cauchySeq_iff_tendsto — Mathlib

English

If u is CauchySeq and f is injective, then u ∘ f is CauchySeq.

Русский

Если u — КошиSeq, и f инъективен, то u ∘ f — КошиSeq.

LaTeX

$$$\\text{CauchySeq}(u) \\Rightarrow \\big( \\text{Injective}(f) \\Rightarrow \\text{CauchySeq}(u\\circ f) \\big).$$$

Lean4

theorem cauchySeq_iff_tendsto [Nonempty β] [SemilatticeSup β] {u : β → α} :
    CauchySeq u ↔ Tendsto (Prod.map u u) atTop (𝓤 α) :=
  cauchy_map_iff'.trans <| by simp only [prod_atTop_atTop_eq, Prod.map_def]

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