Prove by induction that $n!>2^n$ 
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Proof the inequality $n! \geq 2^n$ by induction  

Prove by induction that $n!>2^n$ for all integers $n\ge4$.
I know that I have to start from the basic step, which is to confirm the above for $n=4$, being $4!>2^4$, which equals to $24>16$.
How do I continue though. I do not know how to develop the next step.
Thank you.

(i) When $n=4$, we can easily prove that $\frac{4!}{2^{4}}=\frac{24}{16}>1$.
(ii) Suppose that when $n=k$ $(k\geq4)$, we have that $k!>2^{k}$.
(iii) Now, we need to prove when $n=(k+1)$ $(k\geq4)$, we also have $(k+1)!>2^{k+1}$. 
We transfer the equation that $\frac{k+1}{2}k!>2^{k}$. As (2), we have known that $k!>2^{k}$, now we only need to prove that $\frac{k+1}{2}>1.0$. 
We have known that $k\geq 4$, hence, it easily proves that $\frac{k+1}{2}>1$. In other words, we prove that $(k+1)!>2^{k+1}$.
Thereby, $n!>2^{n}$ for all integers $n\geq 4$. $\square$
