-- 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:
case intro.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
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
E : ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) = ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)
⊢ (⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)).Nonempty

Tactic:
apply Nonempty.mono _ ys

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
E : ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) = ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)
⊢ ⋂ n, closedBall (u (y n)) ((1 / 2) ^ n) ⊆ ⋂ n, ⋂ (_ : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n)

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 * (1 / 2) ^ (n + 1)

Tactic:
have B : x.1 n = y.1 n := mem_cylinder_firstDiff x.1 y.1 n (Nat.pred_lt diff_pos.ne')

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
B : ↑x n = ↑y n
⊢ dist (g x) (g y) ≤ 4 * (1 / 2) ^ (n + 1)

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 ⋯
⊢ ∃ f, Continuous f ∧ Surjective f

Tactic:
have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
  (mem_iInter.1 x.2.some_mem n :)

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
⊢ ∃ 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
⊢ ∃ f, Continuous f ∧ Surjective f

Tactic:
rcases exists_dense_seq α with ⟨u, hu⟩

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
⊢ ∃ 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
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
⊢ ∃ a, g a = y

Tactic:
refine ⟨⟨x, I⟩, ?_⟩

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
⊢ g ⟨x, I⟩ = y

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
⊢ g ⟨x, I⟩ = y

Tactic:
refine dist_le_zero.1 ?_

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
⊢ dist (g ⟨x, I⟩) y ≤ 0