Fall 2020 - Problem 11

Solution.

Since $\set{f_n}_n$ is uniformly bounded, it forms a normal family. Thus, it admits a locally uniform convergent subsequence to a holomorphic function $f$.

Now suppose $\set{f_{n_k}}_k$ is any subsequence of $\set{f_n}_n$. Then this is still a normal family, so it admits a further subsequence which converges locally uniformly to some holomorphic $g$. By assumption, for any $z_0 \in K$, we have $f\p{z_0} = g\p{z_0}$, i.e., $f$ and $g$ agree at infinitely many points in $K$ compact, so it must agree on an accumulation point of $K$. By uniqueness, we see that $f = g$ on $\Omega$, so by the Urysohn subsequence principle, $f_n$ converges locally uniformly to $f$ on $\Omega$. In particular, it converges pointwise, which was what we wanted to show.