I want to prove that
If $f:\mathbb{C}_\infty \to \mathbb{C}_\infty$ is holomorphic, then there exists polynomials $P$ and $Q$ such that $f(z) = \frac{P(z)}{Q(z)}, \forall z \in \mathbb{C}_\infty$
What I want to use is that a meromorphic function in $\mathbb{C}_\infty$ is a meromorphic function in $\mathbb{C}$ who is holomorphic at $\infty$ or have a pole at $\infty$. So, holomorphic functions from $\mathbb{C}_\infty$ are meromorphic functions in $\mathbb{C}_\infty$ because they are holomorphic at $\infty$.
Now, I know from Stein's Complex Analysis book that:
$\textbf{Theorem 3.4} \,$The meromorphic functions in the extended complex plane are the rational functions.
So this theorem implies that holomorphic function in extended complex plane are rational functions.
Is ok to use this to prove my first statement?
$\textbf{Edit:}$ And.. I have a feeling that converse isn't true because if we have rational function in $\mathbb{C}_\infty$ we have that it is meromorphic function and can have poles. If we restrict the domain to $\mathbb{C}_\infty - \{z \in \mathbb{C}_\infty : z \text{ is pole for} f\}$ then we obtain a holomorphic function? Are these 2 statements also true?
