By Cantor's intersection theorem I know that a sequence of nonempty compact sets which are nested has nonempty intersection. But how can I use that to prove that arbitrary intersection of compact sets is compact?
In an finite dimensional metric space (such as the complex numbers), all compact sets are closed and bounded. Then, all of your compact sets are closed and therefore, their intersection is a closed set. Then, because the intersection is closed and contained in any of your compact sets, it is a compact set (This property can be used because metric spaces are, in particular, Hausdorff spaces).