Let $(f_n)$ be a sequence of continuously differentiable functions on an open set $U \subseteq \mathbb{R}^2$. Suppose the three sequences $f_n, \frac{\partial f_n}{\partial x}$ and $\frac{\partial f_n}{\partial y}$ converges compactly (i.e. converges uniformly on every compact subset of domain) to functions $g, h_1$ and $h_2$, respectively. Prove that $\frac{\partial g}{\partial x} = h_1$ and $\frac{\partial g}{\partial y} = h_2$.
Rudin (Theorem 7.17) states that the pointwise convergence of $f_n$ and the uniform convergence of $f_n$'s partial derivatives are enough to conclude this (if I am not reading that wrong), so it is enough to show that $\partial_x f_n \to h_1$ uniformly.
So for every $x \in U$, by openness we can find a open ball $\overline{B(x, R)}$ such that it falls in $U$, so we see that by compact convergence, $\partial_x f_n \to h_1$ uniformly on this compact set.
The problem remains is that, how should I extend it to the whole $U$?
To prove that $\frac {\partial g} {\partial x}=h_1$ you take any point in $U$ and work in a neighborhood of that point. So there is no loss of generality in assuming that $U$ is bounded.