Convergent Sequence in a Metric Space

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

for every $m\in\mathbb{N}$ with $m\ge N$. In this case we write

and call $x$ the limit of $(x_m)$ in $(X,d)$.