Prove that for every convergent serie $\sum^{\infty}_{n=1} a_n$ such that $a_n\ge 0$ for every $n\in \Bbb N$. Exist a sequence ({$b_n$}) such that $\lim_{n\to \infty} b_n=0$ and $\sum^{\infty}_{n=1} a_nb_n$ converge.
I have tried that $\sum^{\infty}_{n=1} a_n$ converge so we know that exist $N$ such that for all $m,n$ with $m>n>N$ we have |$\sum^{m}_n a_n$|<ϵ. But I can not figure out how to proceed.
Take any sequence $(b_n)$ which has positive terms and tends to $0$, e.g. $b_n=\frac1n$. If $n$ is large enough,$0\le b_n<1$ so $$0\le a_nb_n<a_n,$$ and $\sum_{n\ge 1} a_nb_n$ converges by the comparison theorem.