Basic Analysis 1: Introduction to Real Analysis

Jiří Lebl

The book (volume I) starts with analysis on the real line, going through sequences, series, and then into continuity, the derivative, and the Riemann integral using the Darboux approach. There are plenty of available detours along the way, or we can power through towards the metric spaces in chapter 7. The philosophy is that metric spaces are absorbed much better by the students after they have gotten comfortable with basic analysis techniques in the very concrete setting of the real line. As a bonus the book can be used both by a slower paced, less abstract course, and a faster paced more abstract course for future graduate students. The slower paced course never reaches metric spaces. A nice capstone theorem for such a course is the Picard theorem on existence and uniqueness of ordinary differential equations, a proof which brings together everything one has learned in the course. A faster paced course would generally reach metric spaces, and as a reward such students can see a streamlined (but more abstract) proof of Picard.