If $\lim_{n \to \infty }b_n=L$, and $\sum_{n=1}^\infty a_n$ is convergent, then $\sum_{n=1}^\infty a_nb_n$ is convergent.
Prove or disprove: if $\sum\limits_{n=0}^∞a_n$ converges and $\lim\limits_{n\to\infty}b_n =0$, then $\sum\limits_{n=0}^{\infty}a_n b_n$ converges.