Limit Of Product Equals Product Of Limits

The key here is to rewrite $|a_nb_n-AB|$ in a new way so that the quantities $a_n-A$ and $b_n-B$ appear. There are lots of ways to do this. One is to just force them to appear by replacing $a_n$ with $a_n-A+A$, giving us $$ |a_nb_n-AB|= |(a_n-A+A)b_n-AB|= |(a_n-A)b_n+Ab_n-AB|= |(a_n-A)b_n+A(b_n-B)|. $$ At this point, the triangle inequality should help to separate things. Then you'll need to pick $M$ large enough so that both $|(a_n-A)b_n|\leq \frac{\varepsilon}{2}$ and $|A(b_n-B)|<\frac{\varepsilon}{2}$. You'll need to guarantee $|a_n-A||b_n|\leq \frac{\varepsilon}{2}$ and $|b_n-B||A|<\frac{\varepsilon}{2}$. You will need an estimate for $|b_n|$ (did you see the bounded part), and use facts about $(a_n)$ and $(b_n)$ converging to get the needed $M$.