TITLE: Prove that a continuous image of a closed subset of a compact space is a closed subset
QUESTION [5 upvotes]: Suppose $f$ is a continuous mapping from a compact metric space $X$ into a metric space $Y$. Prove that if $F$ is a closed subset of $X$, then $f[F]$ is a closed subset of $Y$.

Here is my idea for the proof: The continuous image of a connected space is connected. Use the intermediate value theorem to show that the image of every continuous real-valued function is an interval, and should return closed sets into closed sets.
Corrections are appreciated!

REPLY [8 votes]: You might want to proceed in the following manner:

Closed subsets of compact sets are compact, thus $F$ is compact;
Continuous image of a compact set is compact, hence $f(F)$ is compact in $Y$;
Compact sets in a metric space are closed, hence $f(F)$ is closed.