Problem with Rudin's proof that $\operatorname{diam}\mkern2mu\overline{\mkern-2muE}=\operatorname{diam}\text{ }E$

Question

Here is a picture of Rudin's proof:

It's the red outlined part that I'm having problems with, because it seems to me that unless both $p$ and $q$ are limit points, points $p'$ and $q'$ might not exist.

Answer 1

Why not? By definition of closure, asserting that $p\in\overline E$ means that $p\in E$ or that $p$ is a limit point of $E$. In both cases, it is true that, for every $\varepsilon>0$, there is a $p\in E$ such that $d(p,p')<\varepsilon$.

Answer 2

The possibilities are that either $p$ or $q$ or both are in $E$ anyway. If they are not they are in $\bar E$ but not in $E$ and they are limit points of $E$.