Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number

Theorem
Let $a \in \R_{> 0}$ be a strictly positive real number such that $0 < a < 1$.

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

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

Then:
 * $\displaystyle \lim_{r \mathop \to 0} f \left({r}\right) = 1$

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

So: