User:Keith.U/Sandbox

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 $r, t \in \Z, s, u \in \Z_{>0}$.

Suppose $\dfrac{r}{s} < \dfrac{t}{u}$.

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

So:

Hence the result.