TITLE: If the sequence $(a_n b_n)$ converges and $a_n \to 0$, when does $(b_n)$ converge too?
QUESTION [9 upvotes]: Given that the sequence $(a_n b_n)$ converges and $a_n \to 0$, are there conditions which can be placed on $(b_n)$ and/or $(a_n b_n)$ so that $(b_n)$ converges as well?

REPLY [2 votes]: Well if you are willing to drop $\langle a_n \rangle$ converging to $0$, then something can be said. 
In general, if $\langle a_n \rangle$ converges to a positive number $a$ and the sequence $\langle b_n \rangle$ is ultimately non-negative, then: $\limsup_{n \to \infty}~a_n b_n=a.\limsup_{n \to \infty}~b_n$. And same for $\liminf$. So, if $\langle a_nb_n \rangle$ does converge, then so will $\langle b_n \rangle$.