mul — Mathlib

English

For a map f: ι → G, Cauchy (map f 𝓕) is equivalent to nontrivial 𝓕 and Tendsto of (f p.2)/(f p.1).

Русский

Для отображения f: ι → G, Cauchy (map f 𝓕) эквивалентно не тривиальному 𝓕 и Tendsto от (f p.2)/(f p.1).

LaTeX

$$$Cauchy(\\mathrm{map}\\ f\\ 𝓕) \\iff 𝓕 \\neq 0 \\land \\mathrm{Tendsto}\\big(\\lambda p. (f\\,p.2)/(f\\,p.1)\\big) (𝓕 \\times 𝓕) (\\mathcal{N}(1))$$$

Lean4

@[to_additive]
theorem mul (hf : TendstoUniformlyOn f g l s) (hf' : TendstoUniformlyOn f' g' l s) :
    TendstoUniformlyOn (f * f') (g * g') l s := fun u hu =>
  ((uniformContinuous_mul.comp_tendstoUniformlyOn (hf.prodMk hf')) u hu).diag_of_prod

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