Lower and Upper Bound of Factorial

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:


 * $\dfrac {n^n} {e^{n - 1} } \le n! \le \dfrac {n^{n + 1} } {e^{n - 1} }$

Proof
We have:

and similarly:

First the inequality:

Then:

Now the inequality:

Then:

Hence the result.