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Ireland-Rosen|exercise_4_5 | Consider a prime $p$ of the form $4 t+3$. Show that $a$ is a primitive root modulo $p$ iff $-a$ has order $(p-1) / 2$. | import Mathlib
open Real
open scoped BigOperators
| theorem exercise_4_5 {p t : β} (hp0 : p.Prime) (hp1 : p = 4*t + 3)
(a : ZMod p) :
IsPrimitiveRoot a p β ((-a) ^ ((p-1)/2) = 1 β§ β (k : β), (0 < k β§ k < (p-1)/2) β (-a)^k β 1) := |
Ireland-Rosen|exercise_4_8 | Let $p$ be an odd prime. Show that $a$ is a primitive root modulo $p$ iff $a^{(p-1) / q} \not \equiv 1(p)$ for all prime divisors $q$ of $p-1$. | import Mathlib
open Real
open scoped BigOperators
| theorem exercise_4_8 {p : β} {a : ZMod p} (hp : Odd p) (hp' : p.Prime) :
IsPrimitiveRoot a p β (β q : β, q β£ (p-1) β q.Prime β Β¬ a^((p-1)/q) = 1) := |
Ireland-Rosen|exercise_5_13 | Show that any prime divisor of $x^{4}-x^{2}+1$ is congruent to 1 modulo 12 . | import Mathlib
open Real
open scoped BigOperators
| theorem exercise_5_13 {p x: β€} (hp : Prime p)
(hpx : p β£ (x^4 - x^2 + 1)) : p β‘ 1 [ZMOD 12] := |
Ireland-Rosen|exercise_5_37 | Show that if $a$ is negative then $p \equiv q(4 a)$ together with $p\not | a$ imply $(a / p)=(a / q)$. | import Mathlib
open Real
open scoped BigOperators
| theorem exercise_5_37 {p q : β} [Fact (p.Prime)] [Fact (q.Prime)] {a : β€}
(ha : a < 0) (h0 : p β‘ q [ZMOD 4*a]) (h1 : Β¬ ((p : β€) β£ a)) :
legendreSym p a = legendreSym q a := |
Ireland-Rosen|exercise_18_4 | Show that 1729 is the smallest positive integer expressible as the sum of two different integral cubes in two ways. | import Mathlib
open Real
open scoped BigOperators
| theorem exercise_18_4 : 1729 = sInf (fun (n : β) => β x y z w : β€,
x^3 + y^3 = n β§ z^3 + w^3 = n β§ x β z β§ x β w β§ y β z β§ y β w) := |
Munkres|exercise_13_1 | Let $X$ be a topological space; let $A$ be a subset of $X$. Suppose that for each $x \in A$ there is an open set $U$ containing $x$ such that $U \subset A$. Show that $A$ is open in $X$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_13_1 (X : Type*) [TopologicalSpace X] (A : Set X)
(h1 : β x β A, β U : Set X, x β U β§ IsOpen U β§ U β A) :
IsOpen A := |
Munkres|exercise_13_4a1 | If $\mathcal{T}_\alpha$ is a family of topologies on $X$, show that $\bigcap \mathcal{T}_\alpha$ is a topology on $X$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| def is_topology (X : Type*) (T : Set (Set X)) :=
univ β T β§
(β s t, s β T β t β T β s β© t β T) β§
(βs, (βt β s, t β T) β sUnion s β T)
theorem exercise_13_4a1 (X I : Type*) (T : I β Set (Set X)) (h : β i, is_topology X (T i)) :
is_topology X (β i : I, T i) := |
Munkres|exercise_13_4b1 | Let $\mathcal{T}_\alpha$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $\mathcal{T}_\alpha$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| def is_topology (X : Type*) (T : Set (Set X)) :=
univ β T β§
(β s t, s β T β t β T β s β© t β T) β§
(βs, (βt β s, t β T) β sUnion s β T)
theorem exercise_13_4b1 (X I : Type*) (T : I β Set (Set X)) (h : β i, is_topology X (T i)) :
β! T', is_topology X T' β§ (β i, T i β T') β§
β T'', is_topology X T'' β (β i, T i β T'') β T' β T'' := |
Munkres|exercise_13_5a | Show that if $\mathcal{A}$ is a basis for a topology on $X$, then the topology generated by $\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\mathcal{A}$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| def is_topology (X : Type*) (T : Set (Set X)) :=
univ β T β§
(β s t, s β T β t β T β s β© t β T) β§
(βs, (βt β s, t β T) β sUnion s β T)
theorem exercise_13_5a {X : Type*}
[TopologicalSpace X] (A : Set (Set X)) (hA : IsTopologicalBasis A) :
generateFrom A = generateFrom (sInter {T | is_topology X T β§ A β T}) := |
Munkres|exercise_13_6 | Show that the lower limit topology $\mathbb{R}_l$ and $K$-topology $\mathbb{R}_K$ are not comparable. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| def lower_limit_topology (X : Type) [Preorder X] :=
