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 $M>\max_{\alpha-1\leq x\leq\alpha+1}|f'(x)|$ .

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

$$\left|f\left(\frac{p}{q}\right)\right|=\frac{\left|a^np^n+a_{n-1}p^{n-1}q+\dots+a_0q^n\right|}{q^n}\geq\frac{1} {q^n}$$

since the numerator is a non-zero integer.

By the mean-value theorem

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