Cauchy sequence

基本列

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A sequence x_n in a metric space is Cauchy if for every epsilon > 0 some N makes d(x_m,x_n) < epsilon whenever m,n >= N.

度量空间中的点列若任给正误差，充分靠后的任意两项距离都小于该误差，则称为基本列。

Example

xn=1/nx_n=1/n satisfies ∣xm−xn∣≤1/N|x_m-x_n|\le1/N for m,n≥Nm,n\ge N.

Usage and distinctions

The condition compares terms to one another, and supplies a limit in the space only when the space is complete.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in One Variable§2.2, equation (2.2.3), printed p.74 · Checked 2026-10-14

Translation sources

  1. 李东风（北京大学）— 金融时间序列分析备课笔记，附录EE.12.1 定义，基本列与完备距离空间 · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.