Problem 2. An infinite increasing sequence a₁ < a₂ < a₃ < · · · of positive integers is called central if for every positive integer n, the arithmetic mean of the first an terms of the sequence is equal to a_n. Show that there exists an infinite sequence b₁, b₂, b₃, . . . of positive integers such that for every central sequence a₁, a₂, a₃, . . ., there are infinitely many positive integers n with a_n = b_n.