Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself.
I'm curious about the converse. For any holomorphic bijection on the sphere, why is it necessarily a Möbius transformation? Is there a proof or reference of why this converse is true? Thanks.
(I would appreciate an explanation at the level of someone whose just self-studying complex analysis for the first time.)