Let $\{f_n\}_{n}$ be a sequence of continuous real-valued functions defined on a closed and bounded interval of the real line, denoted $I$. Moreover, assume that the sequence is uniformly bounded and differentiable with uniformly bounded derivatives.
Then, by Arzela-Ascoli theorem, there exists a subsequence $K \subseteq \mathbb{N}$ such that $f_n$ converges uniformly to a Lipschitz continuous function $f$.
Is it possible to prove something on the convergence of the derivative of $f_n$. For instance something such as: $$ \{\dot{f_n}\} \text{ weakly converge to } \dot{f} \text{ in } L^1(I,\mathbb{R}^k) $$