Prove/disprove if f1(n) = O(g1(n)) and f2(n) = O(g2(n)), then (f1(n))/(f2(n)) = O((g1(n))/(g2(n)))

Question

Here is the problem:

Prove or disprove if $f_1(n) = O(g_1(n))$ and $f_2(n) = O(g_2(n))$, then $\frac{f_1(n)}{f_2(n)} = O(\frac{g1(n)}{g2(n)})$.

My intuition is that I have to disprove this. That somehow I can get $\frac{f_1(n)}{f_2(n)} = \frac{g_1(n)}{g_2(n)}$ to be one number and show that it's not equal to $O(\frac{g1(n)}{g2(n)})$ but I'm not sure how to implement it.

Any help would be greatly appreciated!

Answer 1

$O(\cdot)$ means functions can be of different orders, e.g. $f_1 = \frac{1}{n}, g_1 = 1$. Now take $f_2 = \frac{1}{n^2}$ and $g_2 = 1$