Swanson Irena. Introduction to Analysis

Unpublished, 2017. — 327 p.These notes were written expressly for Mathematics 112 at Reed College, with first usage in the spring of 2013. The title of the course is “Introduction to Analysis”. The prerequisite is calculus. Recently used textbooks have been Steven R. Lay’s “Analysis, With an Introduction to Proof ” (Prentice Hall, Inc., Englewood Cliffs, NJ, 1986, 4th edition), and Ray Mayer’s in-house notes “Introduction to Analysis” (2006, available at http://www.reed.edu/~mayer/math112.html/index.html).In Math 112, students learn to write proofs while at the same time learning about binary operations, orders, fields, ordered fields, complete fields, complex numbers, sequences, and series. We also review limits, continuity, differentiation, and integration. My aim for these notes is to constitute a self-contained book that covers the standard topics of a course in introductory analysis, that handles complex-valued functions, sequences, and series, that has enough examples and exercises, that is rigorous, and is accessible to Reed College undergraduates.
I currently maintain two versions of these notes, one in which the natural, rational and real numbers are constructed and the Least upper bound theorem is proved for the ordered field of real numbers, and one version in which the Least upper bound property is assumed for the ordered field of real numbers. You are currently reading the shorter, latter version. See my Math 112 webpage www.reed.edu/~iswanson/112.html for links to both versions.