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theorem exercise_1_1b (x : ℝ) (y : ℚ) (h : y ≠ 0) : ( irrational x ) -> irrational ( x * y ) :=

If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $rx$ is irrational.

theorem exercise_1_4 (α : Type*) [partial_order α] (s : set α) (x y : α) (h₀ : set.nonempty s) (h₁ : x ∈ lower_bounds s) (h₂ : y ∈ upper_bounds s) : x ≤ y :=

Let $E$ be a nonempty subset of an ordered set; suppose $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$.

theorem exercise_1_8 : ¬ ∃ (r : ℂ → ℂ → Prop), is_linear_order ℂ r :=

Prove that no order can be defined in the complex field that turns it into an ordered field.

theorem exercise_1_12 (n : ℕ) (f : ℕ → ℂ) : abs (∑ i in finset.range n, f i) ≤ ∑ i in finset.range n, abs (f i) :=

If $z_1, \ldots, z_n$ are complex, prove that $|z_1 + z_2 + \ldots + z_n| \leq |z_1| + |z_2| + \cdots + |z_n|$.

theorem exercise_1_14 (z : ℂ) (h : abs z = 1) : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 :=

If $z$ is a complex number such that $|z|=1$, that is, such that $z \bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$.

theorem exercise_1_17 (n : ℕ) (x y : euclidean_space ℝ (fin n)) -- R^n : ‖x + y‖^2 + ‖x - y‖^2 = 2*‖x‖^2 + 2*‖y‖^2 :=

Prove that $|\mathbf{x}+\mathbf{y}|^{2}+|\mathbf{x}-\mathbf{y}|^{2}=2|\mathbf{x}|^{2}+2|\mathbf{y}|^{2}$ if $\mathbf{x} \in R^{k}$ and $\mathbf{y} \in R^{k}$.

theorem exercise_1_18b : ¬ ∀ (x : ℝ), ∃ (y : ℝ), y ≠ 0 ∧ x * y = 0 :=

If $k = 1$ and $\mathbf{x} \in R^{k}$, prove that there does not exist $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$

theorem exercise_2_19a {X : Type*} [metric_space X] (A B : set X) (hA : is_closed A) (hB : is_closed B) (hAB : disjoint A B) : separated_nhds A B :=

If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated.

theorem exercise_2_25 {K : Type*} [metric_space K] [compact_space K] : ∃ (B : set (set K)), set.countable B ∧ is_topological_basis B :=

Prove that every compact metric space $K$ has a countable base.

theorem exercise_2_27b (k : ℕ) (E P : set (euclidean_space ℝ (fin k))) (hE : E.nonempty ∧ ¬ set.countable E) (hP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).nonempty ∧ ¬ set.countable (P ∩ E)}) : set.countable (E \ P) :=

Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that at most countably many points of $E$ are not in $P$.

theorem exercise_2_29 (U : set ℝ) (hU : is_open U) : ∃ (f : ℕ → set ℝ), (∀ n, ∃ a b : ℝ, f n = {x | a < x ∧ x < b}) ∧ (∀ n, f n ⊆ U) ∧ (∀ n m, n ≠ m → f n ∩ f m = ∅) ∧ U = ⋃ n, f n :=

Prove that every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments.

theorem exercise_3_2a : tendsto (λ (n : ℝ), (sqrt (n^2 + n) - n)) at_top (𝓝 (1/2)) :=

Prove that $\lim_{n \rightarrow \infty}\sqrt{n^2 + n} -n = 1/2$.

theorem exercise_3_5 -- TODO fix (a b : ℕ → ℝ) (h : limsup a + limsup b ≠ 0) : limsup (λ n, a n + b n) ≤ limsup a + limsup b :=

For any two real sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, prove that $\limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right) \leq \limsup _{n \rightarrow \infty} a_{n}+\limsup _{n \rightarrow \infty} b_{n},$ provided the sum on the right is not of the form $\infty-\infty$.

theorem exercise_3_7 (a : ℕ → ℝ) (h : ∃ y, (tendsto (λ n, (∑ i in (finset.range n), a i)) at_top (𝓝 y))) : ∃ y, tendsto (λ n, (∑ i in (finset.range n), sqrt (a i) / n)) at_top (𝓝 y) :=

Prove that the convergence of $\Sigma a_{n}$ implies the convergence of $\sum \frac{\sqrt{a_{n}}}{n}$ if $a_n\geq 0$.

theorem exercise_3_13 (a b : ℕ → ℝ) (ha : ∃ y, (tendsto (λ n, (∑ i in (finset.range n), |a i|)) at_top (𝓝 y))) (hb : ∃ y, (tendsto (λ n, (∑ i in (finset.range n), |b i|)) at_top (𝓝 y))) : ∃ y, (tendsto (λ n, (∑ i in (finset.range n), λ i, (∑ j in finset.range (i + 1), a j * b (i - j)))) at_top (𝓝 y)) :=

