Exercise 34, § 2

Let E be a totally ordered set. A (Dedekind) cut is a partition (S, D) of E into two nonempty subsets such that D is of the form $M(A)$. the set of upper bounds for some nonempty subset A of E : a necessary and sufficient condition for D to have a least element x is that $D = (x, \to (, so S = ) \leftarrow , x($. In a totally ordered abelian group G, a cut (S, D) is called proper if for all $e > 0$ in G there exist $x' \in S$ and $x'' \in D$ such that $x'' - x' < e$. Show that (S, D) is proper if and only if D is invertible in the monoid $\mathfrak{M}(G)$ of major subsets of G (VI, p. 35, Ex. 30); all cuts are proper if and only if G is archimedean (VI, p. 35, Ex. 31); and hence \* isomorphic to a subgroup of R (VI, p. 36, Ex. 33). \*