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Nonstandard Analysis: Completeness of R via every limited hyperreal is infinitely close to a real #

conscipost

Member
Jan 26, 2012
39
In a book I am currently reading, the statement "every limited hyperreal is infinitely close to a real #" is shown to imply the completeness of R, that is that any subset A of R bounded above has a least upper bound. What the author offers to do is introduce this construction: for each natural n, let sn be the least k in the integers so that k/n is an upper bound of A. Then we are to take an unlimited N and let L, an element of R, be infinitely close to sN/N.

Without completeness I'm not sure why sn necessarily exists, can anyone give me some hints? Is it just because once I know the set is bounded above, I can start with an integer greater than this upper bound multiplied by n and "count down" so to say, checking whether each integer less than the last is an upper bound until I find one that is not?

Thanks,
 
Last edited:

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
In a book I am currently reading, the statement "every limited hyperreal is infinitely close to a real #" is shown to imply the completeness of R, that is that any subset A of R bounded above has a least upper bound. What the author offers to do is introduce this construction: for each natural n, let sn be the least k in the integers so that k/n is an upper bound of A. Then we are to take an unlimited N and let L, an element of R, be infinitely close to sN/N.

Without completeness I'm not sure why sn necessarily exists, can anyone give me some hints? Is it just because once I know the set is bounded above, I can start with an integer greater than this upper bound multiplied by n and "count down" so to say, checking whether each integer less than the last is an upper bound until I find one that is not?
I would like to know the name of the text/author.
I assume that by limited hyperreal that author means finite.

If you can find a copy of James Henle's Infinitesimal Calculus, there is a good discussion on this problem on page 114. Although Henle does not use limited hyperreal much of that discussion it does use many of the same ideas.
 

conscipost

Member
Jan 26, 2012
39
I would like to know the name of the text/author.
I assume that by limited hyperreal that author means finite.

If you can find a copy of James Henle's Infinitesimal Calculus, there is a good discussion on this problem on page 114. Although Henle does not use limited hyperreal much of that discussion it does use many of the same ideas.
I'm sorry, I'm new to the topic and didn't think about the possibility of different terms. The book is titled Lectures on the Hyperreals by Robert Goldblatt; this bit is on page 56. Comparing to Henle, limited is actually not finite but rather a # that is bound by two real numbers. infinite is not defined in Goldblatt, and rather infinitesimal and unlimited are used, infinitesimal meaning that the absolute value of the # is less than any positive real #, and unlimited to mean that the abs. is greater that any pos. real #.

One question, do the squares denote anything on page 114-115, or are these just formatting?

Thanks,
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
One question, do the squares denote anything on page 114-115, or are these just formatting?
\(\displaystyle \boxed{N}\) is the standard part of \(\displaystyle N\).

You really have to follow any text very closely.
This subject is relatively new, 1964. So there is absolutely no standard notation.
I do not know of that text book.