User:Dfeuer/sandbox

Lexicographic Orderings

 * User:Dfeuer/Definition:Lexicographic Ordering on Product
 * User:Dfeuer/Definition:Lexicographic Ordering of Finite Sequences


 * User:Dfeuer/Well-Founded Relation Determines Minimal Elements


 * Product of Positive Element and Element Greater than One


 * User:Dfeuer/Strictly Positive Power of Element Greater than One Not Less than Element

scrap:

Lemma
Let $S$ be an ordered set.

Let $F$ be a totally ordered field.

Let $f,g \colon S \to F$.

Let $h\colon S \to F$ with $h(x) = f(x)g(x)$.

Suppose that for some $q \in F$ and some $w \in S$ with $q > 0$, $x \succ w \implies f(x) \ge q$.

Suppose that $g$ is increasing and its image is unbounded above.

Then the image of $h$ is unbounded above.

Useful links
Axiom of Foundation at NLab