sum sequence nonnegative How to show the following:

If we have a convergent sum of nonnegative numbers $a_i$, then show that there exists a sequence of nonnegative numbers strictly monotone increasing positive $b_i$ where $b_i$ converging to $\infty$ such that 

$$\sum_{i=0}^{\infty}{a_i  b_i} &lt; \infty\quad?$$ 

Thank you
Hint:
Since
$\sum_{k=1}^{\infty} a_k$
converges,
$A_n = \sum_{k=n}^{\infty} a_k \to 0$

Choose another sequence
$B_n$ which converges to zero
more slowly than $A_n$
and define $b_n$
in terms of $B_n$.
If the sum converges, then
$$
\lim_{i \to \infty} \frac{a_{i + 1}}{a_i} = c &lt; 1
$$
Take $b_i = c^{-i/2}$, so $b_i \to \infty$, while
$$
\lim_{i \to \infty} \frac{a_{i + 1} b_{i + 1}}{a_i b_i} = c^{1/2} &lt; 1
$$