Prove that the n-dimensional real projective space RP^n is compact

**Prove that the n-dimensional real projective space $\displaystyle \mathbb{RP}^n$ is compact.**

I wouldn't know where to start.

Further, I have another question.

I need to prove that R with its cofinite topology is compact and sequentially compact. Compact seems easy, I did it like this:

Let (U_n) be an open cover for R, no select one (non-empty) of these open sets, say U_k0. Now, R\U_k0 is obviously finite so we can write it like {a_1,...,a_n}. Now, every one of these elements is in an open set. So, U_k0 together with these sets is an open cover.

Is this correct?