Question 115
For an irrational number $x$, let $x'$ be the integer nearest to $x$. Define $\langle x \rangle = |x - x'|$. Show that for every irrational number $y$, the minimum of the numbers $\langle y \rangle, \langle 2y \rangle, \dots, \langle 2001y \rangle$ is less than $\frac{1}{2001}$.