TITLE: Convergence of harmonic functions.
QUESTION [7 upvotes]: I am looking for the proof of the following Theorem : Does anyone know where i can find out ? 
If $\Omega$ is open and connected and $u_k$ be uniformly bounded sequence of harmonic functions . There exists a subsequence that converges uniformly to a harmonic function $u:\Omega \to \mathbb R$ on any compact subset of $\Omega$. 
In case you think that its not hard, i look forward to hints as well. 
I hope the statement is true . 
Thanks.

REPLY [2 votes]: You can see this book
D.H. Armitage, S.J. Gardiner, Classical Potential Theory, Springer, London, 2000.
proposition 1.5.11. This will exactly answer your question.
This is a link of this book in booksgoogle
Enjoy.