Asymptotic growth of factorial and exponential

Question

I know that $n!$ growth faster than $a^n$, (Do factorials really grow faster than exponential functions?).

But what about this: $n!$ and $2^{2^n}$ ($2$ to the power of $2$ to the power of $n$)? Can you please prove?

Answer 1

Hint: Suppose $n$ times of $2^n$: $$2^n\times\cdots\times2^n = 2^{n^2}$$

It's straightforward to show that $n < 2^n$ and growth of $n$ is less than $2^n$. Therefore, growth of $n!$ is less than $2^{n^2}$. Also $n^2 < 2^n$ and for their growth respectively. Hence, growth of $n!$ is less than $2^{2^n}$.