Basic Topology Proof: y in E Closure if E is Closed

[bedi]

Homework Statement

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

Homework Equations

E closure = E' $\cup$ E where E' is the set of all limit points of E.

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' $\subset$ 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?

 

[clamtrox]

Shouldn't you be proving that if y=sup(E), then y ∈ E closure?