02:24,860
+علشان هيك بتسمعونا كتير بنقول in the upper
+
+33
+00:02:24,860 --> 00:02:29,560
+respiratory tract infection there is no need for
+
+34
+00:02:29,560 --> 00:02:33,500
+antibioticلأيش؟ لأن الـ Antibiotic للبكتيريا بما
+
+35
+00:02:33,500 --> 00:02:37,880
+أن الـ Upper Respiratory Tract Infection معظم سبب
+
+36
+00:02:37,880 --> 00:02:46,140
+الانفكشن فيروس فإنه No Need وحسب الأورجانيزم و ال
+
+37
+00:02:46,140 --> 00:02:49,320
+extent of infection the manifestation can range
+
+38
+00:02:49,320 --> 00:02:53,800
+from mild to severe and even life threatening حسب
+
+39
+00:02:53,800 --> 00:02:58,190
+الأورجانيزم و حسب ال extent of infectionبدنا نبدأ
+
+40
+00:02:58,190 --> 00:03:00,590
+في الأول في الـ Common Cold اللي احنا بنقول عنه
+
+41
+00:03:00,590 --> 00:03:03,810
+بلغتنا العامية الرشح الـ Most Common Viral
+
+42
+00:03:03,810 --> 00:03:07,230
+Pathogen للـ Common Cold are Rhinovirus,
+
+43
+00:03:07,390 --> 00:03:10,990
+Parainfluenza Virus Respiratory Sinitial Virus,
+
+44
+00:03:11,110 --> 00:03:15,810
+Adenovirus and Coronavirus طبعاً هذه المحاضرة
+
+45
+00:03:15,810 --> 00:03:20,790
+معمولة قبل ما يحصل الكورونا يعني هذه من أربع أو
+
+46
+00:03:20,790 --> 00:03:23,870
+خمس سنين المحاضرة وكنا ذاكرين فيها الـ Coronavirus
+
+47
+00:03:25,470 --> 00:03:41,750
+وهذه الفيروسات تتراوح في أوقات السنة وبتدخل
+
+48
+00:03:41,750 --> 00:03:48,110
+عن طريق الـ nasal mucosa
+
+49
+00:03:48,110 --> 00:03:52,940
+وsurfaces of the eyesوهذا اللي طول النهار بيحكي
+
+50
+00:03:52,940 --> 00:03:56,600
+فيه يا جماعة تحكيش في وجهك اللي بديه يعتص أو بديه
+
+51
+00:03:56,600 --> 00:04:00,520
+يكح ميكحش في وجه التانين و دايما غسل اليدين عشان
+
+52
+00:04:00,520 --> 00:04:03,820
+لو ايدك أخدت شوية فيروسات و جيت حكيت في منخيرك أو
+
+53
+00:04:03,820 --> 00:04:08,880
+في عينك أو في كذا ممكن ينتجل فيروس they are
+
+54
+00:04:08,880 --> 00:04:12,260
+readily spread from person to person via
+
+55
+00:04:12,260 --> 00:04:17,180
+respiratory secretionsmanifestations of the common
+
+56
+00:04:17,180 --> 00:04:20,760
+cold ممكن يكون rhinitis inflammation of the nasal
+
+57
+00:04:20,760 --> 00:04:25,620
+mucosa rhinitis inflammation of the nasal mucosa
+
+58
+00:04:25,620 --> 00:04:29,840
+sinusitis inflammation of the sinus mucosa و احنا
+
+59
+00:04:29,840 --> 00:04:34,800
+اتعرفنا على ال sinuses و قلنا انه في عندنا كام نوع
+
+60
+00:04:34,800 --> 00:04:40,380
+من ال sinuses في الوجه صح ايش و ايش و ايش ايوة
+
+61
+00:04:40,380 --> 00:04:43,020
+صحيح
+
+62
+00:04:44,800 --> 00:04:51,340
+مكسيري صحيح اثمويدل صحيح سفينويدل صحيح frontal
+
+63
+00:04:51,340 --> 00:04:55,860
+صحيح نعم فى اننا كمان الـ pharyngitis اللى هو
+
+64
+00:04:55,860 --> 00:05:00,020
+التهاب الحلق pharynx and throat ممكن يعملنا
+
+65
+00:05:00,020 --> 00:05:03,400
+headache كمان و nasal discharge and congestion
+
+66
+00:05:03,400 --> 00:05:10,600
+احتقال هذا بالنسبة لل common cold وهذا أسهل أنواع
+
+67
+00:05:10,600 --> 00:05:11,980
+التهابات
+
+68
+00:05:13,840 --> 00:05:17,420
+التنفسية العليا اللي هو الـ Common Cold الرشح أو
+
+69
+00:05:17,420 --> 00:05:23,400
+بما نقول عنه الزكاة طيب ال Influenza و اللي هي
+
+70
+00:05:23,400 --> 00:05:27,600
+برضه بتيجي بصورة كبيرة كمان Influenza و اللي هي
+
+71
+00:05:27,600 --> 00:05:32,860
+الآن منتشرة و بتعملنا بتخربطنا في موضوع هل اللي
+
+72
+00:05:32,860 --> 00:05:38,390
+جبالي أنا مفلوز ولا عنده كورونا ولا مكرونلأن
+
+73
+00:05:38,390 --> 00:05:44,390
+الانفلوانزا الموسينية مشهورة أنه بداية الخريف دخول
+
+74
+00:05:44,390 --> 00:05:48,950
+الشتاء تنتشر بصورة كبيرة انفلوانزا is a viral
+
+75
+00:05:48,950 --> 00:05:52,030
+infection كمان مرة اه viral infection انفلوانزا
+
+76
+00:05:52,030 --> 00:05:56,370
+هذا اللي بنبلبع و بند الأطفال مضادات حيوية عشانها
+
+77
+00:05:56,370 --> 00:05:59,570
+انها هي viral infection ت affect upper respiratory
+
+78
+00:05:59,570 --> 00:06:03,290
+tract infection upper respiratory tract and or
+
+79
+00:06:03,290 --> 00:06:10,220
+lower respiratory tractفي عنده تلت أنواع من الـ
+
+80
+00:06:10,220 --> 00:06:13,440
+forms of influenza virus أو strains بتقول عنها
