 Proof of Liouville approximation theorem

For the degree-$d$ minimal polynomial $P\in \mathbb{Z}[X]$ of $\alpha$, the nonzero integer $q^{d}P(p/q)$ gives $|P(p/q)|\ge q^{-d}$. On a fixed neighbourhood of $\alpha$, the mean value theorem boundsOutside that neighbourhood the desired lower bound is immediate, giving $|\alpha -p/q|>c(\alpha )q^{-d}$ after decreasing the constant.