KöMaL 2026 A943

Let $a_n$, $b_n$ and $c_n$ be sequences of real numbers such that $\vert a_n \vert$, $\vert b_n \vert$, $\vert c_n \vert \leq d$. Is there a real number $c>0$ such that there exists a sequence $s_n$ of signs (i.e. $s_n\in \{1, -1\}$ for all $n$) satisfying $\left\vert\sum_{n=1}^{N}s_na_n \right\vert, \left\vert\sum_{n=1}^{N}s_nb_n\right\vert, \left\vert\sum_{n=1}^{N}s_nc_n\right\vert \leq c$ for every positive integer $N$?