+
+81
+00:06:21,620 --> 00:06:25,920
+أشهر واحد في التلاتة لو الـType-A هو الـMost
+
+82
+00:06:25,920 --> 00:06:30,520
+Common ويسبب فيه الموضوع أكتر من المرضين الـType-A
+
+83
+00:06:30,520 --> 00:06:38,980
+الـInfluenza Virus هو بينتقل
+
+84
+00:06:38,980 --> 00:06:50,390
+بسهولة جدًا جدًابسبب وجود تتعييم
+
+85
+00:06:50,390 --> 00:06:54,130
+انفلوازية مختلف عن السنة السابقة لأنهم بيحطوا
+
+86
+00:06:54,130 --> 00:06:58,810
+قطعين تلاتة أو أربعة قطعان داخل التطعيم هذا ولكن
+
+87
+00:06:58,810 --> 00:07:02,390
+بضطرين لأن نفس الفيروس تبقى السنة اللي فاتت مش هو
+
+88
+00:07:02,390 --> 00:07:05,290
+اللي بيبقى المرة هاي ليش؟ لأنه بيغير المادة
+
+89
+00:07:05,290 --> 00:07:08,990
+الوراثية تبعته وهذا الكلام اللي بيقعدين بنقول أنه
+
+90
+00:07:08,990 --> 00:07:13,940
+عنده تتعييم مختلفGenetic mutation يعني بيغير مادته
+
+91
+00:07:13,940 --> 00:07:20,400
+الوراثية وبعملي نسخة جديدة من هذا الفيروس وبتصيب
+
+92
+00:07:20,400 --> 00:07:30,060
+بطريقة أخرى وطبعا حسب الأماكن اللي بيصيب فيها عشان
+
+93
+00:07:30,060 --> 00:07:32,940
+هيك بيصير عندنا متذكرين أنتوا انفلونزة الطيور
+
+94
+00:07:32,940 --> 00:07:38,310
+وانفلونزة الخنازير وهي الآن بدأنا ندخلفي أنواع
+
+95
+00:07:38,310 --> 00:07:41,650
+تانية غير الـ Influenza في الـ Corona في حاجة
+
+96
+00:07:41,650 --> 00:07:45,110
+اسمها Pandemics الـ Pandemics هو Spreading of
+
+97
+00:07:45,110 --> 00:07:48,250
+Infection Across a Large Region يعني في منطقة
+
+98
+00:07:48,250 --> 00:07:53,050
+كبيرة Of Influenza أرسين كل تمام لعشر سنين بيصير
+
+99
+00:07:53,050 --> 00:07:58,200
+عندنابنديميا أو بنقول عنها endemic أو pandemic
+
+100
+00:07:58,200 --> 00:08:03,600
+وبتبقى serious وهذه نتيجة أنه بيغير هذا ال virus
+
+101
+00:08:03,600 --> 00:08:07,220
+اللي هو ال influenza virus بيغير المادة الجينية
+
+102
+00:08:07,220 --> 00:08:12,260
+تبعته فيعجز الجسم عن محاربته وبتنتشر هذا الفيروس
+
+103
+00:08:12,260 --> 00:08:19,200
+بنتشر بصورة كبيرة بين الناس وهنا شايفين أخوه لل
+
+104
+00:08:19,200 --> 00:08:22,520
+influenza virus اللي هو ال coronavirus إيش عامل في
+
+105
+00:08:22,520 --> 00:08:25,790
+العالمإيش عامل في العالم؟ symptoms of influenza
+
+106
+00:08:25,790 --> 00:08:29,430
+infection headache طبعا احنا شوفنا تو في ال common
+
+107
+00:08:29,430 --> 00:08:33,350
+cold ماكانش فيه headache يعني احنا بنحس إن إن ال
+
+108
+00:08:33,350 --> 00:08:37,110
+influenza تأثير تبعها systemic أكتر يعني على الجسم
+
+109
+00:08:37,110 --> 00:08:41,190
+أكتر مع ال upper respiratory لكن كمان بيأثرلي على
+
+110
+00:08:41,190 --> 00:08:43,770
+الجسم إن حنشوف في ال symptoms لكن ال common cold
+
+111
+00:08:43,770 --> 00:08:48,870
+أو الذكام يعني زي ما بقولها restricted الالتهاب في
+
+112
+00:08:48,870 --> 00:08:52,930
+منطقة المناخية ومنطقة الحلق و هيك يعنيالـ Symptoms
+
+113
+00:08:52,930 --> 00:08:56,370
+of Influenza Infection Headache, Fever, Chills،
+
+114
+00:08:56,370 --> 00:09:00,350
+إيش يعني Chills؟ بيصير عنده نتيجة الحرارة اللي هي
+
+115
+00:09:00,350 --> 00:09:04,630
+ال .. إيش بيقولوها؟ جشعريرة، بيصير عنده muscle
+
+116
+00:09:04,630 --> 00:09:08,310
+aches، بيصير عنده وجع في العضلات، nasal discharge,
+
+117
+00:09:08,470 --> 00:09:11,790
+unproductive cough, un-thor, un-sore throat
+
+118
+00:09:14,450 --> 00:09:17,610
+الإنفلوانزا الانفكار يمكن أن يسبب انفلام الهوائي
+
+119
+00:09:17,610 --> 00:09:18,470
+الهوائي الهوائي الهوائي الهوائي الهوائي الهوائي
+
+120
+00:09:18,470 --> 00:09:18,730
+الهوائي الهوائي الهوائي الهوائي الهوائي الهوائي
+
+121
+00:09:18,730 --> 00:09:22,170
+الهوائي الهوائي الهوائي الهوائي الهوائي الهوائي
+
+122
+00:09:22,170 --> 00:09:22,650
+الهوائي الهوائي الهوائي الهوائي الهوائي الهوائي
+
+123
+00:09:22,650 --> 00:09:23,970
+الهوائي الهوائي الهوائي الهوائي الهوائي الهوائي
+
+124
+00:09:23,970 --> 00:09:27,850
+الهوائي الهوائي الهوائي الهوائي الهوائي الهوائي
+
+125
+00:09:27,850 --> 00:09:32,280
+الهوائي الهوائي الهوائيInfection may lead to co
+
+126
+00:09:32,280 --> 00:09:35,440
+-infection of the respiratory passages with
+
+127
+00:09:35,440 --> 00:09:38,500
+bacteria هذا أحنا مشكلتنا وعشان هي كتير بتلاقي
+
+128
