Variation of omega definition
When you negate $\Omega$ you get $\stackrel{\infty}{O}$, the analog of $\stackrel{\infty}{\Omega}$. I suggest writing the definition of $\Omega$ as a first-order formula and then negating it.

To see the difference between $\Omega$ and $\stackrel\infty\Omega$, consider the functions $f(n) = n$ and $$g(n) = \begin{cases} 1 &amp; \text{$n$ even,} \\ n^2 &amp; \text{$n$ odd.} \end{cases}$$
You can check that $f(n) \neq O(g(n))$ and $f(n) \neq \Omega(g(n))$, whereas $f(n) = \stackrel\infty O(g(n))$ and $f(n) = \stackrel\infty\Omega(g(n))$.