Strictly Positive Integer Power Function is Unbounded Above/General Case
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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, $b < \map f c$.
$\blacksquare$