Power Function on Base greater than 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 $a > 1$.

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
We start by treating the right-sided limit.

Let $0 < r < 1$.

Lemma
From the lemma:
 * $1 < a^r < 1 + a r$

Also:

So from the Squeeze Theorem:
 * $\displaystyle \lim_{ r \mathop \to 0^{+} } a^r = 1$

We now treat the left-sided limit.

Let $-1 < r < 0$.

Also:

So from the Squeeze Theorem:
 * $\displaystyle \lim_{r \mathop \to 0^-} a^r = 1$

From Limit iff Limits from Left and Right:
 * $\displaystyle \lim_{r \mathop \to 0} a^r = 1$

Hence the result.