Product of a decreasing sequence and a diverging series Suppose we have a monotonic decreasing sequence $a_n$ converging to $0$.
Does there always exist a non-negative series $b_n$, such that 
$$\sum_{n=1}^\infty b_n = \infty$$ but 
$$\sum_{n=1}^\infty a_n b_n < \infty$$?
Edit: yes, as answered below. What if we also insist $b_n$ is decreasing?

Choose $n_1<n_2<...$ such that $a_{n_{k}} <\frac 1 {2^{k}}$ and define $b_n=1$ if $n \in \{n_1,n_2,...\}$, $b_n=0$ otherwise. Note that $b_n=1$ for infintely many $n$. Hence $b_n$ does not tend to $0$ which implies $\sum b_n =\infty$.
