lean_workbook_plus_24990

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Prove the given lemma: $x_n$ is a Cauchy sequence in $X$ if and only if for any $\epsilon > 0$ , there exists $N$ such that for all $n \geq N$ , $\rho(x_n, x_N) < \epsilon$ .

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_24990 (X : Type*) [MetricSpace X] (x : ℕ → X) :
  CauchySeq x ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (x n) (x N) < ε   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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