Putnam 1963 B6

Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$ , define $S(A)$ to be the set of points that lie on closed segments joining pairs of points of $A$ (a one-point set should be considered to be a special case of a closed segment). For a given nonempty set $A_0$ , define $A_n =S(A_{n-1})$ for $n=1,2,\ldots$ Prove that $A_2 =A_3 =\ldots.$