Suppose is a simply connected domain in , . Then there exists a holomorphic function , the unit disc, and is injective.

Suppose , then never achieves zero on . Since is simply connected, there exists a holomorphic function such that , .

Choose , then . From open map theorem, such that . Then , otherwise such that . Since is injective, one must have , so , which means . Contradiction.

So one can check

satisfies all the conditions.

- Pay attention to the domain , which is simply connected. Some other proof may have flaws when dealing with this stuition.
- Riemann Mapping Theorem was firstly stated in his PhD thesis. Riemann’s flawed proof depended on the Dirichlet principle (whose name was created by Riemann himself). Karl Weierstrass found that this principle was not universally valid. Later, David Hilbert was able to prove that, to a large extent, the Dirichlet principle is valid under the hypothesis that Riemann was working with However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of
*U*which are not valid for simply connected domains in general. See Riemann Mapping Theorem.