Prove that the Cauchy product of two absolutely convergent series converges absolutely.

theorem exercise_3_21 {X : Type*} [metric_space X] [complete_space X] (E : ℕ → set X) (hE : ∀ n, E n ⊃ E (n + 1)) (hE' : tendsto (λ n, metric.diam (E n)) at_top (𝓝 0)) : ∃ a, set.Inter E = {a} :=

If $\left\{E_{n}\right\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X$, if $E_{n} \supset E_{n+1}$, and if $\lim _{n \rightarrow \infty} \operatorname{diam} E_{n}=0,$ then $\bigcap_{1}^{\infty} E_{n}$ consists of exactly one point.

theorem exercise_4_1a : ∃ (f : ℝ → ℝ), (∀ (x : ℝ), tendsto (λ y, f(x + y) - f(x - y)) (𝓝 0) (𝓝 0)) ∧ ¬ continuous f :=

Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies $\lim_{h \rightarrow 0} f(x + h) - f(x - h) = 0$ for every $x \in \mathbb{R}$. Show that $f$ does not need to be continuous.

theorem exercise_4_3 {α : Type} [metric_space α] (f : α → ℝ) (h : continuous f) (z : set α) (g : z = f⁻¹' {0}) : is_closed z :=

Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$ ) be the set of all $p \in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.

theorem exercise_4_4b {α : Type} [metric_space α] {β : Type} [metric_space β] (f g : α → β) (s : set α) (h₁ : continuous f) (h₂ : continuous g) (h₃ : dense s) (h₄ : ∀ x ∈ s, f x = g x) : f = g :=

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 if $g(p) = f(p)$ for all $p \in P$ then $g(p) = f(p)$ for all $p \in X$.

theorem exercise_4_5b : ∃ (E : set ℝ) (f : ℝ → ℝ), (continuous_on f E) ∧ (¬ ∃ (g : ℝ → ℝ), continuous g ∧ ∀ x ∈ E, f x = g x) :=

Show that there exist a set $E \subset \mathbb{R}$ and a real continuous function $f$ defined on $E$, such that there does not exist a continuous real function $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$.

theorem exercise_4_8a (E : set ℝ) (f : ℝ → ℝ) (hf : uniform_continuous_on f E) (hE : metric.bounded E) : metric.bounded (set.image f E) :=

Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$.

theorem exercise_4_11a {X : Type*} [metric_space X] {Y : Type*} [metric_space Y] (f : X → Y) (hf : uniform_continuous f) (x : ℕ → X) (hx : cauchy_seq x) : cauchy_seq (λ n, f (x n)) :=

Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$ and prove that $\left\{f\left(x_{n}\right)\right\}$ is a Cauchy sequence in $Y$ for every Cauchy sequence $\{x_n\}$ in $X$.

theorem exercise_4_15 {f : ℝ → ℝ} (hf : continuous f) (hof : is_open_map f) : monotone f :=

Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic.

theorem exercise_4_21a {X : Type*} [metric_space X] (K F : set X) (hK : is_compact K) (hF : is_closed F) (hKF : disjoint K F) : ∃ (δ : ℝ), δ > 0 ∧ ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ :=

Suppose $K$ and $F$ are disjoint sets in a metric space $X, K$ is compact, $F$ is closed. Prove that there exists $\delta>0$ such that $d(p, q)>\delta$ if $p \in K, q \in F$.

theorem exercise_5_1 {f : ℝ → ℝ} (hf : ∀ x y : ℝ, | (f x - f y) | ≤ (x - y) ^ 2) : ∃ c, f = λ x, c :=

Let $f$ be defined for all real $x$, and suppose that $|f(x)-f(y)| \leq (x-y)^{2}$ for all real $x$ and $y$. Prove that $f$ is constant.

theorem exercise_5_3 {g : ℝ → ℝ} (hg : continuous g) (hg' : ∃ M : ℝ, ∀ x : ℝ, | deriv g x | ≤ M) : ∃ N, ∀ ε > 0, ε < N → function.injective (λ x : ℝ, x + ε * g x) :=

Suppose $g$ is a real function on $R^{1}$, with bounded derivative (say $\left|g^{\prime}\right| \leq M$ ). Fix $\varepsilon>0$, and define $f(x)=x+\varepsilon g(x)$. Prove that $f$ is one-to-one if $\varepsilon$ is small enough.

theorem exercise_5_5 {f : ℝ → ℝ} (hfd : differentiable ℝ f) (hf : tendsto (deriv f) at_top (𝓝 0)) : tendsto (λ x, f (x + 1) - f x) at_top at_top :=

