tendsto_prod_top_iff — Mathlib

English

If F and F' converge uniformly on p to f and f' respectively, then the map i ↦ (F i, F' i) converges uniformly to (f, f') on the product of filters.

Русский

Если F сходится равномерно к f и F' к f', то пара функций сходится к паре (f,f').

LaTeX

$$TendstoUniformly F f p → TendstoUniformly F' f' p' → TendstoUniformly (fun i => Prod.map (F i) (F' i)) (Prod.map f f') (p ×ˢ p')$$

Lean4

/-- Uniform convergence to a constant function is equivalent to convergence in `p ×ˢ ⊤`. -/
theorem tendsto_prod_top_iff {c : β} : Tendsto ↿F (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformly F (fun _ => c) p :=
  by
  rw [tendstoUniformly_iff_tendstoUniformlyOnFilter]
  exact tendsto_prod_filter_iff

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