Examples and Theorems in Analysis
The author presents a book on analysis in which theorems and examples are equally important. The material contains fundamental notions: sequences, continuous functions, integration, series and some applications chosen to illustrate the usefulness of the previously covered subjects. The last chapter contains an invitation to further study of Fourier series and integrals, distributions and asymptotics. The content of the first six chapters is standard with some modifications in approach to the material. Appendix A (6 pp.) is devoted to the Fubini theorem and Appendix B (25 pp.) contains hints and solutions for exercises. There are further hidden important facts, such as the arithmetic-geometric mean of two positive numbers, the rising sun lemma, the Chebyshev's convexity theorem, Wallis' formula, and hyperbolic functions. Most of the book can be covered in a one-semester course for beginners. It is a good textbook for students that are willing to invest some work to obtain a more complete picture of the material and to master basic methods of work in mathematical analysis.