I'm studying real analysis, and I am wanting to clarify something in the proof of Bolzano-Weierstrass.
The theorem states, as I'm sure you know, that in any metric space, an infinite subset $E$ of a compact set $K$ has a limit point in $K$.
The first step in the proof is to assume no $K$ is a limit point of $E$. Then (this is the part I need help on), every $q \in K$ would have a neighborhood that contains at most one point of $E$.
From my understanding, a point is a limit point of a set if any neighborhood around that point intersects that set. So, if no $K$ is a limit point of $E$, then shouldn't any neighborhood of $q \in K$ not hit $E$ at all, rather than at at most one point?