Suppose $f$ is defined and differentiable for every $x>0$, and $f^{\prime}(x) \rightarrow 0$ as $x \rightarrow+\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x) \rightarrow 0$ as $x \rightarrow+\infty$.

theorem exercise_5_7 {f g : ℝ → ℝ} {x : ℝ} (hf' : differentiable_at ℝ f 0) (hg' : differentiable_at ℝ g 0) (hg'_ne_0 : deriv g 0 ≠ 0) (f0 : f 0 = 0) (g0 : g 0 = 0) : tendsto (λ x, f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x)) :=

Suppose $f^{\prime}(x), g^{\prime}(x)$ exist, $g^{\prime}(x) \neq 0$, and $f(x)=g(x)=0$. Prove that $\lim _{t \rightarrow x} \frac{f(t)}{g(t)}=\frac{f^{\prime}(x)}{g^{\prime}(x)}.$

theorem exercise_5_17 {f : ℝ → ℝ} (hf' : differentiable_on ℝ f (set.Icc (-1) 1)) (hf'' : differentiable_on ℝ (deriv f) (set.Icc 1 1)) (hf''' : differentiable_on ℝ (deriv (deriv f)) (set.Icc 1 1)) (hf0 : f (-1) = 0) (hf1 : f 0 = 0) (hf2 : f 1 = 1) (hf3 : deriv f 0 = 0) : ∃ x, x ∈ set.Ioo (-1 : ℝ) 1 ∧ deriv (deriv (deriv f)) x ≥ 3 :=

Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that $f(-1)=0, \quad f(0)=0, \quad f(1)=1, \quad f^{\prime}(0)=0 .$ Prove that $f^{(3)}(x) \geq 3$ for some $x \in(-1,1)$.

theorem exercise_13_3b : ¬ ∀ X : Type, ∀s : set (set X), (∀ t : set X, t ∈ s → (set.infinite tᶜ ∨ t = ∅ ∨ t = ⊤)) → (set.infinite (⋃₀ s)ᶜ ∨ (⋃₀ s) = ∅ ∨ (⋃₀ s) = ⊤) :=

Show that the collection $$\mathcal{T}_\infty = \{U | X - U \text{ is infinite or empty or all of X}\}$$ does not need to be a topology on the set $X$.

theorem exercise_13_4a2 : ∃ (X I : Type*) (T : I → set (set X)), (∀ i, is_topology X (T i)) ∧ ¬ is_topology X (⋂ i : I, T i) :=

If $\mathcal{T}_\alpha$ is a family of topologies on $X$, show that $\bigcup \mathcal{T}_\alpha$ does not need to be a topology on $X$.

theorem exercise_13_4b2 (X I : Type*) (T : I → set (set X)) (h : ∀ i, is_topology X (T i)) : ∃! T', is_topology X T' ∧ (∀ i, T' ⊆ T i) ∧ ∀ T'', is_topology X T'' → (∀ i, T'' ⊆ T i) → T' ⊆ T'' :=

Let $\mathcal{T}_\alpha$ be a family of topologies on $X$. Show that there is a unique largest topology on $X$ contained in all the collections $\mathcal{T}_\alpha$.

theorem exercise_13_5b {X : Type*} [t : topological_space X] (A : set (set X)) (hA : t = generate_from A) : generate_from A = generate_from (sInter {T | is_topology X T ∧ A ⊆ T}) :=

Show that if $\mathcal{A}$ is a subbasis for a topology on $X$, then the topology generated by $\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\mathcal{A}$.

theorem exercise_13_8a : topological_space.is_topological_basis {S : set ℝ | ∃ a b : ℚ, a < b ∧ S = Ioo a b} :=

Show that the collection $\{(a,b) \mid a < b, a \text{ and } b \text{ rational}\}$ is a basis that generates the standard topology on $\mathbb{R}$.

theorem exercise_16_1 {X : Type*} [topological_space X] (Y : set X) (A : set Y) : ∀ U : set A, is_open U ↔ is_open (subtype.val '' U) :=

Show that if $Y$ is a subspace of $X$, and $A$ is a subset of $Y$, then the topology $A$ inherits as a subspace of $Y$ is the same as the topology it inherits as a subspace of $X$.

theorem exercise_16_6 (S : set (set (ℝ × ℝ))) (hS : ∀ s, s ∈ S → ∃ a b c d, (rational a ∧ rational b ∧ rational c ∧ rational d ∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d})) : is_topological_basis S :=

Show that the countable collection \[\{(a, b) \times (c, d) \mid a < b \text{ and } c < d, \text{ and } a, b, c, d \text{ are rational}\}\] is a basis for $\mathbb{R}^2$.

theorem exercise_18_8a {X Y : Type*} [topological_space X] [topological_space Y] [linear_order Y] [order_topology Y] {f g : X → Y} (hf : continuous f) (hg : continuous g) : is_closed {x | f x ≤ g x} :=

