I'm trying to prove by way of induction that:

$\displaystyle n! > 2^n$ $\displaystyle for$ $\displaystyle n \geq 4$

The base case is obviously true as $\displaystyle 24 > 16$

I then took the inductive step and came up with the following:

$\displaystyle (k + 1)! > 2^{k+1}$

$\displaystyle = (k + 1)*k! > 2*2^k$

I then figured that if $\displaystyle k! > 2^k$ (assumption), then if the amount that the LFH is being multiplied by each time is > the amount that the RHS is being multiplied by each time, then the LHS must always be > RHS and the statement is true.

So if:

$\displaystyle (k + 1) > 2$

then the original statement is true.

Since $\displaystyle n \geq 4$ then the LHS multiple is always > 4 which is > 2 (RHS multiple), therefore the LHS is always greater than the RHS and the statement is true.

Is this a valid proof? What are some other ways I could prove this?

Thanks.