comp_injective — 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

$$$\\text{CauchySeq}(u) \\Rightarrow \\text{CauchySeq}(u\\circ f)$ при инъекции f.$$

Lean4

theorem comp_injective [SemilatticeSup β] [NoMaxOrder β] [Nonempty β] {u : ℕ → α} (hu : CauchySeq u) {f : β → ℕ}
    (hf : Injective f) : CauchySeq (u ∘ f) :=
  hu.comp_tendsto <| Nat.cofinite_eq_atTop ▸ hf.tendsto_cofinite.mono_left atTop_le_cofinite

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