Let $a_n$ be a sequence with the property that whenever the sequence $b_n \longrightarrow 0$ ,the series $\sum_\limits{n=1}^{\infty}a_nb_n < \infty $. Prove that $\sum_\limits{n=1}^{\infty}|a_n| < \infty$
Can someone give me a hint for this?
Thank you in advance!