Is it sufficient to show the following inequality holds to prove $2^{2^n} \in \Omega(n!)$?

Question

I proved this by showing that inequality $0\leq n! \leq 2^{2^n}$ always holds using induction.

Is it sufficient to prove $2^{2^n} \in \Omega(n!)$? Do I need to show that $\displaystyle\lim_{n\to\infty}\frac{2^{2^n}}{n!} > 0$ also holds? If yes, then why is it necessary to show this limit (or why it is not sufficient to show the inequality above holds)?