Principles of Mathematical Analysis - Ch2 Basic Topology
Here I am reading Rudin's Book(3rd).
I encountered some problems in Theorem 2.20, page 32.
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?