This is part one of a twovolume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginningthe construction of the number systems and set theorythen goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twentyfive to thirty lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory. In the second edition, several typos and other errors have been corrected. A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels. Readership Undergraduate and graduate students interested in analysis. Reviews From a review of the first edition: "... it would be an error not to stick very close to the text  it's very well crafted indeed and deviating from the score would mean an unacceptable dissonance. "I hope to use Analysis I, II in an honors course myself, when the opportunity arises."  Michael Berg, for MAA Reviews Table of Contents Volume 1  Introduction
 Starting at the beginning: The natural numbers
 Set theory
 Integers and rationals
 The real numbers
 Limits of sequences
 Series
 Infinite sets
 Continuous functions on \(\mathbf{R}\)
 Differentiation of functions
 The Riemann integral
 Appendix A: The basics of mathematical logic
 Appendix B: The decimal system
