AMCS 214 Introduction to Analysis

This course is considered a first course in mathematical analysis. The main goal is to rigorously prove mathematical results that were assumed to be true and used in previous courses. We begin with a review of mathematical logic and proof strategies, and we examine the elementary set theory including cardinality and the notion of countable and uncountable sets. We move to the structure and axioms of the real numbers. We then investigate statements about the behavior of sequences. We proceed with the study of the topology on the real line, the notion of open and closed sets, and compactness. Subsequently, we address continuity and differentiability of real functions, and the convergence of sequences and series of functions. Finally, we discuss the construction of the Riemann integral.