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

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

From Ordering of Reciprocals:

- $0 < \dfrac 1 a < 1$

From Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer:

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

From Real Number to Negative Power: Positive Integer:

- $\dfrac 1 {a^{n + 1} } < \dfrac 1 {a^n}$

From Ordering of Reciprocals:

- $a^n < a^{n + 1}$

Hence the result.

$\blacksquare$