1. The problem statement, all variables and given/known data Prove that: The union of a set U and the set of its limit points is the closure of U. 2. Relevant equations Definitions: Closure: The closure of U is the smallest closed set that contains U. Limit points: if z is a limit point in U, then any open circle around z intersects U at points other than z. 3. The attempt at a solution My attempt was to prove that: The Closure is contained in the union of U and L(U), and The union of L(U) and U contains it's closure. However my instructor did something that I was never able to understand. He first tried to prove that the union of U and L(U) is closed. To do so, he tries to prove it's complement is open. To prove its complement is open, he used contradiction: This is where I really got lost. I think this is what he said: Suppose it's complement of U is closed, then you can pick a point z (like on the boundary) on the complement of U, such that the any open disk around z will have to intersect U. Now suppose the complement of L(U) is closed, then you can pick a point w (once again on the boundary of this set) that is in a open ball of z, and then any disk around w will have to intersect L(U). But if any of w's open disk must intersect L(U), then w is defined to be the limit point of L(U). This is a contradiction, as we let L(U) to be the set of all of its limit points, yet w is limit point of L(U) that is not in L(U). The question is: While this does prove that the complement of L(U) is open, how does it prove the complement of the union of L(U) and U is open? Of course this does not even finish the proof. In the second part of the proof the instructor attempted to prove that one of these two statements: -The Closure is contained in the union of U and L(U), and -The union of L(U) and U contains it's closure. I listed them both because I can't remember which one he was trying to prove. Any suggestions for the second part? Thank you for reading this long thread.