Introduction to Proofs and Proof Strategies

Real analysis is a branch of mathematics focusing on the study of real numbers and related objects. Sets of real numbers, sequences, functions, and series of real numbers are at the core of the subject. The notions of limits and convergence are central in analysis and are used to investigate such objects. Learning real analysis means, in part, deepening our understanding and studying the theoretical foundations of calculus topics. For these reasons, many view real analysis as a rigorous version of calculus. In this chapter, we look at how limits of sequences and function can be formally defined. The precise definitions may require some effort to grasp, but it is absolutely essential for advanced studies in mathematics and related fields. Formal definitions of limits allow us to not only prove various statements (such as the Extreme and the Intermediate Value Theorems, often proved in a Real Analysis course), but also investigate more complicated functions and sequences. Our experience with proof writing and logical statements will be invaluable for our discussion. We also highlight the use of limits to defining continuity and differentiability of functions.