Prove that for every $x_1, x_2,\ldots,x_n\in [0,1]$, there exist $\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n\in\{1/2,-1/2\}$ such that for all $k=1,2,\ldots,n-1$, $$\left\lvert \sum_{i=1}^k \varepsilon_i x_i-\sum_{i=k+1}^n \varepsilon_i x_i \right\rvert\le 1.$$