Power Function on Base Greater than One is Strictly Increasing/Rational Number

Theorem
Let $a \in \R$ be a real number such that $a > 1$.

Let $f: \Q \to \R$ be the real-valued function defined as:
 * $f \left({q}\right) = a^q$

where $a^q$ denotes $a$ to the power of $q$.

Then $f$ is strictly increasing.

Proof
Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z$ are integers and $s, u \in \Z_{>0}$ are strictly positive integers.

Let $\dfrac r s < \dfrac t u$.

From Ordering of Reciprocals:
 * $0 < \dfrac 1 a < 1$

So:

Hence the result.