generateFrom {S : Set X | β a b, a < b β§ S = Ico a b}
def Rl := lower_limit_topology β
def K : Set β := {r | β n : β, r = 1 / n}
def K_topology := generateFrom
({S : Set β | β a b, a < b β§ S = Ioo a b} βͺ {S : Set β | β a b, a < b β§ S = Ioo a b \ K})
theorem exercise_13_6 :
Β¬ (β U, Rl.IsOpen U β K_topology.IsOpen U) β§ Β¬ (β U, K_topology.IsOpen U β Rl.IsOpen U) := |
Munkres|exercise_13_8b | Show that the collection $\{(a,b) \mid a < b, a \text{ and } b \text{ rational}\}$ is a basis that generates a topology different from the lower limit topology on $\mathbb{R}$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| def lower_limit_topology (X : Type) [Preorder X] :=
generateFrom {S : Set X | β a b, a < b β§ S = Ico a b}
theorem exercise_13_8b (T : Set (Set β))
(hT : T = {S : Set β | β a b : β, a < b β§ S = Ioo βa βb}) :
IsTopologicalBasis T β§
generateFrom T β
lower_limit_topology β := |
Munkres|exercise_16_4 | A map $f: X \rightarrow Y$ is said to be an open map if for every open set $U$ of $X$, the set $f(U)$ is open in $Y$. Show that $\pi_{1}: X \times Y \rightarrow X$ and $\pi_{2}: X \times Y \rightarrow Y$ are open maps. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_16_4 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
(Οβ : X Γ Y β X)
(Οβ : X Γ Y β Y)
(hβ : Οβ = Prod.fst)
(hβ : Οβ = Prod.snd) :
IsOpenMap Οβ β§ IsOpenMap Οβ := |
Munkres|exercise_17_4 | Show that if $U$ is open in $X$ and $A$ is closed in $X$, then $U-A$ is open in $X$, and $A-U$ is closed in $X$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_17_4 {X : Type*} [TopologicalSpace X]
(U A : Set X) (hU : IsOpen U) (hA : IsClosed A) :
IsOpen (U \ A) β§ IsClosed (A \ U) := |
Munkres|exercise_18_8b | Let $Y$ be an ordered set in the order topology. Let $f, g: X \rightarrow Y$ be continuous. Let $h: X \rightarrow Y$ be the function $h(x)=\min \{f(x), g(x)\}.$ Show that $h$ is continuous. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_18_8b {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[LinearOrder Y] [OrderTopology Y] {f g : X β Y}
(hf : Continuous f) (hg : Continuous g) :
Continuous (Ξ» x => min (f x) (g x)) := |
Munkres|exercise_19_6a | Let $\mathbf{x}_1, \mathbf{x}_2, \ldots$ be a sequence of the points of the product space $\prod X_\alpha$. Show that this sequence converges to the point $\mathbf{x}$ if and only if the sequence $\pi_\alpha(\mathbf{x}_i)$ converges to $\pi_\alpha(\mathbf{x})$ for each $\alpha$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_19_6a
{ΞΉ : Type*}
{f : ΞΉ β Type*} {x : β β Ξ a, f a}
(y : Ξ i, f i)
[Ξ a, TopologicalSpace (f a)] :
Tendsto x atTop (π y) β β i, Tendsto (Ξ» j => (x j) i) atTop (π (y i)) := |
Munkres|exercise_21_6a | Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}(x)\right)$ converges for each $x \in[0,1]$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_21_6a
(f : β β Icc 0 1 β β)
(h : β x n, f n x = βx ^ n) :
β x, β y, Tendsto (Ξ» n => f n x) atTop (π y) := |
Munkres|exercise_21_8 | Let $X$ be a topological space and let $Y$ be a metric space. Let $f_{n}: X \rightarrow Y$ be a sequence of continuous functions. Let $x_{n}$ be a sequence of points of $X$ converging to $x$. Show that if the sequence $\left(f_{n}\right)$ converges uniformly to $f$, then $\left(f_{n}\left(x_{n}\right)\right)$ converges to $f(x)$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_21_8
{X : Type*} [TopologicalSpace X] {Y : Type*} [MetricSpace Y]
{f : β β X β Y} {x : β β X}
(hf : β n, Continuous (f n))
(xβ : X)
(hx : Tendsto x atTop (π xβ))
(fβ : X β Y)
(hh : TendstoUniformly f fβ atTop) :
Tendsto (Ξ» n => f n (x n)) atTop (π (fβ xβ)) := |
Munkres|exercise_22_2b | If $A \subset X$, a retraction of $X$ onto $A$ is a continuous map $r: X \rightarrow A$ such that $r(a)=a$ for each $a \in A$. Show that a retraction is a quotient map. | import Mathlib
open Filter Set TopologicalSpace Topology
| theorem exercise_22_2b {X : Type*} [TopologicalSpace X]
