1. The problem statement, all variables and given/known data X and Y are compact sets in R^n that are disjoint. Then there must be positive distance between the elements of these sets. 2. Relevant equations 3. 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.