Please prove $n! > (n/3)^n$ is true, without using mathematical induction

Question

Please prove $n! > \left(\frac{n}{3}\right)^n$ is true, without using mathematical induction. I've proved it using mathematical induction, but our teacher asked us to derive it using limits $n$ pre-calculus. I tried, but I'm stuck.

Answer 1

Okay , now I have got it. $$ \frac {n!}{(n/3)^n} = 3^n \frac {n(n-1)(n-2)....}{n.n.n....} = 3 \left( 3- \frac {3}{n} \right) \left( 3- \frac {6}{n} \right) \left( 3- \frac {9}{n} \right) …… $$ If $n$ be a natural number, then the RHS is clearly more than 1 , hence the result.

Edit

The general factor is, $$ \left( 3- \frac {3k}{n} \right) $$ It is obvious that whether $n→0$ or $n→∞$, the product tends to infinity, i.e. the function $\frac {n!}{(n/3)^n}$ grows monotonically. In case $n=1$, the result is trivial. Hence the result.