If $p$ is an element of $\overline E$ but not a limit point of $E$, then why is there a $p' \in E$ such that $d(p, p') < \varepsilon$?

Dylan answered my question in the question comments above.

In particular, if $p \in \overline{E}$ is not a limit point of $E$, then it has to be in $E$, and so letting $p = p'$, we have $d(p, p') = 0 < \epsilon$ for any $\epsilon > 0$.

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General Topology

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