Real numbers/Cauchy sequence/Definition

Cauchy sequence

A real sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called a Cauchy sequence, if the following condition holds.

For every ${}\epsilon >0$, there exists an ${}n_{0}\in \mathbb {N}$, such that for all ${}n,m\geq n_{0}$, the estimate

${}\vert {x_{n}-x_{m}}\vert \leq \epsilon \,$

holds.