1. The problem statement, all variables and given/known data Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. 2. Relevant equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists a q in X and an M>0 s.t. d(p,q)<M for all p in E. 3. The attempt at a solution Proof (so far): Let E be bounded. Then there exists M/2>0 and x in X s.t. d(p,x)<M/2 for all p in E. Now, take arbitrary p,q in E and observe that: d(p,q) ≤ d(p,x)+d(x,q) < M/2+M/2= M Thus, d(p,q)<M for all p,q in E. Now, I'm getting hung up on the second part of the proof, but I feel as if it shouldn't be hard. I think I've just been staring at this for too long at this point. Any advice as to how I ought to start going in the opposite direction would be greatly appreciated. I considered trying to show that E is compact (and would therefore be closed and, more importantly, bounded), but I'm not sure that's the best route or if it's awfully easy to do. Thanks!