My attempt:
$\operatorname{Char}{F}=p>0$ then $F, E$ are finite, hence $F\cong \mathbb F_{p^n}, E\cong \mathbb F_{p^m}, n\mid m$
and we have extensions tower $\mathbb F_{p}\subset F\subset E$.
$E/\mathbb F_p$ is separable as the splitting field of $f(X)=X^{p^m}-X$ over $\mathbb F_p$.
From a theorem, given this extensions tower, $\mathbb F_{p}\subset F\subset E$,
if $E/\mathbb F_p$ is separable, then so is $E/F$, as requested $\square$
I didn't use the fact that $\operatorname{Char}{F}=p\nmid [E:F]$.
Does my proof still hold?
EDIT:
I'm aware of this solution: Every finite extension of a finite field is separable
However, $\operatorname{Char}{F}=p\nmid [E:F]$ is not mentioned, hence my issue is not solved.
Does pointing out $\operatorname{Char}{F}=p\nmid [E:F]$ has any significance in this problem?
Solution:
$\operatorname{Char}{F}=p\nmid [E:F]$ is a redundant detail for this problem, the statement holds for any $\operatorname{Char}{F}$.