Originally Posted by

**Math Major** 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 $\displaystyle \epsilon $ > 0, there exists a point p in E' such that $\displaystyle p \in N_{\epsilon}(y) $.

But, for any p $\displaystyle \in$ E', there exists a real $\displaystyle \delta $, with $\displaystyle 0 < \delta < \epsilon $ such that, for some point q in E, $\displaystyle q \in N_{\delta}(p) $

But then, $\displaystyle q \in N_{\epsilon}(y) $ too, so y must be a limit point of E', and E' is closed by definition.