# Difference between revisions of "Cauchy sequence"

Let ${\displaystyle (X,d)}$ be a metric space. Then a sequence ${\displaystyle x_{1},x_{2},\ldots }$ of elements in X is a Cauchy sequence if for any real number ${\displaystyle \epsilon >0}$ there exists a positive integer ${\displaystyle N(\epsilon )}$, dependent on ${\displaystyle \epsilon }$, such that ${\displaystyle d(x_{n},x_{m})<\epsilon }$ for all ${\displaystyle m,n>N(\epsilon )}$. In limit notation this is written as ${\displaystyle \mathop {\lim } _{n,m\rightarrow \infty }d(x_{m},x_{n})=0}$.