Exercise 5, § 1

a) An ordered vector space E is said to be archimedean if every $x \in E$ such that the set of $nx$ ($n$ an integer $\geq 0$) is bounded above, is necessarily $\leq 0$ (A, VI, §1, Exer. 31). Show that this condition is equivalent to the following: the intersection of any plane passing through 0 with the convex cone P of elements $\geq 0$ of E is a closed angular sector. Show that for an ordered vector space to be archimedean, it is necessary and sufficient that it be isomorphic to a subspace of a fully lattice-ordered space (cf. A, VI, §1, Exer. 31).

b) Let F be the Riesz space $\mathbf{R}^{\mathbf{R}}$ of real-valued functions defined on $\mathbf{R}$ and let H be the subspace $\mathscr{B}(\mathbf{R})$ of bounded functions on $\mathbf{R}$; H is an isolated subspace of F (Exer. 4). Show that the Riesz space $E = F/H$ (Exer. 4) is not archimedean: more precisely, show that for every element $x \geq 0$ of E, there exists a $y \in E$ such that $y \geq nx$ for every integer $n \geq 0$.