Positive Power Function on Non-negative Reals is Strictly Increasing

Theorem
Let $a \in \Q_{\gt 0}$ be a strictly positive rational number.

Let $f_a: \R_{\ge 0} \to \R$ be the real function defined as:
 * $f_a \left({x}\right) = x^a$

Then $f_a$ is strictly increasing.

Real Index
If $a \in \R_{\gt 0}$ is a strictly positive real number, then the same result applies. Just use the real index variations of the theorems used to prove this one.

However, this result is specifically stated for a rational index, as this page is used to prove something else.

Proof
By the power rule for derivatives:
 * $D_x \left({x^a}\right) = a x^{a-1}$

By power of positive real number is positive, it is seen that:
 * $x \gt 0 \implies x^{a-1} \gt 0$

By Positive Real Numbers Closed under Multiplication, it follows that $D_x \left({x^a}\right) \gt 0$ for all $x \in \left({{0}\,.\,.\,{+\infty}}\right)$.

And so by Derivative of Monotone Function, $f_a$ is strictly increasing