Let $Y$ be an ordered set in the order topology. Let $f, g: X \rightarrow Y$ be continuous. Show that the set $\{x \mid f(x) \leq g(x)\}$ is closed in $X$.

theorem exercise_18_13 {X : Type*} [topological_space X] {Y : Type*} [topological_space Y] [t2_space Y] {A : set X} {f : A → Y} (hf : continuous f) (g : closure A → Y) (g_con : continuous g) : ∀ (g' : closure A → Y), continuous g' → (∀ (x : closure A), g x = g' x) :=

Let $A \subset X$; let $f: A \rightarrow Y$ be continuous; let $Y$ be Hausdorff. Show that if $f$ may be extended to a continuous function $g: \bar{A} \rightarrow Y$, then $g$ is uniquely determined by $f$.

theorem exercise_20_2 [topological_space (ℝ ×ₗ ℝ)] [order_topology (ℝ ×ₗ ℝ)] : metrizable_space (ℝ ×ₗ ℝ) :=

Show that $\mathbb{R} \times \mathbb{R}$ in the dictionary order topology is metrizable.

theorem exercise_21_6b (f : ℕ → I → ℝ ) (h : ∀ x n, f n x = x ^ n) : ¬ ∃ f₀, tendsto_uniformly f f₀ at_top :=

Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}\right)$ does not converge uniformly.

theorem exercise_22_2a {X Y : Type*} [topological_space X] [topological_space Y] (p : X → Y) (h : continuous p) : quotient_map p ↔ ∃ (f : Y → X), continuous f ∧ p ∘ f = id :=

Let $p: X \rightarrow Y$ be a continuous map. Show that if there is a continuous map $f: Y \rightarrow X$ such that $p \circ f$ equals the identity map of $Y$, then $p$ is a quotient map.

theorem exercise_22_5 {X Y : Type*} [topological_space X] [topological_space Y] (p : X → Y) (hp : is_open_map p) (A : set X) (hA : is_open A) : is_open_map (p ∘ subtype.val : A → Y) :=

Let $p \colon X \rightarrow Y$ be an open map. Show that if $A$ is open in $X$, then the map $q \colon A \rightarrow p(A)$ obtained by restricting $p$ is an open map.

theorem exercise_23_3 {X : Type*} [topological_space X] [topological_space X] {A : ℕ → set X} (hAn : ∀ n, is_connected (A n)) (A₀ : set X) (hA : is_connected A₀) (h : ∀ n, A₀ ∩ A n ≠ ∅) : is_connected (A₀ ∪ (⋃ n, A n)) :=

Let $\left\{A_{\alpha}\right\}$ be a collection of connected subspaces of $X$; let $A$ be a connected subset of $X$. Show that if $A \cap A_{\alpha} \neq \varnothing$ for all $\alpha$, then $A \cup\left(\bigcup A_{\alpha}\right)$ is connected.

theorem exercise_23_6 {X : Type*} [topological_space X] {A C : set X} (hc : is_connected C) (hCA : C ∩ A ≠ ∅) (hCXA : C ∩ Aᶜ ≠ ∅) : C ∩ (frontier A) ≠ ∅ :=

Let $A \subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$, then $C$ intersects $\operatorname{Bd} A$.

theorem exercise_23_11 {X Y : Type*} [topological_space X] [topological_space Y] (p : X → Y) (hq : quotient_map p) (hY : connected_space Y) (hX : ∀ y : Y, is_connected (p ⁻¹' {y})) : connected_space X :=

Let $p: X \rightarrow Y$ be a quotient map. Show that if each set $p^{-1}(\{y\})$ is connected, and if $Y$ is connected, then $X$ is connected.

theorem exercise_24_3a [topological_space I] [compact_space I] (f : I → I) (hf : continuous f) : ∃ (x : I), f x = x :=

Let $f \colon X \rightarrow X$ be continuous. Show that if $X = [0, 1]$, there is a point $x$ such that $f(x) = x$. (The point $x$ is called a fixed point of $f$.)

theorem exercise_25_9 {G : Type*} [topological_space G] [group G] [topological_group G] (C : set G) (h : C = connected_component 1) : is_normal_subgroup C :=

Let $G$ be a topological group; let $C$ be the component of $G$ containing the identity element $e$. Show that $C$ is a normal subgroup of $G$.

theorem exercise_26_12 {X Y : Type*} [topological_space X] [topological_space Y] (p : X → Y) (h : function.surjective p) (hc : continuous p) (hp : ∀ y, is_compact (p ⁻¹' {y})) (hY : compact_space Y) : compact_space X :=

Let $p: X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y \in Y$. (Such a map is called a perfect map.) Show that if $Y$ is compact, then $X$ is compact.

