atBot_mul_const' — Mathlib

English

A variant of the previous with explicit negation kept on the right, applicable to ring settings with Archimedean property.

Русский

Вариант предыдущего с явным отрицанием по праву, применимый к кольцам с Архимедовым свойством.

LaTeX

$$$\forall r<0\;\mathrm{Tendsto}\ f\ l\ \mathrm{atTop} \Rightarrow \mathrm{Tendsto}(x \mapsto f(x) \cdot r)\ l\ atBot$$$

Lean4

/-- See also `Filter.Tendsto.atBot_mul_const` for a version of this lemma for
`LinearOrderedField`s which does not require the `Archimedean` assumption. -/
theorem atBot_mul_const' (hr : 0 < r) (hf : Tendsto f l atBot) : Tendsto (fun x => f x * r) l atBot :=
  by
  simp only [← tendsto_neg_atTop_iff, ← neg_mul] at hf ⊢
  exact hf.atTop_mul_const' hr

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