Power Function is Strictly Increasing on Positive Elements

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Theorem

Let $\left({R, +, \circ, \le}\right)$ be an ordered ring.

Let $x, y \in R$.

Let $n \in \N_{>0}$ be a strictly positive integer.

Let $0 < x < y$.


Then:

$0 < \circ^n \left({x}\right) < \circ^n \left({y}\right)$


Proof

The result follows by repeated application of Multiplying Positive Inequalities.


Also see