How to go about induction that deals with inequalitites

Question

The only thing i've been able to do is to prove it for 1. How do i go about prvoing it for k+1?

Answer 1

Assume that it is true for $n-1$, i.e. $I_{n-1} < (n-1)!$, then $I_n < nI_{n-1} < n\cdot (n-1)! = n!$, and it is therefore true for all $n$.