# Limit of a convergent series and a divergent sequence

## Homework Statement

Show that if:

$$lim_{k\to\infty}b_k\to+\infty$$,

$$\sum_{k=1}^\infty a_k$$ converges and,

$$\sum_{k=1}^\infty a_k b_k$$ converges, then

$$lim_{m\to\infty} b_m \sum_{k=m}^\infty a_k = 0$$

## The Attempt at a Solution

I only have an idea why this is true--$$\sum a_k$$ converges, so the tails become small, and $$\sum a_k b_k$$ converges, so $$a_k$$ shrinking dominates $$b_k$$ blowing up. I know that $$|b_k| < \epsilon / |a_k|$$ for any epsilon and large k, and I know that $$\left| \sum_{m=k}^\infty a_m \right| < \epsilon$$ for large k, but I can't find the way to combine these two statements.