English
If a specializes to b in a topological monoid with continuous multiplication, then a^n specializes to b^n for every natural n.
Русский
Если a specializes to b в топологической моноиде с непрерывным умножением, то a^n специализируется на b^n для каждого натурального n.
LaTeX
$$$a \xrightarrow{\mathrm{Specializes}} b \Rightarrow \forall n \in \mathbb{N},\ a^n \xrightarrow{\mathrm{Specializes}} b^n$$$
Lean4
theorem const_mul [PosMulStrictMono 𝕜] (h : Tendsto f l (𝓝[>] c)) : Tendsto (fun a => b * f a) l (𝓝[>] (b * c)) :=
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ ((tendsto_nhds_of_tendsto_nhdsWithin h).const_mul b) <|
(tendsto_nhdsWithin_iff.mp h).2.mono fun _ _ => by rw [Set.mem_Ioi] at *; gcongr; assumption
