Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

First are there 2 cases to consider? (1) A is a proper subset of Y (2) $\displaystyle A=Y\cap U$ or does this not matter?

Anyways, all I have is this:

Since A is closed in Y, $\displaystyle Y-A$ is open in Y. Likewise, $\displaystyle X-Y$ is open in X.

I am not sure what to do next.