Compactness implies closedness

Compactness implies closedness: Let $(X,d)$ be a metric space and let $K\subseteq X$ be compact . Then $K$ is closed in $X$.

This is a key structural property: in metric spaces, compact sets behave like “closed and bounded” objects, and this is one half of that picture.

Proof sketch: Let $(x_n)$ be a sequence in $K$ converging in $X$ to some $x$. By sequential compactness of $K$, there is a subsequence $(x_{n_j})$ converging to some $y\in K$. But subsequences of a convergent sequence converge to the same limit, so $y=x$. Hence $x\in K$, proving $K$ is closed.