Strictly Positive Integer Power Function Strictly Succeeds Each Element

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

Suppose that $\left({R,\le}\right)$ is a Directed Set with no upper bound.

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

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

Then the image of $f$ has elements strictly succeeding each element of $R$.

Proof
Let $b \in R$.

By Directed Set has Strict Successors iff Unbounded Above, there is a $c \in R$ such that $b < c$ and a $d \in R$ such that $1 < d$.

By the definition of a directed set, there is an $e \in R$ such that $d \le e$ and $c \le e$.

By transitivity $b < e$ and $1 < e$.

By Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element, $e \le f(e)$.

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