+00:09:38,500 --> 00:09:41,020
+بتحججوا ناس في ال Influenza أنا بدي مضاد حيوي
+
+129
+00:09:41,020 --> 00:09:44,440
+لحمايته من أنه يصير عنده co-infection أو التهاب
+
+130
+00:09:44,440 --> 00:09:48,400
+بكتيري super infection أو co-infection طبعا يا
+
+131
+00:09:48,400 --> 00:09:53,520
+جماعة يعني في مغلطة كبيرة هنا ما لازم أحنا أدي أنا
+
+132
+00:09:53,520 --> 00:09:58,340
+مضاد حيوي مقدما الآن لو حصل أنه شوفت أنه في إشارات
+
+133
+00:09:58,340 --> 00:10:04,510
+ل bacterial infectionطبعاً Co-infection مع
+
+134
+00:10:04,510 --> 00:10:08,290
+الـViral ساعتها بأدى Antibiotic لكن أن أدى
+
+135
+00:10:08,290 --> 00:10:11,570
+الـViral انفكشن Antibiotic هذه غلطة كبيرة It's
+
+136
+00:10:11,570 --> 00:10:14,830
+also possible for influenza virus to infect the
+
+137
+00:10:14,830 --> 00:10:19,330
+tissues of the lung itself and cause viral
+
+138
+00:10:19,330 --> 00:10:23,820
+infectionنمونيا يعني ممكن تنزل فعلا التهاب
+
+139
+00:10:23,820 --> 00:10:26,980
+الفيروسي ينزل وينزل ع الرئتين ويعمل إيه التهاب
+
+140
+00:10:26,980 --> 00:10:33,320
+الفيروسي ال treatment of influenza طبعا بدرس تراحة
+
+141
+00:10:33,320 --> 00:10:38,260
+fluids سوائل دفع and إذا في يعني زي مثلا في
+
+142
+00:10:38,260 --> 00:10:43,670
+الكورونا الآن في محاولات ال antiviral drugsوطبعاً
+
+143
+00:10:43,670 --> 00:10:48,430
+إحنا بندّي حقيقة اللي هو الـ Influenza Vaccine
+
+144
+00:10:48,430 --> 00:10:51,210
+سنوي في شهر 10 وشهر 11
+
+145
+00:10:53,950 --> 00:11:00,570
+مقارنة بـA وB مخلوقات الفلوانزا التي تتوقع أن تكون
+
+146
+00:11:00,570 --> 00:11:05,090
+مميزة في عام محدد الفلوانزا يجب أن يتم تحرير
+
+147
+00:11:05,090 --> 00:11:08,490
+وإدارة الفلوانزا في عام محدد الفلوانزا لكي تكون
+
+148
+00:11:08,490 --> 00:11:09,170
+مفيدا
+
+149
+00:11:17,530 --> 00:11:22,070
+ولكنه لا يعطي حماية كاملة من الـ Influenza فيه
+
+150
+00:11:22,070 --> 00:11:25,890
+strains في الـ Influenza مش محتوطة في التطعيم هذه
+
+151
+00:11:25,890 --> 00:11:28,150
+طبعا تقدر تصيب الواحد بالـ Influenza وفي منها
+
+152
+00:11:28,150 --> 00:11:32,510
+عشرات دون الالتفات للـ Vaccine اللي صار يعني الـ
+
+153
+00:11:32,510 --> 00:11:38,370
+Vaccine بس بيحمي من الفيروسات اللي موجودة في الـ
+
+154
+00:11:38,370 --> 00:11:42,560
+Strains اللي موجودة في التطعيمالفاكسينة
+
+155
+00:11:42,560 --> 00:11:48,920
+الانفلوانزا تشير بشكل خاص للناس الأكبر من 6 أشهر
+
+156
+00:11:48,920 --> 00:11:50,900
+من العمر للناس الأكبر من 6 أشهر من العمر للناس
+
+157
+00:11:50,900 --> 00:11:52,560
+الأكبر من 6 أشهر من العمر للناس الأكبر من 6 أشهر
+
+158
+00:11:52,560 --> 00:11:54,520
+من العمر للناس الأكبر من 6 أشهر من العمر للناس
+
+159
+00:11:54,520 --> 00:11:55,160
+الأكبر من 6 أشهر من العمر للناس الأكبر من 6 أشهر
+
+160
+00:11:55,160 --> 00:11:55,300
+من العمر للناس الأكبر من 6 أشهر من العمر للناس
+
+161
+00:11:55,300 --> 00:11:57,340
+الأكبر من 6 أشهر من العمر للناس الأكبر من 6 أشهر
+
+162
+00:11:57,340 --> 00:12:00,660
+من
+
+163
+00:12:00,660 --> 00:12:02,480
+العمر للناس الأكبر
+
+164
+00:12:12,950 --> 00:12:20,210
+وحقيقة أن الـ Respiratory Tract عادة مش بسهولة أن
+
+165
+00:12:20,210 --> 00:12:24,750
+الواحد ممكن ينصاب بالـ Lower Respiratory Tract
+
+166
+00:12:24,750 --> 00:12:27,450
+Infection يعني نصير عنده اتهاب في الرقى لابد أن
+
+167
+00:12:27,450 --> 00:12:32,640
+يكون فيه هناك عواملأضعفت الجسم، أضعفت الجسم اللي
+
+168
+00:12:32,640 --> 00:12:39,500
+هو الوضع المناعي تبع البني آدم وهي أثرت أنه انصاب
+
+169
+00:12:39,500 --> 00:12:44,800
+بالتهاب بكتيري أو التهاب فيروسي للـ Lower
+
+170
+00:12:44,800 --> 00:12:50,820
+Respiratory Tract Infection طيب، ليش؟ لأنه أصلاً
+
+171
+00:12:50,820 --> 00:12:57,040
+هناك آليات Defense Mechanisms آليات حماية للجسم
+
+172
+00:12:58,900 --> 00:13:04,760
+وخماية من؟ الـ Lungs بالذات، عشان البكتيريا هذه أو
+
+173
+00:13:04,760 --> 00:13:07,220
+الفيروس هذه ما تصل الرئتين
+
+174
+00:13:12,720 --> 00:13:17,180
+طبعاً هناك عوام الأخرى مدى شدة البكتيريا نفس
+
+175
+00:13:17,180 --> 00:13:22,080
+البكتيريا تكون virulent تكون هي نفسها متعبة كمان
+
+176
+00:13:22,080 --> 00:13:26,740
+بالعضو يبقى من الـ Host Defense Barriers بس بي ..
