Product of 2 little oh function

Question

Good day, I wanted to ask about some properties of the little oh notation (or I guess properties of functions that are little oh)

If $f=o(h)$ and $g=o(h)$ then is it true that $fg=o(h)$?

Answer 1

You need to be clear that you are taking the limit as $h \to 0$, not as $h \to \infty$. Once you say that, it is true because (for small enough $h$) you have $f(h)\lt h, g(h)\lt h,$ so $f(h)g(h) \lt h^2 \lt h$