ISL 1988 Problem 75

Let $S$ be an infinite set of integers containing zero, and such that the distances between successive number never exceed a given fixed number. Consider the following procedure: Given a set $X$ of integers we construct a new set consisting of all numbers $x \pm s,$ where $x$ belongs to $X$ and s belongs to $S.$ Starting from $S_0 = \{0\}$ we successively construct sets $S_1, S_2, S_3, \ldots$ using this procedure. Show that after a finite number of steps we do not obtain any new sets, i.e. $S_k = S_{k_0}$ for $k \geq k_0.$