Thread: Principles of Mathematical Analysis - Ch2 Basic Topology

Hi,

Here I am reading Rudin's Book(3rd).
I encountered some problems in Theorem 2.20, page 32.

Theorem 2.20
If p is a limit point of a set E, then every neighborhood of p contains infinitely many points of E.

By intutition or imagination, I can understand the theorem, but I just don't understand why it says,

"The neighborhood Nr(p) contains no point q of E such that p=/=q (not equal), so that p is not a limit point of E." in the proof.

I think it would be that Nr(p) contains some q when r is in some range, but in other range of r, there is no q of E such that p=/=q. In other words, not every neighborhood of p contains a q of E, such that p=/=q.

Is there anyone who has the book on hands?

So he has supposed there is only a finite number of points in a neighborhood around a limit point of the set, and basically taken a smaller neighborhood such that those points aren't there anymore. Since we don't know a priori whether or not p is an element of E, the statement you are questioning is simply stating we exhibit a neighborhood of p with no elements of E other than possibly p, which contradicts the definition of a limit point.

If there are only a finite number of points of E in a neighborhood of p, the there is a closest such point. Take to be half that distance.

Originally Posted by siclar

So he has supposed there is only a finite number of points in a neighborhood around a limit point of the set, andbasically taken a smaller neighborhood such that those points aren't there anymore. Since we don't know a priori whether or not p is an element of E, the statement you are questioning is simply stating we exhibit a neighborhood of p with no elements of E other than possibly p, which contradicts the definition of a limit point.

So he took another r' smaller than the min(r)? Because by the text I couldn't tell. As for the statement, it was obviously volative of the definition of a limit point. I have no idea why I made it I. I guess I was just so confused by the r.

Thank you very much! I've known what my problem is!

Originally Posted by HallsofIvy

If there are only a finite number of points of E in a neighborhood of p, the there is a closest such point. Take to be half that distance.

Thank you very much. I got it!