Prove that ∀n ∈ N, 2 n−1 ≤ n!

Answer 1

Use mathematical induction

for n=0, it's true as 2(0)-1 = -1 < 0! = 1

(of course, 0 is a whole number, but not a natural number)

for n=1, it's true, as 2(1)-1 = 1 = 1! = , 1<1

assume 2n-1 < n! then show 2(n+1)-1= 2n-1 < (n+1)!

2n-1 = 2n+2 - 1 = 2(n+1)- 1 < n! < (n+1)!

so

2(n+1) < (n+1)! and

2(n+1)<(n+1)!

one little problem though

for n=2,

2(2)-1 = 4-1 = 3 > 2! = 2x1 = 2

so it's not true for all natural numbers.

2(3)-1 =5 < 3! = 6

2(4)-1=7 < 4! = 24

It's true that n= any natural number except 2, that 2n-1<n!