Every Möbius transformation is a biholomorphism of the Riemann sphere

On every open set where $cz+d\mathrm{\ne }0$, the finite-chart expression $z\mapsto (az+b)\mathrm{/}(cz+d)$ is a rational function with nonvanishing denominator, hence holomorphic; and if $c\mathrm{\ne }0$, then near the finite pole $p=-d\mathrm{/}c$ the target infinity-chart expression is $(cz+d)\mathrm{/}(az+b)$, which is holomorphic because $az+b$ does not vanish at $p$.

At $\mathrm{\infty }$, if $c\mathrm{\ne }0$ then the source infinity-chart expression is $(a+bw)\mathrm{/}(c+dw)$, which is holomorphic at $0$; if $c=0$, then $M(\mathrm{\infty })=\mathrm{\infty }$ and the target infinity-chart expression is $dw\mathrm{/}(a+bw)$, again holomorphic at $0$. Thus $M$ is holomorphic at every sphere point.

By [L2], the inverse map is again Möbius, so the same two chart computations apply to $M^{-1}$ as well. Therefore $M$ is a biholomorphism of the sphere.