Simple Real Analysis Problem - Using comparison test to prove a series diverges.

Question

Let $a_n > 0$ and suppose that $\sum a_n$ diverges. Prove that $\sum a_n b_n$ diverges for all sequences $\{b_n\}_n$ with $\liminf_n b_n >0$.

I know this is a simple problem. I already proved using the fact that $a_n b_n$ does not converge to 0 and thus the series must diverge. However, I am not sure how to prove this using the comparison test with $\sum a_n$. We would need to deduce that $a_n < a_nb_n$ but how can we do that? Does $\liminf_n b_n >0$ implies this? What if $b_n=0.01$ for all n?

Answer 1

Hint:

If $\liminf b_n = \alpha >0 $ then there exists $N$ such that $ b_n > \alpha/2$ for all $n >N$

Answer 2

First few terms of a series do not affect convergence or divergence of the series. There exist $b >0$ and $m$ such that $b_n \geq b$ for all $n \geq m$. Hence $\sum a_nb_n$ is divergent.