How are you defining "closed set"? In Royden, for example, he defines a closed set as a set that equals its closure (or, a set that equals its limit points). The first proposition after defining "closed set" is that the closure of the closure of a set is equal to the closure of the set. (The operation of obtaining the closure of a set is idempotent.) It seems to me that this is the proposition you're trying to prove; is this correct?