Claim (from Rudin 2.37): If E is an infinite subset of a compact set K, then E has a limit point in K.

Attempt: I follow the proof from the book, but have trouble understanding the first statement. 

It seems clear that proof by contradiction is easy to do. I can start (as the book does) by assuming E has no limit point in K.

The book now claims that for each q belonging to K, there is a neighborhood V\_q which contains at most 1 element of E.

I don't understand how this statement comes from q not being a limit point of E. 

I know the definition of a limit point of a set is that for **every** neighborhood of the point, there is **at least** one element belonging to said set. 

So shouldn't the book (after claiming there is no limit point in K) say that for every point in K, there is at least one neighborhood for each that does not contain an element of E?

Maybe someone can clarify? Thanks.