Top critical review
Takes a challenging subject and makes it needlessly complicated
on November 27, 2013
I loved my Intro to Analysis class, despite this book. The textbook took a challenging subject and made it needlessly complicated. There aren't enough examples and proofs, especially in the first two chapters (which is where they're needed most, because we're just starting).
After the first two chapters the proofs improve and the examples are marginally more plentiful, but the author struggles with definitions throughout the entire book. He usually does a decent job presenting theorems (with some big failures, like Theorem 4.3), but for some reason he can't define new terms in a clear way. He often fails to link the definition to his following example: The example doesn't explain how it uses the definition, so it doesn't help us understand the definition or how to use it. Sometimes the problem is a simple design/formatting issue: the "definition" is a single minimalist sentence enclosed in a text box, but information that is crucial to understanding the definition is on the previous page or in the next paragraph, not in the text box where it should be.
The most embarrassing gaffe is in Chapter 4.1 where the author "proves" Example 4.7 in a single line. The proof starts, "By the Power Rule (see Exercise 4.2.7), the answer is"... QED. But the Power Rule isn't introduced until the next chapter. Worse still, Exercise 4.2.7 is a homework problem that is left to the student, so you can't follow along and see how the author got his answer. I would love to use that method on a test: "Proof: By a theorem you don't know yet, the answer is 42. If you have questions, just refer to a more advanced problem you haven't seen yet. Once you solve that problem, use it to understand this basic concept."
Finally, the notation is confusing and makes it hard to study and organize ideas. Is Theorem 4.3 in Chapter 4.3? No! Theorem 4.3 is in Chapter 4.1, along with Example 4.7. How about the homework exercises - do they follow the same naming convention? Not at all! They start with Exercise 4.1.0.
OK, rant over. But professors, please consider using a different textbook. According to my professor, the 1st edition of Wade's book was very well written, but every edition since then has been a step backwards. I don't think they'll continue to use Wade in following years. Also, why so many editions? Has analysis changed that much, or is it just an excuse for Pearson to sell more overpriced textbooks?
I have Stephen Abbot's "Understanding Analysis" and I find it presents the material better - using intuitive arguments followed up by rigorous proofs. But it's not the text I used for the class, so I didn't spend as much time reading it - maybe it has other shortcomings. One thing Wade offers that Abbot doesn't is multi-dimensional theory. Still, it is worth investigating.
As far as other options, a number of reviews have presented alternative texts. Unfortunately, anyone who has had the opportunity to compare & contrast multiple Intro Analysis textbooks is probably not a beginner to real analysis, so their point of view might differ from a beginner's.
If I were a professor looking for a new Intro Analysis textbook, I'd bring a copy of each of the texts I liked to the prerequisite class for Intro to Analysis. If the students in that class can self-study chapter 1 of the textbook, you've found a winner.