I have a sequence $(f_n)$ of $C^2$ functions defined on a bounded set $\overline{\Omega}\subseteq\mathbb{R}^n$, and there is a constant $L$, uniform in $n$, such that $|f_n(x)-f_n(y)|\leq L|x-y|$ and that $|f_n(x)|\leq L$ for all $x, y\in\Omega$. Then, by Arzela-Ascoli theorem, we know that there is a function $f$ such that $(f_n)$ converges to $f$ uniformly on $\overline{\Omega}$ (upto a subsequence), and thus $f$ is Lipschitz as well. By Rademacher theorem, we know that $f$ is differentiable almost everywhere.
My question is, can we say that $Df_n$ converges pointwisely to $Df$ almost everywhere?
Thanks.