# Power Function on Base Greater than One is Strictly Increasing/Positive Integer

## Theorem

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

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 increasing.

## Proof

Fix $n \in \Z_{\ge 0}$.

$0 < \dfrac 1 a < 1$
$\paren {\dfrac 1 a}^{n + 1} < \paren {\dfrac 1 a}^n$
$\dfrac 1 {a^{n + 1} } < \dfrac 1 {a^n}$
$a^n < a^{n + 1}$

Hence the result.

$\blacksquare$