# Power Function on Base between Zero and One is Strictly Decreasing/Rational Number

## Theorem

Let $a \in \R$ be a real number such that $0 < a < 1$.

Let $f: \Q \to \R$ be the real-valued function defined as:

- $\map f q = a^q$

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

Then $f$ is strictly decreasing.

## Proof

Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z, s, u \in \Z_{>0}$.

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

Then:

\(\displaystyle r u\) | \(<\) | \(\displaystyle t s\) | $\quad$ Real Number Ordering is Compatible with Multiplication | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^{r u}\) | \(>\) | \(\displaystyle a^{t s}\) | $\quad$ Power Function on Base between Zero and One is Strictly Decreasing: Integer | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {a^r}^u\) | \(>\) | \(\displaystyle \paren {a^t}^s\) | $\quad$ Product of Indices of Real Number: Integers | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sqrt [u] {\paren {a^r}^u}\) | \(>\) | \(\displaystyle \sqrt [u] {\paren {a^t}^s }\) | $\quad$ Root is Strictly Increasing | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^r\) | \(>\) | \(\displaystyle \sqrt [u] {\paren {a^t}^s }\) | $\quad$ Definition of $u$th Root | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sqrt [s] {\paren {a^r} }\) | \(>\) | \(\displaystyle \sqrt [s] {\sqrt [u] {\paren {a^t}^s } }\) | $\quad$ Root is Strictly Increasing | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sqrt [s] {\paren {a^r} }\) | \(>\) | \(\displaystyle \sqrt [u] {\sqrt [s] {\paren {a^t}^s } }\) | $\quad$ Root is Commutative | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sqrt [s] {\paren {a^r} }\) | \(>\) | \(\displaystyle \sqrt [u] {\paren {a^t} }\) | $\quad$ Definition of $s$th Root | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^{r / s}\) | \(>\) | \(\displaystyle a^{t / u}\) | $\quad$ Definition of Rational Power | $\quad$ |

$\blacksquare$