Strictly Positive Integer Power Function Strictly Succeeds Each Element
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Theorem
Let $\struct {R, +, \circ, \le}$ be an ordered ring with unity.
Let $\struct {R, \le}$ be a directed set with no upper bound.
Let $n \in \N_{>0}$.
Let $f: R \to R$ be defined by:
- $\forall x \in R: \map f x = \circ^n x$
Then the image of $f$ has elements strictly succeeding each elements 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:
- $\exists e \in R: d \le e, c \le e$
$\struct {R, +, \circ, \le}$ is an ordered ring, so $\le$ is by definition a transitive relation.
Hence by transitivity:
- $b < e$
and:
- $1 < e$
By Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element:
- $e \le \map f e$
Thus by transitivity:
- $b < \map f e$
$\blacksquare$