theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
    (hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 := by
  -- Assume that f is not continuous at 0
  by_contra h
  -- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
  -- and reformulate non-continuity in terms of these bases
  rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩
  simp only [id.def] at bE
  have bE' : (𝓝 (0 : E)).HasBasis (fun x : ℕ => x ≠ 0) fun n : ℕ => (n : 𝕜)⁻¹ • b n := by
    refine' bE.1.to_hasBasis _ _
    · intro n _
      use n + 1
      simp only [Ne.def, Nat.succ_ne_zero, not_false_iff, Nat.cast_add, Nat.cast_one, true_and_iff]
      -- `b (n + 1) ⊆ b n` follows from `Antitone`.
      have h : b (n + 1) ⊆ b n := bE.2 (by simp)
      refine' _root_.trans _ h
      rintro y ⟨x, hx, hy⟩
      -- Since `b (n + 1)` is balanced `(n+1)⁻¹ b (n + 1) ⊆ b (n + 1)`
      rw [← hy]
      refine' (bE1 (n + 1)).2.smul_mem _ hx
      have h' : 0 < (n : ℝ) + 1 := n.cast_add_one_pos
      rw [norm_inv, ← Nat.cast_one, ← Nat.cast_add, IsROrC.norm_natCast, Nat.cast_add, Nat.cast_one,
        inv_le h' zero_lt_one]
      simp
    intro n hn
    -- The converse direction follows from continuity of the scalar multiplication
    have hcont : ContinuousAt (fun x : E => (n : 𝕜) • x) 0 :=
      (continuous_const_smul (n : 𝕜)).continuousAt
    simp only [ContinuousAt, map_zero, smul_zero] at hcont
    rw [bE.1.tendsto_left_iff] at hcont
    rcases hcont (b n) (bE1 n).1 with ⟨i, _, hi⟩
    refine' ⟨i, trivial, fun x hx => ⟨(n : 𝕜) • x, hi hx, _⟩⟩
    simp [← mul_smul, hn]
  rw [ContinuousAt, map_zero, bE'.tendsto_iff (nhds_basis_balanced 𝕜' F)] at h
  push_neg at h
  rcases h with ⟨V, ⟨hV, -⟩, h⟩
  simp only [id.def, forall_true_left] at h
  -- There exists `u : ℕ → E` such that for all `n : ℕ` we have `u n ∈ n⁻¹ • b n` and `f (u n) ∉ V`
  choose! u hu hu' using h
  -- The sequence `(fun n ↦ n • u n)` converges to `0`
  have h_tendsto : Tendsto (fun n : ℕ => (n : 𝕜) • u n) atTop (𝓝 (0 : E)) := by
    apply bE.tendsto
    intro n
    by_cases h : n = 0
    · rw [h, Nat.cast_zero, zero_smul]
      refine' mem_of_mem_nhds (bE.1.mem_of_mem <| by triv)
    rcases hu n h with ⟨y, hy, hu1⟩
    convert hy
    rw [← hu1, ← mul_smul]
    simp only [h, mul_inv_cancel, Ne.def, Nat.cast_eq_zero, not_false_iff, one_smul]
  -- The image `(fun n ↦ n • u n)` is von Neumann bounded:
  have h_bounded : IsVonNBounded 𝕜 (Set.range fun n : ℕ => (n : 𝕜) • u n) :=
    h_tendsto.cauchySeq.totallyBounded_range.isVonNBounded 𝕜
  -- Since `range u` is bounded it absorbs `V`
  rcases hf _ h_bounded hV with ⟨r, hr, h'⟩
  cases' exists_nat_gt r with n hn
  -- We now find a contradiction between `f (u n) ∉ V` and the absorbing property
  have h1 : r ≤ ‖(n : 𝕜')‖ := by
    rw [IsROrC.norm_natCast]
    exact hn.le
  have hn' : 0 < ‖(n : 𝕜')‖ := lt_of_lt_of_le hr h1
  rw [norm_pos_iff, Ne.def, Nat.cast_eq_zero] at hn'
  have h'' : f (u n) ∈ V := by
    simp only [Set.image_subset_iff] at h'
    specialize h' (n : 𝕜') h1 (Set.mem_range_self n)
    simp only [Set.mem_preimage, LinearMap.map_smulₛₗ, map_natCast] at h'
    rcases h' with ⟨y, hy, h'⟩
    

Proof state:
case intro.intro
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : UniformSpace F
inst✝⁶ : FirstCountableTopology E
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsROrC 𝕜'
inst✝¹ : Module 𝕜' F
inst✝ : ContinuousSMul 𝕜' F
σ : 𝕜 →+* 𝕜'
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : HasAntitoneBasis (𝓝 0) fun i => b i
bE' : HasBasis (𝓝 0) (fun x => x ≠ 0) fun n => (↑n)⁻¹ • b n
V : Set F
hV : V ∈ 𝓝 0
u : ℕ → E
hu : ∀ (ia : ℕ), ia ≠ 0 → u ia ∈ (↑ia)⁻¹ • b ia
hu' : ∀ (ia : ℕ), ia ≠ 0 → f (u ia) ∉ V
h_tendsto : Tendsto (fun n => ↑n • u n) atTop (𝓝 0)
h_bounded : IsVonNBounded 𝕜 (Set.range fun n => ↑n • u n)
r : ℝ
hr : 0 < r
n : ℕ
hn : r < ↑n
h1 : r ≤ ‖↑n‖
hn' : ¬n = 0
y : F
hy : y ∈ V
h' : (fun x => ↑n • x) y = ↑n • f (u n)
⊢ f (u n) ∈ V

Next tactic:
apply_fun fun y : F => (n : 𝕜')⁻¹ • y at h'