Cauchy Sequence in a Metric Space

Let $(X,d)$ be a metric space, and let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$. We say that $(x_m)$ is a Cauchy sequence in $(X,d)$ if for every real number $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that

for every $m,\ell\in\mathbb{N}$ with $m\ge N$ and $\ell\ge N$.