X to the x is not of Exponential Order

Theorem
Let $f: \R \to \R$ be defined on $\hointr 0 \to$ with $\map f x = x^x$.

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

That is, it grows faster than any exponential.

Lemma
By the definition of power:
 * $\map f t = \map \exp {t \ln t}$

The theorem is equivalent to that there do not exist strictly positive real constants $M$, $K$, $a$ such that:


 * $\forall t \ge M: \size {\map f t} < K e^{a t}$

such constants $M$, $K$ and $a$ exist.

From the lemma, there exists a constant $C$ such that:
 * $\forall t > C: \size {\map f t} > K e^{a t}$

For any $M$ chosen, $M + C > C$, thus from the lemma:
 * $\size {\map f {M + C} } > K e^{a t}$

However, $M + C > M$, thus from the assumption:
 * $\size {\map f {M + C} } < K e^{a t}$

Thus a contradiction has arisen.

It follows by Proof by Contradiction that our assumption was false.

Hence the result.