smul_isLittleO — Mathlib

English

For any c ≠ 0, (f =O_l (λ x => c • f' x)) is equivalent to (f =O_l f').

Русский

Для любого c ≠ 0: f =O_l (λ x, c • f'(x)) эквивалентно f =O_l f'.

LaTeX

$$$ (f =O[l] (\lambda x => c • f' x)) \iff f =O[l] f'. $$$

Lean4

theorem smul_isLittleO (h₁ : k₁ =O[l] k₂) (h₂ : f' =o[l] g') : (fun x => k₁ x • f' x) =o[l] fun x => k₂ x • g' x :=
  by
  simp only [IsLittleO_def] at *
  intro c cpos
  rcases h₁.exists_pos with ⟨c', c'pos, hc'⟩
  exact (hc'.smul (h₂ (div_pos cpos c'pos))).congr_const (mul_div_cancel₀ _ (ne_of_gt c'pos))

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