l3lab/ntp-mathlib huggingface dataset:
/-- If the domain of a function `f : α → β` is a Fréchet-Urysohn space, then convergence
is equivalent to sequential convergence. See also `Filter.tendsto_iff_seq_tendsto` for a version
that works for any pair of filters assuming that the filter in the domain is countably generated.

This property is equivalent to the definition of `FrechetUrysohnSpace`, see
`FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto`. -/
theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} :
    Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) := by
  refine'
    ⟨fun hf u hu => hf.comp hu, fun h =>
      ((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 _⟩
  rintro s ⟨hbs, hsc⟩
  refine' ⟨closure (f ⁻¹' s), ⟨mt _ hbs, isClosed_closure⟩, fun x => mt fun hx => subset_closure hx⟩
  rw [← seqClosure_eq_closure]
  rintro ⟨u, hus, hu⟩
  

Proof state:
case intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : FrechetUrysohnSpace X
f : X → Y
a : X
b : Y
h : ∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b)
s : Set Y
hbs : b ∉ s
hsc : IsClosed s
u : ℕ → X
hus : ∀ (n : ℕ), u n ∈ f ⁻¹' s
hu : Tendsto u atTop (𝓝 a)
⊢ b ∈ s

Next tactic:
exact hsc.mem_of_tendsto (h u hu) (eventually_of_forall hus)
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
⊢ FrechetUrysohnSpace X

Next tactic:
refine ⟨fun s x hcx => ?_⟩
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  refine ⟨fun s x hcx => ?_⟩
  

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
⊢ x ∈ seqClosure s

Next tactic:
by_cases hx : x ∈ s
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  refine ⟨fun s x hcx => ?_⟩
  by_cases hx : x ∈ s;
  · 

Proof state:
case pos
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∈ s
⊢ x ∈ seqClosure s

Next tactic:
exact subset_seqClosure hx
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  refine ⟨fun s x hcx => ?_⟩
  by_cases hx : x ∈ s;
  · exact subset_seqClosure hx
  · 

Proof state:
case neg
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
⊢ x ∈ seqClosure s

Next tactic:
obtain ⟨u, hux, hus⟩ : ∃ u : ℕ → X, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ x in atTop, u x ∈ s
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  refine ⟨fun s x hcx => ?_⟩
  by_cases hx : x ∈ s;
  · exact subset_seqClosure hx
  · obtain ⟨u, hux, hus⟩ : ∃ u : ℕ → X, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ x in atTop, u x ∈ s
    · 

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
⊢ ∃ u, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ (x : ℕ) in atTop, u x ∈ s

Next tactic:
simpa only [ContinuousAt, hx, tendsto_nhds_true, (· ∘ ·), ← not_frequently, exists_prop,
        ← mem_closure_iff_frequently, hcx, imp_false, not_forall, not_not, not_false_eq_true,
        not_true_eq_false] using h (· ∉ s) x
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  refine ⟨fun s x hcx => ?_⟩
  by_cases hx : x ∈ s;
  · exact subset_seqClosure hx
  · obtain ⟨u, hux, hus⟩ : ∃ u : ℕ → X, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ x in atTop, u x ∈ s
    · simpa only [ContinuousAt, hx, tendsto_nhds_true, (· ∘ ·), ← not_frequently, exists_prop,
        ← mem_closure_iff_frequently, hcx, imp_false, not_forall, not_not, not_false_eq_true,
        not_true_eq_false] using h (· ∉ s) x
    

Proof state:
case neg.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
u : ℕ → X
hux : Tendsto u atTop (𝓝 x)
hus : ∃ᶠ (x : ℕ) in atTop, u x ∈ s
⊢ x ∈ seqClosure s

Next tactic:
rcases extraction_of_frequently_atTop hus with ⟨φ, φ_mono, hφ⟩
/-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. -/
theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
    (h : ∀ (f : X → Prop) (a : X),
      (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
    FrechetUrysohnSpace X := by
  refine ⟨fun s x hcx => ?_⟩
  by_cases hx : x ∈ s;
  · exact subset_seqClosure hx
  · obtain ⟨u, hux, hus⟩ : ∃ u : ℕ → X, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ x in atTop, u x ∈ s
    · simpa only [ContinuousAt, hx, tendsto_nhds_true, (· ∘ ·), ← not_frequently, exists_prop,
        ← mem_closure_iff_frequently, hcx, imp_false, not_forall, not_not, not_false_eq_true,
        not_true_eq_false] using h (· ∉ s) x
    rcases extraction_of_frequently_atTop hus with ⟨φ, φ_mono, hφ⟩
    

Proof state:
case neg.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h :
  ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a
s : Set X
x : X
hcx : x ∈ closure s
hx : x ∉ s
u : ℕ → X
hux : Tendsto u atTop (𝓝 x)
hus : ∃ᶠ (x : ℕ) in atTop, u x ∈ s
φ : ℕ → ℕ
φ_mono : StrictMono φ
hφ : ∀ (n : ℕ), u (φ n) ∈ s
⊢ x ∈ seqClosure s

