Continuous functions map compact sets to closed and bounded sets in R^k

Let $(X,d)$ be a metric space , let $K\subseteq X$ be compact , and let $f:K\to\mathbb{R}^k$ be continuous .

Proposition: The image $f(K)\subseteq\mathbb{R}^k$ is closed and bounded .

This is the Euclidean specialization of “continuous image of compact is compact ” plus the Heine–Borel characterization of compact sets in $\mathbb{R}^k$.

Proof sketch: Since $f$ is continuous and $K$ is compact, $f(K)$ is compact in $\mathbb{R}^k$. By Heine–Borel, compact subsets of $\mathbb{R}^k$ are closed and bounded. Therefore $f(K)$ is closed and bounded.