What one should know before starting with the basics of real analysis?

Question

So I just wanna learn how can one construct the well-known sets N, Z, Q and etc and to prove all those properties on them we learn in school. But before that I wanna know the requirements.

I already know sentential and predicate logic(I mean all the basic "laws" and the ways of proving statements given some other statements). I've also been learning set theory for like 7 months. Can't say I mastered all the things I learned though... It gets progressively difficult and with each page you have to remember more and more definitions. So what are the prerequisits? Should I know like all the set theory or not? I know, it probably won't give much information, but I'm on page 95 of 240 pages textbook, I didn't even started with cardinality, ordinals and that other stuff which I don't know anything about.

Answer 1

This is a common philosophy that people want to start everything from the very bottom, but I think it may be more efficient to learn the set theoretical tools when you need them(if you are not interested in set theory to its own nature).

You probably want to read the Naive theory by Halmos which talks about almost all basic set theoretical results that you may be gonna use in Real analysis including the construction of N. For the construction of R you May refer to the beginning chapter of the The principals of mathematical analysis by Rudin. The construction of Z relatively is easy, which you can search online, and the construction of Q is similar to the construction of Z.

Answer 2

A serious understanding of real analysis requires a deeper appreciation of the way order relations work.

Consider the following two statements :

• There is no greatest natural number.
• There is no smallest positive rational number.

If you understand the above statements and can explain them to someone else without using any mathematical symbols then you already have the pre-requisites for taking an introductory real-analysis course.

Trivial parts of the theory are built on those two statements and the non-trivial parts are built on completeness of real numbers. That's what you should learn at the start of your real-analysis course.

The ideas of set theory and logic are necessary but intuitive ideas learnt in high school are sufficient for the purpose at hand. You don't need to be a master of these topics to deal with real-analysis.