isLittleO_sub_self_inv — Mathlib

English

Same as above, equality for supremum under monotonic u and nonzero m.

Русский

То же самое для верхнего роста: при монотонной u и m ≠ 0 имеем равенство.

LaTeX

$$$\operatorname{linearGrowthSup}(n \mapsto u(mn)) = m \cdot \operatorname{linearGrowthSup}(u)$$$

Lean4

/-- If `f : 𝕜 → E` is bounded in a punctured neighborhood of `a`, then `f(x) = o((x - a)⁻¹)` as
`x → a`, `x ≠ a`. -/
theorem isLittleO_sub_self_inv {𝕜 E : Type*} [NormedField 𝕜] [Norm E] {a : 𝕜} {f : 𝕜 → E}
    (h : IsBoundedUnder (· ≤ ·) (𝓝[≠] a) (norm ∘ f)) : f =o[𝓝[≠] a] fun x => (x - a)⁻¹ :=
  by
  refine (h.isBigO_const (one_ne_zero' ℝ)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr ?_)
  simp only [Function.comp_def, norm_inv]
  exact (tendsto_norm_sub_self_nhdsNE a).inv_tendsto_nhdsGT_zero

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