US-TST 2016 Problem 4

Let $\sqrt{3}=1 . b_{1} b_{2} b_{3} \cdots(2)$ be the binary representation of $\sqrt{3}$. Prove that for any positive integer $n$, at least one of the digits $b_{n}, b_{n+1}, \ldots, b_{2 n}$ equals 1 .