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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. enter image description here

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

Thanks.

Mike_bb
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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.

Mikhail Katz
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  • I don't fully understand your proof. Why do you say about inequality? – Mike_bb Sep 04 '23 at 08:05
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    @Mike_bb, you wanted to know why the two sequences give distinct hyperreal numbers, as claimed by Goldblatt. That's what I tried to explain. – Mikhail Katz Sep 04 '23 at 09:51
  • I want to know how to prove that infinitesimal or large numbers and rates of convergence are related. – Mike_bb Sep 04 '23 at 10:01
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    The idea is that the infinitely large number corresponding to a sequence (via the ultrapower construction) is (more precisely, can be viewed as; alternatively, "represents a measure of") the rate of convergence. The fact that $1,2,3,4,\ldots$ goes to infinity slower than $2,4,6,8,\ldots$ is supposed to be illustrated by the fact that the hyperreal corresponding to the sequence $1,2,3,4,\ldots$ is smaller than the hyperreal corresponding to the sequence $2,4,6,8,\ldots$. @Mike_bb – Mikhail Katz Sep 04 '23 at 10:03
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    Incidentally, I have a set of lecture notes on the hyperreals that is based on Goldblatt and is supposed to be more accessible than Goldblatt, including an introductory chapter with a discussion at the level of infinitesimal calculus (which Goldblatt does not have). This can be seen at https://u.math.biu.ac.il/~katzmik/coursenotes.pdf – Mikhail Katz Sep 04 '23 at 10:15