X to the x is not of Exponential Order/Lemma

Lemma
Let $f: \R \to \R$ be defined on $\left [{0 \,.\,.\, \to} \right)$ with $f \left({x}\right) = x^x$.

Aiming for a contradiction to the main theorem, assume there exist strictly positive real constants $M$, $K$, $a$ such that:


 * $\forall t \ge M: \left\vert {f \left({t}\right)} \right \vert < K e^{a t}$

Then, there exists a constant $C$ such that:
 * $\forall t > C: \left\vert {f \left({t}\right)} \right \vert > K e^{a t}$

Proof
By the definition of power:
 * $f \left({t}\right) = \exp \left({t \ln t}\right)$

By Exponential of Real Number is Strictly Positive, we can reduce the lemma into the existence of $C$ such that:
 * $\forall t > C: f \left({t}\right) > K e^{a t}$

We will divide into two cases.

Case 1: $K > 1$
Assume that $t > K e^a$.

Here, $C = K e^a$.

Case 2: $K \le 1$
Assume that $t > e^a$.

Here, $C = e^a$.