-- Mathlib/Topology/MetricSpace/PiNat.lean
theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
    [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
    ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f

Proof state:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
y : α
x : ℕ → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty
J : ∀ (n : ℕ), dist (g ⟨x, I⟩) y ≤ (1 / 2) ^ n + (1 / 2) ^ n
L : Tendsto (fun n => (1 / 2) ^ n + (1 / 2) ^ n) atTop (𝓝 (0 + 0))
⊢ dist (g ⟨x, I⟩) y ≤ 0

Tactic:
rw [add_zero] at L

Result:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
y : α
x : ℕ → ℕ
hx : ∀ (n : ℕ), y ∈ closedBall (u (x n)) ((1 / 2) ^ n)
I : (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty
J : ∀ (n : ℕ), dist (g ⟨x, I⟩) y ≤ (1 / 2) ^ n + (1 / 2) ^ n
L : Tendsto (fun n => (1 / 2) ^ n + (1 / 2) ^ n) atTop (𝓝 0)
⊢ dist (g ⟨x, I⟩) y ≤ 0

Proof state:
case intro
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
g_surj : Surjective g
⊢ ∃ f, Continuous f ∧ Surjective f

Tactic:
have s_closed : IsClosed s := by
  refine isClosed_iff_clusterPt.mpr fun x hx ↦ ?_
  have L : Tendsto (fun n : ℕ => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0) := by
    have : Tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) atTop (𝓝 (2 * 0)) :=
      (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).const_mul _
    rw [mul_zero] at this
    exact
      squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this
  refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_closedBall)
    (fun n => isBounded_closedBall) (fun N ↦ ?_) L
  obtain ⟨y, hxy, ys⟩ : ∃ y, y ∈ ball x ((1 / 2) ^ N) ∩ s :=
    clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))
  have E :
    ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) =
      ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n) := by
    refine iInter_congr fun n ↦ iInter_congr fun hn ↦ ?_
    have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn
    rw [this]
  rw [E]
  apply Nonempty.mono _ ys
  apply iInter_subset_iInter₂

Result:
case intro
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
g_surj : Surjective g
s_closed : IsClosed s
⊢ ∃ f, Continuous f ∧ Surjective f

Proof state:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
n : ℕ
hn : n ≤ N
⊢ closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)

Tactic:
have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn

Result:
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this✝ : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
L : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)
N : ℕ
y : ℕ → ℕ
hxy : y ∈ ball x ((1 / 2) ^ N)
ys : y ∈ s
n : ℕ
hn : n ≤ N
this : x n = y n
⊢ closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)

Proof state:
case intro
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
g_cont : Continuous g
y : α
n j : ℕ
hj : dist y (u j) < (1 / 2) ^ n
⊢ ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n)

Tactic:
exact ⟨j, hj.le⟩

Result:
no goals

Proof state:
case inr
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
diff_pos : 0 < firstDiff ↑x ↑y
hn : firstDiff ↑x ↑y = n + 1
⊢ dist (g x) (g y) ≤ 4 * dist x y

Tactic:
rw [dist', dist_eq_of_ne hne', hn]

Result:
case inr
α : Type u_2
inst✝³ : MetricSpace α
inst✝² : CompleteSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : Nonempty α
this : MetricSpace (ℕ → ℕ) := metricSpaceNatNat
I0 : 0 < 1 / 2
I1 : 1 / 2 < 1
u : ℕ → α
hu : DenseRange u
s : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}
g : ↑s → α := fun x => Set.Nonempty.some ⋯
A : ∀ (x : ↑s) (n : ℕ), dist (g x) (u (↑x n)) ≤ (1 / 2) ^ n
y x : ↑s
hxy : dist x y < 1
hne : x ≠ y
hne' : ↑x ≠ ↑y
dist' : dist x y = dist ↑x ↑y
n : ℕ := firstDiff ↑x ↑y - 1
diff_pos : 0 < firstDiff ↑x ↑y
hn : firstDiff ↑x ↑y = n + 1
⊢ dist (g x) (g y) ≤ 4 * (1 / 2) ^ (n + 1)