Proving an inequality of reduction formula by induction

Question

If $I_n = \int_0^1 t^n e^{-t} dt$ for $n \geq 0$ then $I_n=nI_{n-1} - e^{-1}$ for all $n \geq 1$.

Prove by induction that, for all positive integers $n$,

$I_n < n!$

How do I prove this ??

Answer 1

The result is true for $n=0$, obviously.(Here we use the convention $0!=1$.)

Suppose $I_k<k!$ for some $k\geq 0$, then consider the case $n=k+1$, we have $$I_{k+1}=(k+1)I_k-e^{-1}<(k+1)I_k<(k+1)\cdot k!=(k+1)!$$

Hence $I_n<n!$ for all $n\geq 0$ by induction. So it is true for all positive integers.