+
+177
+00:13:26,740 --> 00:13:33,160
+بنحكي عنها اللي بتضعفها اللي هي cigarette smoking
+
+178
+00:13:33,160 --> 00:13:40,340
+التدخين من الحاجات اللي بتضعفنا الجهاز المناعي و
+
+179
+00:13:40,340 --> 00:13:46,980
+بتعمل weakeningللـ Respiratory Defence Barriers زي
+
+180
+00:13:46,980 --> 00:13:51,900
+ما قلنا ليش؟ لأنها بتعمل Paralysing للـCilia
+
+181
+00:13:51,900 --> 00:13:55,780
+فاكرين الأهداف اللي حكينا عنها؟ بتعملها Paralysing
+
+182
+00:13:55,780 --> 00:14:00,160
+و بتعملها Damaging وهذه الـCilias هي كانت المسؤولة
+
+183
+00:14:00,160 --> 00:14:04,200
+في لحظة من اللحظات إن نتيجة حركتها إلى فوق
+
+184
+00:14:04,200 --> 00:14:11,650
+الشعيرات هذه أو الأهداف هذه إنها تخرج كلما يسبب
+
+185
+00:14:11,650 --> 00:14:17,570
+بلغم أو يسبب secretions أو يسبب particles أو micro
+
+186
+00:14:17,570 --> 00:14:21,490
+organism هي بترضه فلمّا أجت سيجارة smoking على
+
+187
+00:14:21,490 --> 00:14:25,950
+المدى الطويل عملت damaging للـcelias هاي بطلت
+
+188
+00:14:25,950 --> 00:14:29,650
+تشتغل بتصير ال particles زي البكتيريا و الحاجات
+
+189
+00:14:29,650 --> 00:14:36,430
+هاي تنزل دون حماية لمين للريقتينطبعاً لابد أنه كما
+
+190
+00:14:36,430 --> 00:14:43,230
+قلنا الـ virulence اللي هي شدة الـ شدة البكتيريا
+
+191
+00:14:43,230 --> 00:14:47,890
+هاي و قوتها بتلعب دور و مش بس الأجواء المحيطة
+
+192
+00:14:47,890 --> 00:14:52,270
+بتلعب دور زي الـ cold weather عسب اللي تعرف من
+
+193
+00:14:52,270 --> 00:14:57,010
+برضه الـ defence mechanisms اللي هو
+
+194
+00:15:00,630 --> 00:15:05,590
+السيرفيس كلها بدأت من الأنف والحلق كلها مويست و
+
+195
+00:15:05,590 --> 00:15:12,450
+عليها زي نوع من الـ mucous وهذه تتلقط الأورجانيزم
+
+196
+00:15:12,450 --> 00:15:15,970
+و الـ particles فيه إن الـ isosomes و الـ cell
+
+197
+00:15:15,970 --> 00:15:19,750
+service IGA فيه إن زي ما حكينا قبل شوية الـ
+
+198
+00:15:19,750 --> 00:15:23,210
+ciliated epithelium عنا كمان حاجة ربنا أدانا إياها
+
+199
+00:15:23,210 --> 00:15:28,040
+هي الكحةاليّة أو ميكانيكية الكحّة أو ردّة فعل
+
+200
+00:15:28,040 --> 00:15:30,500
+الكحّة كل ما ييجي يدخل إشي يعني زي الواحد اللي
+
+201
+00:15:30,500 --> 00:15:34,020
+ميجي تشردق يدخل إشي على الرئتين على طول واحد بكح
+
+202
+00:15:34,020 --> 00:15:37,980
+فبطلعها هذه نعمة من ربنا الكحّة نعمة من ربنا
+
+203
+00:15:37,980 --> 00:15:43,400
+الدنياها to prevent aspiration شردقة of particles
+
+204
+00:15:43,400 --> 00:15:48,700
+and irritants into the lower airways وفيه طبعا أن
+
+205
+00:15:48,700 --> 00:15:54,290
+أنواع زي اللي هو الـ Pulmonary Macrophageالـ
+
+206
+00:15:54,290 --> 00:16:00,710
+Microfage الكبير اللي بتحاول تعمل فجوط صيتز الـ
+
+207
+00:16:00,710 --> 00:16:06,210
+foreign particles أو الـ organisms طبعاً
+
+208
+00:16:06,210 --> 00:16:09,470
+لما أنا بقول lung tissue infection بقصد فيها
+
+209
+00:16:09,470 --> 00:16:15,250
+pneumonia pneumonia ال P ممكن أنه ما ألفزهاش لفظ
+
+210
+00:16:15,250 --> 00:16:18,410
+كامل pneumonia في ناس بتقول عن ابنومونيا لكن هي
+
+211
+00:16:18,410 --> 00:16:23,160
+pneumonia pneumonia is a condition thatالتي تتعلق
+
+212
+00:16:23,160 --> 00:16:27,340
+بالإنفلام من الـ Lower Lung Structures مثل الـ
+
+213
+00:16:27,340 --> 00:16:32,700
+Alveoli و الـ Interstitial Spaces يعني كل ال .. ال
+
+214
+00:16:32,700 --> 00:16:35,060
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+215
+00:16:35,060 --> 00:16:35,760
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+216
+00:16:35,760 --> 00:16:35,860
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+217
+00:16:35,860 --> 00:16:36,220
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+218
+00:16:36,220 --> 00:16:36,380
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+219
+00:16:36,380 --> 00:16:36,480
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+220
+00:16:36,480 --> 00:16:37,160
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+221
+00:16:37,160 --> 00:16:37,200
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+222
+00:16:37,200 --> 00:16:37,440
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+223
+00:16:37,440 --> 00:16:43,140
+.. ال .. ال .. ال .. ال .. ال ..
+
+224
+00:16:43,140 --> 00:16:44,360
+ال
+
+225
+00:16:53,800 --> 00:16:56,560
+طبعاً الـ Pneumonia الـ prevalence تبعها و الـ
+
+226
+00:16:56,560 --> 00:16:59,660
+severity of pneumonia have been heightened in
+
+227
+00:16:59,660 --> 00:17:05,220
+recent years due to the emergence of HIV فاكرين
+
+228
+00:17:05,220 --> 00:17:09,940
+إيش الـ HIV؟ صحيح الـ AIDS as well as antibiotic
+
+229
+00:17:09,940 --> 00:17:17,000
+resistance نتيجة إنه صار فيه كتير HIV وهذا اسمه
+
+230
+00:17:17,000 --> 00:17:24,410
+ضعف المناع المكتسبونتيجة إنه صار في عنا تكون
+
+231
+00:17:24,410 --> 00:17:29,750
+مقاومة للمضادات الحيوية نتيجة استعمال غير الرشيد
+
+232
+00:17:29,750 --> 00:17:34,650
+للمضادات الحيوية فزادت الـ severity و ال
+
+233
+00:17:34,650 --> 00:17:39,210
+prevalence of pneumonia و ال classification تبع
+
+234
+00:17:39,210 --> 00:17:42,170
+الـ pneumonia حسب ممكن نقولها حسب الـ pathogen حسب
+
+235
+00:17:42,170 --> 00:17:45,430
+المادة اللي عملت التهاب يعني viral pneumonia أو
