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[Answers] Does $O(1) * o(1)$ equal a $o(1)$ function?

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Problem Detail: 

Say I have two functions $f(x) = O(1)$ and $g(x) = o(1)$. Let $h(x) = f(x)g(x)$. Is $h(x) = o(1)$?

By definition of small-o $g(x)$ must approach 0 as $x \rightarrow \infty$, so I think yes. However, how can I formally prove it? Is my intuition correct?

Asked By : David Faux

Answered By : Shaull

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 $f(n)g(n)\in o(1)$.

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