Let P (n) be the statement that n! < nn where n is an integer greater than 1.

a) What is the statement P (2) ?

b) Show that P (2) is true, completing the basis step of theproof.

c) What is the inductive hypothesis?

d) What do you need to prove in the inductive step?

e) Complete the inductive step.

f) Explain why these steps show that this formula is true whenevern is an integer greater than 1.

Answer: See the step by step explanation.

Step-by-step explanation:

a) First, Let P (n) be the statement that n! < n^n

where n ≥ 2 is an integer (This is because we want the statement of P (2).

In this case the statement would be (n = 2) : P (2) = 2! < 2^2

b) Now to prove this, let's complet the basis step:

We know that 2! = 2 * 1 = 2

and 2^2 = 2 * 2 = 4

Therefore: 2 < 4

c) For this part, we'll say that the inductive hypothesis would be assuming that k! < k^k for some k ≥ 1

d) In this part, the only thing we need to know or prove is to show that P (k+1) is also true, given the inductive hypothesis in part c.

e) To prove that P (k+1) is true, let's solve the inductive hypothesis of k! < k^k:

(k + 1) ! = (k + 1) k!

(k + 1) k! < (k + 1) ^k < (k + 1) (k + 1) ^k

Since k < k+1 we have:

= (k + 1) ^k+1

f) Finally, as the base and inductive steps are completed, the inequality is true for any integer for any n ≥ 1. If we had shown P (4)

as our basis step, then the inequality would only be proven for n ≥ 4.