{A : Set X} (r : X β A) (hr : Continuous r) (h : β x : A, r x = x) :
QuotientMap r := |
Munkres|exercise_23_2 | Let $\left\{A_{n}\right\}$ be a sequence of connected subspaces of $X$, such that $A_{n} \cap A_{n+1} \neq \varnothing$ for all $n$. Show that $\bigcup A_{n}$ is connected. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_23_2 {X : Type*}
[TopologicalSpace X] {A : β β Set X} (hA : β n, IsConnected (A n))
(hAn : β n, A n β© A (n + 1) β β
) :
IsConnected (β n, A n) := |
Munkres|exercise_23_4 | Show that if $X$ is an infinite set, it is connected in the finite complement topology. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
set_option checkBinderAnnotations false | theorem exercise_23_4 {X : Type*} [TopologicalSpace X] [CofiniteTopology X]
(s : Set X) : Infinite s β IsConnected s := |
Munkres|exercise_23_9 | Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X \times Y)-(A \times B)$ is connected. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_23_9 {X Y : Type*}
[TopologicalSpace X] [TopologicalSpace Y]
(Aβ Aβ : Set X)
(Bβ Bβ : Set Y)
(hA : Aβ β Aβ)
(hB : Bβ β Bβ)
(hA : IsConnected Aβ)
(hB : IsConnected Bβ) :
IsConnected ({x | β a b, x = (a, b) β§ a β Aβ β§ b β Bβ} \
{x | β a b, x = (a, b) β§ a β Aβ β§ b β Bβ}) := |
Munkres|exercise_24_2 | Let $f: S^{1} \rightarrow \mathbb{R}$ be a continuous map. Show there exists a point $x$ of $S^{1}$ such that $f(x)=f(-x)$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_24_2 {f : (Metric.sphere 0 1 : Set β) β β}
(hf : Continuous f) : β x, f x = f (-x) := |
Munkres|exercise_25_4 | Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_25_4 {X : Type*} [TopologicalSpace X]
[LocPathConnectedSpace X] (U : Set X) (hU : IsOpen U)
(hcU : IsConnected U) : IsPathConnected U := |
Munkres|exercise_26_11 | Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed connected subsets of $X$ that is simply ordered by proper inclusion. Then $Y=\bigcap_{A \in \mathcal{A}} A$ is connected. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_26_11
{X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
(A : Set (Set X)) (hA : β (a b : Set X), a β A β b β A β a β b β¨ b β a)
(hA' : β a β A, IsClosed a) (hA'' : β a β A, IsConnected a) :
IsConnected (ββ A) := |
Munkres|exercise_27_4 | Show that a connected metric space having more than one point is uncountable. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_27_4
{X : Type*} [MetricSpace X] [ConnectedSpace X] (hX : β x y : X, x β y) :
Β¬ Countable (univ : Set X) := |
Munkres|exercise_28_5 | Show that X is countably compact if and only if every nested sequence $C_1 \supset C_2 \supset \cdots$ of closed nonempty sets of X has a nonempty intersection. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| def countably_compact (X : Type*) [TopologicalSpace X] :=
β U : β β Set X,
(β i, IsOpen (U i)) β§ ((univ : Set X) β β i, U i) β
(β t : Finset β, (univ : Set X) β β i β t, U i)
theorem exercise_28_5
(X : Type*) [TopologicalSpace X] :
countably_compact X β β (C : β β Set X), (β n, IsClosed (C n)) β§
(β n, C n β β
) β§ (β n, C (n + 1) β C n) β β x, β n, x β C n := |
Munkres|exercise_29_1 | Show that the rationals $\mathbb{Q}$ are not locally compact. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_29_1 : Β¬ LocallyCompactSpace β := |
Munkres|exercise_29_10 | Show that if $X$ is a Hausdorff space that is locally compact at the point $x$, then for each neighborhood $U$ of $x$, there is a neighborhood $V$ of $x$ such that $\bar{V}$ is compact and $\bar{V} \subset U$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_29_10 {X : Type*}
[TopologicalSpace X] [T2Space X] (x : X)
(hx : β U : Set X, x β U β§ IsOpen U β§ (β K : Set X, U β K β§ IsCompact K))
(U : Set X) (hU : IsOpen U) (hxU : x β U) :
β (V : Set X), IsOpen V β§ x β V β§ IsCompact (closure V) β§ closure V β U := |
Munkres|exercise_30_13 | Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_30_13 {X : Type*} [TopologicalSpace X]