+
+236
+00:17:45,430 --> 00:17:51,210
+bacterial pneumoniaوممكن لكن نقول عنها كمان
+
+237
+00:17:51,210 --> 00:17:56,570
+التفريق التاني أقول عنها hospital أو community
+
+238
+00:17:56,570 --> 00:18:02,670
+pneumonia أو hospital nosocomial pneumonia ال
+
+239
+00:18:02,670 --> 00:18:07,330
+individuals most at risk of pneumonia الالدرلي
+
+240
+00:18:07,330 --> 00:18:11,870
+الناس اللي عندها viral infection ال chronic ill ال
+
+241
+00:18:11,870 --> 00:18:15,690
+HIV أو ال AIDS والimmune suppressant patient
+
+242
+00:18:15,690 --> 00:18:21,330
+smokersبشيء مثل الـ Bronchial Asthma بشيء ممكن
+
+243
+00:18:21,330 --> 00:18:26,330
+كمان مرة في عنا نمونيا اسمها community acquired
+
+244
+00:18:26,330 --> 00:18:30,250
+community acquired يعني من المجتمع وفي عنا
+
+245
+00:18:30,250 --> 00:18:34,410
+aspiration pneumonia الشردقة ينزل حاجة على الرئتين
+
+246
+00:18:34,410 --> 00:18:38,730
+وفي عنا ال hospital acquired فرجوها قالوا في
+
+247
+00:18:38,730 --> 00:18:42,620
+hospital acquired pneumonia وفي ventilatorبنتليتر
+
+248
+00:18:42,620 --> 00:18:46,180
+جوز جهزتنا في الصناعي associated pneumonia وفيه
+
+249
+00:18:46,180 --> 00:18:49,320
+health care associated pneumonia زي ما هنحكي بعد
+
+250
+00:18:49,320 --> 00:18:55,040
+شوية ال
+
+251
+00:18:55,040 --> 00:18:59,680
+potential pathogens اللي احنا بنعرفه يعني خلنا
+
+252
+00:18:59,680 --> 00:19:05,230
+نقول ممكن نتعرف على اتنين تلاتة من كل واحدوهي الـ
+
+253
+00:19:05,230 --> 00:19:06,850
+atypical pneumonia الـ atypical pneumonia هي
+
+254
+00:19:06,850 --> 00:19:10,310
+النمونيا العادية المتوقعة زي الـ Streptococcus
+
+255
+00:19:10,310 --> 00:19:13,770
+pneumonia و الـ Hemophilus influenza و الـ
+
+256
+00:19:13,770 --> 00:19:15,790
+Klebsiella pneumonia و الـ Mycobacterium
+
+257
+00:19:15,790 --> 00:19:18,350
+catarrhalis و الـ atypical زي الـ Chlamydia
+
+258
+00:19:18,350 --> 00:19:22,670
+pneumonia و Legionella pneumonia و Mycoplasma
+
+259
+00:19:22,670 --> 00:19:30,190
+pneumonia هذه الصورة بتوضح لنا أن بعض الأنواع
+
+260
+00:19:32,690 --> 00:19:39,850
+والـ classification للـ pneumonia هنرجعلها
+
+261
+00:19:39,850 --> 00:19:47,710
+بنحكي الآن عن ال typical و ال atypical إيش بنقصد
+
+262
+00:19:47,710 --> 00:19:54,090
+فيهم؟ كال pneumonia حسب
+
+263
+00:19:56,930 --> 00:20:01,990
+حسب اللي هو قلنا فيه typical و atypical حسب
+
+264
+00:20:01,990 --> 00:20:06,090
+specific structure لأصيبة في الـ lung الـ typical
+
+265
+00:20:06,090 --> 00:20:12,470
+عادة بتبقى بكتيريا in origin و organisms replicate
+
+266
+00:20:12,470 --> 00:20:17,090
+يعني عملية توالد أو تجدد ال organisms بتصير كلها
+
+267
+00:20:17,090 --> 00:20:21,210
+في ال alveoli ال manifestation تبعها يبقى
+
+268
+00:20:21,210 --> 00:20:25,410
+inflammation of and fluid accumulation are seen in
+
+269
+00:20:25,410 --> 00:20:29,680
+the alveoliبالتالي White Cell Infiltration and
+
+270
+00:20:29,680 --> 00:20:33,380
+Exudation Can Be Seen On Chest Radiograph لو عملنا
+
+271
+00:20:33,380 --> 00:20:39,700
+X-Ray بنلاقي إنه في عندنا Infiltration بنلاقي High
+
+272
+00:20:39,700 --> 00:20:43,940
+Fever، Chest Pain، Chills، Malaise، تعب ماليز
+
+273
+00:20:44,680 --> 00:20:55,180
+بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت
+
+274
+00:20:55,180 --> 00:20:56,100
+بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت
+
+275
+00:20:56,100 --> 00:20:57,340
+بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت
+
+276
+00:20:57,340 --> 00:21:03,360
+بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت
+
+277
+00:21:03,360 --> 00:21:03,380
+بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت
+
+278
+00:21:03,380 --> 00:21:04,200
+بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت بيرولنت
+
+279
+00:21:04,200 --> 00:21:07,380
+بيرولنت
+
+280
+00:21:07,380 --> 00:21:13,520
+بيرولنتوالعضلات تتعادل في المقاطع حول الـ alveoli
+
+281
+00:21:13,520 --> 00:21:18,240
+قبل قليل كانت في المقاطع حول الـ alveoli جوة الـ
+
+282
+00:21:18,240 --> 00:21:22,520
+alveoli لكن هنا حوالين الـ alveoli بنلاقي إن الـ
+
+283
+00:21:22,520 --> 00:21:25,520
+atypical أو الـ viral هي mild
+
+284
+00:21:29,250 --> 00:21:32,570
+وبنلاقيش في الـ alveoli infiltration زي ما احنا
+
+285
+00:21:32,570 --> 00:21:36,550
+بنشوفها و lack of fluid accumulation in the
+
+286
+00:21:36,550 --> 00:21:40,730
+alveoli not usually evident on radiographs ممكن
+
+287
+00:21:40,730 --> 00:21:44,650
+أعمل X-ray وما لاجيهاش، ماشوفهاش و may make the
+
+288
+00:21:44,650 --> 00:21:48,390
+patient susceptible to bacterial pneumonia نبقى
+
+289
+00:21:48,390 --> 00:21:51,270
+هنا احنا حكينا عن واحد من أنواع ال classification
+
+290
+00:21:51,270 --> 00:21:56,500
+الـtypical و الـatypical pneumoniaفي عندنا كمان
+
+291
+00:21:56,500 --> 00:21:59,720
+حاجة احنا بنعرفها اللي هي الـ opportunistic
+
+292
+00:21:59,720 --> 00:22:05,080
+organisms يعني احنا كنا bacterial و viral لأننا
+
+293
+00:22:05,080 --> 00:22:08,920
+ندخل على opportunistic organisms وهي عبارة عن
+
+294
+00:22:08,920 --> 00:22:11,460
+number of organisms not commonly associated with
+
+295
+00:22:11,460 --> 00:22:15,460
