tendsto_atTop_mono' — Mathlib

English

If f1 ≤ᶠ[l] f2 and Tendsto f1 l atTop, then Tendsto f2 l atTop.

Русский

Если f1 ≤ᶠ[l] f2 и Tendsto f1 l приTop, то Tendsto f2 l приTop.

LaTeX

$$$$\text{Tendsto}_{l}(f_1)\, \text{atTop} \Rightarrow (f_1 \le^\mathrm{F} f_2) \Rightarrow \text{Tendsto}_{l}(f_2)\, \text{atTop}. $$$$

Lean4

theorem tendsto_atTop_mono' [Preorder β] (l : Filter α) ⦃f₁ f₂ : α → β⦄ (h : f₁ ≤ᶠ[l] f₂) (h₁ : Tendsto f₁ l atTop) :
    Tendsto f₂ l atTop :=
  tendsto_atTop.2 fun b => by filter_upwards [tendsto_atTop.1 h₁ b, h] with x using le_trans

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