Q: Proof or Riemann Mapping Theorem

Question

I am working with this proof of the Riemann mapping theorem:

http://people.reed.edu/~jerry/311/rmt.pdf

How do you know that the function they found satisfies the condition of having Real and positive derivative at $z_0$? Because its never really proven there. (Maybe its obvious but i do not see it)

And I have one more doubt, but its more about efficiency. Instead of proving that the family is equicontinous so you can use Arzela-Ascoli, can you just use Montel's theorem instead? Since al functions in $\mathcal{F}$ are bounded.

Thanks in advanced.

Answer 1

It might not. Any analytic bijection $f : \Omega \to D$ with $f(z_0) = 0$ followed by a rotation is again such an analytic bijection. The point is that up to rotation it is unique - any two such analytic bijections only differ by a rotation.