English
Let e be an order isomorphism. Then Tendsto (λx, e(f(x))) toBot iff Tendsto f toBot.
Русский
Пусть e — порядок-изоморфизм. Тогда Tendsto (λx, e(f(x))) к нижнему пределу эквивалентно Tendsto f к нижнему пределу.
LaTeX
$$$\\forall l:\\\\Filter \\gamma, \\forall f:\\\\gamma \\to \\alpha,\\; \\operatorname{Tendsto}(\\lambda x. e(f(x)))\,l\\,\\operatorname{atBot} \\iff \\operatorname{Tendsto} f\,l\\,\\operatorname{atBot}$$$
Lean4
/-- Infinitely many natural numbers are equal to `d` mod `n`. -/
theorem frequently_modEq {n : ℕ} (h : n ≠ 0) (d : ℕ) : ∃ᶠ m in atTop, m ≡ d [MOD n] :=
((tendsto_add_atTop_nat d).comp (tendsto_id.nsmul_atTop h.bot_lt)).frequently <|
Frequently.of_forall fun m => by simp [Nat.modEq_iff_dvd]
