1. The problem statement, all variables and given/known data Prove that the union of a collection of indexed sets has finite diameter if the intersection of the collection is non-empty, and every set in the collection is bounded by a constant A. 3. The attempt at a solution The picture I have is if they all intersect (and assuming there are infinitely many sets), then all the sets must be 'localized' in some way so that there's some overlap. I'm not yet sure how to make this rigorous though.