/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
      let δ := ‖d‖ * ‖y‖ / 4
      have δpos : 0 < δ := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by norm_num)
      have : a + d • y ∈ ball a ε := by
        simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]
      rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₁, z₁im, h₁⟩
      rcases (mem_image _ _ _).1 z₁im with ⟨x₁, hx₁, xz₁⟩
      rw [← xz₁] at h₁
      rw [mem_ball, dist_eq_norm, sub_zero] at hx₁
      have : a ∈ ball a ε := by
        simp only [mem_ball, dist_self]
        exact εpos
      rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₂, z₂im, h₂⟩
      rcases (mem_image _ _ _).1 z₂im with ⟨x₂, hx₂, xz₂⟩
      rw [← xz₂] at h₂
      rw [mem_ball, dist_eq_norm, sub_zero] at hx₂
      let x := x₁ - x₂
      have I : ‖f x - d • y‖ ≤ 2 * δ :=
        calc
          ‖f x - d • y‖ = ‖f x₁ - (a + d • y) - (f x₂ - a)‖ := by
            congr 1
            simp only [f.map_sub]
            abel
          _ ≤ ‖f x₁ - (a + d • y)‖ + ‖f x₂ - a‖ := (norm_sub_le _ _)
          _ ≤ δ + δ := by
            apply add_le_add
            · rw [← dist_eq_norm, dist_comm]
              exact le_of_lt h₁
            · rw [← dist_eq_norm, dist_comm]
              exact le_of_lt h₂
          _ = 2 * δ := (two_mul _).symm
      have J : ‖f (d⁻¹ • x) - y‖ ≤ 1 / 2 * ‖y‖ :=
        calc
          ‖f (d⁻¹ • x) - y‖ = ‖d⁻¹ • f x - (d⁻¹ * d) • y‖ := by
            rwa [f.map_smul _, inv_mul_cancel, one_smul]
          _ = ‖d⁻¹ • (f x - d • y)‖ := by rw [mul_smul, smul_sub]
          _ = ‖d‖⁻¹ * ‖f x - d • y‖ := by rw [norm_smul, norm_inv]
          _ ≤ ‖d‖⁻¹ * (2 * δ) := by
            apply mul_le_mul_of_nonneg_left I
            rw [inv_nonneg]
            exact norm_nonneg _
          _ = ‖d‖⁻¹ * ‖d‖ * ‖y‖ / 2 := by
            simp only
            ring
          _ = ‖y‖ / 2 := by
            rw [inv_mul_cancel, one_mul]
            simp [norm_eq_zero, hd]
          _ = 1 / 2 * ‖y‖ := by ring
      rw [← dist_eq_norm] at J
      

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
d : 𝕜
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this✝ : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : ‖x₁‖ < ↑n
xz₁ : f x₁ = z₁
this : a ∈ ball a ε
z₂ : F
z₂im : z₂ ∈ ⇑f '' ball 0 ↑n
x₂ : E
h₂ : dist a (f x₂) < δ
hx₂ : ‖x₂‖ < ↑n
xz₂ : f x₂ = z₂
x : E := x₁ - x₂
I : ‖f x - d • y‖ ≤ 2 * δ
J : dist (f (d⁻¹ • x)) y ≤ 1 / 2 * ‖y‖
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
have K : ‖d⁻¹ • x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖ :=
        calc
          ‖d⁻¹ • x‖ = ‖d‖⁻¹ * ‖x₁ - x₂‖ := by rw [norm_smul, norm_inv]
          _ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖ * (n + n) := by
            refine' mul_le_mul dinv _ (norm_nonneg _) _
            · exact le_trans (norm_sub_le _ _) (add_le_add (le_of_lt hx₁) (le_of_lt hx₂))
            · apply mul_nonneg (mul_nonneg _ (norm_nonneg _)) (norm_nonneg _)
              exact inv_nonneg.2 (le_of_lt (half_pos εpos))
          _ = (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖ := by ring