Introduction to Real analysis

Most of Mathematics 281 deals with some fundamental ideas about and properties of the real numbers and continuous functions. These ideas and properties include sequences and series (which are studied in some first-year calculus), the least upper bound property, elementary topology of R and R d , convergence and uniform convergence of series of functions, and term-by-term differentiation and integration of series of functions. These notes will serve as a text for this part of the course. For approximately twenty-five years ending with the 1996/97 academic year the last seven or so weeks of the second-year honours calculus course (Mathematics 220) dealt with the topics just described. In the 1970's there was no suitable text dealing with these topics and the first version of these notes was written by R. D. Norman; this version was really an outline and omitted some proofs and contained few examples or exercises. Over the years various other instructors added examples, comments, and other additions, with the result that the already terse notes became even harder to read. In 1992 I began revising these notes with the idea of adding exercises, an index, a table of contents, and more examples and exercises. This has proceeded since then with the assistance of Professor Norman and with students Peter Attia and Christina In the Winter term of 2016, I taught MATH/MTHE 281 for the first time, and thought it a good idea to take the L A T E X for the original version of these notes, and make minor revisions. Also, the intention is to arrive at a " living " version of these course notes that can evolve, while still maintaining the suitability of the original form as a course text for MATH/MTHE 281.