Consider two sequences of random variables, $\,\{a_n\}$ and $\{b_n\}$ and suppose that $\,a_n\stackrel{p}{\longrightarrow}0$. Provided that $\,b_n\stackrel{d}{\longrightarrow}Z$ with $Z\sim F$ for a given distribution $F$, I understood that Slutsky's theorem dictates that $$a_n\,b_n\stackrel{d}{\longrightarrow}0\,Z$$such that effectively $$a_n\,b_n\stackrel{p}{\longrightarrow}0.$$ However, when defining "constant" random variables $\,a_n=1/n$ and $b_n=n$, it clearly follows that $$a_n\,b_n\stackrel{p}{\longrightarrow}1.$$ I'd very much appreciate, if someone could briefly delineate a set of high-level conditions on $\{b_n\}$ that would be sufficient for the result that$$a_n\,b_n\stackrel{p}{\longrightarrow}0\quad\text{provided that }\,\,\,a_n\stackrel{p}{\longrightarrow}0.$$ E.g. does it suffice to assume that $b_n$ is $O_p(1)$ - which would clearly be violated in my counter example.
Thank you very much for your time.
Best regards,
Jon