Power of Strictly Positive Real Number is Strictly Positive/Positive Integer

Theorem

Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Then:

$x^n > 0$

where $x^n$ denotes the $n$th power of $x$.

Proof

Proof by Mathematical Induction:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

$\forall x \in \R_{>0}: x^n > 0$

$\map P 0$ is true, as this just says:

\(\ds x^0\)

\(=\)

\(\ds 1\)

Definition of Integer Power

\(\ds \)

\(>\)

\(\ds 0\)

Basis for the Induction

$\map P 1$ true, as this just says:

\(\ds x^1\)

\(=\)

\(\ds x\)

Definition of Integer Power

\(\ds \)

\(>\)

\(\ds 0\)

This is our basis for the induction.

Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 0 $, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

$\forall x \in \R_{>0}: x^k > 0$

Then we need to show:

$\forall x \in \R_{>0}: x^{k + 1} > 0$

Inductive Step

This is our induction step:

\(\ds x^k\)

\(>\)

\(\ds 0\)

Induction Hypothesis

\(\ds \leadsto \ \ \)

\(\ds x^{k + 1}\)

\(>\)

\(\ds 0\)

Multiply both sides by $x > 0$

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

Therefore:

$\forall n \in \Z_{\ge 0}: \forall x \in \R_{>0}: x^n > 0$

$\blacksquare$