Archimedean property written multiplicatively
My question: Let $(\Gamma,+,\leq)$ be an ordered abelian group. We know that archimedean property can be stated as: for all $a,b\in\Gamma$ with $a>0,b\geq0$, there exists $n\geq0$ such that $b\leq na$. However, if we consider the multiplicative case, namely $(\Gamma,\cdot,\leq)$ is the ordered abelian group. Is there exists Archimedean property written multiplicatively? I think there is. And I state that as follows: for all $a,b\in\Gamma$ with $b<1,a\leq1$, there exists $n\geq0$ such that ...