ProofOfLiouvilleApproximationTheorem 2002-12-26 07:56:57 2003-02-11 02:54:42 Proof Liouville approximation theorem 0 % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $$\frac{1}{q^n}\leq \left|f\left(\frac{p}{q}\right)-f(\alpha)\right|=\left|\left(\frac{p}{q}-\alpha\right)f'(x)\right|<M\left|\frac{p}{q}-\alpha\right|.$$