To Clarify A Question on Limits

Today I went to my advanced calculus professor's office hours. I didn't really have any questions but I knew that other people would be there so maybe one of them would ask a question I had failed to think of myself. Sadly this did not happen, however, one person did ask a question for which I was able to think of two different solutions.

Let be the set of limit points of a set Prove that is closed. The proof the professor gave assumed knowledge of convergent sequences. This is actually the simplest proof I have seen, so it will now be the one documented in this post.


My proof: Consider the set and a point So clearly is not a limit point of thus there is a neighborhood of which contains only finitely many points of Now if this neighborhood contained any point then there is a neighborhood of such that it is a subset of the neighborhood of But then since is a limit point of every one of it's neighborhoods contains infinitely many points of which contradicts the facts that the neighborhood of only has a finite number of points from . Therefore this neighborhood of is a subset of thus making it an open set. At this point we are done.

Now that wasn't too bad. I think this is the proof that Rudin was looking for since this problem was asked before the introduction of sequences. Just a side note, if you replace "finitely many" with "no" in the above proof then you will run into some issues. Just consider , then , so just choose and you see where the problem arises.