Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 12F / i

The Liouville approximation theorem states that if a real irrational algebraic number $\alpha$ has degree $d$, there is $c>0$ such that $|\alpha -p/q|\ge c/q^d$ for every integer $p$ and positive integer $q$.

Choose its irreducible polynomial $F\in \mathbb{Z}[X]$ of degree $d$. Since $\alpha$ is irrational, $F(p/q)\ne 0$; after multiplication by $q^d$ it is a nonzero integer. Hence $|F(p/q)|\ge q^{-d}$. On $[\alpha -1,\alpha +1]$ the polynomial derivative has a finite bound $M>0$. For $|p/q-\alpha |<1$, the mean value theorem gives $|F(p/q)|\le M|p/q-\alpha |$, proving the required bound with $c=1/M$. For more distant rationals, the bound holds with $c\le 1$. Taking $c=\min \{1,1/M\}$ covers every rational and proves the theorem.