X to the x is not of Exponential Order/Lemma

Lemma
Let $f: \R_{>0} \to \R$ be defined as:
 * $\forall x \in \R_{>0}: \map f x = x^x$

Let there exist strictly positive real constants $M, K, a \in \R_{> 0}$ such that:


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

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

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

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

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:
 * $C = e^a$