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. 

If $a_n \to L_1 \neq 0$ then $\frac{1}{a_n} \to \frac{1}{L_1}$
From hypothesis $a_nb_n \to L_2 \in \mathbb{R}$
Thus $b_n =({a_nb_n})\frac{1}{a_n} \to L_2 \frac{1}{L_1}$
