Heine–Cantor in $\mathbb{R}^n$

Theorem. A continuous $f:K\to\mathbb{R}^m$ on a compact $K\subseteq\mathbb{R}^n$ is uniformly continuous.

Proof. Assume not: $\exists\varepsilon>0$ with $\|x_k-y_k\|\to0$ but $\|f(x_k)-f(y_k)\|\ge\varepsilon$. Compactness gives a subsequence $x_{k_j}\to x$. Then $y_{k_j}\to x$ too. Continuity: $f(x_{k_j})\to f(x)$ and $f(y_{k_j})\to f(x)$, contradiction.

Conclusion. Heine–Cantor holds in $\mathbb{R}^n$.