  A celebrated theorem due to Bannai-Bannai-Stanton says that if $A$ is a set
of points in $\mathbb{R}^{d}$, which determines $s$ distinct distances, then
$$|A| \leq {d+s \choose s}.$$ In this note, we give a new simple proof of this
result by combining Sylvester's Law of Inertia for quadratic forms with the
proof of the so-called Croot-Lev-Pach Lemma from additive combinatorics.
