Power of Positive Real Number is Positive/Integer

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Theorem

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

Let $n \in \Z$ be an integer.


Then:

$x^n > 0$

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


Proof

By Power of Positive Real Number is Positive: Natural Number, the theorem is already proven for non-negative integers.

Suppose $n \in \Z_{< 0}$.

When $n < 0$, by Order of Real Numbers is Dual of Order Multiplied by Negative Number:

$-n > 0$

Then, by Power of Positive Real Number is Positive: Natural Number:

$x^{-n} > 0$

Therefore, by Reciprocal of Strictly Positive Real Number is Strictly Positive:

$x^n = \dfrac 1 {x^{-n} } > 0$

$\blacksquare$