Exercise 34, § 1

Let $G = \mathbf{Z}^\mathbf{N}$ be the product of a countably infinite family of totally ordered groups $\mathbf{Z}$, and let $H = \mathbf{Z}^{(\mathbf{N})}$ be the filtered convex subgroup, the direct sum of the factors of G. Show that the lattice ordered group $G/H$ (VI, p. 30, Ex. 4) is not archimedean, although G and H are completely lattice ordered.