Extremely hard and stimulating (undergraduate) real analysis $problems$

Solution 1:

You could try the book "Selected problems in real analysis" by Makarov and 3 co-authors: http://www.amazon.com/Selected-Problems-Translations-Mathematical-Monographs/dp/0821845594

This is a very beautiful book, and some of the problems are quite hard.

Solution 2:

I recommend G. H. Hardy, "A Course of Pure Mathematics", a classic with many challenging problems. http://ebookee.org/A-Course-of-Pure-Mathematics-Centenary-edition_731929.html

Solution 3:

Here are two suggestions:

-- A link to Vaughan Jones's RA course at Berkeley. In the introductory remarks he acknowledges the difficulty of the HW problems (links on the page)

https://math.berkeley.edu/~vfr/MATH10411/index.html

-- Pugh's "Real Mathematical Analysis," a great book in its own right, has over 500 problems, with many from Berkeley qualifiers.

http://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/0387952977

Solution 4:

Here is a list of undergrad level analysis problems intended to prepare you (in part) for an analysis graduate qualifying exam.

You can find lots of undergrad level problems in qualifying exams from many different departments. For instance, you can get Wisconsin's here.