atTop_mul_const — Mathlib

English

The pure filter at a point x is disjoint from the bottom filter atBot.

Русский

Чистый фильтр при x несовместим с нижним фильтром atBot.

LaTeX

$$$\mathrm{Disjoint}(\mathrm{pure}(x), \mathrm{atBot})$$$

Lean4

/-- If a function `f` tends to infinity along a filter, then `f` multiplied by a positive
constant (on the right) also tends to infinity. For a version working in `ℕ` or `ℤ`, use
`Filter.Tendsto.atTop_mul_const'` instead. -/
theorem atTop_mul_const (hr : 0 < r) (hf : Tendsto f l atTop) : Tendsto (fun x => f x * r) l atTop :=
  (tendsto_mul_const_atTop_of_pos hr).2 hf

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