1. The problem statement, all variables and given/known data Let A,B be two disjoint, non-empty, compact subsets of a metric space (X,d). Show that there exists some r>0 such that d(a,b) > r for all a in A, b in B. Hint provided was: Assume the opposite, consider a sequence argument. 2. Relevant equations N/A 3. The attempt at a solution I've tried a few different characterizations of compactness, but neither one has led me anywhere particularly useful. I'm missing something such that the teacher's hint isn't helping me much. If I assume the opposite, Assume that for all r>0, there exists some pair (a,b) such that d(a,b) <= r I sense that I am supposed to reach a contradiction regarding the property that any sequence in a compact A or B should have a convergent subsequence. But how do I show that I would reach such a contradiction. Can anyone give me a push in the right direction?