a series of functions that converges uniformly, converges to a harmonic function and so its derivative

Question

let $u_{n}:G\rightarrow\mathbb{R}$ be a series of harmonic on a domain G, and $u_{n}\rightarrow u$ uniformly. Then

1. u must be also harmonic. it was written here, but not formalize:Does a uniformly sequence of harmonic function converge to a harmonic function?
2. $\frac{\partial u_{n}}{\partial y}\rightarrow\frac{\partial u}{\partial y}\land\frac{\partial v_{n}}{\partial x}\rightarrow\frac{\partial v}{\partial x}$

Answer 1

Let $z \in G$ then since $u_n$ are harmonic they satisfy the mean value property.

That is, $u_n(z) = \frac{1}{2 \pi}\int_0^{2 \pi} u_n(z+re^{i \theta})d\theta$ for some $r > 0$. Now use uniform convergence to show that $|u_n (z) - \frac{1}{2 \pi}\int_0^{2 \pi} u(z+re^{i \theta})d\theta |\to 0$ and conclude that $u$ is harmonic because it satisfies the mean value property and because the limit is unique.