Basic analysis. I : introduction to real analysis, volume I

"This is a course in undergraduate real analysis, also known as advanced calculus. The book works for both basic courses for students who do not necessarily wish to go to graduate school and also more advanced courses that prepare students for graduate study and cover topics such as metric spaces. A prerequisite for the course is a basic proof course. This book 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 you can power through toward 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 slower-paced, more concrete courses, as well as a faster-paced, more abstract courses for future graduate students"--BCcampus website.