CWMO 2014 Problem 3

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$ , there are only finitely many values of $j$ such that $A_j\subseteq A_i$ . Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$ .