isCauSeq_iff_cauchySeq — Mathlib

English

If a normed field is such that CauSeq is complete, then the space is complete as a metric space by converging Cauchy sequences to limits.

Русский

Если CauSeq полно, то пространство полностью по метрике через предельные пределы последовательностей Коши.

LaTeX

$$$ \text{CompleteSpace } \beta \;\text{ follows from CauSeq.IsComplete }\beta \text{ via } \text{complete_of_cauchySeq_tendsto}. $$$

Lean4

/-- In a normed field, `CauSeq` coincides with the usual notion of Cauchy sequences. -/
theorem isCauSeq_iff_cauchySeq {α : Type u} [NormedField α] {u : ℕ → α} : IsCauSeq norm u ↔ CauchySeq u :=
  ⟨fun h => CauSeq.cauchySeq ⟨u, h⟩, fun h => h.isCauSeq⟩
    -- see Note [lower instance priority]

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