I presume the OP had in the mind the definition that the closure of S is the union of S and the set of its limit points. In this case:
Denote by S' the closure of S. Then we wish to show that S' is closed. Suppose x is in the complement of S'. Then x is not in S and is not a limit point of S. So there is an open ball around x that doesn't intersect S. This open ball cannot contain any limit point of S since if y is inside it, then there is a smaller ball centered at y contained in the bigger - and so there is an open ball around y that doesn't intersect S, so y is not a limit point of S. It follows that the open ball around x does not intersect S'. Therefore the complement of S' is open; so S' is closed.