$E$ is dense in $X$ if and only if every $x\in X$ is a limit point of $E$.

Question

My book makes the following definitions:

Let $X$ be a normed linear space.

Let $E\subseteq X$. Then $x\in X$ is a limit point of $E$ if there exists $x_n\in E$ with all $x_n\neq x$ such that $x_n\to x$.

$\overline E=\cap\left\{F:F\text{ is closed and }E\subseteq F\right\}$.

A subset $E\subseteq X$ is dense in $X$ if $\overline E=X$.

Later, it asks me to prove the following claim:

$E$ is dense in $X$ if and only if every $x\in X$ is a limit point of $E$.

I have a problem with that claim: if $x\in E$ is isolated, then it cannot be a limit point of $E$.

Am I overlooking something?

Answer 1

Suppose that $x\in E$ is an isolated point of $E$. Then there exists some radius $\epsilon$ so that $B(x,\epsilon)\cap E=\{x\}$.

Let $y\in B(x,\epsilon)\setminus\{x\}$, then there is some $\epsilon'$ so that $B(y,\epsilon')\subseteq B(x,\epsilon)$ and $x\not\in B(y,\epsilon')$.

Then, $y$ is not a limit point of $E$ since $B(y,\epsilon')$ contains no points of $E$. Hence, $E$ is not dense since any sequence of points in $E$ must remain at least $\epsilon'$ away from $y$ (and cannot get arbitrarily close to $y$).