Proof: By Induction
(related to Proposition: Inequality between Powers of $2$ and Factorials)
We provide a proof by induction for all natural numbers $n\in\mathbb N.$
 Base case: $n=4$
 Induction step $n\to n+1$
 Assume, $$2^n < n !$$ is correct for an $n\ge 4.$
 Then $$\begin{align}2^(n+1)&=2\cdot 2 n\nonumber\\
& < 2\cdot n !\quad\text{(by assumption)}\nonumber\\
& < (n+1)\cdot n ! \nonumber\\
& = (n+1) ! \nonumber
\end{align}$$
 Thus, the powers of $2$ are smaller than the corresponding factorials for $n\ge 4.$
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983