How can we prove that $g = O(f) $ (Big-Oh notation)
I think you have the wrong definition for $\Omega$. When we write $g=Ω(f)$, informally this means that $g$ is bounded below by ${\displaystyle f}$  asymptotically. So, it is basically just the reverse way of saying $f=O(g)$, which means that $|f|$ is bounded above by ${\displaystyle g}$ (up to constant factor) asymptotically. 

What you are describing is closest to the definition of Big theta $\Theta$.