Heine–Cantor theorem

Theorem. Continuous $f:K\to Y$ with $K$ compact is uniformly continuous.

Proof. Suppose not: $\exists\varepsilon>0$ with $x_n,y_n\in K$, $d(x_n,y_n)\to0$ but $d(f(x_n),f(y_n))\ge\varepsilon$. Compactness of $K$ gives a subsequence $x_{n_k}\to x$; then $y_{n_k}\to x$. Continuity: $f(x_{n_k}),f(y_{n_k})\to f(x)$, contradiction.

Conclusion. Heine–Cantor holds in general metric spaces.