Prove the following $\frac{\Omega(f(n))}{\Omega(g(n))} \subseteq \Omega(\frac{f(n)}{g(n)})$

Question

I want to prove the following: $$\frac{\Omega(f(n))}{\Omega(g(n))} \subseteq \Omega(\frac{f(n)}{g(n)})$$ I wonder if its true?
What about using $n$ and $n^2$?
Any suggestions? Thanks!

Answer 1

The statement is false.

$$f(n)=n^4,g(n)=n^2\\ n^4\in \Omega(f(n)),n^3\in \Omega(g(n))\\\frac{n^4}{n^3} \subseteq \Omega(\frac{n^4}{n^2})\\\\n \nsubseteq \Omega(n^2)$$