-- 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
g_surj : Surjective g
x : ℕ → ℕ
hx : ClusterPt x (𝓟 s)
⊢ Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)

Tactic:
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 _

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)
this : Tendsto (fun n => 2 * (1 / 2) ^ n) atTop (𝓝 (2 * 0))
⊢ Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0)

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
n : ℕ
⊢ 0 < (1 / 2) ^ n

Tactic:
simp

Result:
no goals

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
⊢ ∃ f, Continuous f ∧ Surjective f

Tactic:
have I1 : (1 / 2 : ℝ) < 1 := by norm_num

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
⊢ ∃ 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
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
h : firstDiff ↑x ↑y ≤ 0
⊢ ↑x 0 = ↑y 0

Tactic:
apply apply_eq_of_dist_lt _ le_rfl

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
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
h : firstDiff ↑x ↑y ≤ 0
⊢ dist ↑x ↑y < (1 / 2) ^ 0

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
⊢ dist (g x) (g y) ≤ 4 * dist x y

Tactic:
let n := firstDiff x.1 y.1 - 1

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
⊢ dist (g x) (g y) ≤ 4 * dist x 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
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
h : firstDiff ↑x ↑y ≤ 0
⊢ ↑x (firstDiff ↑x ↑y) = ↑y (firstDiff ↑x ↑y)

Tactic:
rw [Nat.le_zero.1 h]

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
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
h : firstDiff ↑x ↑y ≤ 0
⊢ ↑x 0 = ↑y 0

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
⊢ ∃ a, g a = y

Tactic:
have : ∀ n : ℕ, ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n) := fun n ↦ by
  rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
  exact ⟨j, hj.le⟩

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
this : ∀ (n : ℕ), ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n)
⊢ ∃ a, g a = y