The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs

This book began its life as the author’s undergraduate thesis project. The idea was that “real analysis is hard” (a direct quote from p. 3). In particular, real analysis out of baby Rudin at Princeton is hard. The Real Analysis Life Saver is not meant to be a textbook in its own right. It has, for instance, no sections devoted to exercises. It features a bibliography-meets-review section of other real analysis sources — beginning with Rudin (which the author admits is where most of the definitions and proofs in this text originate). Also included are recommendations for Lay, Abbott, and Ross with even the Schaum Outlines receiving an honorable mention.

Instead of competing with other existing textbooks, then, The Real Analysis Life Saver is meant to be a simultaneously more detailed and very laid-back accompaniment. The author takes a colloquial approach, and recognizes that the notation and material may be overwhelming to some students. He likes to joke with his readers; as an example, at one point with regards to suprema he says, “If you hate them and want them to go away, [try] saying ‘soupy’ whenever you read \( \sup E\); it might help you feel better.” (p. 34)

The author emphasizes drawing pictures and does a nice job of using graphics to motivate both definitions and proofs. This first figures appear in early sections on set intersections and unions; they are perhaps most helpful, however, in the chapters on point-set topology. His diagrams in the proofs of Theorems 9.23 (every neighborhood about a point in a metric space is an open set) and 11.7 (every compact subset of a metric space is closed) are good examples of this.

Another asset of this book is its use of examples. While examples are not proof, frequently in textbooks examples are provided after a proof to highlight the utility of the theorem. But Grinberg uses examples in a slightly different way; he uses them to show why certain hypotheses are needed. For instance, before delving into the main proof, Chapter 12 (on the Heine-Borel Theorem) gives an example of a closed and bounded subset of \(\mathbb Q\) which is not compact — highlighting the need for the subset to lie in \( \mathbb R\).

A more unique aspect to this book is the “fill-in-the-blank” proofs (which perhaps were influenced by Lay). This is the closest thing to an exercise the Life Saver offers (and, as in most advanced math textbooks, you’re not going to find the answers in the back). While most real analysis classes assume students have had some exposure to proof, what Grinberg highlights is that real analysis proofs may be trickier than those previously seen. Not to fear — every proof that is “assigned” mimics in style and/or content a proof immediately preceding it, which was worked out in fair detail.

Katherine Thompson is an Assistant Professor of Mathematics at DePaul University in Chicago. Her research area is number theory; in particular, she is interested in quadratic and modular forms. Thompson is passionate about both research and teaching outreach opportunities; for three summers beginning in 2015 she has run number theory REUs, and since 2016 she has been an online course instructor for The Art of Problem Solving.