This text serves as the first volume of a two-part series on real analysis, aimed at senior undergraduate mathematics students who have a foundational understanding of calculus. The focus of this book is on the rigorous principles and foundational concepts of analysis. It starts with the development of number systems and set theory, before moving on to core analysis topics such as limits, series, continuity, differentiation, and the Riemann integral. The discussion extends to power series, multivariable calculus, and Fourier analysis, concluding with the Lebesgue integral. Most of the material is grounded in the familiar context of the real line and Euclidean spaces, though some sections explore abstract metric and topological spaces. Additionally, the book includes appendices covering mathematical logic and the decimal system. The entire content (excluding some less essential topics) is structured to be delivered over two quarters, with each quarter comprising 25 to 30 lectures.

The course is designed to integrate exercises closely with the material, encouraging students to engage actively with the concepts by proving key theoretical results, thereby honing their rigorous thinking and writing skills.