Real Number between Zero and One is Greater than Power/Natural Number/Proof 1

Proof
For all $n \in \N$, let $\map P n$ be the proposition:
 * $0 < x < 1 \implies 0 < x^n \le x$

Basis for the Induction
$\map P 1$ is true, since $0 < x < 1 \implies 0 < x^1 \le x$ by definition of exponent of $1$.

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:


 * $0 < x < 1 \implies 0 < x^k \le x$

Then we need to show:


 * $0 < x < 1 \implies 0 < x^{k + 1} \le x$

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:
 * $\forall n \in \N: 0 < x < 1 \implies 0 < x^n \le x$

Hence the result.