2018 - Understanding Real Analysis - 2e - Paul Zorn - SPE
Preface
This Edition: News and oldies.
The second edition conserves the essential of mathematical and pedagogical strategies of the first edition, as outlined below. For example, this edition retains — and perhaps increases — the first edition’s strong focus on helping students acquire and use mathematical language. Many students at the level envisioned for this text find the linguistic challenge of parsing complicated definitions and theorems just as difficult as the mathematics itself. This edition, therefore, contains additional problems and exercises that ask students to unpack, instantiate, or even “perturb” the statements of definitions and theorems. (See, e.g., Problem7, page 112.)
This book, like any second edition, incorporates repair of first edition typos (and possibly the introduction of some brand-new ones). The narrative has also been“smoothed” or (in the author’s view) otherwise improved in places identified either by the author or (thanks!) by other users.
Following are some more significant new features and content.New sections on topology and compactness. Two entirely new sections, Sec tions 3.5 and 3.6, introduce basic ideas of topology and compactness. Some general definitions and principles are discussed, but the setting is almost always the real line, treated (when helpful) as a metric space. Open covers define compactness(which pose their own linguistic challenges!) but there is strong emphasis on closed and bounded sets in R, via the Heine–Borel theorem.
This material is very occasionally alluded to, but not really depended on, in later sections. In this sense these sections are essentially self-contained, and could be used for independent study or enrichment projects.New material on series of functions and “Taylor stuff.” Treatment of function series in general and of Taylor series in particular has been beefed up, both in Section 4.4 and in the new Section 4.5, on Taylor series. Section4.5 joins Section 2.7(on upper and lower limits) in the mode of “guided discovery”. Both sections are designed for students to encounter as projects or other forms of enrichment. Their content, though mathematically important, is not explicitly required in the sequel.ix
New problems and exercises
This edition englobes some new problems and exercises in almost all the sections. Many new problems focus on helping students understand and “unpack” (in the sense discussed above) definitions and theorems. Students are often asked to prove special cases, explore concrete instances of general results, and the like. Many problem sets have also been more carefully ordered to distinguish between odd - and even - numbered exercises; most of the former have hints or solutions in the back.
2018 - Understanding Real Analysis - 2e - Paul Zorn - SPE
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