Preface

This textbook, containing detailed solutions of problems in mathematical analysis, is the first part in a series of three, covering the material that students of science encounter in the first two or three semester courses of analysis. It was prepared on the basis of my experience of many years of teaching this challenging subject in the Department of Physics at the University of Warsaw. Some exercises were inspired by educational materials, which have long been used by the staff of the Department of Mathematical Methods in Physics.

This set of problems is distinct from other books available on the market and should be complementary to them. The basic assumption is that all problems (apart from those which are intended for the reader’s own work) are solved in detail, even if it requires several pages—solved, that no topic is left unexplained, and no question, which could arise when studying solutions, remain unanswered. I am aware that this intention can be successful only partially. However, I would be satisfied if students after having carefully analyzed the solutions could say that they understood a given problem as if they had participated in the university exercise classes. For this reason, a lot of space in this book was devoted to thorough demonstration of each logical step and detailed—for some readers probably even too elementary—transformations of formulas.

Such profile of the book leads, however, to certain limitations. First of all, it cannot contain too many problems or else the book would be too cumbersome. For the same reason, the formal theoretical introductions, which are normally located at the beginning of each chapter of a conventional analysis problems book, are maximally reduced. I assume that students know the theoretical issues from their lecture or have a high-quality mathematical analysis textbook. Some definitions and theorems (only when really necessary) are recalled in a more informal way within the solutions of specific problems. My teaching practice shows that such a system is more likely to be accepted, or even expected, by students who would rather than study a few pages of theoretical and abstract considerations, prefer to be given their practical application as soon as possible. This arrangement of the book has the advantage of allowing the reader to start studying problems without getting acquainted with the theoretical subsections.

There is also the question of language used in this book. I tried to maximally simplify it and—in place of abstract terms—use the notions, which are intuitively clear (and even used in everyday life). Someone may, and would be right, formulate the objection that they are not precise enough. However, my intention was to present the issues in such a way that the student, without much effort, could translate difficult concepts to notions that are more understandable and assimilable. This observation comes from many years of work at universities. The students’ understanding depends to a large extent on the choice of a simple language, especially in the first few years of study. To increase the level of abstraction, there will be time in their further course of study. At the beginning, it is helpful to make the students aware that many new concepts can be mastered with their present knowledge and intuition.

With the hope that this set will help to better understand (from a practical point of view) certain issues of mathematical analysis, I encourage the reader to use other textbooks that provide exercises for independent work and that certainly cannot be replaced by this set.

Tomasz Radożycki

Warsaw, Poland

Definitions and Notation

Below, some definitions and notation are gathered in order to avoid their repetition in each applicable chapter.

- Positive integers (i.e., without 0) will be marked with $\mathbb {N}$ :and will be called “naturals.” If we wish to include the 0 in this set, we will simply write $\mathbb {N}\cup \{0\}$ , and call it “naturals with zero.”$\displaystyle \begin{aligned}\mathbb{N} = \{1, 2, 3, \ldots\} \;, \end{aligned}$
The real, rational, and positive integer numbers will be denoted as $\mathbb {R}_+$ , $\mathbb {Q}_+$ , and $\mathbb {Z}_+$ , respectively. One naturally has $\mathbb {Z}_+ =\mathbb {N}$ . Similarly, symbols $\mathbb {R}_-$ , $\mathbb {Q}_-$ , and $\mathbb {Z}_-$ will refer to negative numbers.

If a special notation is not introduced in a particular problem, the symbol

*X*will mean the whole space.In all problems, apart from those contained in Chap. 3 and the last problem of Sect. 6.1 , the Euclidean metric is used as default, based on the Pythagorean theorem, discussed in detail in Problem 1 of Sect. 3.1 . In the case of the set $\mathbb {R}$ , it reduces to “natural metric,” so that the distance of the two numbers

*x*and*y*is given by*d*(*x*,*y*) = |*x*−*y*|.It is assumed that a ball is open. For example, the ball centered at some point

*x*_{0}and of a radius*r*is a set of points*x*satisfying the condition:*d*(*x*_{0},*x*) <*r*. If, in any problem, a closed ball is needed, it will be written explicitly.The function

*f*as a mapping of a set*X*into*Y*, formally speaking, apart from the assignment itself (e.g., the formula for*f*(*x*)) requires also the definitions of sets*X*and*Y*. We accept the rule that if in the specific exercise they are not given, the largest sets for which the formula*y*=*f*(*x*) makes sense is taken. What is meant always results from the context of the discussed issues. For example, in the textbook, which is generally concerned with real numbers, we will certainly not expand the logarithmic function to the complex plane. Similarly, if the set*Y*is not given, we assume it to be identical to the image of the function, i.e.,*f*(*X*).The domain of a function will generally be denoted as

*D*or sometimes as*X*.The symbol $ \log $ denotes natural logarithm: $ \log x= \log _e x$ .

The symbols := or =: will be used when the equality is a definition or a new designation, and we want to particularly emphasize it.

- The symbol ≃ is used as a shortcut for “behaves like.” If, for example, for very large
*x*,*a*(*x*) ≃*b*(*x*), this is to be understood that$\displaystyle \begin{aligned}\lim_{x\rightarrow\infty} \frac{a(x) } {b(x) } = 1 . \end{aligned}$

Contents