Exercise 1, § 3

A linearly ordered group $G$ (not necessarily commutative) is said to be Archimedean if the set $I$ of elements $\geq e$ (the identity element of $G$) satisfies axiom (GR_{IV}) of § 2. Show that a linearly ordered group $G$ is isomorphic to a subgroup of the additive group $\mathbf{R}$ if and only $G$ is Archimedean (distinguish two cases according as the set of elements $> e$ in $G$ has or has not a smallest element, and use Proposition 1 of § 2).