Is this correct in term of big-oh notation: given $g = O(f)$ and $h = O(f)$ can we say $g = O(h)$? We have two equations $g = O(f)$ and $h = O(f)$ , then can we derive that $g = O(h)$.
I came up with following proof but i dont know it's correct or not.
$$g = O(f)$$
$$g \le c_1*f $$
$$h \le c_2*f $$

Now we have to prove,
$$g = O(h)$$
$$g \le c_3*h$$
$$g \le c_3*c_2*f  $$
$$ c1 \le c3*c2 $$
We can find $c_3$ such that last equation $ c1 \le c3*c2 $ will be true.


  Is this correct or not? If not what will be correct proof for this?
$f(n)=n^2$, $g(n)=n$, $h(n)=1$.