Assignment 1 Question 4

The assignment has come and gone but is there anybody who can explain using the induction principle formulated in question 4 to prove n! > 2^n for all integers n >= 4. I'd like to know if I was on the right track when answering the question.

steven

Here is what I did hope it comes out on the forums nicely:

i) P is the smallest subset of ℤ such that:
• 4 ∈ P and
• if k ∈ P then (k+1)∈P

ii) Induction principle
If subset A of P contains the element 4 and is such that if it contains the element (k+1)
whenever it contains the element k, then this subset equals P
iii)

Step one: A = {n∈P | if n≥4 then n! > 2^n}

Step two: check if 4∈A
4! > 2^4
24 > 16 which is true so 4∈A

Step three assume k∈A thus k! > 2^k
check if (k+1)∈A
(k+1)! = (k+1)k!
(k+1)! > (k+1)2^k --------from the assumption k! > 2^k
(k+1)! > 2 . 2^k -----------since (k+1) > 2 whenever k ≥4
(k+1)! > 2^(k+1)

hence A = P and the statement (n! > 2^n ) is true for every element in P