Proving that a pointset is $\Sigma^0_2$ iff it is a countable union of closed sets.

Question

I'm (independently) going through Moschovakis' text on DST, and I'm trying to solve problem 1B.4, which asks the reader to prove that for any pointset, $P$, $P\in \Sigma_2^0$ if and only if $P = \bigcup_{i=0}^\infty F_i$ where all $F_i$ are closed. For the forward direction, I've managed to demonstrate that $P$ is a projection over $\omega$ of the intersection of closed sets, but I'm unsure of how I would finish this directionof the proof. Am I approaching this the right way, if so what steps do I take next? Otherwise, how should I approach this problem?