Every rational function which is holomorphic on Riemann Sphere($\mathbb{C}_{\infty}$)

Question

could any one give me a hint how to show Every rational function which is holomorphic on every point of Riemann Sphere( $\mathbb{C}_{\infty}$) must be constant?(with out applying Maximum Modulas Theorem). Thank you.

Answer 1

If a function is analytic on the sphere at $\infty$, it is bounded in an neighborhood of $\infty$. Consequently, it is bounded globally, since the complement of a neighbhorhood at $\infty$ is compact. Now invoke Liouville's theorem; the functon must be constant.