# Thread: induction inequality

1. ## induction inequality

I have to prove that $2^n < n! for all n>4$ using induction.

So,um I started like this,

$p(n) : 2^n < n !$
so $p(n+1) : 2^(n+1) < (n+1)!$
=> $2^n * 2 < (n+1) * n!$
=> $2^n < (n+1)/2 * n!
$

so um idk what to do next.How do I prove that this prove holds for all n>4

2. Originally Posted by NidhiS

I have to prove that $2^n < n! for all n>4$ using induction.

So,um I started like this,

$p(n) : 2^n < n !$
so $p(n+1) : 2^(n+1) < (n+1)!$
=> $2^n * 2 < (n+1) * n!$
=> $2^n < (n+1)/2 * n!
$

so um idk what to do next.How do I prove that this prove holds for all n>4
Compare the factors at both sides:

$2^n * 2 < (n+1) * n!~\implies~\underbrace{2^n3}$