Looking for a rigorous analysis book

Rudin's text is good and has almost everything you want. But I feel that Rudin + some other book may suit your purposes better.

• 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.
• Rudin no doubt has very good exercises and if you get stuck at any of them, there are solutions available online in a pdf and very helpful companion notes-here for understanding theory better with an exercise set at the end of each chapter preparing you for Rudin's exercises.
  Though reading it can be sometimes be very frustrating what with lack of examples. I usually advise people to first read through a gentler text like Sherbert then come back to it.
  Browse through all the books first and if you feel you're ready for Rudin's, go for it.
• After you're done with whatever analysis text you choose to read, this three problem book set published by AMS is very good. Read more about it here- Problems in Mathematical Analysis.
• Other good books I've heard of but personally have no experience in-Serge Lang's undergraduate analysis, Charles Pugh's Real Mathematical Analysis, Stephen Abbott's Understanding Analysis.

The reason why I shall never write a Calculus textbook is because Michael Spivak's Calculus is a masterpiece written at a level that I would never be able to attain.

If you find it too advanced, I suggest that you read first another book by Spivak: The Hitchhiker's Guide to Calculus.

Spivak's Calculus is still the best book for a rigorous foundation of Calculus and introduction to Mathematical Analysis. It includes, in its last chapter, very interesting topics, such as construction of transcendental number and the proof that e is transcendental, and the proof that $\pi$ is irrational. It also includes, in the Appendix, a rigorous construction of the set of real numbers by Dedekind cuts.

It is, in my opinion, by far the best Calculus book, if one wants to understand well the $\delta-\varepsilon$ definitions, and be able to solve challenging problems, which require these definitions. One of my favourite Spivak problems of this kind is the following:

Let $f:\mathbb R\to\mathbb R$ be a function $($not necessarily continuous$)$, which has a real limit at every point. Set $$ g(x)=\lim_{y\to x}f(y),\quad x\in\mathbb R. $$
Show that $g$ is continuous.

However, Spivak's book treats only one-dimensional Calculus.

Second reading, right after Spivak: Principles of Mathematical Analysis, by W. Rudin. Apart from a good introduction of the Metric Space Theory (to learn what is open, closed, compact, perfect and connected set), there is a number of results on convergence of sequences of functions, multivariate calculus, introduction of $k-$forms and introduction to Lebesgue measure.

As a sequel, one should consider the great little classic, Spivak's Calculus on Manifolds, which provides an elegant and concise introduction of $k-$forms and proof of Stokes Theorem in Euclidean spaces and manifolds.