User:Dfeuer/Archimedean Totally Ordered Group is Abelian/Lemma 1
Theorem
Let $G$ be an Archimedean totally ordered group with a smallest strictly positive element, $\epsilon$.
Then $G$ is the infinite cyclic group generated by $\epsilon$, with the natural ordering.
Proof
Let $e$ be the identity element of $G$.
Let $x\in G$.
Since $G$ is totally ordered, either $x=e$, $x > e$, or $x < e$.
$e$ is in the group generated by $\epsilon$ by ....
Suppose that $x>e$.
Then $x\ge \epsilon$ by the premise.
Since $G$ is Archimedean, there is a smallest $n \in \N_{>0}$ such that $\epsilon^n \ge x$.
Suppose for the sake of contradiction that $\epsilon^n > x$.
Then $\epsilon^n x^{-1} > e$.
Thus by our choice of $\epsilon$, $\epsilon^n x^{-1} \ge \epsilon$, so $\epsilon^{n-1} \ge x$, a contradiction.
Thus we conclude that $x = \epsilon^n$
Suppose instead that $x < e$.
Then $x^{-1} > e$, so for some $n \in\N_{>0}$, $x^{-1}=\epsilon^n$.
Thus $x = \epsilon^{-n}$.
Finally, since $\epsilon > e$, $x\epsilon > x$ for each $x\in G$, so $G$ has the natural ordering.
$\blacksquare$