Convergent 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, let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $X$, and let $x\in X$.

(i) $(x_{m})$ converges to $x$ in $(X,d)$, written $x_{m}\to x$, if for every real number $\varepsilon>0$ there is $N\in\mathbb{N}$ with $d(x_{m},x)<\varepsilon$ for every $m\in\mathbb{N}$ with $m\ge N$.

(ii) $(x_{m})$ is convergent in $(X,d)$ if it converges to some point of $X$.