I am finishing up a class on discrete mathematics and I am interested in skipping my schools transition courses in order to take a rigorous theory course next semester (topology, analysis, abstract algebra). What are some good transition books for me to read that offer problems and some solutions so I can monitor my progress, as well as being very , almost laboriously, detailed in each step of proof including theorem applications. For example, I have found abbots Understanding Analysis to be fairly cogent but Laczkovich Conjecture and Proof to be lacking some information necessary for me to understand some proofs as much as I would like. Thanks for any help
Do persist with Laczkovich's Conjecture and Proof. This is a first class research mathematician who has written an elegant and beautiful account of how mathematicians think. If you have an opportunity to learn from the "gods" always take it even though you may need to supplement it with easier stuff.
The Abbott book would be considered "analysis light" although it appears to be well-written and useful at that level. Here is Bartle's review of that book, which should indicate to you that it is preparatory to the more advanced courses that you will be taking where they throw you into the deep end for sure.
The author has written an interesting book designed as a text for a one-semester course for students (1) who may not intend to pursue graduate study in mathematics, and (2) whose previous courses are intuitive rather than rigorous. He regards his tasks to be: to demonstrate the need for rigor and precision in analysis, to teach the students what constitutes a rigorous proof, and to expose them to "the tantalizing complexities of the real line, to the subtleties of different flavors of convergence, and to the intellectual delights hidden in the paradoxes of the infinite''.
Although the chapter headings are entirely predictable, the content is not. Page 1 starts with a proof that $\sqrt2$ is not rational; the first topic in Chapter 2 is the rearrangement of infinite (and double) series; Chapter 3 starts with a discussion of the Cantor set. Having gotten the students' attention, the author does some serious analysis, but always with a light touch. He also expects the students to do their part by leaving to them (with copious hints) the details of many of the steps in the proofs. This enables him to cover a surprising amount of material in an attractive and stimulating manner.
Students who work their way through this text will have seen a lot of interesting analysis and have developed a good understanding of the material. (Some of them may even decide to pursue graduate study in mathematics.)
Reviewed by R. G. Bartle in Math Reviews