theorem exercise_28_4 {X : Type*} [topological_space X] (hT1 : t1_space X) : countably_compact X ↔ limit_point_compact X :=

A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. Show that for a $T_1$ space $X$, countable compactness is equivalent to limit point compactness.

theorem exercise_28_6 {X : Type*} [metric_space X] [compact_space X] {f : X → X} (hf : isometry f) : function.bijective f :=

Let $(X, d)$ be a metric space. If $f: X \rightarrow X$ satisfies the condition $d(f(x), f(y))=d(x, y)$ for all $x, y \in X$, then $f$ is called an isometry of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism.

theorem exercise_29_4 [topological_space (ℕ → I)] : ¬ locally_compact_space (ℕ → I) :=

Show that $[0, 1]^\omega$ is not locally compact in the uniform topology.

theorem exercise_30_10 {X : ℕ → Type*} [∀ i, topological_space (X i)] (h : ∀ i, ∃ (s : set (X i)), countable s ∧ dense s) : ∃ (s : set (Π i, X i)), countable s ∧ dense s :=

Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset.

theorem exercise_31_1 {X : Type*} [topological_space X] (hX : regular_space X) (x y : X) : ∃ (U V : set X), is_open U ∧ is_open V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅ :=

Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint.

theorem exercise_31_3 {α : Type*} [partial_order α] [topological_space α] (h : order_topology α) : regular_space α :=

Show that every order topology is regular.

theorem exercise_32_2a {ι : Type*} {X : ι → Type*} [∀ i, topological_space (X i)] (h : ∀ i, nonempty (X i)) (h2 : t2_space (Π i, X i)) : ∀ i, t2_space (X i) :=

Show that if $\prod X_\alpha$ is Hausdorff, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty.

theorem exercise_32_2c {ι : Type*} {X : ι → Type*} [∀ i, topological_space (X i)] (h : ∀ i, nonempty (X i)) (h2 : normal_space (Π i, X i)) : ∀ i, normal_space (X i) :=

Show that if $\prod X_\alpha$ is normal, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty.

theorem exercise_33_7 {X : Type*} [topological_space X] (hX : locally_compact_space X) (hX' : t2_space X) : ∀ x A, is_closed A ∧ ¬ x ∈ A → ∃ (f : X → I), continuous f ∧ f x = 1 ∧ f '' A = {0} :=

Show that every locally compact Hausdorff space is completely regular.

theorem exercise_34_9 (X : Type*) [topological_space X] [compact_space X] (X1 X2 : set X) (hX1 : is_closed X1) (hX2 : is_closed X2) (hX : X1 ∪ X2 = univ) (hX1m : metrizable_space X1) (hX2m : metrizable_space X2) : metrizable_space X :=

Let $X$ be a compact Hausdorff space that is the union of the closed subspaces $X_1$ and $X_2$. If $X_1$ and $X_2$ are metrizable, show that $X$ is metrizable.

theorem exercise_43_2 {X : Type*} [metric_space X] {Y : Type*} [metric_space Y] [complete_space Y] (A : set X) (f : X → Y) (hf : uniform_continuous_on f A) : ∃! (g : X → Y), continuous_on g (closure A) ∧ uniform_continuous_on g (closure A) ∧ ∀ (x : A), g x = f x :=

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f \colon A \rightarrow Y$ is uniformly continuous, then $f$ can be uniquely extended to a continuous function $g \colon \bar{A} \rightarrow Y$, and $g$ is uniformly continuous.

theorem exercise_1_2 : (⟨-1/2, real.sqrt 3 / 2⟩ : ℂ) ^ 3 = -1 :=

Show that $\frac{-1 + \sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1).

theorem exercise_1_4 {F V : Type*} [add_comm_group V] [field F] [module F V] (v : V) (a : F): a • v = 0 ↔ a = 0 ∨ v = 0 :=

Prove that if $a \in \mathbf{F}$, $v \in V$, and $av = 0$, then $a = 0$ or $v = 0$.

theorem exercise_1_7 : ∃ U : set (ℝ × ℝ), (U ≠ ∅) ∧ (∀ (c : ℝ) (u : ℝ × ℝ), u ∈ U → c • u ∈ U) ∧ (∀ U' : submodule ℝ (ℝ × ℝ), U ≠ ↑U') :=

Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbf{R}^2$.

theorem exercise_1_9 {F V : Type*} [add_comm_group V] [field F] [module F V] (U W : submodule F V): ∃ U' : submodule F V, (U'.carrier = ↑U ∩ ↑W ↔ (U ≤ W ∨ W ≤ U)) :=

Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other.

theorem exercise_3_8 {F V W : Type*} [add_comm_group V] [add_comm_group W] [field F] [module F V] [module F W] (L : V →ₗ[F] W) : ∃ U : submodule F V, U ⊓ L.ker = ⊥ ∧ linear_map.range L = range (dom_restrict L U):=

