Analysis 1

Abstract

Introduction The aim of this course is to bridge the gap between the knowledge of analysis accumu- lated in high school and the fundamentals that will form one of the pillars of mathematical analysis training in the bachelor’s degree. Given that recruitment to the first year of anal- ysis is fairly heterogeneous, it seems a good idea to start by recalling the basic notions that will be used throughout this course, so as not to lose anyone along the way. Where necessary, at the start of each chapter, we’ll remind you of what you’re supposed to know by the end of the course. As far as possible, we’ll also try to provide the essential results of each chapter on a single page, so as to summarize the knowledge you need to master in order to move on to the next chapter. We will provide as many examples and figures as necessary afin order to achieve a better understanding of the course. We’ll also try to point out the pitfalls that everyone can fall into, either through inattention or poor mastery of the course. This course brings together the notes from the analysis module dedicated to first-year students in the mathematics and computer science stream. It includes set theory, real se- quences, real functions of one real variable, real functions of one real variable: continuity, real functions of one real variable: derivability and elementary functions their relations. The general idea behind the first chapter is to handle real numbers properly, you first need to know a few basic rules. These rules all have a name afin order to identifier them right away, so you know what you’re talking about. A bit like a map, this lets us know what we’re seeing, where we’re going and how we’re getting there. Let’s start with the simple, familiar rules. In the second chapter we give the notion of a sequence is central in Analysis. Among sequences, convergent sequences form a special important class. They are introduced and studied in this chapter. Further, based on this, the notion of a convergent series is intro- duced. For the third chapter: In this chapter the notion of The basics of functions, and Some properties of functions are presented The forth chapter: the key notion of a continuous function is introduced, followed by several important theorems about continuous functions. The fifth chapter:The notion of a differentiable function is developed, and several impor- tant theorems about differentiable functions are presented. For the last chapter: We use power series to strictly define the Exponential, Logarithmic, and Trigonometric functions and describe their properties. Each chapter is followed by a list of exercises, usually illustrating a point made earlier. It goes without saying that these exercises complement the course