Exercise 33, § 1

a) A totally ordered group G is archimedean if and only if, for any pair $x > 0, y > 0$ of elements of G, there exists an integer $n > 0$ such that $y \leq nx$. \* b) Any totally ordered, archimedean, completely lattice ordered group G is isomorphic to $\{0\}, \mathbf{Z}$ or $\mathbf{R}$ (ruling out the first two cases, fix $a > 0$ in G, and make each $x \in G$ correspond to the infimum of the rational numbers $p/q$ such that $qx \leq pa$). c) Deduce that every archimedean totally ordered group G is isomorphic to a subgroup of $\mathbf{R}$ (notice that $\mathscr{M}(G)$ is totally ordered). d) The lexicographic product $\mathbf{Z} \times \mathbf{Z}$ is totally ordered and not archimedean.