1. The problem statement, all variables and given/known data Show that If S is totally bounded in ℂ, then the S closure is also totally bounded in ℂ. 2. Relevant equations 3. The attempt at a solution Assume S is totally bounded. then for very ε>0 there are finitely many discs (O=Union of finitely many discs) that covers S let x be a limit points of S that is in S closure. but not in S. hence x is in O (how can I show this?) So x is in O for all x in S closure. Hence S closure is totally bounded. Am I on the right track?