Real Analysis

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.