Real Analysis I
Course Content
Chapter I: The Real Line System The set of natural numbers The set of rational numbers and its dense property Algebraic numbers Absolute value The set of real numbers and its completeness property Finite sets and compact sets
Chapter II: Topology of the Real Line: Open and closed sets definitions and theorems Compact sets
Chapter III: Sequences Definition of the limit of a sequence and its theorems Divergent sequences Monotone sequences and Cauchy sequences Subsequences
Chapter IV: Functions, Limits and Continuity Definition of the limit of a function and its theorems Definition of a continuous function and its theorems Properties of continuous functions including the boundedness theorem, the max-min theorem and the intermediate value theorem Uniform continuity
Chapter V: Differentiation Definition and some theorems Differentiability and continuity Rolle's theorem, mean value theorem and their applications Course Portfolio Map of contents Chapter I: The Real Line System Homework Problems for Chapter 1 More problems on chapter I Chapter II: Topology of the Real Line Homework Problems for Chapter II A Sample for the First Eexam Chapter III: Sequences of Real Numbers Homework Problems for Chapter III Chapter IV: Functions, Limits and Continuity Homework Problems for Chapter IV A Sample for the Second Exam Chapter V: Differentiation Homework Problems for Chapter V A Sample for the final exam