Sequentially Compact Subset of a Metric Space

Let $(X,d)$ be a metric space, and let $K\subseteq X$.

We say that $K$ is sequentially compact in $(X,d)$ if for every sequence $(x_m)_{m\in\mathbb{N}}$ in $X$ with $x_m\in K$ for every $m\in\mathbb{N}$, there exist a point $x\in K$ and a strictly increasing sequence $(n_k)_{k\in\mathbb{N}}$ in $\mathbb{N}$, in the sense of Subsequence of a Sequence in a Set, such that the subsequence $(x_{n_k})_{k\in\mathbb{N}}$ converges to $x$ in $(X,d)$.