Strictly Positive Integer Power Function is Unbounded Above/General Case

Theorem
Let $\left({R,+,\circ,\le}\right)$ be a totally ordered ring with unity.

Suppose that $R$ has no upper bound.

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

Let $f: R \to R$ be defined by $f \left({x}\right) = \circ^n x$.

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, $c \le f(c)$.

Thus by transitivity, $b < f(c)$.