Criterion for closed

I have this theorem.

If $X$ is a metric space and $Y \subset X$ . $D \subset Y$ is closed in $Y$ iff $D = C \cap Y$ for some closed set $C$ in $X$ .

(I have proved a very similar theorem for open sets.)

However I cant seem to make any progress for the above theorem. (In either direction)

Going forward, we have $D$ is closed.

So $Y-D$ is open in $Y$ .

By using similar theorem for open sets.

$Y-D = E \cap Y$ for some open set $E$ in $X$ .

But trying to get back to $D$ by taking complements, I get

$D = X-E \cup X-Y$

Which doesn't seem to help me at all.

Also using $D$ has all its limit points doesn't give me a lot to work with either.

This leads me to believe that the theorem is not true.

Is there something I am not considering? Even going backwards, I can't figure it out.

Thank you for your help.