Exercise 18, § 1

Let $(G_\iota)_{\iota \in I}$ be a family of totally ordered groups indexed by a well-ordered set $I$; define an ordering on the direct sum $G'$ of the $G_\iota$ by taking the strictly positive elements to be the nonzero $(x_\iota)$ such that $x_\iota > 0$ for the first index $\iota$ such that $x_\iota \neq 0$. Show that $G'$ is totally ordered.