Multiplication property of equality for infinitesimals

Question

I am to prove this property of *$\Bbb R$: If $x \approx y$ and $u \approx v$ and $u,x$ are finite then $xu \approx yv$. My question is can I just use the transfer principle for the multiplication property of equality? Also since, $x - y \approx 0$ and $x$ is finite, wouldn't it be true that $y$ is finite as well?

Answer 1

Simply write $y=x+\epsilon$ and $v=u+\delta$ with infinitesimals $\epsilon,\delta$, and see what you get for $yv$.

Answer 2

One cannot use transfer directly in this case because the relation $\approx$ of infinite proximity is not an internal relation. However the proof is elementary and uses merely the fact that an infinitesimal multiplied by a finite hyperreal is still infinitesimal. The latter fact is immediate from the definition of an infinitesimal.