Product of infinitesimal by inverse of non infinitesimal is infinitesimal

Question

This is probably very simple, though I am getting confused because of my inexperience with infinitesimals. Let $K$ be an ordered real-closed field. We say that $a \in K$ is finite if it's absolute value is less than $n$ for some positive integer $n$, and $a$ is infinitesimal if $ |a|<1/n$ for each positive integer $n$. I want to show that if $a,b$ are non-zero, $a$ infinitesimal and $b$ finite but not infinitesimal, then $a/b$ is infinitesimal. I tried using the fact that $|b|<n$ implies $1/|b| > 1/n$, however it seems that I need an inequality that goes the other way around.

Answer 1

The important thing is that $b$ is not infinitesimal, so there is some integer $m$ such that $|b| \geq \frac{1}{m}$, or rather that $\frac{1}{|b|} \leq m$. That gives you an inequality the other way around to use.