If I grasp the problem correctly,the condition turns to" for every positive integer $ i$,$ a_i=p_1^{\alpha_1}*...*p_k^{\alpha_k},then,\sum \alpha_i \le 1987$ We replace $ 1987$ by $ n$ and consider an induction on $ n$,the foundation is trivial,we assume it is true for all positive integer $ <n$,now consider the case $ n$ Note that if there are infinite terms share a common prime divisor $ p$,we can just divide these terms by $ p$ and use induction hypothesis and reach our goal So we can simply assume that there exsists no prime $ p$such that we can find infinitely many terms which is a multiple of $ p$ $ ......(*)$ Thus,we pick out $ a_1$,by $ (*)$ we can find infinitely many terms which is relatively prime to $ a_1$,denote the infinite sequence $ \{a_i'\}$ then we pick out $ a_2$ as what we did with $ a_1$...Finally we can pick out an infinite subsquence $ \{b_i\}$such that $ (b_i,b_j)=1$ complete our induction. QED