Hey, I was just wondering if anyone could tell me if this was about right.

Let E' be the set of limit points of E. Then E' is closed.

Proof:

Take an arbitrary limit point of E', say y.

Then, for any real > 0, there exists a point p in E' such that .

But, for any p E', there exists a real , with such that, for some point q in E,

But then, too, so y must be a limit point of E', and E' is closed by definition.