Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 3 / 7B / i

If all three points are finite, define the Möbius transformationIts numerator vanishes at $\alpha$, its pole is at $\gamma$, and substitution of $\beta$ gives $1$. The three points are distinct, so the numerator and denominator are not proportional and the associated $2\times 2$ matrix has nonzero determinant.

If one point is infinite, the limiting formulas areIn each case the finite values and the limit at infinity give $g(\alpha )=0$, $g(\beta )=1$, $g(\gamma )=\infty$, and the corresponding determinant is nonzero. Distinctness permits at most one of the points to be infinite. Hence every ordered triple of distinct points on the Riemann sphere can be normalized to $(0,1,\infty )$.