Power Function is Strictly Increasing on Positive Elements

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.

Suppose that $0 < x < y$

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

 * Odd Power Function is Strictly Increasing