I see examples online that proves that if $f(n)\in O(h(n))$ and $g(n)\in O(h(n))$, then $f(n)+g(n)\in O(h(n))$ using the definition of O notation.
However, if given $f(n)\in O(h(n))$ and $g(n)\in O(h(n))$, then how can I prove/disprove:
$$f(n)*g(n)\in O(h(n))?$$
So far, I have gotten to defining (from the definition of O notation):
$$f(n)\ \le\ c_1h(n)\ for\ n\ \geq\ n_0^\prime$$ $$g(n)\ \le\ c_2h(n)\ for\ n\ \geq\ n_0^{\prime\prime}$$
where $c_1, c_2, n_0^\prime, n_0^{\prime\prime}$ are all positive constants, but I'm not quite sure what to do next. Do I multiply the constants together?
Any tips would be appreciated!
You cannot prove it, because it is false.
Counterexample: $f(n)=g(n)=h(n)=n$.
Then clearly $f(n)\in O(h(n))$, $g(n)\in O(h(n))$, but $f(n)g(n) = n^2 \not\in O(h(n)) = O(n)$.