You are given something that looks like a proof You need to

You are given something that looks like a proof. You need to decide if it correct or not. (It\'s most likely not). If it is not correct you need to clearly identify the error and explain why it is an error, then give a correct proof.

Prove n! > 2^n (All n) >= 4
Let P(n) be the proposition that n! > 2n
P(4) says 4! > 2^4 or 24 > 16 so P(4) is true.
Show P(n) (arrow) (if) P(n + 1) We know
(n + 1)! > n! > 2n; and
2^(n+1) > 2^n Therefore
(n + 1)! > 2^(n+1)

Solution

Show P(n) (arrow) (if) P(n + 1) We know
(n + 1)! > n! > 2n; and
2^(n+1) > 2^n Therefore
(n + 1)! > 2^(n+1)

In the above part of your proof, a slight adjustment is required.

The highlighted step how you could conclude from the previous step.

The correct proof should be

P(n) is true. Hence n!>2ⁿ

Multiply both the sides by n+1. As n+1 is positive, inequality is preserved\\

n!(n+1) > (n+1)2ⁿ

Or (n+1)! > (n+1)2ⁿ...(i)

As n >4, n+1 >5 and hence n+1 >2

Multiply by 2 power n. (n+1) 2ⁿ> 2ⁿ(2) ... (ii)

Use this result in our inequality (i)

(n+1)! > (n+1)2ⁿ>2ⁿ(2)

Or (n+1)!>2ⁿ⁺¹

Hence true for all natural numbers n >4