Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 12I (Topics in Analysis)

Liouville approximation theorem says that if algebraic irrational $\alpha$ has degree $d\ge 2$, then $|\alpha -p/q|>C(\alpha )q^{-d}$ for all rationals $p/q$.

Taking determinants of the convergent matrices gives $q_n{p}_{n-1}-q_{n-1}{p}_n=(-1{)}^n$. The complete-quotient formula then yields $|\alpha -p_n/q_n|<1/(q_n{q}_{n+1})<1/q_n^2$. Comparing two consecutive errors shows at least one is below $1/(2q^2)$; hence both stated inequalities occur infinitely often (the second is the elementary precursor of Hurwitz's sharper $1/\sqrt{5}$ bound).

Here $q_{n+1}\ge a_{n+1}{q}_n=q_n^{n+1}$, so $|\alpha -p_n/q_n|<q_n^{-(n+2)}$. If $\alpha$ were algebraic of fixed degree $d$, Liouville's theorem would contradict this for $n>d$. Thus $\alpha$ is transcendental.