Analysis: With an Introduction to Proof (4th Edition)

By introducing logic and by emphasizing the structure and nature of the arguments used, this book helps readers transition from computationally oriented mathematics to abstract mathematics with its emphasis on proofs. Uses clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers. Offers a new boxed review of key terms after each section. Rewrites many exercises. Features more than 250 true/false questions. Includes more than 100 practice problems. Provides exceptionally high-quality drawings to illustrate key ideas. Provides numerous examples and more than 1,000 exercises. A thorough reference for readers who need to increase or brush up on their advanced mathematics skills.

Then the book goes into a very nice introduction to topology. Basic concepts like open/closed sets, accumulation points, compact sets, etc. Topology can be a little intimidating simply because it's _so_ abstract, but this book makes the basic concepts very easy to understand, and prepares one for a more advanced course in topology. Alot of (good) Elementary Analysis books leave topology out, but I'm glad this book contained it. It is a very interesting subject.

All the material in the book is explained probably about as easily as the concepts CAN be explained. If you still have trouble with it, you might consider a different major. Not to say that this book transforms a very difficult subject into a pathetically easy piece of cake because that's impossible, but the material is presented probably as easily as it can be in order to maintain precision and detail (which is the whole point of Analysis).

The book is definitely not running short in the examples or end-of-section problems department, so that is another plus. The problems at the end of each section range in difficulty from problems that almost exactly match an example worked in detail in the section, to fairly challenging problems. With enough time though the average student could probably do every problem at the end of every section.

I'd recommend this book for self study as well as a supplement to any introductory analysis course. If you have already have exposure to rigorous proof of calculus theorems, then this book will probably be too basic for you.

The reason this book got 4 stars instead of 5 is because of its utterly ridiculous price. Just as good is Elementary Analysis: The Theory of Calculus, ISBN: 038790459X, except that it doesn't include the section on Topology ...