Cauchy Sequence

tags:definition |topology

Cauchy Sequence

Statements

Definition (@Eck23, Definition 5.1). Let be a $(X,d)$ be a metric space. A sequence $\{x_i{\}}_{i\in \mathbb{N}}\subset X$ is said to be a Cauchy sequence if, for every $\epsilon >0$, there exists an integer $N>0$ such that if $n$ and $m$ are integers greater than or equal to $N$ then the inequality $d(x_n,x_m)<\epsilon$ holds.

Remark

A Cauchy sequence is a sequence where the terms of the sequence are getting closer and closer to each other.