A textbook for advanced undergraduates and graduate students. Chapter 1. Set Theory Chapter 2. Real Numbers. Fields Chapter 3. Vector Spaces. Metric Spaces Chapter 4. Function Limits and Continuity Chapter 5. Differentiation and Antidifferentiation All Chapters contain problem sets. This text is an outgrowth of lectures given at the University of Windsor, Canada. One of our main objectives is updating the undergraduate analysis as a rigorous postcalculus course. While such excellent books as Dieudonn'e's Foundations of Modern Analysis are addressed mainly to graduate students, we try to simplify the modern Bourbaki approach to make it accessible to sufficiently advanced undergraduates. On the other hand, we endeavor not to lose contact with classical texts.