(h : β (s : Set X), Countable s β§ Dense s) (U : Set (Set X))
(hU : β (x y : Set X), x β U β y β U β x β y β x β© y = β
)
(hUopen : β (u : Set X), u β U β IsOpen u) :
Countable U := |
Munkres|exercise_31_2 | Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_31_2 {X : Type*}
[TopologicalSpace X] [NormalSpace X] {A B : Set X}
(hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) :
β (U V : Set X), IsOpen U β§ IsOpen V β§ A β U β§ B β V β§ closure U β© closure V = β
:= |
Munkres|exercise_32_1 | Show that a closed subspace of a normal space is normal. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_32_1 {X : Type*} [TopologicalSpace X]
(hX : NormalSpace X) (A : Set X) (hA : IsClosed A) :
NormalSpace {x // x β A} := |
Munkres|exercise_32_2b | Show that if $\prod X_\alpha$ is regular, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_32_2b
{ΞΉ : Type*} {X : ΞΉ β Type*} [β i, TopologicalSpace (X i)]
(h : β i, Nonempty (X i)) (h2 : RegularSpace (Ξ i, X i)) :
β i, RegularSpace (X i) := |
Munkres|exercise_32_3 | Show that every locally compact Hausdorff space is regular. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_32_3 {X : Type*} [TopologicalSpace X]
(hX : LocallyCompactSpace X) (hX' : T2Space X) :
RegularSpace X := |
Munkres|exercise_33_8 | Let $X$ be completely regular, let $A$ and $B$ be disjoint closed subsets of $X$. Show that if $A$ is compact, there is a continuous function $f \colon X \rightarrow [0, 1]$ such that $f(A) = \{0\}$ and $f(B) = \{1\}$. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_33_8
(X : Type*) [TopologicalSpace X] [CompletelyRegularSpace X]
(A B : Set X) (hA : IsClosed A) (hB : IsClosed B)
(hAB : Disjoint A B)
(hAc : IsCompact A) :
β (f : X β Icc 0 1), Continuous f β§ f '' A = {0} β§ f '' B = {1} := |
Munkres|exercise_38_6 | Let $X$ be completely regular. Show that $X$ is connected if and only if the Stone-Δech compactification of $X$ is connected. | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| theorem exercise_38_6 {X : Type*} [TopologicalSpace X] [CompletelyRegularSpace X] :
IsConnected (univ : Set X) β IsConnected (univ : Set (StoneCech X)) := |
Pugh|exercise_2_12a | Let $(p_n)$ be a sequence and $f:\mathbb{N}\to\mathbb{N}$. The sequence $(q_k)_{k\in\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is an injection, the limit of a sequence is unaffected by rearrangement. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_2_12a {Ξ± : Type*} [TopologicalSpace Ξ±]
(f : β β β) (p : β β Ξ±) (a : Ξ±)
(hf : Injective f) (hp : Tendsto p atTop (π a)) :
Tendsto (Ξ» n => p (f n)) atTop (π a) := |
Pugh|exercise_2_29 | Let $\mathcal{T}$ be the collection of open subsets of a metric space $\mathrm{M}$, and $\mathcal{K}$ the collection of closed subsets. Show that there is a bijection from $\mathcal{T}$ onto $\mathcal{K}$. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_2_29 (M : Type*) [MetricSpace M]
(O C : Set (Set M))
(hO : O = {s | IsOpen s})
(hC : C = {s | IsClosed s}) :
β f : O β C, Bijective f := |
Pugh|exercise_2_41 | Let $\|\cdot\|$ be any norm on $\mathbb{R}^{m}$ and let $B=\left\{x \in \mathbb{R}^{m}:\|x\| \leq 1\right\}$. Prove that $B$ is compact. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_2_41 (m : β) [NormedSpace β ((Fin m) β β)] :
IsCompact (Metric.closedBall (0 : EuclideanSpace β (Fin m)) 1) := |
Pugh|exercise_2_57 | Show that if $S$ is connected, it is not true in general that its interior is connected. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_2_57 {X : Type*} [TopologicalSpace X]
: β (S : Set X), IsConnected S β§ Β¬ IsConnected (interior S) := |
Pugh|exercise_2_126 | Suppose that $E$ is an uncountable subset of $\mathbb{R}$. Prove that there exists a point $p \in \mathbb{R}$ at which $E$ condenses. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_2_126 {E : Set β}
(hE : Β¬ Set.Countable E) : β (p : β), ClusterPt p (π E) := |
Pugh|exercise_3_4 | Prove that $\sqrt{n+1}-\sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_3_4 (n : β) :