+respiratory illness in otherwise healthy
+
+296
+00:22:15,460 --> 00:22:18,540
+individuals can cause severeالـ Respiratory
+
+297
+00:22:18,540 --> 00:22:23,540
+Infection باختصار هدولة نوع من أنواع الـ Organism
+
+298
+00:22:23,540 --> 00:22:27,800
+زي الـ Histoplasma و الـ Pneumocystis carinae يعني
+
+299
+00:22:27,800 --> 00:22:30,760
+الـ Fungal Infection مع الـ Protozoa اللي هو
+
+300
+00:22:30,760 --> 00:22:34,280
+Pneumocystis carinae و الـ Mycobacteria في الوضع
+
+301
+00:22:34,280 --> 00:22:39,220
+الطبيعي في الـ Healthy Body بيعملوش حاجة ها بتجد
+
+302
+00:22:39,220 --> 00:22:44,640
+عنهم لما بدعف الجسم عشان هيك بنقول عنهم المستنفعين
+
+303
+00:22:44,640 --> 00:22:49,290
+opportunistic organismمستنفعين، these organisms
+
+304
+00:22:49,290 --> 00:22:53,750
+include زي ما قلنا Mycobacteria أو fungus زي
+
+305
+00:22:53,750 --> 00:22:57,090
+الهيستو بلازما أو Protisuga زي الـ Gnemosis carini
+
+306
+00:22:57,090 --> 00:23:00,590
+treatment of these organisms require specific drug
+
+307
+00:23:00,590 --> 00:23:06,510
+therapy and in case of Protisuga بنحتاجه طبعاً
+
+308
+00:23:06,510 --> 00:23:10,470
+بتبقى صعبة علاجها لأن هي بتبقى توحشت نتيجة إن
+
+309
+00:23:10,470 --> 00:23:14,670
+الimmune نتيجة إن الimmune system ضعيف عند ال
+
+310
+00:23:14,670 --> 00:23:18,560
+patientهذه ما بيقولها opportunistic organisms
+
+311
+00:23:18,560 --> 00:23:22,620
+حكينا
+
+312
+00:23:22,620 --> 00:23:24,960
+عن infection، هندخل على الـ Obstructive
+
+313
+00:23:24,960 --> 00:23:28,080
+Respiratory Disorders إيش يعني Obstructive؟ فيه
+
+314
+00:23:28,080 --> 00:23:31,900
+تضيق Respiratory Disorders الأمراض المرتبطة
+
+315
+00:23:31,900 --> 00:23:37,580
+بالتضيق طبعا
+
+316
+00:23:37,580 --> 00:23:44,110
+أهمها لـ Bronchial Asthmaاللي بنقول عنها احنا اللي
+
+317
+00:23:44,110 --> 00:23:47,310
+هي حساسية السدر في ناس بتقول عنها النهجة في ناس
+
+318
+00:23:47,310 --> 00:23:52,070
+بتقول عنها الأزمة في ناس بتقول عنها الربو أزمة is
+
+319
+00:23:52,070 --> 00:23:54,650
+a condition characterized by reversible
+
+320
+00:23:54,650 --> 00:23:58,310
+bronchospasm reversible يعني بيروحوا بي .. ممكن
+
+321
+00:23:58,310 --> 00:24:03,350
+يروحوا ييجي bronchospasm تضيق في البرونكيولز and
+
+322
+00:24:03,350 --> 00:24:07,290
+chronic inflammation of airway passages وكمان مش
+
+323
+00:24:07,290 --> 00:24:13,120
+بسبتدينا inflammation مش infection inflammation of
+
+324
+00:24:13,120 --> 00:24:16,980
+airway passages the incidence of asthma has been
+
+325
+00:24:16,980 --> 00:24:20,400
+steadily increasing in the recent years فعلا بتزيد
+
+326
+00:24:20,400 --> 00:24:27,320
+نتيجة ال environmental pollution although the
+
+327
+00:24:27,320 --> 00:24:31,800
+extent the exact etiology is still uncertain there
+
+328
+00:24:31,800 --> 00:24:36,440
+appears to be a definite genetic predisposition to
+
+329
+00:24:36,440 --> 00:24:38,020
+the development of asthma
+
+330
+00:24:41,660 --> 00:24:45,520
+لكن كمان في عندنا الـ Genetic Predisposition
+
+331
+00:24:49,030 --> 00:24:53,330
+وورثياً جينياً إنها يصير عندها اكي كومبوننط of
+
+332
+00:24:53,330 --> 00:24:57,770
+أزمة appears to be airway hyper reactivity اللي هي
+
+333
+00:24:57,770 --> 00:25:02,530
+زيادة حساسية اللي هو ال .. ال .. ال .. ال airway
+
+334
+00:25:02,530 --> 00:25:06,890
+اللي هي المسالك التنفسية in affected individuals
+
+335
+00:25:06,890 --> 00:25:11,110
+exposure to certain triggers can include marked
+
+336
+00:25:11,110 --> 00:25:15,670
+bronchospasm and airway inflammationفي مستخدم
+
+337
+00:25:15,670 --> 00:25:15,870
+مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم
+
+338
+00:25:15,870 --> 00:25:18,210
+مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم
+
+339
+00:25:18,210 --> 00:25:18,470
+مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم
+
+340
+00:25:18,470 --> 00:25:21,930
+مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم
+
+341
+00:25:21,930 --> 00:25:27,570
+مستخدم مستخدم مستخدم مستخدم مستخدم مستخدم
+
+342
+00:25:27,570 --> 00:25:30,310
+مستخدم
+
+343
+00:25:38,690 --> 00:25:41,310
+فبصير عندنا صورة اللي هي الـ bronchial asthma
+
+344
+00:25:41,310 --> 00:25:45,110
+Individuals with asthma appear to produce large
+
+345
+00:25:45,110 --> 00:25:51,970
+amounts of antibody IGE that attach to the mast
+
+346
+00:25:51,970 --> 00:25:57,790
+cells present in many tissues هلجيت ال exposure to
+
+347
+00:25:57,790 --> 00:26:02,210
+a trigger such as pollen اللي هي الحاجات اللي
+
+348
+00:26:02,210 --> 00:26:05,880
+بتطلع من الشجروالرزولت في الـ Allergy-Binding
+
+349
+00:26:05,880 --> 00:26:10,740
+Muscles-Pound IgE اللي يسبب في إطلاق إطلاق مدينة
+
+350
+00:26:10,740 --> 00:26:14,200
+انفلامات مثل الهستامين والليكوترين
+
+351
+00:26:14,200 --> 00:26:19,100
+والأيزوفينيكيكيموتاكتك فاكتور الاتجاه للطبيعي مع
+
+352
+00:26:19,100 --> 00:26:23,620
+الأزمة لتلك التغييرات ممكن أقول عنه في منه early
+
+353
+00:26:23,620 --> 00:26:28,570
+phase و في late phaseمن الـ triggers اللي لازم
+
+354
+00:26:28,570 --> 00:26:36,110
+طبعا ابحثوا عن بعضهم ممكن تبحثوا تجروا الألراجين
+
+355
