SE-SKOLORNAS 1967 Problem 4

The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges. Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.