Abstract Algebra: Theory and Applications · Problem 2.4.3

Groups — Thomas Judson

Mathematics — Abstract Algebra · induction-factorial-vs-power

Problem

Prove that

n! > 2^n

for

n \geq 4

. Hint. The base case,

S(4) : 4! = 24 > 16 = 2^4

is true. Assume

S(k) : k! > 2^k

is true. Then

(k+1)! = k!(k+1) > 2^k \cdot 2 = 2^{k+1}

, so

S(k+1)

is true. Thus,

S(n)

is true for all positive integers

n