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

tactic: have := hl.1
state_before: X : Type u_1
Y : Type ?u.18740
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, x ∈ s ∧ l ≤ 𝓝 x

tactic: rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩
state_before: X : Type u_1
Y : Type ?u.18740
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, x ∈ s ∧ l ≤ 𝓝 x

tactic: choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n)
state_before: case intro
X : Type u_1
Y : Type ?u.18740
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, x ∈ s ∧ l ≤ 𝓝 x

tactic: have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx =>
  @hWV n (x, y) ⟨x, refl_mem_uniformity <| hW _, hx⟩
state_before: case intro
X : Type u_1
Y : Type ?u.18740
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, x ∈ s ∧ l ≤ 𝓝 x

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)⟩
state_before: case intro
X : Type u_1
Y : Type ?u.18740
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, x ∈ s ∧ l ≤ 𝓝 x

tactic: choose u hu using fun n => Filter.nonempty_of_mem (htl n)
state_before: case intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
⊢ ∃ x, x ∈ s ∧ l ≤ 𝓝 x

tactic: have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ =>
    ⟨N, fun m hm n hn => hWV' _ <| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩
state_before: case intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
⊢ ∃ x, x ∈ s ∧ l ≤ 𝓝 x

tactic: rcases hs.exists_tendsto (fun n => hts n (hu n)) huc with ⟨x, hxs, hx⟩
state_before: case intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
⊢ ∃ x, x ∈ s ∧ l ≤ 𝓝 x

tactic: refine ⟨x, hxs, (nhds_basis_uniformity' hV.toHasBasis).ge_iff.2 fun N _ => ?_⟩
state_before: case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
⊢ ∃ x, x ∈ s ∧ l ≤ 𝓝 x

tactic: obtain ⟨n, hNn, hn⟩ : ∃ n, N ≤ n ∧ u n ∈ ball x (W N)
state_before: case intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
N : ℕ
x✝ : True
⊢ ball x (V N) ∈ l

tactic: refine mem_of_superset (htl n) fun y hy => hWV N ⟨u n, hn, htW N ?_⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
N : ℕ
x✝ : True
n : ℕ
hNn : N ≤ n
hn : u n ∈ ball x (W N)
⊢ ball x (V N) ∈ l

tactic: exact ⟨ht_anti hNn (hu n), ht_anti hNn hy⟩
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
N : ℕ
x✝ : True
n : ℕ
hNn : N ≤ n
hn : u n ∈ ball x (W N)
y : X
hy : y ∈ t n
⊢ (u n, (x, y).snd) ∈ t N ×ˢ t N

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
state_before: X : Type u_1
Y : Type ?u.18740
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

tactic: choose t htl htW hts using this
state_before: X : Type u_1
Y : Type ?u.18740
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
this : ∀ (n : ℕ), ∃ t, t ∈ l ∧ t ×ˢ t ⊆ W n ∧ t ⊆ s
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

tactic: have : ∀ n : ℕ, (⋂ k ≤ n, t k) ⊆ t n := fun n => by apply iInter₂_subset; rfl
state_before: X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

tactic: 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)⟩
state_before: X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
this : ∀ (n : ℕ), (⋂ (k : ℕ) (_ : k ≤ n), t k) ⊆ t n
⊢ ∃ t, Antitone t ∧ (∀ (n : ℕ), t n ∈ l) ∧ (∀ (n : ℕ), t n ×ˢ t n ⊆ W n) ∧ ∀ (n : ℕ), t n ⊆ s

tactic: rw [le_principal_iff] at hls
state_before: X : Type u_1
Y : Type ?u.18740
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, t ∈ l ∧ t ×ˢ t ⊆ W n ∧ t ⊆ s

tactic: have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls)
state_before: X : Type u_1
Y : Type ?u.18740
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, t ∈ l ∧ t ×ˢ t ⊆ W n ∧ t ⊆ s

tactic: simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff,
  prod_self_subset_prod_self, and_assoc] using this
state_before: X : Type u_1
Y : Type ?u.18740
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
this : ∀ (n : ℕ), W n ∩ s ×ˢ s ∈ l ×ˢ l
⊢ ∀ (n : ℕ), ∃ t, t ∈ l ∧ t ×ˢ t ⊆ W n ∧ t ⊆ s

tactic: apply iInter₂_subset
state_before: X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
n : ℕ
⊢ (⋂ (k : ℕ) (_ : k ≤ n), t k) ⊆ t n

tactic: rfl
state_before: case j
X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
n : ℕ
⊢ n ≤ n

tactic: exact ((eventually_ge_atTop N).and (hx <| ball_mem_nhds x (hW N))).exists
state_before: X : Type u_1
Y : Type ?u.18740
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 : ℕ → Set X
ht_anti : Antitone t
htl : ∀ (n : ℕ), t n ∈ l
htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n
hts : ∀ (n : ℕ), t n ⊆ s
u : ℕ → X
hu : ∀ (n : ℕ), u n ∈ t n
huc : CauchySeq u
x : X
hxs : x ∈ s
hx : Tendsto (fun n => u n) atTop (𝓝 x)
N : ℕ
x✝ : True
⊢ ∃ n, N ≤ n ∧ u n ∈ ball x (W N)