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I am enrolled in real analysis course in university. Our professor started right from the beginning defining the real numbers and all the usual operations on real numbers (like addition, multiplication, etc.) quite rigourously. I liked that as it is what a mathematics course should teach specially a course like real analysis. But to supplement my study I could not find a single book doing that. A course in pure mathematics did to an extent but that is very short introduction. I am requesting for the recommendation for a book which deals with rigourous definition of real numbers and also treats the subject rigourously.

dfeuer
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ankur singh
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    Unfortunately, most "rigorous" definitions of the reals I know require a mathematical maturity well beyond that usually found among students of first year analysis, and this is the reason this isn't usually done "rigorously" at that stage but later. – DonAntonio Sep 19 '13 at 14:39
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    Rudin's Principles of Mathematical Analysis & Pugh's Real Mathematical Analysis both have good treatments on Dedekind Cuts. –  Sep 19 '13 at 14:50
  • @DonAntonio: why do you say so? Some variants of the Dedekind cut approach (in particular, lower sets of rationals without maxima) are quite intuitive, no? – dfeuer Sep 19 '13 at 14:56
  • I don't think so, @dfeuer...never mind that in my university they were the second chapter in our first year text book: nobody taught that at that time. – DonAntonio Sep 19 '13 at 15:59
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    @dfeuer: I agree that the idea behind Dedekind cuts can be made pretty natural intuitively, and it’s easy to prove that you get the order-completion; the problem is that the necessary technical details make the definitions of the arithmetic operations rather messy. Equivalence classes of Cauchy sequences are less intuitive to start with, I think, and they don’t give you the order properties in a very nice way, but they give you the arithmetic properties very easily. I prefer Dedekind cuts, but it probably really is six of one, half a dozen of the other. – Brian M. Scott Sep 19 '13 at 18:44

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Plenty of books include constructions of the real numbers from the natural numbers. See for instance Rudin's Principles of Mathematical Analysis.

Try also these books:

lhf
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In addition to the books that lhf listed, look at Edmund Landau's Foundations of Analysis.

The German original was published in 1930, an English translation was published in 1951, and various other translations and reprintings have appeared since 1930.

I'm surprised that no one has mentioned Landau's book yet, since his book is famous for being perhaps the simplest and most straightforward treatment ever published for what you are asking about. In fact, even the The MacTutor History of Mathematics archive has a web page devoted to Landau's book.

Dave L. Renfro
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Actually, this is one of the topics that is usually covered in many different courses in undergraduate level and that is bad. In my own university, we studied Peano's axioms for natural numbers in a naive set theory course and we learned how to construct integers from natural numbers by defining an equivalence relation and then we proved properties of integers from scratch. Then in abstract algebra we studied how to construct the field of fractions for integral domains like $\mathbb{Z}$ and then in Analysis I we studied Dedekind cuts and Cauchy sequences.

You can find the Dedekind cuts approach in Rudin's Principles of Mathematical Analysis at the end of chapter 1. The Cauchy sequence approach is not directly discussed in Rudin's book, but you can find it in the problem set of chapter 3. Problems #23,#24 and #25 talk about constructing a completion for a metric space by using Cauchy sequences.

You can see the construction of field of fractions for an integral domain in an abstract algebra book like Herstein's Abstract Algebra. And construction of integers from natural numbers is discussed in many books about naive set theory.

user66733
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You can take look at Topology book by Munkres.

Arash
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