Euler's Number as Limit of n over nth Root of n Factorial

Theorem

 * $\displaystyle e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$

where:
 * $e$ denotes Euler's number
 * $n!$ denotes $n$ factorial.

Proof
From Limit of Root of Positive Real Number:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {2 \pi}^{1 / 2 n} = 1$

and from Limit of Integer to Reciprocal Power:
 * $n^{1 / n} = 1$

Hence the result.