Factorial is not of Exponential Order

Theorem
Let $\Gamma$ denote the gamma function.

Let $\map f t = \map \Gamma {t + 1} = t!$.

Then:
 * $f$ is not of exponential order.

That is, it grows faster than any exponential.

Proof
From Gamma Function is Continuous on Positive Reals, $f$ is continuous for $t \ge 0$.

Set $t > 0$.

From Stirling's Formula:


 * $t! \sim \sqrt {2 \pi t} \paren {\dfrac t e}^t$

where $\sim$ denotes asymptotic equality.

That is,

$t^t \, e^{-t}$ is of exponential order $a$.

Then, for $t$ sufficiently large, there exists a $K > 0$ such that:

This implies that $t^t$ is of exponential order, which is false.

From this contradiction it follows that $t!$ is also not of exponential order.