Note that . So, they're clearly contracting. Also, We know that is closed and open, thus is the intersection of two closed sets, thus closed. Lastly, note that and so

Which Cantor intersection theorem? The one that deals with complete spaces or the one that deals with compact spaces? Noting the insistence on bounded subspaces I assume the latter.(b) Use the Cantor intersection theorem to deduce that if does not equal an empty set for every n,

Merely note from the above that given a finite subclass of that where and since by assumption this is non-empty we see that is a class of closed subsets of the compact space with the FIP. It follows that