Continuous image of compact

Proof. Let $K$ compact and $f:K\to Y$ continuous. Given a sequence $(f(x_n))$ in $f(K)$, extract a subsequence $x_{n_k}\to x\in K$. Then $f(x_{n_k})\to f(x)\in f(K)$.

Conclusion. $f(K)$ is compact; if $Y=\mathbb{R}$, extrema are attained.