Compact convergence and partial differentiability

Question

Suppose that $\{f_n\}_{n \in \mathbb{N}}$ is a family of sufficiently smooth functions, defined on a domain $D \subseteq \mathbb{R}^m$, where they converge to a function $f$ in the topology of compact convergence. Then, suppose that their partial derivatives $\left\lbrace\frac{\partial f_n}{\partial i}\right\rbrace_{n \in \mathbb{N}}$ also converge to a function $g$ uniformly on compacts on the domain $D$. Can we conclude that $\frac{\partial f}{\partial i} = g$?

Answer 1

For $\Bbb{R}^n$ you can take a look at a proof here in case you struglle to generalize Rudin's arguments.