Let G be a simply connected domain, $G \not \neq \mathbb{C}$ and $z_0 \in G$, I got to show that for every $n \in \mathbb{N}$ there is a holomorphic and injective mapping: $f_n:G \rightarrow D_1(0)$, the open unit disk. That is $f_n(z_0)=1-\frac{1}{n}$. Then I got to show that this $f_n$ converges to locally uniformly to a non-injective function.
I think the first assertion I can show with the Riemann mapping theorem, right? I just I don't understand why it connverges to a non-injective function, as all $f_n$ are biholomorphic by the Riemann mapping theorem. Am I totally wrong?