Tendsto (Ξ» n => (sqrt (n + 1) - sqrt n)) atTop (π 0) := |
Pugh|exercise_3_63b | Prove that $\sum 1/k(\log(k))^p$ diverges when $p \leq 1$. | import Mathlib
open Filter Real Function
open scoped Topology
| theorem exercise_3_63b (p : β) (f : β β β) (hp : p β€ 1)
(h : f = Ξ» (k : β) => (1 : β) / (k * (log k) ^ p)) :
Β¬ β l, Tendsto f atTop (π l) := |
Putnam|exercise_1998_a3 | Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that
$f(a) \cdot f^{\prime}(a) \cdot f^{\prime \prime}(a) \cdot f^{\prime \prime \prime}(a) \geq 0$. | import Mathlib
open scoped BigOperators
| theorem exercise_1998_a3 (f : β β β) (hf : ContDiff β 3 f) :
β a : β, (f a) * (deriv f a) * (iteratedDeriv 2 f a) * (iteratedDeriv 3 f a) β₯ 0 := |
Putnam|exercise_2000_a2 | Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers. | import Mathlib
open scoped BigOperators
| theorem exercise_2000_a2 :
β N : β, β n : β, n > N β§ β i : Fin 6 β β, n = (i 0)^2 + (i 1)^2 β§
n + 1 = (i 2)^2 + (i 3)^2 β§ n + 2 = (i 4)^2 + (i 5)^2 := |
Putnam|exercise_2010_a4 | Prove that for each positive integer $n$, the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime. | import Mathlib
open scoped BigOperators
| theorem exercise_2010_a4 (n : β) (hn : n > 0) :
Β¬ Nat.Prime (10^10^10^n + 10^10^n + 10^n - 1) := |
Putnam|exercise_2017_b3 | Suppose that $f(x)=\sum_{i=0}^{\infty} c_{i} x^{i}$ is a power series for which each coefficient $c_{i}$ is 0 or 1 . Show that if $f(2 / 3)=3 / 2$, then $f(1 / 2)$ must be irrational. | import Mathlib
open scoped BigOperators
| theorem exercise_2017_b3 (f : β β β) (c : β β β)
(hf : f = Ξ» x => (β' (i : β), (c i) * x^i))
(hc : β n, c n = 0 β¨ c n = 1)
(hf1 : f (2/3) = 3/2) :
Irrational (f (1/2)) := |
Putnam|exercise_2018_b2 | Let $n$ be a positive integer, and let $f_{n}(z)=n+(n-1) z+$ $(n-2) z^{2}+\cdots+z^{n-1}$. Prove that $f_{n}$ has no roots in the closed unit disk $\{z \in \mathbb{C}:|z| \leq 1\}$. | import Mathlib
open scoped BigOperators
| theorem exercise_2018_b2 (n : β) (hn : n > 0) (f : β β β β β)
(hf : β n : β, f n = Ξ» (z : β) => (β i : Fin n, (n-i)* z^(i : β))) :
Β¬ (β z : β, βzβ β€ 1 β§ f n z = 0) := |
Putnam|exercise_2020_b5 | For $j \in\{1,2,3,4\}$, let $z_{j}$ be a complex number with $\left|z_{j}\right|=1$ and $z_{j} \neq 1$. Prove that $3-z_{1}-z_{2}-z_{3}-z_{4}+z_{1} z_{2} z_{3} z_{4} \neq 0 .$ | import Mathlib
open scoped BigOperators
| theorem exercise_2020_b5 (z : Fin 4 β β) (hz0 : β n, βz nβ = 1)
(hz1 : β n, z n β 1) :
3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) β 0 := |
Rudin|exercise_1_1a | If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $r+x$ is irrational. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_1a
(x : β) (y : β) (hy : y β 0) :
( Irrational x ) -> Irrational ( x + y ) := |
Rudin|exercise_1_2 | Prove that there is no rational number whose square is $12$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_2 : Β¬ β (x : β), ( x ^ 2 = 12 ) := |
Rudin|exercise_1_5 | Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $\inf A=-\sup (-A)$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_5 (A minus_A : Set β) (hA : A.Nonempty)
(hA_bdd_below : BddBelow A) (hminus_A : minus_A = {x | -x β A}) :
sInf A = - sSup minus_A := |
Rudin|exercise_1_11a | If $z$ is a complex number, prove that there exists an $r\geq 0$ and a complex number $w$ with $| w | = 1$ such that $z = rw$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_11a (z : β) :
β (r : β) (w : β), r β₯ 0 β§ abs w = 1 β§ z = r * w := |
Rudin|exercise_1_13 | If $x, y$ are complex, prove that $||x|-|y|| \leq |x-y|$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_13 (x y : β) :
|(abs x) - (abs y)| β€ abs (x - y) := |
Rudin|exercise_1_16a | Suppose $k \geq 3, x, y \in \mathbb{R}^k, |x - y| = d > 0$, and $r > 0$. Prove that if $2r > d$, there are infinitely many $z \in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_16a
(n : β)
(d r : β)
(x y z : EuclideanSpace β (Fin n)) -- R^n
