# How to tackle Makarovs “Selected Problems in Real Analysis”?

The book "Selected Problems in Real Analysis" was recommended in this article:

http://www.ams.org/notices/200510/comm-fowler.pdf

For those of you who don't know the book: It is a book that only consists of mathematical problems, starting from problems about set theorie and ending with problems about measure theorie. It seems to be rather challenging. I already attended Analysis I and Analysis II, currently I'm hearing Analysis III.

Now, I'm wondering how to start with a book like this. He merely gives a background about the questions he asks, which means that there might be tasks where I simply don't have a good foundation. For example, the first task is to prove that the power set of $\Bbb N$ and the set of all possible binary sequences have the same cardinality.

Assuming that I didn't have any clue about what he is talking there (which is not true in this case, but there might be cases like this), how would one approach a problem like this?

Normally, one has the foundation given by the lectures, but that kind of style is completely different of course.

• I suppose you would only do problems that are of interest to you, likely those would only be ones where you already know the terminology used. If the power set problem interested you but you didn't know what "power set" was, you would have to look it up on, say, wikipedia. If this is a book, I would hope the author would give at least a one-sentence "reminder" of what a power set is, but some authors do not do that. – Michael Nov 7 '16 at 15:04
• I guess that book is only for at least intermediate or even advanced level mathematics students who are already familiar with most of the definitions and concepts. – Julian Nov 7 '16 at 15:35

Makarov's Selected problems in real analysis gathers mostly challenging and thought-provoking problems (as opposed to routine exercises). The reader is rarely guided in any of the problems, and hints (forget about solutions) are extremely terse and concise.

Although you're sometimes tasked with well-known or classical problems (such as divergence of $\sum \frac{1}{p_n}$) the lack of guidance can make it understandably frustrating for beginners and undergraduates.

You should attempt this book only if

• you have enough time available to ponder each problem (a problem is sometimes divided into multiple equally difficult questions)
• you have completed a course in undergraduate real analysis and acquired good intuition on sequences, series, functions, ...

If you want to get a taste of the general level of this book:

A game with $\delta$, $\epsilon$ and uniform continuity.

Convergence of $\sum_n \frac{|\sin(n^2)|}{n}$

$\sum \frac{a_n}{\ln a_n}$ converges $\implies \sum \frac{a_n}{\ln (1+n)}$ converges

And these are some of the most accessible problems.

• Thanks for your answer! Is there something similar to Makarov that is not as hard? I already finished two real analysis courses, but I wouldn't consider myself to be "excellent" at it, plus, our tasks were much easier. But I like the style of it somehow, just giving out problems without telling too much about it. – Julian Nov 14 '16 at 10:11
• I know 3 other problem books aimed towards undergraduates with concise questions and full-fledged solutions: Aksoy's A Problem Book in Real Analysis, Shakarchi's Problems and Solutions for Undergraduate Analysis, Kaczor's Problems in Mathematical Analysis (first two volumes). Two more books with an olympic mindset: Andreescu's Problems in Real Analysis: Advanced Calculus on the Real Axis and Putnam and Beyond. – Gabriel Romon Nov 14 '16 at 11:53