Suppose that $V$ is finite dimensional and that $T \in \mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that $U \cap \operatorname{null} T=\{0\}$ and range $T=\{T u: u \in U\}$.

theorem exercise_5_1 {F V : Type*} [add_comm_group V] [field F] [module F V] {L : V →ₗ[F] V} {n : ℕ} (U : fin n → submodule F V) (hU : ∀ i : fin n, map L (U i) = U i) : map L (∑ i : fin n, U i : submodule F V) = (∑ i : fin n, U i : submodule F V) :=

Suppose $T \in \mathcal{L}(V)$. Prove that if $U_{1}, \ldots, U_{m}$ are subspaces of $V$ invariant under $T$, then $U_{1}+\cdots+U_{m}$ is invariant under $T$.

theorem exercise_5_11 {F V : Type*} [add_comm_group V] [field F] [module F V] (S T : End F V) : (S * T).eigenvalues = (T * S).eigenvalues :=

Suppose $S, T \in \mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.

theorem exercise_5_13 {F V : Type*} [add_comm_group V] [field F] [module F V] [finite_dimensional F V] {T : End F V} (hS : ∀ U : submodule F V, finrank F U = finrank F V - 1 → map T U = U) : ∃ c : F, T = c • id :=

Suppose $T \in \mathcal{L}(V)$ is such that every subspace of $V$ with dimension $\operatorname{dim} V-1$ is invariant under $T$. Prove that $T$ is a scalar multiple of the identity operator.

theorem exercise_5_24 {V : Type*} [add_comm_group V] [module ℝ V] [finite_dimensional ℝ V] {T : End ℝ V} (hT : ∀ c : ℝ, eigenspace T c = ⊥) {U : submodule ℝ V} (hU : map T U = U) : even (finrank U) :=

Suppose $V$ is a real vector space and $T \in \mathcal{L}(V)$ has no eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.

theorem exercise_6_3 {n : ℕ} (a b : fin n → ℝ) : (∑ i, a i * b i) ^ 2 ≤ (∑ i : fin n, i * a i ^ 2) * (∑ i, b i ^ 2 / i) :=

Prove that $\left(\sum_{j=1}^{n} a_{j} b_{j}\right)^{2} \leq\left(\sum_{j=1}^{n} j a_{j}{ }^{2}\right)\left(\sum_{j=1}^{n} \frac{b_{j}{ }^{2}}{j}\right)$ for all real numbers $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$.

theorem exercise_6_13 {V : Type*} [inner_product_space ℂ V] {n : ℕ} {e : fin n → V} (he : orthonormal ℂ e) (v : V) : ‖v‖^2 = ∑ i : fin n, ‖⟪v, e i⟫_ℂ‖^2 ↔ v ∈ span ℂ (e '' univ) :=

Suppose $\left(e_{1}, \ldots, e_{m}\right)$ is an or thonormal list of vectors in $V$. Let $v \in V$. Prove that $\|v\|^{2}=\left|\left\langle v, e_{1}\right\rangle\right|^{2}+\cdots+\left|\left\langle v, e_{m}\right\rangle\right|^{2}$ if and only if $v \in \operatorname{span}\left(e_{1}, \ldots, e_{m}\right)$.

theorem exercise_7_5 {V : Type*} [inner_product_space ℂ V] [finite_dimensional ℂ V] (hV : finrank V ≥ 2) : ∀ U : submodule ℂ (End ℂ V), U.carrier ≠ {T | T * T.adjoint = T.adjoint * T} :=

Show that if $\operatorname{dim} V \geq 2$, then the set of normal operators on $V$ is not a subspace of $\mathcal{L}(V)$.

theorem exercise_7_9 {V : Type*} [inner_product_space ℂ V] [finite_dimensional ℂ V] (T : End ℂ V) (hT : T * T.adjoint = T.adjoint * T) : is_self_adjoint T ↔ ∀ e : T.eigenvalues, (e : ℂ).im = 0 :=

Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.

theorem exercise_7_11 {V : Type*} [inner_product_space ℂ V] [finite_dimensional ℂ V] {T : End ℂ V} (hT : T*T.adjoint = T.adjoint*T) : ∃ (S : End ℂ V), S ^ 2 = T :=

Suppose $V$ is a complex inner-product space. Prove that every normal operator on $V$ has a square root. (An operator $S \in \mathcal{L}(V)$ is called a square root of $T \in \mathcal{L}(V)$ if $S^{2}=T$.)

theorem exercise_1_30 {n : ℕ} : ¬ ∃ a : ℤ, ∑ (i : fin n), (1 : ℚ) / (n+2) = a :=

Prove that $\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ is not an integer.

