/-- Any nonempty complete second countable metric space is the continuous image of the
fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
`exists_nat_nat_continuous_surjective_of_polishSpace`. -/
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 := by
  /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
    this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
    Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
    balls `closedBall (u xₙ) (1/2^n)` have a nonempty intersection. This set is closed,
    and we define `f x` there to be the unique point in the intersection.
    This function is continuous and surjective by design. -/
  letI : MetricSpace (ℕ → ℕ) := PiNat.metricSpaceNatNat
  have I0 : (0 : ℝ) < 1 / 2 := by norm_num
  have I1 : (1 / 2 : ℝ) < 1 := by norm_num
  rcases exists_dense_seq α with ⟨u, hu⟩
  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
  let g : s → α := fun x => x.2.some
  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 : _)
  have g_cont : Continuous g := by
    refine continuous_iff_continuousAt.2 fun y => ?_
    refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
    rcases eq_or_ne x y with (rfl | hne)
    · simp
    have hne' : x.1 ≠ y.1 := Subtype.coe_injective.ne hne
    have dist' : dist x y = dist x.1 y.1 := rfl
    let n := firstDiff x.1 y.1 - 1
    have diff_pos : 0 < firstDiff x.1 y.1 := by
      by_contra! h
      apply apply_firstDiff_ne hne'
      rw [le_zero_iff.1 h]
      apply apply_eq_of_dist_lt _ le_rfl
      rw [pow_zero]
      exact hxy
    have hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm
    rw [dist', dist_eq_of_ne hne', hn]
    have B : x.1 n = y.1 n := mem_cylinder_firstDiff x.1 y.1 n (Nat.pred_lt diff_pos.ne')
    calc
      dist (g x) (g y) ≤ dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) :=
        dist_triangle_right _ _ _
      _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := (add_le_add (A x n) (A y n))
      _ = 4 * (1 / 2) ^ (n + 1) := by ring
  have g_surj : Surjective g := fun y ↦ by
    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⟩
      

Proof state:
case intro
E : ℕ → Type u_1
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 | Set.Nonempty (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n))}
g : ↑s → α := fun x => Set.Nonempty.some (_ : ↑x ∈ s)
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)

Next tactic:
exact ⟨j, hj.le⟩