Let $\{a_n\}_{n=0} ^\infty $ be a sequence of positive numbers so that $\sum_{n=1}^{\infty} a_n $ converges.
a) show that there exists an nondecreasing sequence $\{b_n\}$ so that $\lim_{n\to\infty}b_n = \infty$ and $\sum_{n=1}^{\infty} a_n b_n < \infty. $
How can I use the partial sums of $\sum_{n=1}^{\infty} a_n $ to define $\{b_n\}$?