# Prove $\sum a_n b_n$ diverges if $a_n$ diverges, $a_n>0$, and $\lim\inf_n b_n >0$

I'm having trouble starting this proof. From the initial hypothesis, we know for some $$n>N$$, $$b_n>0$$. Since $$a_n>0$$ and $$a_n$$ diverges, for some $$n>N'$$, $$a_n\geq \varepsilon$$. Then $$a_n b_n>0$$, so if $$\sum a_n b_n$$ is a series of positive numbers. I'm not sure how to show it diverges however, so any help would be appreciated!

• @qbert isn't the answer below correct regardless? – clark Dec 3 '18 at 3:03

Let $$\liminf_{n\to \infty}b_n=2\delta>0$$ Then, for large enough $$N$$, $$b_n\geq \delta$$ for all $$n\geq N$$. Then, $$\sum_{n=N}^\infty b_na_n\geq \delta\sum_{n=N}^\infty a_n=+\infty$$ since $$\lim_{n\to \infty}a_n\ne 0$$.
• Thank you! My only question is, where does the 2 go in the $2\delta$ term? – t.perez Dec 5 '18 at 2:12