Let be a real number. If use the fact that is open to show that is not a limit point of It will then follow that if is a limit point of then proving that contains all its limit points and so is closed.
Another way is to note that R-(R-S)=S, that is S is the complement of an open set. It helps to think better, sometimes, to write X-A as A^c. You will see the DeMorgan laws quicker as well as other stuff.