"Honest" introductory real analysis book

Question

I was asked if I could suggest an "honest" introductory real analysis book, where "honest" means:

• with every single theorem proved (that is, no "left to the reader" or "you can easily see");
• with every single problem properly solved (that is, solved in a formal (exam-like) way).

I've studied using Rudin mostly and I liked it, but it really doesn't fit the description, so I don't know what book I should suggest. Do you have any recommendations?

Update: I need to clarify that my friend has just started to study real analysis and the course starts from the very basics, deals with real valued functions of one variable, but introduces topological concepts and metric spaces too.

Answer 1

Get two volumes of Zorich.$ $

Answer 2

Possibly Abbott, Understanding Analysis

Answer 3

A bit of self publicity, but the reason that A Primer on Hilbert Space Theory was written is precisely to give what you refer to as an 'honest' introduction to the foundations of analysis.

Edit: OP's comment below clarifies this book is not at the intended introductory level.

Answer 4

I have had the pleasure to teach introductory real analysis from couple of excellent texts, which I would also recommend.

• Elementary Analysis: The Theory of Calculus by Kenneth A. Ross
• Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert

Answer 5

Advanced Calculus by Patrick M. Fitzpatrick is a great text that starts from the very basics and goes up through point-set topology and metric spaces.

It starts with field axioms and builds from there so it doesn't "cheat" in that regard (edit: just to clarify, this means almost NO proofs are "left to the reader," the only exceptions being very small special cases that are then presented in exercises), but it doesn't contain solutions--I think you'd be hard pressed to find a textbook at that level that had complete solutions to every single problem. However, it's popular enough that most of the solutions can be found on this site or elsewhere on the internet.

Starting from general one-dimensional stuff and moving up to metric spaces in a single course seems like quite a task though.

Answer 6

I used Rosenlicht's Introduction to Analysis in my real analysis course in undergrad. It's cheap and is a lot easier to digest than Rudin.

Answer 7

For an introduction more 'foot on the ground' analysis I recommend Elementary Classical Analysis. It's a shame that the partial visualization is not available. But you will not regret in search for this book in the library of a good university.

See too Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. By John Hubbard and Barbara Burke. More in Amazon site.

Answer 8

Bartle's Introduction to Real Analysis has a small number of "left to the reader" proofs, from I have seen so far.

Answer 9

Terence Tao's Analysis I and Analysis II. These books are expanded and cleaned up versions of lecture notes which you can find here and here.

Answer 10

Yet Another Introduction to Analysis by Bryant is my favourite. It probably doesn't meet all the criteria you listed. However, it is the most intuitive first book on the topic I know, and once you have read it, other analysis books become much easier.