Theorem 1.3.1. Stieljes-Osgood Theorem.
A locally uniformly bounded sequences of holomorphic functions on a domain \(A \subset \C\text{,}\) has a locally uniformly convergent subsequence.
Section 1.3 (optional) A proof of the Riemann Mapping Theorem
The Riemann Mapping Theorem asserts that a simply connected domain \(A\) is conformally equilvalent to the complex unit disk \(\D\text{.}\) To prove this is going to take a bit of work, so roll up your sleeves! We'll proceed in pieces.
A sequence of functions \(f_n\) is said to be locally uniformly bounded on a domain \(A\) if, for any \(z_0 \in A\text{,}\) there exists a neighborhood \(U\) of \(z_0\) and a constant \(M\) so that \(\abs{f_n} \lt M\) on \(U\) for all \(n\text{.}\) Equivalently, a sequence of functions is locally uniformly bounded if the family of functions is uniformly bounded on each compact subset of the domain.
Proof.
Let \(A\) be the domain of the sequence of locally uniformly bounded functions \(\{f_n\}_{n=1}^\infty\text{.}\)
It is enough to show that for each closed disk \(D\) in \(A\) that the sequence \(f_n\) converges uniformly on \(D\text{.}\) Suppose we could do so. Then we could take a countable cover of \(A\) by open disks whose closures \(cl(D_k) \subset A\) (say by using points with rational real and imaginary parts as centers), which can be arranged into a sequence \(\{D_k\}\text{.}\) Then by supposition, the sequence \(f_n\) has a subsequence \(f_{1,n}\) converging uniformly on \(D_1\text{.}\) Applying the same argument to \(f_{1,n}\) on \(D_2\) gives a subsequence converging uniformly on \(D_2\) (and \(D_1\)). Proceed inductively. The sequence \(f_{k,n}\) will be a subsequence of every previous subsequence, including \(f_{n}\text{.}\) Now take the sequence of elements \(f_{n,n}\text{.}\) It is a subsequence of \(f_n\) and it is locally uniformly convergent in \(A\text{.}\) (This is a form of a “diagonal” argument, ala Cantor.)
We can reduce the problem even further. Consider one of the disks, say \(D_k\text{.}\) Because \(cl(D_k)\) is a subset of then open set \(A\text{,}\) it is contained in an open disk \(U\) also contained in \(A\) and with \(cl(U) \subset A\) on which \(f_n\) is uniformly bounded.
