comp_cauchySeq — Mathlib

English

If f is uniformly continuous and u is CauchySeq, then f ∘ u is CauchySeq.

Русский

Если f непрерывна по однородной равномерности, то f ∘ u — КошиSeq.

LaTeX

$$$\\text{UniformContinuous}(f) \\Rightarrow (\\text{CauchySeq}(u) \\Rightarrow \\text{CauchySeq}(\\,f\\circ u\\,)).$$$

Lean4

theorem comp_cauchySeq {γ} [UniformSpace β] [Preorder γ] {f : α → β} (hf : UniformContinuous f) {u : γ → α}
    (hu : CauchySeq u) : CauchySeq (f ∘ u) :=
  hu.map hf

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