Show by induction that $(\frac{n}{3})^n \leq \frac{n!}{3}$

Question

Show by induction that $(\frac{n}{3})^n \leq \frac{n!}{3}$

In oder to prove that, I must prove that $(\frac{n+1}{3})^{n+1} \leq \frac{(n+1)!}{3}$

But how can i show that?

Answer 1

Applying the induction hypothesis, and the fact that $(1+1/n)^n\uparrow e$ and that $e<3$, $$\left(\frac{n+1}3\right)^{n+1}=\frac{n+1}3\left(1+\frac1n\right)^n\left(\frac n3\right)^n\le\frac{n+1}3\cdot e\cdot\frac{n!}3\le\frac{(n+1)!}3$$