I am trying to solve this old qual problem:

> 
> Suppose $\{u\_n\}\_{n=1}^\infty$ is a sequence of functions harmonic on an open set $U \subset \mathbb{C}$ and uniformly bounded by 1. Suppose there is a function $u : U \to \mathbb{R}$ such that $u\_n \to u$ pointwise. Show $u$ is harmonic on $U$.
> 
> 
> 

I would like to solve this using only elementary complex analysis. My idea is this:

Pick a point $z \in U$ and a disk $D\_r(z) \subset U$. Since the disk is simply connected, each $u\_n$ is the real part of some holomorphic $f\_n$. Now, we would like to show that $f\_n$ converges uniformly on compact subsets...

I haven't gotten any further than this.

Will this approach work? Other ideas which use complex analysis are appreciated.