# Good transition books

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

• Try Spivak's Calculus – Simon S Nov 7 '15 at 19:46

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.