Cauchy Sequence in a Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space, and let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $X$.

(i) $(x_{m})$ is a Cauchy sequence in $(X,d)$ if for every real number $\varepsilon>0$ there is $N\in\mathbb{N}$ with $d(x_{m},x_{\ell})<\varepsilon$ for all $m,\ell\in\mathbb{N}$ with $m\ge N$ and $\ell\ge N$.