Compact space and Hausdorff space
Combine the following facts:
1) A closed subspace of a compact space is compact.
2) A continuous map always maps compact spaces onto compact spaces.
3) Compact subspaces of Hausdorff spaces are closed.
If $f:X\to Y$ is continuous and $K\subseteq X$ is compact then $f(K)$ is compact.
Now use that closed subsets of compact sets are compacts and that compact subsets of Hausdorff spaces are closed.