theorem exercise_2_4 {a : ℤ} (ha : a ≠ 0) (f_a :=

If $a$ is a nonzero integer, then for $n>m$ show that $\left(a^{2^{n}}+1, a^{2^{m}}+1\right)=1$ or 2 depending on whether $a$ is odd or even.

theorem exercise_2_27a : ¬ summable (λ i : {p : ℤ // squarefree p}, (1 : ℚ) / i) :=

Show that $\sum^{\prime} 1 / n$, the sum being over square free integers, diverges.

theorem exercise_3_4 : ¬ ∃ x y : ℤ, 3*x^2 + 2 = y^2 :=

Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers.

theorem exercise_3_10 {n : ℕ} (hn0 : ¬ n.prime) (hn1 : n ≠ 4) : factorial (n-1) ≡ 0 [MOD n] :=

If $n$ is not a prime, show that $(n-1) ! \equiv 0(n)$, except when $n=4$.

theorem exercise_4_4 {p t: ℕ} (hp0 : p.prime) (hp1 : p = 4*t + 1) (a : zmod p) : is_primitive_root a p ↔ is_primitive_root (-a) p :=

Consider a prime $p$ of the form $4 t+1$. Show that $a$ is a primitive root modulo $p$ iff $-a$ is a primitive root modulo $p$.

theorem exercise_4_6 {p n : ℕ} (hp : p.prime) (hpn : p = 2^n + 1) : is_primitive_root 3 p :=

If $p=2^{n}+1$ is a Fermat prime, show that 3 is a primitive root modulo $p$.

theorem exercise_4_11 {p : ℕ} (hp : p.prime) (k s: ℕ) (s :=

Prove that $1^{k}+2^{k}+\cdots+(p-1)^{k} \equiv 0(p)$ if $p-1 \nmid k$ and $-1(p)$ if $p-1 \mid k$.

theorem exercise_5_28 {p : ℕ} (hp : p.prime) (hp1 : p ≡ 1 [MOD 4]): ∃ x, x^4 ≡ 2 [MOD p] ↔ ∃ A B, p = A^2 + 64*B^2 :=

Show that $x^{4} \equiv 2(p)$ has a solution for $p \equiv 1(4)$ iff $p$ is of the form $A^{2}+64 B^{2}$.

theorem exercise_12_12 : is_algebraic ℚ (sin (real.pi/12)) :=

Show that $\sin (\pi / 12)$ is an algebraic number.

theorem exercise_1_13b {f : ℂ → ℂ} (Ω : set ℂ) (a b : Ω) (h : is_open Ω) (hf : differentiable_on ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).im = c) : f a = f b :=

Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Im}(f)$ is constant, then $f$ is constant.

theorem exercise_1_19a (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ) (h : s = (λ n, ∑ i in (finset.range n), i * z ^ i)) : ¬ ∃ y, tendsto s at_top (𝓝 y) :=

Prove that the power series $\sum nz^n$ does not converge on any point of the unit circle.

theorem exercise_1_19c (z : ℂ) (hz : abs z = 1) (hz2 : z ≠ 1) (s : ℕ → ℂ) (h : s = (λ n, ∑ i in (finset.range n), i * z / i)) : ∃ z, tendsto s at_top (𝓝 z) :=

Prove that the power series $\sum zn/n$ converges at every point of the unit circle except $z = 1$.

theorem exercise_2_2 : tendsto (λ y, ∫ x in 0..y, real.sin x / x) at_top (𝓝 (real.pi / 2)) :=

Show that $\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$.

theorem exercise_2_13 {f : ℂ → ℂ} (hf : ∀ z₀ : ℂ, ∃ (s : set ℂ) (c : ℕ → ℂ), is_open s ∧ z₀ ∈ s ∧ ∀ z ∈ s, tendsto (λ n, ∑ i in finset.range n, (c i) * (z - z₀)^i) at_top (𝓝 (f z₀)) ∧ ∃ i, c i = 0) : ∃ (c : ℕ → ℂ) (n : ℕ), f = λ z, ∑ i in finset.range n, (c i) * z ^ n :=

Suppose $f$ is an analytic function defined everywhere in $\mathbb{C}$ and such that for each $z_0 \in \mathbb{C}$ at least one coefficient in the expansion $f(z) = \sum_{n=0}^\infty c_n(z - z_0)^n$ is equal to 0. Prove that $f$ is a polynomial.

theorem exercise_3_4 (a : ℝ) (ha : 0 < a) : tendsto (λ y, ∫ x in -y..y, x * real.sin x / (x ^ 2 + a ^ 2)) at_top (𝓝 (real.pi * (real.exp (-a)))) :=

Show that $ \int_{-\infty}^{\infty} \frac{x \sin x}{x^2 + a^2} dx = \pi e^{-a}$ for $a > 0$.

