PL-OM 2023 Problem 3

Given a positive integer $n \geq 2$ and real numbers $a_1, a_2, \ldots, a_n \in [0,1]$ . Prove that there exist real numbers $b_1, b_2, \ldots, b_n \in \{0,1\}$ , such that for all $1\leq k\leq l \leq n$ we have $$\left| \sum_{i=k}^l (a_i-b_i)\right| \leq \frac{n}{n+1}.$$