Strictly Positive Integer Power Function Strictly Succeeds Each Element

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Theorem

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

Let $\left({R, \le}\right)$ be a Directed Set with no upper bound.

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

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

$\forall x \in R: f \left({x}\right) = \circ^n x$.


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:

$\exists c \in R: b < c$
$\exists d \in R: 1 < d$

By the definition of a directed set:

$\exists e \in R: d \le e, c \le e$

By transitivity:

$b < e$

and:

$1 < e$

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

$e \le f \left({e}\right)$

Thus by transitivity:

$b < f \left({e}\right)$

$\blacksquare$