Prove that there exists a decreasing sequence $b_n \rightarrow 0$ such that $\sum_{n=1}^{\infty} a_n b_n$ diverges.
If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges?