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?

$F\supset F'$ because  $F $ is closed.   $F'\supset E'$ because  $F\supset E $, by assumption. Therefore $F\supset E' $.
