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 $f\left(\frac{p}{q}\right)\neq 0$ .

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

since the numerator is a non-zero integer.

By the mean-value theorem

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

Title	proof of Liouville approximation theorem
Canonical name	ProofOfLiouvilleApproximationTheorem
Date of creation	2013-03-22 13:19:22
Last modified on	2013-03-22 13:19:22
Owner	lieven (1075)
Last modified by	lieven (1075)
Numerical id	6
Author	lieven (1075)
Entry type	Proof
Classification	msc 11J68