Question 88
Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \neq y$ and $x$ and $y$ are both in $S$.
Prove that the elements of $S$ may be arranged in an arithmetic progression. This means that there are numbers $a$ and $d$ such that $S=\{a, a+d, a+2d, a+3d, \ldots, a+kd, \ldots\}$.
Prove that the elements of $S$ may be arranged in an arithmetic progression. This means that there are numbers $a$ and $d$ such that $S=\{a, a+d, a+2d, a+3d, \ldots, a+kd, \ldots\}$.