Does $O(1) * o(1)$ equal a $o(1)$ function?
The formal proof would proceed along these lines:
Let $C$ be a constant such that $f(n)\le C$ for all $n$ (such a constant exists since $f(n)\in O(1)$).
Then 
$$\lim_{n\to \infty}f(n)\cdot g(n)\le C\cdot \lim_{n\to \infty}g(n)=0$$
So $f(n)g(n)\in o(1)$.