(hβ : n β₯ 3)
(hβ : βx - yβ = d)
(hβ : d > 0)
(hβ : r > 0)
(hβ
: 2 * r > d)
: Set.Infinite {z : EuclideanSpace β (Fin n) | βz - xβ = r β§ βz - yβ = r} := |
Rudin|exercise_1_18a | If $k \geq 2$ and $\mathbf{x} \in R^{k}$, prove that there exists $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$ | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_18a
(n : β)
(h : n > 1)
(x : EuclideanSpace β (Fin n)) -- R^n
: β (y : EuclideanSpace β (Fin n)), y β 0 β§ (inner x y) = (0 : β) := |
Rudin|exercise_1_19 | Suppose $a, b \in R^k$. Find $c \in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_1_19
(n : β)
(a b : EuclideanSpace β (Fin n))
: (β c r, r > 0 β§ (β x, βx - aβ = 2 * βx - bβ β βx - cβ = r)) β§
(β c r, (r > 0 β§ (β x, βx - aβ = 2 * βx - bβ β βx - cβ = r)) β
(3 β’ c = 4 β’ b - a β§ 3 * r = 2 * βb - aβ)) := |
Rudin|exercise_2_24 | Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_2_24 {X : Type*} [MetricSpace X]
(hX : β (A : Set X), Infinite A β β (x : X), x β closure A) :
SeparableSpace X := |
Rudin|exercise_2_27a | Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that $P$ is perfect. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_2_27a (k : β) (E P : Set (EuclideanSpace β (Fin k)))
(hE : Β¬ Set.Countable E)
(hP : P = {x | β U β π x, Β¬ Set.Countable (U β© E)}) :
Perfect P := |
Rudin|exercise_2_28 | Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_2_28 (X : Type*) [MetricSpace X] [SeparableSpace X]
(A : Set X) (hA : IsClosed A) :
β Pβ Pβ : Set X, A = Pβ βͺ Pβ β§ Perfect Pβ β§ Set.Countable Pβ := |
Rudin|exercise_3_1a | Prove that convergence of $\left\{s_{n}\right\}$ implies convergence of $\left\{\left|s_{n}\right|\right\}$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_3_1a
(f : β β β)
(h : β (a : β), Tendsto (Ξ» (n : β) => f n) atTop (π a))
: β (a : β), Tendsto (Ξ» (n : β) => |f n|) atTop (π a) := |
Rudin|exercise_3_3 | If $s_{1}=\sqrt{2}$, and $s_{n+1}=\sqrt{2+\sqrt{s_{n}}} \quad(n=1,2,3, \ldots),$ prove that $\left\{s_{n}\right\}$ converges, and that $s_{n}<2$ for $n=1,2,3, \ldots$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| noncomputable def f : β β β
| 0 => sqrt 2
| (n + 1) => sqrt (2 + sqrt (f n))
theorem exercise_3_3
: β (x : β), Tendsto f atTop (π x) β§ β n, f n < 2 := |
Rudin|exercise_3_6a | Prove that $\lim_{n \rightarrow \infty} \sum_{i<n} a_i = \infty$, where $a_i = \sqrt{i + 1} -\sqrt{i}$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| noncomputable def g (n : β) : β := sqrt (n + 1) - sqrt n
theorem exercise_3_6a
: Tendsto (Ξ» (n : β) => (β i in range n, g i)) atTop atTop := |
Rudin|exercise_3_8 | If $\Sigma a_{n}$ converges, and if $\left\{b_{n}\right\}$ is monotonic and bounded, prove that $\Sigma a_{n} b_{n}$ converges. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_3_8
(a b : β β β)
(h1 : β y, (Tendsto (Ξ» n => (β i in (range n), a i)) atTop (π y)))
(h2 : Monotone b)
(h3 : Bornology.IsBounded (Set.range b)) :
β y, Tendsto (Ξ» n => (β i in (range n), (a i) * (b i))) atTop (π y) := |
Rudin|exercise_3_20 | Suppose $\left\{p_{n}\right\}$ is a Cauchy sequence in a metric space $X$, and some sequence $\left\{p_{n l}\right\}$ converges to a point $p \in X$. Prove that the full sequence $\left\{p_{n}\right\}$ converges to $p$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_3_20 {X : Type*} [MetricSpace X]
(p : β β X) (l : β+) (r : X)
(hp : CauchySeq p)
(hpl : Tendsto (Ξ» n => p (l * n)) atTop (π r)) :
Tendsto p atTop (π r) := |
Rudin|exercise_3_22 | Suppose $X$ is a nonempty complete metric space, and $\left\{G_{n}\right\}$ is a sequence of dense open sets of $X$. Prove Baire's theorem, namely, that $\bigcap_{1}^{\infty} G_{n}$ is not empty. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_3_22 (X : Type*) [MetricSpace X] [CompleteSpace X] [Nonempty X]
(G : β β Set X) (hG : β n, IsOpen (G n) β§ Dense (G n)) :
β x, β n, x β G n := |
