Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma

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

Let $r \in \Q_{> 0}$ be a strictly positive rational number such that $r < 1$.

Then:
 * $1 < a^r < 1 + a r$

Proof
Define a real function $g_r: \R_{> 0} \to \R$ as:
 * $g_r \left({a}\right) = 1 + a r - a^r$

Then differentiating $a$ gives:
 * $D_a g_r \left({a}\right) = r \left({1 - a^{r - 1}}\right)$

We show now that the derivative of $g_r$ is positive for all $a > 1$:

So $D_a g_r \left({a}\right)$ is positive for all $a > 1$.

Whence, by Derivative of Monotone Function, $g_r$ is increasing for all $a > 1$.

Now, $g_r \left({1}\right) = r > 0$.

So $g_r \left({a}\right)$ is positive for all $a > 1$.

That is:
 * $0 < 1 + a r - a^r$

Adding $a^r$ to both sides of the above yields:
 * $a^r < 1 + a r$

Finally:

So, for $0 < r < 1$:
 * $1 < a^r < 1 + a r$