Proving that the group of holomorphic automorphisms of the Riemann Sphere $\mathbb{C}_\infty$ are the Möbius Transformations

Question

Is the following proof correct?

Proof: Let $F\in\text{Aut}(\mathbb{C}_\infty)$. Let $L:\mathbb{C}_\infty\to\mathbb{C}_\infty$ be a Möbius transformation that maps $F^{-1}(\infty)$ to $\infty$. For example, one can take $$ L(z) = \dfrac{1}{z-F^{-1}(\infty)}. $$ Then $G=F\circ L\in\text{Aut}(\mathbb{C}_\infty)$ and $G(\infty)=\infty$. This means that the restriction of G to the complex plane $\mathbb{C}$ is a holomorphic automorphism of $\mathbb{C}$ and hence $$ G(z) = (F\circ L)(z) = az+b, $$ for some $a,b\in\mathbb{C}$ with $a\neq 0$. This means $$F = aL^{-1}(z) + b $$ which is a Möbius transformation.

Answer 1

It is correct. First I thought maybe you needed this fact to calculate automorphisms of the plane (a fact you use in this proof), but it's not.