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. 
 A: 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.
