Product of two sequences convergence proof Suppose that a sequence $\{a_n\}$ converges to a nonzero number and a sequence $\{b_n\}$ is such that $\{a_nb_n\}$ converges. Prove that $\{b_n\}$ must also converge. 

hint: let $a_n \to A \ne 0, a_nb_n \to B$, we have: $\left|b_n - \dfrac{B}{A}\right|= \dfrac{1}{|A|}\left|Ab_n-B\right|\le \dfrac{1}{|A|}\left(\left|b_n||a_n-A| \right|+ |a_nb_n - B|\right)$. Secondly you can write: $|b_n| = \dfrac{|a_nb_n|}{|a_n|}< \dfrac{M}{\frac{|A|}{2}}= \dfrac{2M}{|A|}, n \ge N_0$. Can you finish it ?
