A la Halmos: textbooks on, not with, problems

Solution 1:

Here are two recommendations. Although the first is not a dedicated problem book it might nevertheless come close to your expectations and it is connected with the second recommendation.

• In A Radical approach to Real Analysis by David M. Bressoud the course does not follow the traditional development, namely starting with a discussion of properties of real numbers, then moving on to continuity, then differentiability and so forth. He instead takes the reader along the often devious chronological paths of development and forces him this way to think about concepts and so to grasp essential ideas.

  From the preface: ... the first part of this book ... starts with infinite series ... illustrating the great successes that led the early pioneers onward, as well as the obstacles that stymied even such luminaries as Euler and Lagrange. There is an intentional emphasis on the mistakes that have been made. These highlight difficult conceptual points. ... The student needs time with them. The highly refined proofs that we know today leave the mistaken impression that the road of discovery in mathematics is straight and sure. It is not. Experimentation and misunderstandng have been essential components in the growth of mathematics.

This book addresses especially what you stated as forcing the reader to think along the way, almost to construct Analysis himself.

The problems in this book are stated as exercises and they do not just form some kind of appendix, but are a dominant, integral part of each section. So, you are challenged all along by hundreds of examples and exercises when doing this radical approach.

• Many of the exercises in Bressouds book are taken from the following book (as he explicitly states in his foreword):

  Problems in Mathematical Analysis I-III by W.J. Kaczor and M.T. Nowak. These three volumes (Vol. I: Real Numbers, Sequences and Series; Vol. II: Continuity and Differentiation; Vol. III: Integration) are classical problem books consisting of a problem and a solution part.

Solution 2:

Alan Clark’s Elements of Abstract Algebra is a great book, which heavily relies on problems. Clark writes brief articles (often less than a page), then lets the reader do the rest by solving several problems, some very easy and others reasonably difficult. The text covers enough algebra for an undergraduate course, including topics like the Sylow theorems and some Galois theory. The book is also cheaply available, since it is published by Dover.