How to prove that $ n < n! - 1 $ for $n > 2.$?

Question

How to prove that $ n < n! - 1 $ for $n > 2.$?

I have tried it by induction but I got stucked in the induction step in proving $ n +1< (n + 1)! - 1 $ for $n + 1> 2$.

Could anyone help me please?

Answer 1

Assume $n<n!-1$; i.e., $n+1<n!$.

If we can show $n!<(n+1)!-1$, then we're done showing $n+1<(n+1)!-1$.

$n!<(n+1)!-1 $ is equivalent to $1<(n+1)!-n! = (n+1)n!-n!=(n+1-1)n!=n\times n!$.

Well, if $n>2$ then clearly $1<n\times n!$, so we're done.

Answer 2

Hint: $n! \ge 2n$ for $n>2$.