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 Real Number Ordering is Compatible with Multiplication: Negative Factor:
- $-n > 0$
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:
- $x^n = \dfrac 1 {x^{-n} } > 0$
$\blacksquare$