TITLE: Recommended Problem books for undergraduate Real Analysis
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: 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.