Problems in Mathematical Analysis I belongs to the great tradition of Eastern European problem books. Although covering only three topics, it boasts over 600 problems, which include all the traditional results usually studied in the first part of an analysis course. Such an impressive collection fills a gap in the materials available for the study of mathematics to students in the US. The book has complete solutions to all the problems, making it also useful for individual study.
However, this is not a book for the "calculus-phobic." There are no exercises like "list the first five terms of the sequence," and very few of the problems would be suitable for a 50-minute midterm exam. Most of the problems assume that the reader is skillful in writing proofs, and they are meant to emphasize concepts. The statements are straightforward, without gradual questions designed to break the problems into easier parts or give hints as to which approach is better.
Inside, the student will find all the classical results that one should learn when studying analysis — for example, one problem asks us to show that if an tends to plus or minus infinity, then (1+1/an)an tends to e, and another asks us to show that the sum of the reciprocals of the squares is equal to pi squared over six. The solutions, even when labeled "elementary" by the authors, are not short and easy (nor could they be), on the contrary, they require a solid background in calculus, algebra, and trigonometry. "Obvious" steps are usually skipped from the proofs.
Bottom line: if you love mathematics and are really serious about understanding analysis, this book is a must. Note that a second volume is currently being translated.
Ioana Mihaila (firstname.lastname@example.org) is assistant professor of mathematics at Coastal Carolina University, SC. Her research area is analysis and she has a special interest in student math contests at all grade levels.