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Real analysis lies at the core of modern mathematics, yet many students struggle with the transition from computational calculus to rigorous proof-based thinking.
This book offers a systematic development of convergence and integration, treating sequences, series, sequences of functions, and integrals within a unified conceptual framework. The exposition begins with numerical sequences and series, proceeds through the Riemann and improper Riemann integrals, and then introduces the Stieltjes and Lebesgue integrals together with the necessary measure-theoretic foundations. Detailed proofs, explicit discussion of proof strategies, and carefully chosen counterexamples reveal the logical structure of the subject and motivate the extension from classical to modern integration.
With 165 worked examples and 547 exercises, the text is ideal for advanced undergraduates and beginning graduate students in mathematics, statistics, and related fields who require a rigorous yet accessible pathway from calculus to contemporary real analysis.
René Erlín Castillo obtained his Ph.D. from Ohio University. He has been Professor in Universidad de Oriente, Venezuela, visiting Professor in Ohio University and nowadays he is Professor in Universidad Nacional de Colombia. His research, spread across 108 papers, is done mainly in functional analysis, real analysis, complex analysis, harmonic analysis, operator theory, potential theory and partial differential equations. He has authored 6 books four in English and two in Spanish. He has supervised 2 doctoral theses, 6 master's theses, and 3 undergraduate theses.
Julio C. Ramos-Fernández received his Ph.D. from the Department of Mathematical Analysis at the University of La Laguna, Tenerife, Spain. He is Associate Professor at the Universidad Distrital Francisco José de Caldas in Bogotá, Colombia. His research interests include real and complex analysis, functional and harmonic analysis, operator theory, differential and integral equations, and numerical analysis. He is the author of 79 research papers published in indexed journals and co-author of three books. He has been an invited speaker at 23 international conferences and has supervised 47 theses in mathematics (33 undergraduate, 10 master’s, and 4 doctoral). He has extensive teaching experience in courses related to mathematical analysis at the undergraduate, master’s, and doctoral levels, and serves as a referee for several specialized journals in mathematics and Mathematical Reviews/MathSciNet.
Edixon M. Rojas S. is Associate Professor of Mathematics at the Universidad Nacional de Colombia in Bogotá. His research focuses on nonlinear analysis and its applications, including operator theory, integral equations, and ordinary and partial differential equations. He has authored numerous peer reviewed articles and books, led and participated in multiple research projects, and carried out research visits to institutions in Europe. He founded and leads a research group in analysis recognized at the national level. He is an active referee for international journals and Mathematical Reviews/MathSciNet, serves on thesis and academic evaluation committees, and contributes to curriculum development in mathematical analysis. He has extensive experience teaching undergraduate and graduate courses in analysis, operator theory, calculus, and differential equations, supervising MSc and BSc theses, and engaging in outreach and advisory activities in science and education.