Real Number Sequences: Introduction

This lecture introduces the central topic of real number sequences, which will serve as the main tool for understanding the concept of limits, upon which all other analysis notions rely. The discussion covers examples, recurrence-defined sequences, terminology, and properties of real number sequences. It also explores the concept of limits, the tools for establishing their existence, and the connection to infimum and supremum, fundamental in defining real numbers. Through examples, the lecture illustrates how irrational numbers can be constructed as limits of rational number sequences, leading to a deeper understanding of real numbers in terms of rational number sequences, setting the stage for the next chapter's discussion.