Prove that a holomorphic map from $\mathbb{P}^{1}$ to itself is either constant or a rational function. Prove that a holomorphic map of degree 1 from $\mathbb{P}^{1}$ to itself is a Möbius transformation.

Show that, for every finite set of distinct points $z_{1}, z_{2}, \ldots, z_{N}$ in $\mathbb{P}^{1}$ and any values $w_{1}, w_{2}, \ldots, w_{N} \in \mathbb{P}^{1}$, there is a holomorphic function $f: \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}$ with $f\left(z_{n}\right)=w_{n}$ for $n=1,2, \ldots, N$.