theorem exercise_3_14 {f : ℂ → ℂ} (hf : differentiable ℂ f) (hf_inj : function.injective f) : ∃ (a b : ℂ), f = (λ z, a * z + b) ∧ a ≠ 0 :=

Prove that all entire functions that are also injective take the form $f(z) = az + b$, $a, b \in \mathbb{C}$ and $a \neq 0$.

theorem exercise_5_1 (f : ℂ → ℂ) (hf : differentiable_on ℂ f (ball 0 1)) (hb : bounded (set.range f)) (h0 : f ≠ 0) (zeros : ℕ → ℂ) (hz : ∀ n, f (zeros n) = 0) (hzz : set.range zeros = {z | f z = 0 ∧ z ∈ (ball (0 : ℂ) 1)}) : ∃ (z : ℂ), tendsto (λ n, (∑ i in finset.range n, (1 - zeros i))) at_top (𝓝 z) :=

Prove that if $f$ is holomorphic in the unit disc, bounded and not identically zero, and $z_{1}, z_{2}, \ldots, z_{n}, \ldots$ are its zeros $\left(\left|z_{k}\right|<1\right)$, then $\sum_{n}\left(1-\left|z_{n}\right|\right)<\infty$.

theorem exercise_2018_a5 (f : ℝ → ℝ) (hf : cont_diff ℝ ⊤ f) (hf0 : f 0 = 0) (hf1 : f 1 = 1) (hf2 : ∀ x, f x ≥ 0) : ∃ (n : ℕ) (x : ℝ), iterated_deriv n f x = 0 :=

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function satisfying $f(0)=0, f(1)=1$, and $f(x) \geq 0$ for all $x \in$ $\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x)<0$.

theorem exercise_2018_b4 (a : ℝ) (x : ℕ → ℝ) (hx0 : x 0 = a) (hx1 : x 1 = a) (hxn : ∀ n : ℕ, n ≥ 2 → x (n+1) = 2*(x n)*(x (n-1)) - x (n-2)) (h : ∃ n, x n = 0) : ∃ c, function.periodic x c :=

Given a real number $a$, we define a sequence by $x_{0}=1$, $x_{1}=x_{2}=a$, and $x_{n+1}=2 x_{n} x_{n-1}-x_{n-2}$ for $n \geq 2$. Prove that if $x_{n}=0$ for some $n$, then the sequence is periodic.

theorem exercise_2014_a5 (P : ℕ → polynomial ℤ) (hP : ∀ n, P n = ∑ (i : fin n), (n+1) * X ^ n) : ∀ (j k : ℕ), j ≠ k → is_coprime (P j) (P k) :=

Let

theorem exercise_2001_a5 : ∃! a n : ℕ, a > 0 ∧ n > 0 ∧ a^(n+1) - (a+1)^n = 2001 :=

Prove that there are unique positive integers $a, n$ such that $a^{n+1}-(a+1)^n=2001$.

theorem exercise_1999_b4 (f : ℝ → ℝ) (hf: cont_diff ℝ 3 f) (hf1 : ∀ (n ≤ 3) (x : ℝ), iterated_deriv n f x > 0) (hf2 : ∀ x : ℝ, iterated_deriv 3 f x ≤ f x) : ∀ x : ℝ, deriv f x < 2 * f x :=

Let $f$ be a real function with a continuous third derivative such that $f(x), f^{\prime}(x), f^{\prime \prime}(x), f^{\prime \prime \prime}(x)$ are positive for all $x$. Suppose that $f^{\prime \prime \prime}(x) \leq f(x)$ for all $x$. Show that $f^{\prime}(x)<2 f(x)$ for all $x$.

theorem exercise_1998_b6 (a b c : ℤ) : ∃ n : ℤ, n > 0 ∧ ¬ ∃ m : ℤ, sqrt (n^3 + a*n^2 + b*n + c) = m :=

Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+a n^2+b n+c}$ is not an integer.

theorem exercise_2_26 {M : Type*} [topological_space M] (U : set M) : is_open U ↔ ∀ x ∈ U, ¬ cluster_pt x (𝓟 Uᶜ) :=

Prove that a set $U \subset M$ is open if and only if none of its points are limits of its complement.

theorem exercise_2_32a (A : set ℕ) : is_clopen A :=

Show that every subset of $\mathbb{N}$ is clopen.

theorem exercise_2_46 {M : Type*} [metric_space M] {A B : set M} (hA : is_compact A) (hB : is_compact B) (hAB : disjoint A B) (hA₀ : A ≠ ∅) (hB₀ : B ≠ ∅) : ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a : M) (b : M), a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b :=

Assume that $A, B$ are compact, disjoint, nonempty subsets of $M$. Prove that there are $a_0 \in A$ and $b_0 \in B$ such that for all $a \in A$ and $b \in B$ we have $d(a_0, b_0) \leq d(a, b)$.