I'm trying generalize this corollary for the case which the sequence of functions $\{ f_n \}_{n \in \mathbb{N}}$ are defined on a bounded domain (open and connected) $U \subset \mathbb{R}^m$ ($m \geq 2$). I would like to know how I can prove this or if I need put some restrictions on more to this statement be true.
Thanks in advance!
After do some research, I found this lecture notes, which pratically solves the problem as we can see by the proofs of the lemma and of the proposition $2$ of this notes. Applying the Arzela Ascoli theorem a countable number of times as it was applied on proposition $2$, we develop a countable collection of subsequences of our original sequence and remains apply a standard diagonal argument (the same argument which is used on the proof of Arzela Ascoli theorem) in order to ensure that the uniform limit of the sequence $(f_n)$ is differentiable, i.e., if $f_n \rightarrow f$, then $f \in \mathcal{C}^{\infty}(U)$.