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!