theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
    (h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
  obtain ⟨c, c_pos, hc⟩ := h.exists_pos
  obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
  obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
  clear h hc z_mem
  cases' n with n
  · exact norm_eq_zero.mp (by
      -- porting note: the symmetric difference of the `simpa only` sets:
      -- added `Nat.zero_eq, zero_add, pow_one`
      -- removed `zero_pow', Ne.def, Nat.one_ne_zero, not_false_iff`
      simpa only [Nat.zero_eq, fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
        mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
  · refine' Or.elim (Classical.em (y = 0))
      (fun hy => by simpa only [hy] using p.map_zero) fun hy => _
    replace hy := norm_pos_iff.mpr hy
    refine' norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => _) (norm_nonneg _))
    have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
    obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
      (lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
    have h₁ : ‖k • y‖ < δ := by
      rw [norm_smul]
      exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
        mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
    have h₂ :=
      calc
        ‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
          -- porting note: now Lean wants `_root_.`
          simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
          --simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
        _ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
          -- porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
          simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
          -- porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
          ring
    have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
      inv_mul_cancel_right₀ h₀.ne.symm ε ▸
        mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
    calc
      ‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
        simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
          Finset.card_fin] using
          congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
      _ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
      _ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
        rw [← mul_assoc]
        

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : ContinuousMultilinearMap 𝕜 (fun i => E) F
ht : ∀ x ∈ t, ‖p fun x_1 => x‖ ≤ c * ‖‖x‖ ^ (Nat.succ n + 1)‖
hy : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
h₀ : 0 < c * ‖y‖ ^ (Nat.succ n + 1)
k : 𝕜
k_pos : 0 < ‖k‖
k_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (Nat.succ n + 1))⁻¹)
h₁ : ‖k • y‖ < δ
h₂ : ‖p fun x => k • y‖ ≤ ‖k‖ ^ Nat.succ n * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)))
h₃ : ‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)) < ε
⊢ ‖k⁻¹ ^ Nat.succ n‖ * ‖k‖ ^ Nat.succ n * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1))) =
    ‖(k⁻¹ * k) ^ Nat.succ n‖ * (‖k‖ * (c * ‖y‖ ^ (Nat.succ n + 1)))

Next tactic:
simp [norm_mul, mul_pow]