proof of Liouville approximation theorem

Let $\\alpha$ satisfy the equation $f(\\alpha)=a_{n}\\alpha^{n}+a_{n-1}\\alpha^{n-1}+\\dots+a_{0}=0$ where the $a_{i}$ are integers. Choose $M$ such that $M>\\max_{\\alpha-1\\leq x\\leq\\alpha+1}|f^{\\prime}(x)|$ .

Suppose $\\frac{p}{q}$ lies in $(\\alpha-1,\\alpha+1)$ and $f\\left(\\frac{p}{q}\\right)\eq 0$ .

\\left|f\\left(\\frac{p}{q}\\right)\\right|=\\frac{\\left|a^{n}p^{n}+a_{n-1}p^{n-1}q+%\\dots+a_{0}q^{n}\\right|}{q^{n}}\\geq\\frac{1}{q^{n}}

since the numerator is a non-zero integer.

By the mean-value theorem

\\frac{1}{q^{n}}\\leq\\left|f\\left(\\frac{p}{q}\\right)-f(\\alpha)\\right|=\\left|%\\left(\\frac{p}{q}-\\alpha\\right)f^{\\prime}(x)\\right|<M\\left|\\frac{p}{q}-\\alpha%\\right|.