/-- **Tietze extension theorem** for real-valued bounded continuous maps, a version with a closed
embedding and bundled composition. If `e : C(X, Y)` is a closed embedding of a topological space
into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous function, then there exists
a bounded continuous function `g : Y →ᵇ ℝ` of the same norm such that `g ∘ e = f`. -/
theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X, Y))
    (he : ClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f := by
  /- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
    between the previous approximation and `f`. -/
  choose F hF_norm hF_dist using fun f : X →ᵇ ℝ => tietze_extension_step f e he
  set g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
  have g0 : g 0 = 0 := rfl
  have g_succ : ∀ n, g (n + 1) = g n + F (f - (g n).compContinuous e) := fun n =>
    Function.iterate_succ_apply' _ _ _
  have hgf : ∀ n, dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖ := by
    intro n
    induction' n with n ihn
    · simp [g0]
    · rw [g_succ n, add_compContinuous, ← dist_sub_right, add_sub_cancel', pow_succ, mul_assoc]
      refine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))
      rwa [← dist_eq_norm']
  have hg_dist : ∀ n, dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n := by
    intro n
    calc
      dist (g n) (g (n + 1)) = ‖F (f - (g n).compContinuous e)‖ := by
        rw [g_succ, dist_eq_norm', add_sub_cancel']
      _ ≤ ‖f - (g n).compContinuous e‖ / 3 := (hF_norm _)
      _ = 1 / 3 * dist ((g n).compContinuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]
      _ ≤ 1 / 3 * ((2 / 3) ^ n * ‖f‖) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))
      _ = 1 / 3 * ‖f‖ * (2 / 3) ^ n := by ac_rfl
  have hg_cau : CauchySeq g := cauchySeq_of_le_geometric _ _ (by norm_num1) hg_dist
  have :
    Tendsto (fun n => (g n).compContinuous e) atTop
      (𝓝 <| (limUnder atTop g).compContinuous e) :=
    ((continuous_compContinuous e).tendsto _).comp hg_cau.tendsto_limUnder
  have hge : (limUnder atTop g).compContinuous e = f := by
    refine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)
    refine' squeeze_zero (fun _ => dist_nonneg) hgf _
    rw [← zero_mul ‖f‖]
    refine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num1
  refine' ⟨limUnder atTop g, le_antisymm _ _, hge⟩
  · 

Proof state:
case refine'_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : ClosedEmbedding ⇑e
F : (X →ᵇ ℝ) → Y →ᵇ ℝ
hF_norm : ∀ (f : X →ᵇ ℝ), ‖F f‖ ≤ ‖f‖ / 3
hF_dist : ∀ (f : X →ᵇ ℝ), dist (compContinuous (F f) e) f ≤ 2 / 3 * ‖f‖
g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0
g0 : g 0 = 0
g_succ : ∀ (n : ℕ), g (n + 1) = g n + F (f - compContinuous (g n) e)
hgf : ∀ (n : ℕ), dist (compContinuous (g n) e) f ≤ (2 / 3) ^ n * ‖f‖
hg_dist : ∀ (n : ℕ), dist (g n) (g (n + 1)) ≤ 1 / 3 * ‖f‖ * (2 / 3) ^ n
hg_cau : CauchySeq g
this : Tendsto (fun n => compContinuous (g n) e) atTop (𝓝 (compContinuous (limUnder atTop g) e))
hge : compContinuous (limUnder atTop g) e = f
⊢ ‖limUnder atTop g‖ ≤ ‖f‖

Next tactic:
rw [← dist_zero_left, ← g0]