+00:26:36,110 --> 00:26:40,810
+زي الـ pollen، الpit، الdander، الفنجاي، ال dust
+
+356
+00:26:40,810 --> 00:26:46,250
+bites، cold air، pollutants، cigarette smoking،
+
+357
+00:26:46,250 --> 00:26:49,490
+strong emotion، exercise، respiratory tract
+
+358
+00:26:49,490 --> 00:26:53,520
+infection، كل هاي برنكا الأزمةأو potential trigger
+
+359
+00:26:53,520 --> 00:26:56,220
+طبعاً لا يعني أنه ما فيش اللي هي ممكن كل واحد إله
+
+360
+00:26:56,220 --> 00:27:00,660
+ال triggers تبعته اللي ممكن تعمله الموضوع هذا why
+
+361
+00:27:00,660 --> 00:27:04,680
+asthma makes it hard to breathe طبعاً زي ما احنا
+
+362
+00:27:04,680 --> 00:27:11,640
+شايفين هي ال two bronchial tube الجهة اللي فيها
+
+363
+00:27:11,640 --> 00:27:17,920
+صار عندنا affection نتيجة ألرجي معين عملتنا
+
+364
+00:27:17,920 --> 00:27:22,800
+inflamed bronchial tubeوالـ Obstruction مع
+
+365
+00:27:22,800 --> 00:27:25,400
+Inflammation في الجهة التانية Normal Bronchial
+
+366
+00:27:25,400 --> 00:27:31,200
+Tube واسع وبقدر واحد يتنفس زي الناس فيه طبعا عندنا
+
+367
+00:27:31,200 --> 00:27:37,500
+حكينا عن الـ Early Phase و الـ Late Phase خلّيني
+
+368
+00:27:37,500 --> 00:27:39,360
+أحكيه ما وجهته وبعدين نرجع لـ Clinical
+
+369
+00:27:39,360 --> 00:27:44,060
+Classification Early Phase of Asthma characterized
+
+370
+00:27:44,060 --> 00:27:46,820
+by mild
+
+371
+00:27:49,530 --> 00:27:53,950
+أو marked construction of bronchial airways بيصير
+
+372
+00:27:53,950 --> 00:28:00,090
+إديمة in the airway وفي عندنا production of excess
+
+373
+00:28:00,090 --> 00:28:03,630
+mucus يبقى marked construction of bronchial
+
+374
+00:28:03,630 --> 00:28:06,530
+airways bronchospasm إديمة in the airway
+
+375
+00:28:06,530 --> 00:28:11,890
+وproduction of excess mucus
+
+376
+00:28:13,910 --> 00:28:17,950
+وطبعاً الـ Bronchospasm جاء نتيجة أول حاجة Direct
+
+377
+00:28:17,950 --> 00:28:23,050
+نتيجة الـ inflammation لكن كمان نتيجة اللي هو الـ
+
+378
+00:28:23,050 --> 00:28:28,210
+الـ الـ الـ الـ mucous building الـ Late phase
+
+379
+00:28:28,210 --> 00:28:32,990
+طبعا
+
+380
+00:28:32,990 --> 00:28:35,230
+الـ Mediators الأساسية كانت في الـ Early phase
+
+381
+00:28:35,230 --> 00:28:40,470
+اللي هي Histamine, Prostaglandin and Predicanin في
+
+382
+00:28:40,470 --> 00:28:45,620
+الـ Early في ال Late phaseهذا بيجي طبعا بعد بضع
+
+383
+00:28:45,620 --> 00:28:49,240
+ساعات after the initial onset of symptoms and
+
+384
+00:28:49,240 --> 00:28:53,440
+manifests mainly as inflammatory response خدوا
+
+385
+00:28:53,440 --> 00:28:57,380
+بالكم الأولاني كان Bronchospasm ما inflammation،
+
+386
+00:28:57,380 --> 00:29:01,680
+الآن بس inflammatory response وهذا الفكرة بتفيدنا
+
+387
+00:29:01,680 --> 00:29:06,080
+في قضية العلاج بصورة عامة The primary mediators of
+
+388
+00:29:06,080 --> 00:29:10,440
+inflammation during the asthmatic response are the
+
+389
+00:29:10,440 --> 00:29:15,140
+white blood cells اللي هي الـAzonophilsيبقى في الـ
+
+390
+00:29:15,140 --> 00:29:19,260
+Late Phase of Asthma الـ eosinophils لعبت دور كبير
+
+391
+00:29:19,260 --> 00:29:23,400
+في الموضوع وsubsequent infiltration of the airway
+
+392
+00:29:23,400 --> 00:29:27,260
+tissue with white blood cells such as neutrophils
+
+393
+00:29:27,260 --> 00:29:29,800
+و الـ lymphocytes also contribute to the overall
+
+394
+00:29:29,800 --> 00:29:34,020
+inflammatory response of the Late Phase of Asthma
+
+395
+00:29:34,020 --> 00:29:37,220
+يبقى بدنا نفرج بين إيش بيصير في الـ Early Phase
+
+396
+00:29:37,220 --> 00:29:46,150
+وإيش بيصير في ال Late Phaseفي عنا طبعاً symptoms،
+
+397
+00:29:46,150 --> 00:29:49,110
+manifestation الـ symptoms اللي بتصير عنا في الـ
+
+398
+00:29:49,110 --> 00:29:52,630
+bronchial asthma اللي هي coughing، wheezing، إيش
+
+399
+00:29:52,630 --> 00:29:55,670
+يعني wheezing؟ بيصير يسفر السدر، يزمر، difficulty
+
+400
+00:29:55,670 --> 00:29:59,810
+breathing, rapid and shallow breathing, increased
+
+401
+00:29:59,810 --> 00:30:02,470
+respiratory rate, excess mucus production and
+
+402
+00:30:02,470 --> 00:30:05,430
+significant anxiety، مافيش حرارة، خدوا بالكم، هاد
+
+403
+00:30:05,430 --> 00:30:11,510
+هو ال manifestation of bronchial asthmaفي عنا طبعا
+
+404
+00:30:11,510 --> 00:30:17,090
+حاجة اسمها staging of
+
+405
+00:30:17,090 --> 00:30:23,100
+acute asthma attackأو حد على المستشفى مجرد ما
+
+406
+00:30:23,100 --> 00:30:26,200
+شوفته أنا عرفت أن البرونكيل أزمة بدأ أقيمه هو
+
+407
+00:30:26,200 --> 00:30:30,200
+stage one ولا two ولا three ولا four طب إيش one؟
+
+408
+00:30:30,200 --> 00:30:35,300
+one mild، two moderate، three severe، and four
+
+409
+00:30:35,300 --> 00:30:38,420
+respiratory failure يعني بده عناية مركزة عمليًا هو
+
+410
+00:30:38,420 --> 00:30:41,900
+والتلاتة ممكن يحتاجوا عناية مركزة stage تلاتة وال
+
+411
+00:30:41,900 --> 00:30:45,650
+respiratory failure stage أربعةطب إيش بيصير في الـ
+
+412
+00:30:45,650 --> 00:30:49,310
+Mild بيصير ديسنيا خفيفة diffuse wheezing يعني في
+
+413
+00:30:49,310 --> 00:30:56,390
+تسفيرة عنده ولكن نفسه كويس لو حطيته على جهاز ال
+
+414
+00:30:56,390 --> 00:31:00,310
