1. The problem statement, all variables and given/known data So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything to support this. Where does this come from? 2. Relevant equations 3. The attempt at a solution I keep thinking that maybe it can be approached by the set being totally bounded since it is compact. So it seems for a given ε can cover the subset in open balls of radius ε. Then the centers of these balls is a finite set, then somehow choosing a corresponding convex set might let you represent any of the elements of the original compact set as the sum of elements of these new sets. But I don't know, I just don't have any basis from which to approach this.