# Looking for a rigorous analysis book

I’m a mathematics undergrad student who finished his first university year succesfully. I got courses of calculus, but these weren’t very rigorous. I did learn about stuff like epsilon and delta proofs but we never made exercises on those things. The theory I saw contained proofs but the main goal of the course was to succesfully learn to solve integrals (line integrals, surface integrals, double integrals, volume integrals, …), solve differential equations, etc.

I already took proof based courses like linear algebra and group theory, so I think I am ready to start to learn rigorous real analysis, so I’m looking for a book that suits me.

I want the book to contain the following topics:

The usual analysis stuff:

• a construction of $\mathbb{R}$ or a system that takes $\mathbb{R}$ axiomatically for granted
• rigorous treatment of limits, sequences, derivatives, series, integrals
• the book can be about single variable analysis, but this is no requirement
• exercises to practice (I want certainly be able to prove things using epsilon and delta definitions after reading and working through the book)

Other requirements:

• The book must be suited for self study (I have 3 months until the next school year starts, and I want to be able to prepare for the analysis courses).

I have heard about the books ‘Real numbers and real analysis’ by Ethan D. Block and ‘Principles of mathematical analysis’ by Walter Rudin, and those seem to be good books.

• Terence Tao’s Analysis- 1 describes construction of $\mathbb{R}$ very well. Read the first answer to a question that I asked a while ago here–Good First Course in real analysis book for self study
• Rudin has rigorous development of limits, continuity, etc.. but so do Bartle, Sherbert’s Introduction to real analysis and Thomas Bruckner’s Elementary real analysis. The latter two deal with single variable only and contain really elementary examples of proving limits, continuity using $\epsilon-\delta$ definition, I don’t remember Rudin’s text having such solved examples. Worth checking out in my opinion.