Basic topology proof

1. The problem statement, all variables and given/known data

Let E be a nonempty set of real numbers which is bounded above. Let y=sup E. Then y [itex]\in[/itex] E closure. Hence y [itex]\in[/itex] E if E is closed.


2. Relevant equations

E closure = E' [itex]\cup[/itex] E where E' is the set of all limit points of E.

3. The attempt at a solution

By the definition of closure, y is either in E' or E (maybe in both). If E is closed, E' [itex]\subset[/itex] E and we know that the union of a set and its subset is the set itself. Therefore E closure = E.

Is this a valid proof?