Definitions are your friends when it comes to proofs like this.
$V$ is a metric space...what does that tell you about $\;X \subseteq V\,$?
Read your question again (from your text or problem sheet): I suspect that you are also given that $X$ is closed (and $V$ compact), and/or that $X$ is compact. Otherwise, I'm afraid, you face a losing battle.
What do you need to know whether a set is compact?
With that, what does it mean to know that $\;f:X \to Y\;$ is continuous?
What does this imply in terms of the properties of the image of $f$ in $Y\,$? (Image: the set $f(X) \subset Y$). What do you need to know about $X$ and $f$ to conclude that $f(X)$ it is compact in $Y$?
If you work through these questions, unpack the definitions, use some handy theorems, and answer the questions with these "tools", you'll pretty much have your proof.