Uniform convergence on compact subsets

Question

$\omega$ is a region, $f_n$ is holomorphic on $\omega$ for n=1,2,3,.., $u_{n}$ is the real part of $f_n$, $u_{n}$ converges uniformly on compact subsets of $\omega$, {$f_{n}(z)$} converges for atleast one $z \space \epsilon \space \omega$. Then how to prove that {$f_{n}$} converges uniformly on compact subsets of $\omega$? I think Harnack's theorem has to be used.