Analysis with an Introduction to Proof

by Steven R. Lay

For courses in undergraduate Analysis and Transition to Advanced Mathematics. Analysis with an Introduction to Proof, Fifth Edition helps fill in the groundwork students need to succeed in real analysis-often considered the most difficult course in the undergraduate curriculum. By introducing logic and emphasizing the structure and nature of the arguments used, this text helps students move carefully from computationally oriented courses to abstract mathematics with its emphasis on proofs. Clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers make this text readable, student-oriented, and teacher- friendly.

Genres: Mathematics, Textbooks, Science

368 pages, Paperback

First published January 1, 1986

Reviews

[Cristina · Rating 4 out of 5]

I was scared of this class but I had a good professor and this book was pretty good. I found the proofs to be detailed enough (in general) that I feel like I learned how to construct them pretty well. I've looked through other advanced calculus/intro to real analysis texts and I like the order in which the material in Lay is presented as compared to others. Particularly now that I'm taking a graduate level analysis course, I feel like the Lay book was consistent with material I'm covering in my current class and prepared me well for it.

[Raj · Rating 4 out of 5]

Much easier to follow Lay's arguments than Rudin's.

[Gloria · Rating 5 out of 5]

First edition is full of errors. Fourth still has few but it contains much more material than the first and has an improved format. It's the book you'd wish you could have read before high school geometry proofs though you probably would not have been at that level anyway. But it is consider baby logic and proofs. It give a good foundation for writing proofs. Excellent resource.

If one day soon I could finish this book it would be a help for my course work. Can't find the time.

[Matt · Rating 4 out of 5]

This is a satisfactory introduction to real analysis, emphasis on 'satisfactory' and 'introduction'. Most of the essential topics are covered that one would expect:

• logical operators, quantifiers, and methods of proof;
• set theory and set theoretical treatment of relations, functions, and countability;
• the real number system and the topology of the real line, including the Heine-Borel and Bolzano-Weierstrass theorems;
• sequences and their limits, monotonicity, boundedness, and Cauchy sequences;
• limits and continuity of functions through the epsilon-delta definition;
• differentiation (univariate);
• Riemann integration (univariate);
• infinite series and convergence tests; and finally,
• sequences/series of functions, as well as pointwise vs. uniform convergence.

Additionally, supplementary sections provide a bridge to other abstract topological spaces such as metric spaces of arbitrary dimension and arbitrary metric.

As stressed above, this text is meant to be an introductory textbook to analysis, and as such it tries to present the material at a digestible pace and not overwhelm the reader with more advanced topics (and when it does, it notifies them that they may be skipped, e.g. the sections on metric spaces).

While I was reading Lay's book, I was also skimming through Lewin's and Lewin's An Introduction to Mathematical Analysis (2nd edition), which was much more thorough at times, more technically challenging in how it presented the material, and differed with the supplementary material presented - no treatment of metric spaces, instead offering a discussion of the problems of Riemann integration together with a bridge to measure theory. However, on the material that overlapped between Lay's and Lewins', they were more or less identical, so the difference is mostly over style as opposed to substance. (I also intend to give Trench's An Introduction to Real Analysis a read, mainly for his chapters on differentiation and integration of vector and multivalued functions, as well as a whole chapter dedicated to metric spaces, but that is for another time.)

Overall a 3.5 out of 5.

[Blerta · Rating 3 out of 5]

It was ok I guess. Unfortunately everything makes sense until I’m forced to prove it.

Did fuck with sequence and function convergence proofs tho, those were light work. But proving stuff involving supremums and infimums… yea wrap it up. Also wish the book gave more examples, like why in the convergence sections would it give like two at most? Excuse me??? Irritating

[Ryan Pennell · Rating 5 out of 5]

I don't have a strong enough background in Real Analysis to be able to compare this Textbooks to others. I thought that Lay did a good job of mixing proofs with easy examples as well. The book could have been a little longer but what is there is pretty good. I think for an introductory textbook this is a good book and the notation is still current unlike other books I have read.

I think there needs to be at least another chapter at the end about measure theory, lebesgue measure, and lebesgue integration.

[Rachel C. · Rating 4 out of 5]

I liked this book for the most part. Proof is definitely a weak spot for me, but I felt this was a well-written and straight forward book that helped me a lot as I was working through the course that I used it for.