KöMaL 2017 B4874

Let $\|x\|$ denote the distance of the real number $x$ from the closest integer. Let $a_1,a_2,\ldots,a_n$ be positive integers. Prove that there exists a real number $x$ such that $\|a_ix\|\ge \frac{1}{2n}$ for all $1\le i\le n$.