Rudin's Theorem 2.20 states:
If $p$ is a limit point of a set $E$, then every neighborhood of $p$ contains infinitely many points of $E$.
He assumes for a contradiction that some neighborhood $N$ of $p$ contains only finitely many elements of $E$, and labels those elements (which he claims are distinct from $p$) of $N \cap E$ by $q_1, \ldots, q_n$.
Here's what I don't understand. How do we know, even after the assumption toward a contradiction, that any such $q_i$ exist? The empty set is still finite, of course, so $N$ may contain no points of $E$. It surely contains $p$ itself, but $p$ may not be an element of $E$ itself. Further, how can he assert that these points are distinct from $p$? The definition of a limit point says that any neighborhood contains one distinct $p$, but we might have $p \in E$ and that be the only point of $E$.
I might be misunderstanding. How can I resolve these issues?
Best Answer
The definition of a limit point $p\in X$ of a subset $E \subseteq X$ says that every neighborhood of $p$ contains at least one point $q\in E$ such that $p\neq q.$ So there is no issue.