Show that the set of $x\in\mathbb{R}$ such that there are infinitely many fractions p/q with p,q relatively prime integers and $|x-p/q|\le1/q^3$ has Lebesgue measure zero.
I know how to show this for $x\in[0,1]$ by using Borel-Cantelli. Now my questions are:
- How can I show this for $x\in\mathbb{R}$?
- Is there a way to show this without using Borel-Cantelli?
Thanks for your help!