Recommended Problem books for undergraduate Real Analysis

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!

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)

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.