Baby Rudin Theorem 2.27c I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:
If $X$ is a metric space and $E\subset X$, then $\overline{E}\subset F$ for every closed set $F\subset X$ such that $E\subset F$. Note: $\overline{E}$ denotes the closure of $E$; in other words, $\overline{E} = E \cup E'$, where $E'$ is the set of limit points of $E$.
Proof: If $F$ is closed and $F \supset E$, then $F\supset F'$, hence $F\supset E'$. Thus $F \supset \overline{E}$.
What I'm confused about is how we know $F \supset E'$ from the previous facts?

If $x$ is a limit point of $E$ then $x = \lim x_n$ for some sequence $x_n \in E \setminus \{x\}$. If $E \subseteq F$ then $x_n \in F \setminus \{x\}$ so we can also say that $x$ is a limit point of $F$. Therefore
$$ E' \subseteq F' \subseteq F. $$
