# Math Help - group homomorphism

1. ## group homomorphism

Let $G\text{ and }H$ be additive subgroups of $\mathbb{R}$ both containing 1. Let $\eta:G\rightarrow H$ be an order preserving group homomorphism such that $\eta(1)=1$. Show that $\eta(g)=g$ for all $g\in G$.

The idea that I am working with is to let $s,t\in G$ such that
$s-t>1$ then there exists an integer $m$ such that
$s>m>t$.

now apply $\eta$ and get the following
$\eta(s)-\eta(t)>1$ and $\eta(s)>\eta(m)>\eta(t)$
and suppose that $\eta(g)>g$ for all $g\in G$. In this way I hope to get some contradiction

Or is there another way?

2. This problem is still bugging me... does anybody have any ideas?