The same question has been asked here
Some authors define $\Omega$ in a slightly different way: let’s use $ \overset{\infty}{\Omega}$ (read “omega infinity”) for this alternative definition. We say that $f(n) = \overset{\infty}{\Omega}(g(n))$ if there exists a positive constant $c$ such that $f(n) \geq c\cdot g(n) \geq 0$ for infinitely many integers $n$, whereas the usual $\Omega$ requires that this holds for all integers greater than a certain $n_0$.
Show that for any two functions $f(n)$ and $g(n)$ that are asymptotically nonnegative, either $f(n) = O(g(n))$ or $f(n)= \overset{\infty}{\Omega}(g(n))$ or both, whereas this is not true if we use $\Omega$ in place of $\overset{\infty}{\Omega}$
The solution given in the link hints that we can negate the definition of $ \overset{\infty}{\Omega}$ which gives us the definition of big O.
But negation of the definition of $ {\Omega}$ also gives us the same result, doesn't it?
I cannot wrap my head on how the condition of infinitely many integers makes difference rather than saying for all n greater than certain $n_0$.
Please help me understand this.