Proving Convergence to 0

1. Sep 29, 2012

sitia

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution
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.