Theorem: Given a metric space [itex]\left(X,d\right)[/itex], the set of all limit points of a subset [itex]E\subset X[/itex], denoted [itex]E'[/itex] is a closed set.
I have an Analysis Exam tomorrow and have been studying for quite awhile and last week, my professor gave us a list of Theorems to know the proofs of. This is one of them. I perused my notes and could not find the proof. I also searched through three or four Analysis texts and could not find a good proof. It doesn't seem like it would be hard to prove, but I can't think of a really rigorous one.
I have the following ideas:
- Show that [itex](E')^c[/itex] is open.
- I let there be a [itex]p \in (E')^c[/itex] and a [itex]q\in E'[/itex]
- From there I thought that possibly considering the two cases of [itex]q \in E[/itex] and [itex]q \notin E[/itex].
- From there I am stuck. I think I might need to use something about an open neighborhood and use distances between points, but I am not sure.
I really appreciate any helpful clues or hints. I am just looking to be pushed in a good direction, not for the solution to be given out. I would ask my professor, but I am not on campus today and he doesn't have office hours today.