Continuous function from a compact space to a Hausdorff space is a closed function
Let $C$ be a closed subset of $X$; you want to prove that $f(C)$ is closed in $Y$.
We use three basic facts about compact spaces:
- a closed subset of a compact space is compact;
- the image of a compact subset under a continuous function is compact;
- a compact subset of a Hausdorff space is closed.
Now the proof of your statement.
Since $X$ is compact, $C$ is compact as well; therefore $f(C)$ is compact. A compact subset of a Hausdorff space is closed. Hence $f(C)$ is closed.