Variations of Omega and Omega infinity
Hint: If $f(n) \notin \overset{\infty}{\Omega}(g(n))$ and $g(n)$ is asymptotically non-negative, then for all positive constants $c$, $f(n) \leq c \cdot g(n)$ for large enough $n$. This follows by ignoring the condition $c \cdot g(n) \geq 0$ and negating the definition of $f(n) \in \overset{\infty}{\Omega}(g(n))$. In fact, this way you get the stronger result that either $f(n) \in \overset{\infty}{\Omega}(g(n))$ or $f(n) \in o(g(n))$ (but not both).

Further hint: You can start by showing that the negation of "$P(n)$ for infinitely many $n$" is "$\lnot P(n)$ for large enough $n$".