mul_const — Mathlib

English

For any a ∈ M, the map b ↦ a·b is continuous.

Русский

Для любого фиксированного a ∈ M отображение b ↦ a·b непрерывно.

LaTeX

$$$a \in M \Rightarrow \text{Continuous}(b \mapsto a b)$$$

Lean4

theorem mul_const [MulPosStrictMono 𝕜] (h : Tendsto f l (𝓝[>] c)) : Tendsto (fun a => f a * b) l (𝓝[>] (c * b)) :=
  tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ ((tendsto_nhds_of_tendsto_nhdsWithin h).mul_const b) <|
    (tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Ioi] at *; gcongr; assumption

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