On ordered abelian groups containing $\mathbb{Z}$

Question

Let $\Delta$ be an ordered abelian group containing $\mathbb{Z}$ as a subgroup of index $e$. I need to show that for any positive element $\delta \in \Delta$, we have $e\delta \geq 1$.

I have no clue as how to do this?

Answer 1

$\Delta/Z=Z/eZ$, let $\delta\in \Delta, \delta>0$ and $p:\Delta\rightarrow \Delta/Z$, $ep(\delta)=p(e\delta)=0$, this implies $e\delta\in Z$, since $\delta>0, e\delta>0$ and $e\delta\in Z$ implies $e\delta\geq 1$.