Analysis

This book offers a rigorous yet accessible introduction to mathematical analysis for undergraduate students in mathematics, physics, computer science, and engineering. It is designed as a textbook for a first university course in real analysis and calculus. The exposition begins with fundamental proof techniques, in particular mathematical induction, before constructing the real numbers as an ordered complete field, which serves as the foundation for all subsequent developments. Based on this framework, the core topics of analysis are developed in a clear and logically coherent manner, including sequences and limits, infinite series, continuous functions, differential calculus, and integral calculus. Further chapters cover uniform convergence, power series, Taylor expansions, and central convergence theorems. Throughout the book, emphasis is placed on mathematical rigor and conceptual understanding. Definitions, theorems, and proofs are presented in a structured and transparent way, highlighting their logical connections and underlying ideas. A large number of carefully selected examples and fully worked exercises support comprehension and encourage independent study. The book is suitable both as a companion to university lectures and for self-study, and it provides a solid foundation for advanced courses in mathematical analysis.