Let $U$ be uniform distributed in $[0,1]$ . Show that with probability $1$ there's maximum a finite amount of $n \in \mathbb N$, so that the inequality
$\small| U-\frac{m}{n}\small| \leq \frac{1}{n^3}$
is true for $m \in \mathbb N$.
I have tried for two days and I couldn't solve it.