tendsto_inv_smul_nhds_zero — Mathlib

English

For an indexed family f'' and g'' with a fixed index x, f'' =o[pure x] g'' if and only if f''(x) = 0.

Русский

Для фиксированного индекса x справедливо f'' =o[pure x] g'' тогда и только тогда, когда f''(x) = 0.

LaTeX

$$$ (f'' =o[\\text{pure }x] g'') \\iff f''(x) = 0 $$$

Lean4

theorem tendsto_inv_smul_nhds_zero [Module 𝕜 E'] [NormSMulClass 𝕜 E'] {f : α → E'} {g : α → 𝕜} {l : Filter α}
    (h : f =o[l] g) : Tendsto (fun x => (g x)⁻¹ • f x) l (𝓝 0) := by
  simpa only [div_eq_inv_mul, ← norm_inv, ← norm_smul, ← tendsto_zero_iff_norm_tendsto_zero] using
    h.norm_norm.tendsto_div_nhds_zero

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