If $b_n$ is a bounded sequence and $\lim (a_n)= 0$ , show that $\lim(a_nb_n)=0$ and explain why the next Theorem cannot be used.
Theorem: If $x_n$ converges to $x$ and $y_n$ converges to $y$ then $\lim(x_ny_n)=xy.$
To use the theorem both sequences should be convergents. Sequence $b_n$ is a bounded sequence but may not be convergent.
I know $|x_n|<\varepsilon, \varepsilon>0 $ and by definition of bounded sequence $|b_n|\leq B, B>0$ . How can I continue with my proof on this?