More students today than ever before take calculus in high school. This comes at a cost, however: fewer and fewer take a rigorous course in Euclidean geometry. Moreover, the calculus course taken by almost all students, whether in high school or college, avoids proofs, and often does not even give a formal definition of a limit. Indeed some students enter the university having never read or written a proof by induction, or encountered a mathematical proof of any kind.
As a consequence, teachers of upper level undergraduate mathematics courses in linear algebra, abstract algebra, analysis and topology have to work extremely hard inculcating the concept of proof while simultaneously trying to cover the syllabus. This problem has been addressed at many universities by introducing a bridge course, with a title like "Foundations for Higher Mathematics", taken by students who have completed the regular calculus sequence. Some of these students plan to become mathematics majors. Others just want to learn some more mathematics; but if what they are exposed to is interesting and satisfying, many will choose to major or double major in mathematics.
This book is written for students who have taken calculus and want to learn what "real mathematics" is. We hope you will find the material engaging and interesting, and that you will be encouraged to learn more advanced mathematics.