Proof of Liouville approximation theorem
= Proof of Liouville approximation theorem
For the degree-$d$ minimal polynomial $P\in\mathbb Z[X]$ of $\alpha$, the nonzero integer $q^dP(p/q)$ gives $|P(p/q)|\geq q^{-d}$. On a fixed neighbourhood of $\alpha$, the mean value theorem bounds
$$
|P(p/q)|\leq\max|P'|\,|p/q-\alpha|.
$$
Outside that neighbourhood the desired lower bound is immediate, giving $|\alpha-p/q|>c(\alpha)q^{-d}$ after decreasing the constant.