Power Function on Base between Zero and One is Strictly Decreasing

Theorem

Positive Integer

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

Let $f: \Z_{\ge 0} \to \R$ be the real-valued function defined as:

$\map f n = a^n$

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

Then $f$ is strictly decreasing.

Integer

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

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

$\map f k = a^k$

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

Then $f$ is strictly decreasing.

Rational Number

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.

Real Number

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

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

$\map f x = a^x$

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

Then $f$ is strictly decreasing.