Does bn Converge to Zero if an Diverges and an*bn Converges?

[sitia]

Homework Statement

If sequence an diverges to infinity and sequence an*bn converges then how do I prove that sequence bn must converge to zero?

Homework Equations

The Attempt at a Solution

I really don't know how to go about this so any help would be so appreciated.
Thanks

 

[HallsofIvy]

If $a_nb_n$ converges, to, say L, then, given any $\epsilon> 0$, there exist $N_1$ such that if $n> N_1$, $|a_nb_n- L|< \epsilon$.

Since $a_n$ diverges to infinity, then, given any X>0, there exist $N_2$ such that if $n> N_2$, $a_n> X$.

Take n greater than the larger of $N_1$ and $N_2$ and use both of those.