Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element

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Theorem

Let $\struct {R, +, \circ, \le}$ be an ordered ring with unity.

Let $x \in R$ with $x > 1$ and $x > 0$.

Let $n \in \N_{>0}$.

Then:

$\circ^n x \ge x$

Proof

The proof proceeds by induction:

If $n = 1$, then $\circ^n x = x$.

So:

$\circ^n x \ge x$

Suppose that $\circ^n x \ge x$.

Then since $x > 1$:

$\circ^n x > 1$

By Product of Positive Element and Element Greater than One:

$x \circ \paren {\circ^n x} > x$

Hence:

$\circ^{n + 1} x \ge x$

$\blacksquare$