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So I am taking an analysis class in my university and I want a problem book for it.

The topics included in the teaching plan are

Real Numbers: Introduction to the real number field, supremum, infimum, completeness axiom, basic properties of real numbers, decimal expansion, construction of real numbers.

Sequences and Series: Convergence of a sequence, Cauchy sequences and subsequences, absolute and conditional convergence of an infinite series, Riemann's theorem, various tests of convergence.

Point-set Topology of: : Open and closed sets; interior, boundary and closure of a set; Bolzano-Weierstrass theorem; sequential definition of compactness and the Heine-Borel theorem.

Limit of a Function: Limit of a function, elementary properties of limits.

Continuity: Continuous functions, elementary properties of continuous functions, intermediate value theorem, uniform continuity, properties of continuous functions defined on compact sets, set of discontinuities.

I am already following up Michael J. Schramm's Introduction to Real Analysis for my theory

But a problem book with varied questions on the concepts would help me a lot.

Please recommend some problem books.

Thanks

P.S : I have already asked my professor to recommend some books but he always recommends baby Rudin and also doesn't provide a lot of assignments. I am not compatible with Rudin's book. Also his tests are very tough as he wants us to cook up counter examples and I am very poor in that. So I need a good problem book to master real analysis.

Pratik Patnaik
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3 Answers3

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Try these books:

  • Problems in Mathematical analysis I, II and III : W.J. Kaczor and M.T.Nowak

Book I deals with sequences and series, II deals with continuity and diffrentiabilty and III deals with integration

  • A problem book in real analysis: Asuman G. Aksoy an Mohamed A. Kahmsi

This book contains $11$ chapters and it covers almost all topics in analysis

  • Berkeley problems in Mathematics: P. N. D Souza and J. N. Silva

This book contains some interesting problems in Real analysis also!

For General Topology, try this:

  • Elementary Topology Problem Textbook: O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov

You should also try the following for general topology. This book contain lot of problems with sufficient hints

  • Topology of Metric spaces: Kumaresan

Enjoy!

Chinnapparaj R
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What follows are from my bookshelves, not from an extensive search, so it's likely you may find others by googling some of these titles. Although I've restricted this list to what you're actually asking for, I hope you realize that there have been well over 100 undergraduate level real analysis texts published in the last 50 some years, many of which are likely in your university library, and the problem sets (and text examples) in these books should not be overlooked if you later find yourself wanting to conduct an especially thorough search on a certain specific topic.

[1] Robert L. Brabenec, Resources for the Study of Real Analysis (2004)

[2] Raffi Grinberg, The Real Analysis Lifesaver (2017) [See my comments here.]

[3] W. J. Kaczor and M. T. Nowak, Problems in Mathematical Analysis I. Real Numbers, Sequences and Series (2000)

[4] W. J. Kaczor and M. T. Nowak, Problems in Mathematical Analysis II. Continuity and Differentiation (2001)

[5] W. J. Kaczor and M. T. Nowak, Problems in Mathematical Analysis III. Integration (2003)

[6] Sergiy Klymchuk, Counterexamples in Calculus (2010) [The title says "Calculus", but this book would also be useful in a beginning real analysis course.]

[7] B. M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selected Problems in Real Analysis (1992) [Most of this will be too advanced, but there are some problems in the first few chapters that might be appropriate.]

[8] Joanne E. Snow and Kirk E. Weller, Exploratory Examples for Real Analysis (2003)

[9] Murray R. Spiegel, Schaum’s Outline of Theory and Problems of Real Variables (1969)

Dave L. Renfro
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  • What chapters of makarov's text are appropriate for the undergrad level? – sun_Jiaoliao Dec 05 '24 at 07:51
  • @sun_Jiaoliao: All of Chapters I-V contain problems suitable for the (U.S.) undergraduate level, and to a lesser extent Chapters VI & VII (there are 10 chapters in the book), but probably even the earliest chapters contain problems possibly outside the background knowledge an undergraduate would typically be expected to have, and many problems even within this background knowledge are outside the ability level of a large majority of undergraduates. I suppose the book can be described as being at a more advanced level and having a higher-ability-targeted audience than the Kaczor/Nowak books. – Dave L. Renfro Dec 05 '24 at 13:49
  • i find chapters 1-3 are of background knowledge level for undergrads? would you agree? – sun_Jiaoliao Dec 06 '24 at 01:54
  • @sun_Jiaoliao: "undergrads" for (U.S.) mathematics majors (and I assume we're talking about math majors) consists of 4 years, and thus ranges from beginning calculus (perhaps begin with one semester of precalculus for weaker students) through (following is for pure mathematics abstract algebra and 2nd level linear algebra and real analysis (continued) – Dave L. Renfro Dec 06 '24 at 05:29
  • and complex analysis and topology, where I've chosen texts that would be appropriate for the (U.S.) undergraduate level. Even the first 3 chapters of the Makarov book would not be suitable for students in the first two years of undergraduate study (elementary calculus sequence, first course in linear algebra, first course in differential equations, maybe a "transition to higher mathematics / how to prove it" type of course), unless they were very exceptional. (continued) – Dave L. Renfro Dec 06 '24 at 05:30
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    My earlier comments more appropriately apply to 3rd and 4th year undergraduates, and also mainly to those who plan to attend graduate school in math. Although many of the topics in the first 3 chapters sound like things that a 1st or 2nd year undergraduate would encounter (e.g. sequences and series are covered in a U.S. 2nd semester calculus course), the abstractness and sophistication and difficulty level of the problems (maybe with VERY few exceptions) are well beyond what would be appropriate for non-exceptional 1st or 2nd year students. – Dave L. Renfro Dec 06 '24 at 05:38
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I am personally fond of "A Primer of Real Functions" by Ralph Boas. It's a little book, but I think it covers most of the topics you have outlined (maybe not the construction of the real numbers). Although not a problem book per se, there are many exercises, and I think it is much more amenable to self-study than Rudin. Most of the exercises have answers in the back.

In addition to the usual topics, Boas includes some interesting material not commonly covered, such as the Universal Chord Theorem. There are only three chapters: Sets, Functions, and Integration. Each chapter starts off very gently and then moves more rapidly, with the more advanced topics toward end of the chapter.

Disclaimer: I personally have only skimmed the material in the third chapter, Integration, because when I first read the book it only had the first two chapters. The third chapter was added in a later edition.

awkward
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