Next tactic:
exact ⟨u ∘ φ, hφ, hux.comp φ_mono.tendsto_atTop⟩
/-- Every Fréchet-Urysohn space is a sequential space. -/
instance (priority := 100) FrechetUrysohnSpace.to_sequentialSpace [FrechetUrysohnSpace X] :
    SequentialSpace X :=
  ⟨fun s hs => by 

Proof state:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : FrechetUrysohnSpace X
s : Set X
hs : IsSeqClosed s
⊢ IsClosed s

Next tactic:
rw [← closure_eq_iff_isClosed, ← seqClosure_eq_closure, hs.seqClosure_eq]
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  

Proof state:
X : Type u_1
Y : Type u_2
inst✝ : UniformSpace X
s : Set X
h : IsSeqCompact s
⊢ TotallyBounded s

Next tactic:
intro V V_in
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  

Proof state:
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, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}

Next tactic:
unfold IsSeqCompact at h
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  

Proof state:
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, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}

Next tactic:
contrapose! h
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  

Proof state:
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), Set.Finite t → ¬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)

Next 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, ball_image_iff, not_and] using seq_of_forall_finite_exists h
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  obtain ⟨u, u_in, hu⟩ : ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ ∀ n m, m < n → u m ∉ ball (u n) V := by
    

Proof state:
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), Set.Finite t → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}
⊢ ∃ u, (∀ (n : ℕ), u n ∈ s) ∧ ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V

Next tactic:
simp only [not_subset, mem_iUnion₂, not_exists, exists_prop] at h
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  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
    

Proof state:
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), Set.Finite t → ∃ a ∈ s, ∀ (x : X), ¬(x ∈ t ∧ a ∈ {x_1 | (x_1, x) ∈ V})
⊢ ∃ u, (∀ (n : ℕ), u n ∈ s) ∧ ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V

Next tactic:
simpa only [forall_and, ball_image_iff, not_and] using seq_of_forall_finite_exists h
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  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, ball_image_iff, not_and] using seq_of_forall_finite_exists h
  

Proof state:
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), Set.Finite t → ¬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)

Next tactic:
refine' ⟨u, u_in, fun x _ φ hφ huφ => _⟩
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  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, ball_image_iff, not_and] using seq_of_forall_finite_exists h
  refine' ⟨u, u_in, fun x _ φ hφ huφ => _⟩
  

Proof state:
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), Set.Finite t → ¬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

Next tactic:
obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  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, ball_image_iff, not_and] using seq_of_forall_finite_exists h
  refine' ⟨u, u_in, fun x _ φ hφ huφ => _⟩
  obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V
  

Proof state:
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), Set.Finite t → ¬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, ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V
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), Set.Finite t → ¬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

Next tactic:
exact huφ.cauchySeq.mem_entourage V_in
/-- A sequentially compact set in a uniform space is totally bounded. -/
protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by
  intro V V_in
  unfold IsSeqCompact at h
  contrapose! h
  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, ball_image_iff, not_and] using seq_of_forall_finite_exists h
  refine' ⟨u, u_in, fun x _ φ hφ huφ => _⟩
  obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V
  exact huφ.cauchySeq.mem_entourage V_in
  

Proof state:
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), Set.Finite t → ¬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

Next tactic:
exact hu (φ <| N + 1) (φ N) (hφ <| lt_add_one N) (hN (N + 1) N N.le_succ le_rfl)
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
have := hl.1
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
⊢ ∃ x ∈ s, l ≤ 𝓝 x

Next tactic:
obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
    choose t htl htW hts using this
    have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl
    exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h =>
      biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n =>
      (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n =>
      (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

Next tactic:
have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
      simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
        prod_self_subset_prod_self, and_assoc] using this
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : l ≤ 𝓟 s
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
⊢ ∀ (n : ℕ), ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s

Next tactic:
rw [le_principal_iff] at hls
/-- A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. -/
protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by
  have := hl.1
  rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
  choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
  have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
    @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
  obtain ⟨t, ht_anti, htl, htW, hts⟩ :
      ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by
    have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by
      rw [le_principal_iff] at hls
      

Proof state:
X : Type u_1
Y : Type u_2
inst✝¹ : UniformSpace X
s : Set X
inst✝ : IsCountablyGenerated (𝓤 X)
hs : IsSeqCompact s
l : Filter X
hl : Cauchy l
hls : s ∈ l
this : NeBot l
V : ℕ → Set (X × X)
hV : HasAntitoneBasis (𝓤 X) V
W : ℕ → Set (X × X)
hW : ∀ (n : ℕ), W n ∈ 𝓤 X
hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n
hWV' : ∀ (n : ℕ), W n ⊆ V n
⊢ ∀ (n : ℕ), ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s

Next tactic:
have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)