Real Number to Negative Power/Positive Integer

Theorem
Let $r \in \R_{> 0}$ be a (strictly) positive real number. Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $r^n$ be defined as $r$ to the power of $n$.

Then:


 * $r^{-n} = \dfrac 1 {r^n}$

Proof
Proof by induction on $m$:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $r^{-n} = \dfrac 1 {r^n}$

$\map P 0$ is the case:

Basis for the Induction
$\map P 1$ is the case:

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:
 * $r^{- k} = \dfrac 1 {r^k}$

Then we need to show:
 * $r^{- \paren {k + 1}\ } = \dfrac 1 {r^{k + 1} }$

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:
 * $\forall n \in \Z_{\ge 0}: r^{-n} = \dfrac 1 {r^n}$