Theorem: IsSeqCompact.totallyBounded
File: Mathlib/Topology/Sequences.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
h : IsSeqCompact s
⊢ TotallyBounded s
state_after: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
h : IsSeqCompact s
V : Set (X × X)
V_in : V ∈ 𝓤 X
⊢ ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
tactic: intro V V_in
state_before: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
h : IsSeqCompact s
V : Set (X × X)
V_in : V ∈ 𝓤 X
⊢ ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
state_after: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
h : ∀ ⦃x : ℕ → X⦄, (∀ (n : ℕ), x n ∈ s) → ∃ a ∈ s, ∃ φ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a)
V : Set (X × X)
V_in : V ∈ 𝓤 X
⊢ ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
tactic: unfold IsSeqCompact at h
state_before: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
h : ∀ ⦃x : ℕ → X⦄, (∀ (n : ℕ), x n ∈ s) → ∃ a ∈ s, ∃ φ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a)
V : Set (X × X)
V_in : V ∈ 𝓤 X
⊢ ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
state_after: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
⊢ ∃ x, (∀ (n : ℕ), x n ∈ s) ∧ ∀ a ∈ s, ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)
tactic: contrapose! h
state_before: X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
⊢ ∃ x, (∀ (n : ℕ), x n ∈ s) ∧ ∀ a ∈ s, ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)
state_after: case intro.intro
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
u : ℕ → X
u_in : ∀ (n : ℕ), u n ∈ s
hu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V
⊢ ∃ x, (∀ (n : ℕ), x n ∈ s) ∧ ∀ a ∈ s, ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)
tactic: obtain ⟨u, u_in, hu⟩ : ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ ∀ n m, m < n → u m ∉ ball (u n) V := by
    simp only [not_subset, mem_iUnion₂, not_exists, exists_prop] at h
    simpa only [forall_and, forall_mem_image, not_and] using seq_of_forall_finite_exists h
state_before: case intro.intro
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
u : ℕ → X
u_in : ∀ (n : ℕ), u n ∈ s
hu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V
⊢ ∃ x, (∀ (n : ℕ), x n ∈ s) ∧ ∀ a ∈ s, ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)
state_after: case intro.intro
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
u : ℕ → X
u_in : ∀ (n : ℕ), u n ∈ s
hu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V
x : X
x✝ : x ∈ s
φ : ℕ → ℕ
hφ : StrictMono φ
huφ : Tendsto (u ∘ φ) atTop (𝓝 x)
⊢ False
tactic: refine ⟨u, u_in, fun x _ φ hφ huφ => ?_⟩
state_before: case intro.intro
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
u : ℕ → X
u_in : ∀ (n : ℕ), u n ∈ s
hu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V
x : X
x✝ : x ∈ s
φ : ℕ → ℕ
hφ : StrictMono φ
huφ : Tendsto (u ∘ φ) atTop (𝓝 x)
⊢ False
state_after: case intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
u : ℕ → X
u_in : ∀ (n : ℕ), u n ∈ s
hu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V
x : X
x✝ : x ∈ s
φ : ℕ → ℕ
hφ : StrictMono φ
huφ : Tendsto (u ∘ φ) atTop (𝓝 x)
N : ℕ
hN : ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V
⊢ False
tactic: obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V :=
    huφ.cauchySeq.mem_entourage V_in
state_before: case intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
V : Set (X × X)
V_in : V ∈ 𝓤 X
h : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
u : ℕ → X
u_in : ∀ (n : ℕ), u n ∈ s
hu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V
x : X
x✝ : x ∈ s
φ : ℕ → ℕ
hφ : StrictMono φ
huφ : Tendsto (u ∘ φ) atTop (𝓝 x)
N : ℕ
hN : ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V
⊢ False
state_after: no goals
tactic: exact hu (φ <| N + 1) (φ N) (hφ <| lt_add_one N) (hN (N + 1) N N.le_succ le_rfl)