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.

Hint: prove inductively that a product is $> 1$ if each factor is $>1$. Apply that to the product $$\frac{n!}{2^n}\: =\: \frac{4!}{2^4} \frac{5}2  \frac{6}2 \frac{7}2\: \cdots\:\frac{n}2$$
This is a prototypical example of a proof employing multiplicative telescopy. Notice how much simpler the proof becomes after transforming into a form where the induction is obvious, namely: $\:$ a product is $>1$ if all factors are $>1$. Many inductive proofs reduce to  standard inductions.
