Set of limit points is always closed

**Problem:** Let be the set of all limit points of . Prove that is closed.

I think I can do it if the set has the least-upper-bound property.

**Attempt: **Let be a limit point of . (If the set of is finite or empty then is closed.) Then for every there exists a point such that . Let be the set of points of such that . This is bounded above by , which is the lub of . Hence and by the least-upper-bound property. Since was arbitrary, this proves that is closed.

Firstly, does this sound okay? If so, how could I prove this without using the least-upper-bound property?

I don't have anybody to look at my work, so I may be making huge fallacies or assumptions, and I'd greatly appreciate any input on this. Thanks.