Natural Number Power is of Exponential Order Epsilon

Theorem

Let $n \in \N$ be a natural number.

Then:

$t \mapsto t^n$

is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.

Proof

The proof proceeds by induction on $n$, where $n$ is the degree of the polynomial.

Basis for the Induction

When $n = 0$, the mapping is a constant mapping.

By Constant Function is of Exponential Order Zero and Raising Exponential Order, the mapping is of exponential order $\epsilon$.

This is the basis for the induction.

Induction Hypothesis

Fix $k \in \N$ with $k \ge 0$.

Assume:

$t^k \in \EE_\epsilon$

That is,

$\forall t \ge M : \size {t^k} < K e^{\epsilon t}$

For some $M > 0$, $K > 0$, and for any $\epsilon > 0$ arbitrarily small.

This is our induction hypothesis.

Induction Step

We have:

$x^k$ is of exponential order epsilon by the induction hypothesis

Identity is of Exponential Order Epsilon

Product of Functions of Exponential Order

Therefore, $x^{k + 1}$ is also of exponential order epsilon.

The result follows by the Principle of Mathematical Induction.

$\blacksquare$