real analysis rigourous definition of real numbers 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.
 A: 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:


*

*Number Systems and the Foundations of Analysis by Mendelson

*The Number System by Thurston

*The Structure of Number Systems by Parker

A: 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.
A: 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.
A: You can take look at Topology book by Munkres.
