Cluster Point of a Sequence in a Metric Space

Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ indexed by the natural numbers with the order $\le$, and let $x\in X$.

We say that $x$ is a cluster point of $(x_m)_{m\in\mathbb{N}}$ in $(X,d)$ if for every real number $\varepsilon>0$ and every $N\in\mathbb{N}$ there exists $m\in\mathbb{N}$ with $N\le m$ and