Strictly Positive Integer Power Function is Unbounded Above/General Case

Theorem

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

Suppose that $R$ has no upper bound.

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

Let $f: R \to R$ be defined by:

$\map f x = \circ^n x$

Then the image of $f$ is unbounded above in $R$.

Proof

Let $1_R$ be the unity of $R$.

Let $b \in R$.

We will show that $b$ is not an upper bound of the image of $f$.

Since $R$ is totally ordered and unbounded above, there is an element $c \in R$ such that $b < c$ and $1_R < c$.

By Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element, $c \le \map f c$.

Thus by transitivity of ordering, $b < \map f c$.

$\blacksquare$