Prove by induction that n! > (n/3)^n
to complete your induction use the identity: $\displaystyle (n+1) \left(\frac{n}{3} \right)^n=\frac{3}{(1+\frac{1}{n})^n}\left(\frac{n +1}{3} \right)^{n+1} $ and this fact that: $\displaystyle \left(1+ \frac{1}{n} \right)^n < e < 3.$
note that the solution is actually suggesting a stronger inequality, i.e. $\displaystyle \forall n \in \mathbb{N}: \ n! > \left(\frac{n}{e} \right)^n.$