How to prove that infinitely small and large numbers can be used as measures of rates of convergence?

Question

I read "Lectures on the Hyperreals: An Introduction to Nonstandard Analysis" Robert Goldblatt.

He wrote that infinitely small and large numbers can be used as measures of rates of convergence.

It's intuitively clear for me. But how to prove this in rigorously way? Is it possible?

Thanks.

Answer 1

The sequences $x=(1,2,3,4,\ldots)$ and $y=(2,4,6,8,\ldots)$ satisfy the relation $y=2x$ and therefore so do their equivalence classes $[x]$ and $[y]$, so that $[y]=2[x]$ in the hyperreals. But we know that all nonzero real numbers $r$ satisfy $r\not=2r$. Therefore by the transfer principle, the inequality holds for all hyperreals, and in particular $[x]\not=[y]$.

Note that Goldblatt is not a historian and his historical claims should be taken with a grain of salt, including the assertion that

According to Cauchy, the sequence $1,\frac12,\frac13,\frac14,\ldots$ is an infinitesimal.

Cauchy specifically does not say that. Rather, Cauchy says in his Cours d'Analyse that such a sequence becomes an infinitesimal (implying a change of nature). For further details, see British Journal for the History of Mathematcs and the references cited there.