Sequentially Compact Subset of 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, and let $K\subseteq X$.

¶ $K$ 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 are a subsequence $(x_{\sigma(k)})_{k\in\mathbb{N}}$ of $(x_{m})$ and a point $x\in K$ such that $(x_{\sigma(k)})_{k\in\mathbb{N}}$ converges to $x$ in $(X,d)$.