Proof by induction - $3.2 *10^{-6} * n! - 352.5 \le n!$

Question

This is a topic I find very hard. Could somebody please help me with this proof?

Prove by induction:

$3.2 *10^{-6} * n! - 352.5 \le n!$ for all $ n \ge 0 $

Thanks in advanced.

Answer 1

This example is not the best one to use and learn induction. The case $n=0$ is obviously fulfilled. Assume the inequality holds for $n\in \mathbf N_{>0}$. Then by the induction hypothesis $$ \frac{3.2}{10^6}\cdot (n+1)! -352.5 = \frac{3.2}{10^6}\cdot n! -352.5 + \frac{3.2}{10^6}(n+1) \stackrel{\text{I.H.}}{\leq} n! + \frac{3.2}{10^6}(n+1) \stackrel{(\star)}{<} n! + n+ 1< (n+1)!$$ In $(\star)$ we used $\frac{3.2}{10^6} <1$. The last inequality can be proofed again by using induction but there might be better ways.