MathDB

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.

Problem Statement