Uniqueness of Positive Root of Positive Real Number/Positive Exponent/Proof 1
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Theorem
Let $x \in \R$ be a real number such that $x > 0$.
Let $n \in \Z$ be an integer such that $n > 0$.
Then there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.
Proof
Let the real function $f: \hointr 0 \to \to \hointr 0 \to$ be defined as:
- $\map f y = y^n$
First let $n > 0$.
By Identity Mapping is Order Isomorphism, the identity function $I_\R$ on $\hointr 0 \to$ is strictly increasing.
We have that:
- $\map f y = \paren {\map {I_\R} y}^n$
By Product of Positive Strictly Increasing Mappings is Strictly Increasing, $f$ is strictly increasing on $\hointr 0 \to$.
From Strictly Monotone Mapping with Totally Ordered Domain is Injective:
- there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.
$\blacksquare$