l3lab/ntp-mathlib huggingface dataset:
/-- A metric space structure on a subset `s` of a metric space, designed to make it complete
if `s` is open. It is given by `dist' x y = dist x y + |1 / dist x sᶜ - 1 / dist y sᶜ|`, where the
second term blows up close to the boundary to ensure that Cauchy sequences for `dist'` remain well
inside `s`.

Porting note: the definition changed to ensure that the `TopologicalSpace` structure on
`TopologicalSpace.Opens.CompleteCopy s` is definitionally equal to the original one. -/
-- Porting note: in mathlib3 this was only a local instance.
instance instMetricSpace : MetricSpace (CompleteCopy s) := by
  refine @MetricSpace.ofT0PseudoMetricSpace (CompleteCopy s)
    (.ofDistTopology dist (fun _ ↦ ?_) (fun _ _ ↦ ?_) (fun x y z ↦ ?_) fun t ↦ ?_) _
  · simp only [dist_eq, dist_self, one_div, sub_self, abs_zero, add_zero]
  · simp only [dist_eq, dist_comm, abs_sub_comm]
  · calc
      dist x z = dist x.1 z.1 + |1 / infDist x.1 sᶜ - 1 / infDist z.1 sᶜ| := rfl
      _ ≤ dist x.1 y.1 + dist y.1 z.1 + (|1 / infDist x.1 sᶜ - 1 / infDist y.1 sᶜ| +
            |1 / infDist y.1 sᶜ - 1 / infDist z.1 sᶜ|) :=
        add_le_add (dist_triangle _ _ _) (dist_triangle (1 / infDist _ _) _ _)
      _ = dist x y + dist y z := add_add_add_comm ..
  · refine ⟨fun h x hx ↦ ?_, fun h ↦ isOpen_iff_mem_nhds.2 fun x hx ↦ ?_⟩
    · rcases (Metric.isOpen_iff (α := s)).1 h x hx with ⟨ε, ε0, hε⟩
      exact ⟨ε, ε0, fun y hy ↦ hε <| (dist_comm _ _).trans_lt <| (dist_val_le_dist _ _).trans_lt hy⟩
    · rcases h x hx with ⟨ε, ε0, hε⟩
      simp only [dist_eq, one_div] at hε
      have : Tendsto (fun y : s ↦ dist x.1 y.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist y.1 sᶜ)⁻¹|)
        (𝓝 x) (𝓝 (dist x.1 x.1 + |(infDist x.1 sᶜ)⁻¹ - (infDist x.1 sᶜ)⁻¹|))
      · refine (tendsto_const_nhds.dist continuous_subtype_val.continuousAt).add
          (tendsto_const_nhds.sub <| ?_).abs
        refine (continuousAt_inv_infDist_pt ?_).comp continuous_subtype_val.continuousAt
        rw [s.isOpen.isClosed_compl.closure_eq, mem_compl_iff, not_not]
        exact x.2
      simp only [dist_self, sub_self, abs_zero, zero_add] at this
      

Proof state:
case refine_4.refine_2.intro.intro
α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set (CompleteCopy s)
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : CompleteCopy s), dist x y < ε → y ∈ t
x : CompleteCopy s
hx : x ∈ t
ε : ℝ
ε0 : ε > 0
hε : ∀ (y : CompleteCopy s), dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹| < ε → y ∈ t
this : Tendsto (fun y => dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹|) (𝓝 x) (𝓝 0)
⊢ t ∈ 𝓝 x

Next tactic:
exact mem_of_superset (this <| gt_mem_nhds ε0) hε
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
⊢ CompleteSpace (CompleteCopy s)

Next tactic:
refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
⊢ ∀ (n : ℕ), 0 < (fun x => (1 / 2) ^ x) n

Next tactic:
simp
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
have A : CauchySeq fun n => (u n).1
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · 

Proof state:
case A
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
⊢ CauchySeq fun n => ↑(u n)

Next tactic:
refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · 

Proof state:
case A.refine_1
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
n m N : ℕ
hNn : N ≤ n
hNm : N ≤ m
⊢ dist ↑(u n) ↑(u m) ≤ (fun n => (1 / 2) ^ n) N

Next tactic:
exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · 

Proof state:
case A.refine_2
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
⊢ Tendsto (fun n => (1 / 2) ^ n) atTop (𝓝 0)

Next tactic:
exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
⊢ 0 ≤ 1 / 2

Next tactic:
norm_num
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
⊢ 1 / 2 < 1

Next tactic:
norm_num
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  

Proof state:
case intro
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
by_cases xs : x ∈ s
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∈ s
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
⊢ ∃ C, ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C

Next tactic:
refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
n : ℕ
⊢ 1 / infDist (↑(u n)) (↑s)ᶜ < (1 / 2) ^ 0 + 1 / infDist (↑(u 0)) (↑s)ᶜ

