Limit of sequences and integers If $a$ is a non zero real number , $x \ge 1$ is a rational number and $(r_n)$ is a sequence of positive integers such that $\lim _{n \to \infty}ax^n-r_n=0$ , then is it true that $x$ is an integer ? 

According to the Wikipedia article on Pisot–Vijayaraghavan numbers, this is true. In fact, if $x$ is algebraic (not necessarily rational) then it has to be a Pisot number, and in particular an algebraic integer. A rational integer is of course a bona fide integer.
