X and Y are compact sets in R^n that are disjoint. Then there must be positive distance between the elements of these sets.
The Attempt at a Solution
since X and Y are compact , X X Y is compact. Then, for the distance function d(x in X, y in Y): R^n X R^n -> R, there is a maximum and a minimum.
I think this should be a sufficient proof, although I'm not really sure how exactly to show that X X Y is compact as well.