If $\limsup(b_{n})=+\infty$, is $\limsup_{n}(b_{n} S_{n})=+\infty$?

Question

If $\limsup(b_{n})=+\infty$, $b_{n}$ and $S_{n}$ are nonnegative sequences. Is $\limsup_{n}(b_{n} S_{n})=+\infty$?

Answer 1

Hint: what if $b_n=n$ and $S_n=1/n$?

Answer 2

Or simply $S_{n}=0$, then $\limsup_{n}(b_{n}S_{n})=0$.