For the first one hundred and fifty years after its introduction, calculus saw an explosive development in its applications to mathematical and physical problems, defeating old problems thought of as insoluble, and solving new problems no-one had even thought to consider before. At the same time, it was under a cloud of suspicion: it rested on vague arguments about quantities becoming “infinitely” small or “infinitely” numerous, and though it usually gave correct answers in the end, it was far from the model of logical clarity provided by Euclid’s Elements. In this class, you will prove everything that was taken for granted in introductory calculus, starting from first principles. Aside from providing logical certainty, these techniques of proof provide insight as to the real meaning of “infinitely” small, “infinitely” many, and “limiting” value. These techniques are used almost universally in higher mathematics, and a course in Analysis is the central building block of an undergraduate mathematics degree. In addition, the techniques are also essential to theoretical computer science, so students interested in that field should take this course as well.