Every Sequence in a Compact Subset of a Metric Space Has a Cluster Point There

Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_d)$, and let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ with $x_m\in K$ for every $m\in\mathbb{N}$.

Then there exists $x\in K$ that is a cluster point of $(x_m)_{m\in\mathbb{N}}$ in $(X,d)$.