+monitor بلاقي ال oxygen كويس stage two moderate
+
+415
+00:31:01,460 --> 00:31:05,280
+متوسط يعني Respiratory distress وهو جاعد لأ نفسه
+
+416
+00:31:05,280 --> 00:31:12,980
+تعبان وصورة واضحة صورة الصوت التسفير في جسمه في
+
+417
+00:31:12,980 --> 00:31:17,100
+صدره واضح Stage 3 Severe Marked Respiratory
+
+418
+00:31:17,100 --> 00:31:21,130
+distressأه بدأ يصير أزرق الآن يعني مش بس صار عنده
+
+419
+00:31:21,130 --> 00:31:27,150
+ضيوف في النفس أزرق وحتى بدون يعني بشوف إنه فيه
+
+420
+00:31:27,150 --> 00:31:31,910
+تسفير واضح أو إنه كتم صدره من الكترب وهم سكر صدره،
+
+421
+00:31:31,910 --> 00:31:35,690
+سكر صدره بالظبط absence of breath sounds يعني أنا
+
+422
+00:31:35,690 --> 00:31:39,110
+ماحط السماعة بسمعش ولا حاجة، ليش؟ لأنه اتسكرت
+
+423
+00:31:39,110 --> 00:31:42,070
+الطرق obstructed لإن احنا برنكيل الأزمة قلنا إيش
+
+424
+00:31:42,070 --> 00:31:44,370
+اللي بيصير؟ بيصير يعني obstruction of bronchioles
+
+425
+00:31:44,370 --> 00:31:48,210
+فهو سكرالـ Stage IV الـ Respiratory Failure هو
+
+426
+00:31:48,210 --> 00:31:51,450
+عبارة عن Severe Respiratory Stress، Lethargy، بدأ
+
+427
+00:31:51,450 --> 00:31:54,650
+يتأثر، Confusion، بطل يصل الدم للدماج، فصار يصير
+
+428
+00:31:54,650 --> 00:31:58,270
+عنده Confusion، Prominent Pulses Paradoxical، هذه
+
+429
+00:31:58,270 --> 00:32:02,750
+اللي هي لما يجيني واحد كده أقيمه في الـ Acute
+
+430
+00:32:02,750 --> 00:32:07,070
+Attack، ساعة ما بيجيني الـ Severe، الـ Severity،
+
+431
+00:32:07,070 --> 00:32:09,550
+Staging of the Severity، خدوا بالكم من العنوان،
+
+432
+00:32:09,550 --> 00:32:13,050
+Staging of the Severity of the Acute Asthma
+
+433
+00:32:13,050 --> 00:32:17,940
+Attack،لكن في عنا حاجة تانية احنا كنا فطناها قبل
+
+434
+00:32:17,940 --> 00:32:21,640
+شويه اللي هي Classification من الأزمة هذا بصورة
+
+435
+00:32:21,640 --> 00:32:25,980
+عامة مش الـ Acute أي واحد عنده أزمة وبيجيني هو
+
+436
+00:32:25,980 --> 00:32:30,660
+هيكون واحد من أربع Classification Clinical
+
+437
+00:32:30,660 --> 00:32:35,360
+Classification الـMild Intermittent وMild
+
+438
+00:32:35,360 --> 00:32:38,540
+Persistent، Moderate Persistent وSevere Persistent
+
+439
+00:32:40,550 --> 00:32:44,610
+الـ Mild Intermittent أنه بتجيله نوبات مرتين في
+
+440
+00:32:44,610 --> 00:32:49,990
+الأسبوع أو أقل النوبة اللي حكينا عنها قبل شوية الـ
+
+441
+00:32:49,990 --> 00:32:53,050
+Mild Persistent attacks occur more than two times
+
+442
+00:32:53,050 --> 00:32:57,330
+per week هذا Mild Persistent أكتر من تنتين الـ
+
+443
+00:32:57,330 --> 00:33:02,350
+Moderate Persistent هو attacks occur daily or
+
+444
+00:33:02,350 --> 00:33:08,110
+almost daily يوميا أو تقريبا يومياand are severe
+
+445
+00:33:08,110 --> 00:33:13,210
+enough to affect activity مجعدا، رميا، تعبا، يوميا
+
+446
+00:33:13,210 --> 00:33:17,110
+و الـ severe persistent attacks are very frequent
+
+447
+00:33:17,110 --> 00:33:20,750
+and persistent for a long period of time attacks
+
+448
+00:33:20,750 --> 00:33:24,570
+severely limit activity يبقى هذه اسمها clinical
+
+449
+00:33:24,570 --> 00:33:28,770
+classification of asthma كم نوبة بتيجي عنده خلال
+
+450
+00:33:28,770 --> 00:33:34,690
+الأسبوع أو بصورة عامة اللي حكيناه قبلها اللي هوالـ
+
+451
+00:33:34,690 --> 00:33:38,350
+Staging هذه للـ Acute Attack كيف بقى أقيم كل Acute
+
+452
+00:33:38,350 --> 00:33:41,450
+Attack؟
+
+453
+00:33:41,450 --> 00:33:46,210
+نكمل
+
+454
+00:33:46,210 --> 00:33:48,430
+بس لأنه ضايق سلايدين بس في الـ Bronchial Asthma
+
+455
+00:33:48,430 --> 00:33:53,690
+possible complications of asthma can includeطبعاً
+
+456
+00:33:53,690 --> 00:33:56,790
+status asthmaticus اللي بيقول severe acute أزمة
+
+457
+00:33:56,790 --> 00:33:59,910
+وهذا is a life-threatening condition زي status
+
+458
+00:33:59,910 --> 00:34:06,190
+epilepticus النوبة الصراع الكبرى هذه نوبة الأزمة
+
+459
+00:34:06,190 --> 00:34:11,000
+الكبرى واللي بيحتاج prolongedالبرونكوسبازم هو
+
+460
+00:34:11,000 --> 00:34:14,920
+البرونكوسبازم اللي عادةً لا يتراجع للدواء الترابي
+
+461
+00:34:14,920 --> 00:34:18,500
+كمان بتدي دوة مفيدش كتير، في الآخر بده ايش؟ بده
+
+462
+00:34:18,500 --> 00:34:22,440
+تنفس صناعي، ايش كمان ممكن يحصل؟ بنوعي Motorax،
+
+463
+00:34:22,440 --> 00:34:25,700
+هنحكي عن بنوعي Motorax هي عبارة عن possible
+
+464
+00:34:25,700 --> 00:34:32,800
+consequence of lung pressure increasesيمكن أن
+
+465
+00:34:32,800 --> 00:34:36,300
+ينتهي من الصعوبة الكهربائية المرتبطة في الانتصاف
+
+466
+00:34:36,300 --> 00:34:41,380
+خلال اتكاب أزمة مستمر، فإن فشل الهيبوكسيميا
+
+467
+00:34:41,380 --> 00:34:49,100
+والأسدوزية قد يحدث تعامل
+
+468
+00:34:49,100 --> 00:34:54,170
+الأزمةحقيقة يعني احنا زي ما قلنا برضه مش لازمنا
+
+469
+00:34:54,170 --> 00:34:59,510
+كتير الأدوية لكن لابد انه نتعرف على الحاجات
+
+470
+00:34:59,510 --> 00:35:04,330
+الأساسية تبع ال treatment of asthma وهذا بنتركه لل
+
+471
+00:35:04,330 --> 00:35:07,570
+slide اللي جاي عشان مايصيرش كتير عليكم، يعطيكوا
+
+472
+00:35:07,570 --> 00:35:11,170
+العافية وإيقاع القطاع، ان شاء الله هنكمل ال
+
+473
+00:35:11,170 --> 00:35:17,790
+obstructive respiratory tract disorders
+