I have just proved the following:
However my teacher provided me with the following:
Note that $\Omega$ is open connected subset of $C$.
This is what i don't get. I know that on each compact subset of $\Omega$ there would exist a subsequence of $f_n$ that converges uniformly. But how do I know that this same subsequence converges(need not be uniformly) over $\Omega$ ?
How do I find a subsequence that converges over $\Omega$ and uniformly on compact subsets.