All types implies torsion

We prove the following theorem. Given a positive integer n and a subset -I of z with the following properties: (1) for every (a. an) in .I the inequality i=7 Iai I > 2 holds, and (2) for every (X. Xn) in Rn there exists an (a,. an) in .4 with a x,i > 0 for i = I, n . there exists a subset .10 of .such that zn modulo the subgroup generated by 40 contains a nontrivial torsion element.