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?