Rudin|exercise_4_2a | If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $f(\overline{E}) \subset \overline{f(E)}$ for every set $E \subset X$. ($\overline{E}$ denotes the closure of $E$). | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_2a
{Ξ± : Type} [MetricSpace Ξ±]
{Ξ² : Type} [MetricSpace Ξ²]
(f : Ξ± β Ξ²)
(hβ : Continuous f)
: β (x : Set Ξ±), f '' (closure x) β closure (f '' x) := |
Rudin|exercise_4_4a | Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_4a
{Ξ± : Type} [MetricSpace Ξ±]
{Ξ² : Type} [MetricSpace Ξ²]
(f : Ξ± β Ξ²)
(s : Set Ξ±)
(hβ : Continuous f)
(hβ : Dense s)
: f '' Set.univ β closure (f '' s) := |
Rudin|exercise_4_5a | If $f$ is a real continuous function defined on a closed set $E \subset \mathbb{R}$, prove that there exist continuous real functions $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_5a
(f : β β β)
(E : Set β)
(hβ : IsClosed E)
(hβ : ContinuousOn f E)
: β (g : β β β), Continuous g β§ β x β E, f x = g x := |
Rudin|exercise_4_6 | If $f$ is defined on $E$, the graph of $f$ is the set of points $(x, f(x))$, for $x \in E$. In particular, if $E$ is a set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous on $E$ if and only if its graph is compact. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_6
(f : β β β)
(E : Set β)
(G : Set (β Γ β))
(hβ : IsCompact E)
(hβ : G = {(x, f x) | x β E})
: ContinuousOn f E β IsCompact G := |
Rudin|exercise_4_8b | Let $E$ be a bounded set in $R^{1}$. Prove that there exists a real function $f$ such that $f$ is uniformly continuous and is not bounded on $E$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_8b
(E : Set β) (h : Bornology.IsBounded E) :
β f : β β β, UniformContinuousOn f E β§ Β¬ Bornology.IsBounded (Set.image f E) := |
Rudin|exercise_4_12 | A uniformly continuous function of a uniformly continuous function is uniformly continuous. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_12
{Ξ± Ξ² Ξ³ : Type*} [UniformSpace Ξ±] [UniformSpace Ξ²] [UniformSpace Ξ³]
{f : Ξ± β Ξ²} {g : Ξ² β Ξ³}
(hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous (g β f) := |
Rudin|exercise_4_19 | Suppose $f$ is a real function with domain $R^{1}$ which has the intermediate value property: if $f(a)<c<f(b)$, then $f(x)=c$ for some $x$ between $a$ and $b$. Suppose also, for every rational $r$, that the set of all $x$ with $f(x)=r$ is closed. Prove that $f$ is continuous. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_19
{f : β β β} (hf : β a b c, a < b β f a < c β c < f b β β x, a < x β§ x < b β§ f x = c)
(hg : β r : β, IsClosed {x | f x = r}) : Continuous f := |
Rudin|exercise_4_24 | Assume that $f$ is a continuous real function defined in $(a, b)$ such that $f\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$ for all $x, y \in(a, b)$. Prove that $f$ is convex. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_4_24 {f : β β β}
(hf : Continuous f) (a b : β) (hab : a < b)
(h : β x y : β, a < x β x < b β a < y β y < b β f ((x + y) / 2) β€ (f x + f y) / 2) :
ConvexOn β (Set.Ioo a b) f := |
Rudin|exercise_5_2 | Suppose $f^{\prime}(x)>0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that $g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)} \quad(a<x<b)$. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_5_2 {a b : β}
{f g : β β β} (hf : β x β Set.Ioo a b, deriv f x > 0)
(hg : g = fβ»ΒΉ) :
StrictMonoOn f (Set.Ioo a b) β§
DifferentiableOn β g (Set.Ioo a b) β§
β x β Set.Ioo a b, deriv g (f x) = 1 / deriv f x := |
Rudin|exercise_5_4 | If $C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0,$ where $C_{0}, \ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_5_4 {n : β}
(C : β β β)
(hC : β i in (range (n + 1)), (C i) / (i + 1) = 0) :
β x, x β (Set.Icc (0 : β) 1) β§ β i in range (n + 1), (C i) * (x^i) = 0 := |
Rudin|exercise_5_6 | Suppose (a) $f$ is continuous for $x \geq 0$, (b) $f^{\prime}(x)$ exists for $x>0$, (c) $f(0)=0$, (d) $f^{\prime}$ is monotonically increasing. Put $g(x)=\frac{f(x)}{x} \quad(x>0)$ and prove that $g$ is monotonically increasing. | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_5_6
