Set of derivatives of a normal family of analytic functions is itself a normal family.

Set of derivatives of a normal family of analytic functions is itself a
normal family.

I'm working on a problem that is easily solvable if I can prove the
statement in the title. Here's what I've done so far:
Given $\mathscr{F}$ a normal family of analytic functions, let
$\mathscr{F}'=\{f':f\in \mathscr{F}\}$. Let $\{f_n'\}$ be a sequence in
$\mathscr{F}'$. Then the corresponding sequence $\{f_n\}$ (unique up to
constants) contains a subsequence that converges to some analytic $f$.
This implies that the subsequence of derivatives converges to $f'$, so
every sequence in $\mathscr{F}'$ has a subsequence that converges and
$\mathscr{F}'$ is normal.
Am I missing something here? It seems like a very strong statement that I
wouldn't necessarily expect to be true, but I can't think of a
counterexample either.