A question about a mathematical analysis book

I am a newcomer to Analysis. All knowledge I know about "Analysis" are differentials,limit and integration (basically, what we have been taught in high school)

I am studying Principles of Mathematical Analysis by Walter Rudin, and I must say that this is, by far, the most difficult book I have ever touched. Even my Ph.D Microeconomic Theory by Mas Colell is not as hard as this book. I struggled a lot with Rudin,trying to draw some pictures about open sets, closed sets,perfect sets,..., but I still cannot grasp the "gist" of Analysis. So, I realize that Rudin may not be good enough for self-study due to its superb terseness and decide to switch to other Analysis's book in order to get a good picture of what Real Analysis is. Rudin lacks of ability to do that since it assumes the reader must know some topology to a degree.

Hence, I am looking for a good substitution of Rudin and have seen on Amazon that there is a book called "Mathematical Analysis I" and "Mathematical Analysis II" written by Zorich that can cover Real Analysis that a math major undergraduate needs to know. Is there anyone know how good those books are so I can prepare a very great Analysis courses for myself? I will definitely read Rudin again, but now I think I need some books as great as Spivak's Calculus book so that I can understand Analysis in a very good way. I know Apostol's is a good book, but I want to learn a book that can cover as much material as it can in the shortest time so that I can reach Royden's later on.

I thank you very much for your advice.

EDIT 1: So, after having read all of your advice (and Mr.Dave's), I WANT TO "invent" a way to read and understand Rudin's Analysis. Since Rudin is now causing me difficulty in understanding concepts in chapter 2's topology(perfect sets,open sets, closed set, closure sets, cantor sets,connected sets), I have 3 choices:

1. Spivak's Calculus => Apostol's => Rudin's => Royden's
2. Spivak's Calculus => Zorich (I+II) => Rudin's => Royden's
3. Spivak's Calculus => Munkres's Introduction to Topology => Rudin's => Royden's

Which one do you think it is the most appropriate way? I thank you for your advice.

EDIT 2: I thank you very much for your advice, Mr.Dave. However, I have a big trouble with topology in the very chapter 2 of Rudin's Analysis. After reading Spivak's, does my trouble go away? Because after reading materials in chapter 2, I have A VERY VAGUE understanding of what a limit point, perfect sets,infinite,finite,countable,uncountable, open sets,closed sets,...etc. Topology is really really difficult.

If you're concerned about time, I don't think reading Calculus by Spivak is the best thing to do.

Either Zorich or Apostol is a great choice. I would say that they're "intermediate" in difficulty. Zorich contains more in-depth discussions of topics, and more examples than does Apostol.

If your goal is only to move on to Royden, you'll probably cover the material more quickly in Apostol. Zorich covers a number of topics not addressed in Apostol, such as vector analysis and submanifolds of $\mathbf{R}^n$. These are important topics, but not direct prerequisites for Royden. Still, I think with all the Lagrange multipliers and similar tools people use in economics, the submanifold topic is important if you want to understand the theory very clearly.

Zorich's first volume is quite concrete, whereas Apostol becomes abstract more quickly. This is probably because he doesn't want to duplicate what would be in a rigorous calculus book like his Calculus, although he does this more than Rudin's book does. His analysis book was for second- or third-year North American students, whereas Zorich's is, at the outset, for first-year Russian ones. Russian students have typically had some calculus in high school, but the practical portion of learning calculus continues into their first-year of university, with harder problems. So in Zorich I, you deal with hard problems on real numbers, rather than delving straight into metric spaces as you would in Apsotol's book.

Zorich covers only Riemann integration, whereas Apostol has chapters on Riemann-Stieltjes integration in one variable, Lebesgue integrals on the line, multiple Riemann integrals, and multiple Lebesgue integrals. The treatment of Lebesgue integration is less abstract than in more advanced books. Since it's limited to $\mathbf{R}$ or $\mathbf{R}^n$, it's more elementary, but at the same time there is some loss in clarity compared to the abstract theory on measure spaces. One reason to use Apostol might be a sort of introduction to the Lebesgue theory before returning to it at a higher level and "relearning" certain parts of it. Whether you'd want this is up to you.

The fact that both Rudin and Apostol have chapters on Riemann-Stieltjes, rather than Riemann, integration, indicates to me that they assumed students had already studied Riemann integrals rigorously, and would be ready for a generalized version right from the start. Considering the type of calculus courses most students take these days, this is rarely the case now. Zorich doesn't have this problem.

All in all, for a typical student who is good at math but didn't learn their calculus from a book like Spivak's or Apostol's Calculus, I think Zorich is the better choice because of the more concrete approach in the first volume (this doesn't necessarily mean easier). On the other hand, time constraints might cause you to prefer Apostol's analysis book.

EDIT: An important point that I neglected to mention is that Zorich's book will be much better than Apostol's if you aren't yet acquainted at all with multivariable calculus. A practical knowledge of some multivariable calculus is probably one of the tacit assumptions that Apostol and Rudin make about their readers, which is what allows them to deal with multivariable calculus in a briefer and more abstract way. Compare Apostol's 23-page chapter on multivariable differential calculus to Zorich's 132 (in the Russian version).

EDIT: Based on your later comments, I would suggest that reading

1. Spivak's Calculus,

2. Whichever you prefer of Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis.

would be a reasonable plan.

However, before beginning the multivariable calculus parts of those books, it would be best to learn some linear algebra and multivariable calculus from another source. This could be Volume 2 of Apostol's Calculus. You could instead skip straight to the multivariable part of Volume 1 of Zorich, but you'd have to learn the necessary linear algebra elsewhere first. I don't recommend Spivak's Calculus on Manifolds if you want to learn multivariable calculus for the first time. Also, you won't need Munkres - you'll get enough topology to start with in whichever other book you read.

EDIT: In answer to your additional question, these topics are mostly not discussed in Spivak.

However, Spivak is an excellent introduction to the mathematical way of thinking. That is, although you will not learn all the specific facts that arise in higher-level books (you do learn many, of course), you will learn to read and understand definitions, theorems and proofs the way mathematicians do, and to produce your own proofs. You will become intimately familiar with real numbers, sequences of real numbers, functions of a real variable and limits, so you will have examples in mind for the more general structures introduced in topology. You will also solve difficult problems.

So it is not that you will know topology already when you've read Spivak's book, it's mainly that it ought to be easier for you to learn because you will have improved your way of approaching mathematical questions. Countable sets are in fact discussed in the exercises to Spivak, however.

I can't guarantee that your trouble will "go away," but there is a good chance it will.

Also feel free to use Zorich rather than Rudin or Apostol, after Spivak, or even to jump straight to the multivariable part of Zorich at the end of Volume 1 and start reading from there.

I have found the 3 volumes by Garling "A Course in Mathematical Analysis" published by Cambridge University Press excellent. The author has several years of teaching experience and has thought through the subject in great depth. Vol. I ISBN: 978-1-107-03202-6 Vol. II ISBN: 978-1-107-03203-3 Vol III ISBN: 978-1-107-03204-0