atTop_mul_one_eventuallyLE — Mathlib

English

For an order isomorphism e, Tendsto (λx, e(f(x))) toBot iff Tendsto f toBot.

Русский

Для порядка-изоморфизма e: Tendsto (λx, e(f(x))) к низу эквивалентно Tendsto f к низу.

LaTeX

$$$\\forall l f,\\; \\operatorname{Tendsto}(\\lambda x. e(f(x)))\,l\\operatorname{atBot} \\iff \\operatorname{Tendsto} f\,l\\operatorname{atBot}$$$

Lean4

@[to_additive]
theorem atTop_mul_one_eventuallyLE (hf : Tendsto f l atTop) (hg : 1 ≤ᶠ[l] g) : Tendsto (fun x => f x * g x) l atTop :=
  tendsto_atTop_mono' l (hg.mono fun _ => le_mul_of_one_le_right') hf

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