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

Question

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!

Answer 1

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 $\lim_{n\to \infty}a_n\ne 0$.