Next tactic:
rw [← sub_lt_iff_lt_add]
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    rw [← sub_lt_iff_lt_add]
    

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
n : ℕ
⊢ 1 / infDist (↑(u n)) (↑s)ᶜ - 1 / infDist (↑(u 0)) (↑s)ᶜ < (1 / 2) ^ 0

Next tactic:
calc
      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
      _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
      _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    rw [← sub_lt_iff_lt_add]
    calc
      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
      _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
      _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
  

Proof state:
case neg.intro
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
have Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    rw [← sub_lt_iff_lt_add]
    calc
      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
      _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
      _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
  have Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)
  

Proof state:
case neg.intro
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
have Hmem : ∀ {y}, y ∈ s ↔ 0 < infDist y sᶜ := fun {y} ↦ by
    rw [← s.isOpen.isClosed_compl.not_mem_iff_infDist_pos ⟨x, xs⟩]; exact not_not.symm
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    rw [← sub_lt_iff_lt_add]
    calc
      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
      _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
      _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
  have Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)
  have Hmem : ∀ {y}, y ∈ s ↔ 0 < infDist y sᶜ := fun {y} ↦ by
    

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
y : α
⊢ y ∈ s ↔ 0 < infDist y (↑s)ᶜ

Next tactic:
rw [← s.isOpen.isClosed_compl.not_mem_iff_infDist_pos ⟨x, xs⟩]
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    rw [← sub_lt_iff_lt_add]
    calc
      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
      _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
      _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
  have Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)
  have Hmem : ∀ {y}, y ∈ s ↔ 0 < infDist y sᶜ := fun {y} ↦ by
    rw [← s.isOpen.isClosed_compl.not_mem_iff_infDist_pos ⟨x, xs⟩]; 

Proof state:
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
y : α
⊢ y ∈ s ↔ y ∉ (↑s)ᶜ

Next tactic:
exact not_not.symm
instance instCompleteSpace [CompleteSpace α] : CompleteSpace (CompleteCopy s) := by
  refine Metric.complete_of_convergent_controlled_sequences ((1 / 2) ^ ·) (by simp) fun u hu ↦ ?_
  have A : CauchySeq fun n => (u n).1
  · refine cauchySeq_of_le_tendsto_0 (fun n : ℕ => (1 / 2) ^ n) (fun n m N hNn hNm => ?_) ?_
    · exact (dist_val_le_dist (u n) (u m)).trans (hu N n m hNn hNm).le
    · exact tendsto_pow_atTop_nhds_0_of_lt_1 (by norm_num) (by norm_num)
  obtain ⟨x, xlim⟩ : ∃ x, Tendsto (fun n => (u n).1) atTop (𝓝 x) := cauchySeq_tendsto_of_complete A
  by_cases xs : x ∈ s
  · exact ⟨⟨x, xs⟩, tendsto_subtype_rng.2 xlim⟩
  obtain ⟨C, hC⟩ : ∃ C, ∀ n, 1 / infDist (u n).1 sᶜ < C
  · refine ⟨(1 / 2) ^ 0 + 1 / infDist (u 0).1 sᶜ, fun n ↦ ?_⟩
    rw [← sub_lt_iff_lt_add]
    calc
      _ ≤ |1 / infDist (u n).1 sᶜ - 1 / infDist (u 0).1 sᶜ| := le_abs_self _
      _ = |1 / infDist (u 0).1 sᶜ - 1 / infDist (u n).1 sᶜ| := abs_sub_comm _ _
      _ ≤ dist (u 0) (u n) := le_add_of_nonneg_left dist_nonneg
      _ < (1 / 2) ^ 0 := hu 0 0 n le_rfl n.zero_le
  have Cpos : 0 < C := lt_of_le_of_lt (div_nonneg zero_le_one infDist_nonneg) (hC 0)
  have Hmem : ∀ {y}, y ∈ s ↔ 0 < infDist y sᶜ := fun {y} ↦ by
    rw [← s.isOpen.isClosed_compl.not_mem_iff_infDist_pos ⟨x, xs⟩]; exact not_not.symm
  

Proof state:
case neg.intro
α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → CompleteCopy s
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x => (1 / 2) ^ x) N
A : CauchySeq fun n => ↑(u n)
x : α
xlim : Tendsto (fun n => ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
Hmem : ∀ {y : α}, y ∈ s ↔ 0 < infDist y (↑s)ᶜ
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Next tactic:
have I : ∀ n, 1 / C ≤ infDist (u n).1 sᶜ := fun n ↦ by
    have : 0 < infDist (u n).1 sᶜ := Hmem.1 (u n).2
    rw [div_le_iff' Cpos]
    exact (div_le_iff this).1 (hC n).le