tendsto_mul_zero_of_disjoint_cocompact_right

English

If f tends to 0 and the image under g is disjoint from cocompact, then x ↦ f(x) g(x) tends to 0.

Русский

Если f(x) → 0, а образ g(l) расстроен от кококомпакт, то f(x) g(x) → 0.

LaTeX

$$tendsto_mul_zero_of_disjoint_cocompact_right$$

Lean4

/-- Let `M` be a topological space with a continuous multiplication operation and a `0`.
Let `f : α → M` and `g : α → M` be functions. If `f` tends to zero on a filter `l`
and the image of `l` under `g` is disjoint from the cocompact filter on `M`, then
`fun x : α ↦ f x * g x` also tends to zero on `l`. -/
theorem tendsto_mul_zero_of_disjoint_cocompact_right {f g : α → M} {l : Filter α} (hf : Tendsto f l (𝓝 0))
    (hg : Disjoint (map g l) (cocompact M)) : Tendsto (fun x ↦ f x * g x) l (𝓝 0) :=
  tendsto_mul_nhds_zero_prod_of_disjoint_cocompact hg |>.comp (hf.prodMk tendsto_map)

Похожие утверждения