KöMaL 2017 A708

Let $S$ be a finite set of rational numbers. For each positive integer $k$, let $b_k=0$ if we can select $k$ (not necessarily distinct) numbers in $S$ whose sum is $0$, and $b_k=1$ otherwise. Prove that the binary number $0.b_1b_2b_3\dots$ is a rational number. Would this statement remain true if we allowed $S$ to be infinite?