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

Theorem
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.

Proof
Proof by induction on $n$:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $a^{n + 1} < a^n$

$\map P 0$ is the case:

Basis for the Induction
$\map P 1$ is true, since:

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:


 * $a^{k + 1} < a^k$

Then we need to show:


 * $a^{k + 2} < a^{k + 1}$

Induction Step
This is our induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\ge 0}: a^{n + 1} < a^n$

Hence the result, by definition of strictly decreasing.