{f : β β β}
(hf1 : ContinuousOn f (Set.Ici 0))
(hf2 : DifferentiableOn β f (Set.Ioi 0))
(hf3 : f 0 = 0)
(hf4 : MonotoneOn (deriv f) (Set.Ioi 0)) :
MonotoneOn (Ξ» x => f x / x) (Set.Ioi 0) := |
Rudin|exercise_5_15 | Suppose $a \in R^{1}, f$ is a twice-differentiable real function on $(a, \infty)$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $|f(x)|,\left|f^{\prime}(x)\right|,\left|f^{\prime \prime}(x)\right|$, respectively, on $(a, \infty)$. Prove that $M_{1}^{2} \leq 4 M_{0} M_{2} .$ | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| theorem exercise_5_15 {f : β β β} (a M0 M1 M2 : β)
(hf' : DifferentiableOn β f (Set.Ioi a))
(hf'' : DifferentiableOn β (deriv f) (Set.Ioi a))
(hM0 : M0 = sSup {(|f x|) | x β (Set.Ioi a)})
(hM1 : M1 = sSup {(|deriv f x|) | x β (Set.Ioi a)})
(hM2 : M2 = sSup {(|deriv (deriv f) x|) | x β (Set.Ioi a)}) :
(M1 ^ 2) β€ 4 * M0 * M2 := |
Shakarchi|exercise_1_13a | Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Re}(f)$ is constant, then $f$ is constant. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_1_13a {f : β β β} (Ξ© : Set β) (a b : Ξ©) (h : IsOpen Ξ©)
(hf : DifferentiableOn β f Ξ©) (hc : β (c : β), β z β Ξ©, (f z).re = c) :
f a = f b := |
Shakarchi|exercise_1_13c | Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $|f|$ is constant, then $f$ is constant. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_1_13c {f : β β β} (Ξ© : Set β) (a b : Ξ©) (h : IsOpen Ξ©)
(hf : DifferentiableOn β f Ξ©) (hc : β (c : β), β z β Ξ©, abs (f z) = c) :
f a = f b := |
Shakarchi|exercise_1_19b | Prove that the power series $\sum z^n/n^2$ converges at every point of the unit circle. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_1_19b (z : β) (hz : abs z = 1) (s : β β β)
(h : s = (Ξ» n => β i in (range n), z ^ (i + 1) / (i + 1) ^ 2)) :
β y, Tendsto s atTop (π y) := |
Shakarchi|exercise_1_26 | Suppose $f$ is continuous in a region $\Omega$. Prove that any two primitives of $f$ (if they exist) differ by a constant. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_1_26
(f Fβ Fβ : β β β) (Ξ© : Set β) (h0 : Nonempty Ξ©) (h1 : IsOpen Ξ©) (h2 : IsConnected Ξ©) (hf : ContinuousOn f Ξ©)
(hFβ : DifferentiableOn β Fβ Ξ©) (hFβ : DifferentiableOn β Fβ Ξ©)
(hdFβ : β x β Ξ©, deriv Fβ x = f x) (hdFβ : β x β Ξ©, deriv Fβ x = f x)
: β c : β, β x, Fβ x = Fβ x + c := |
Shakarchi|exercise_2_9 | Let $\Omega$ be a bounded open subset of $\mathbb{C}$, and $\varphi: \Omega \rightarrow \Omega$ a holomorphic function. Prove that if there exists a point $z_{0} \in \Omega$ such that $\varphi\left(z_{0}\right)=z_{0} \quad \text { and } \quad \varphi^{\prime}\left(z_{0}\right)=1$ then $\varphi$ is linear. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_2_9
{f : β β β} (Ξ© : Set β) (b : Bornology.IsBounded Ξ©) (h : IsOpen Ξ©)
(hf : DifferentiableOn β f Ξ©) (z : Ξ©) (hz : f z = z) (h'z : deriv f z = 1) :
β (f_lin : β βL[β] β), β x β Ξ©, f x = f_lin x := |
Shakarchi|exercise_3_3 | Show that $ \int_{-\infty}^{\infty} \frac{\cos x}{x^2 + a^2} dx = \pi \frac{e^{-a}}{a}$ for $a > 0$. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_3_3 (a : β) (ha : 0 < a) :
Tendsto (Ξ» y => β« x in -y..y, Real.cos x / (x ^ 2 + a ^ 2))
atTop (π (Real.pi * (Real.exp (-a) / a))) := |
Shakarchi|exercise_3_9 | Show that $\int_0^1 \log(\sin \pi x) dx = - \log 2$. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_3_9 : β« x in (0 : β)..(1 : β), Real.log (Real.sin (Real.pi * x)) = - Real.log 2 := |
Shakarchi|exercise_3_22 | Show that there is no holomorphic function $f$ in the unit disc $D$ that extends continuously to $\partial D$ such that $f(z) = 1/z$ for $z \in \partial D$. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| theorem exercise_3_22 (D : Set β) (hD : D = ball 0 1) (f : β β β)
(hf : DifferentiableOn β f D) (hfc : ContinuousOn f (closure D)) :
Β¬ β z β (sphere (0 : β) 1), f z = 1 / z := |
Subsets and Splits