Use induction to prove 2^n

We are asked to prove that `2^n<n! ` using mathematical induction. Note that this is only true for `n>=4 ` .

(1) Base case: If n=4 we have ` 2^4=16<24=4!`

(2) Inductive hypothesis: Assume there exists a k>4 such that `2^k<k! ` . We will show that for such a k, `2^(k+1)<(k+1)! ` .

(3) Since k>4 we have:

`2^(k+1)=2*2^k<2*k! ` by the inductive hypothesis and a property of inequalities. So:

`2^(k+1)=2*2^k<2*k!<(k+1)*(k!) = (k+1)! `

Thus by mathematical induction, `2^n<n!, n>=4,n in NN `

Comments

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