Contents

1.Introduction

    1.1BASIC TERMINOLOGY

            1.1.1Definitions and Comments

    1.2FINITE AND INFINITE SETS; COUNTABLY INFINITE AND UNCOUNTABLY INFINITE SETS

    1.3DISTANCE AND CONVERGENCE

            1.3.1Definitions and Comments

    1.4MINICOURSE IN BASIC LOGIC

            1.4.1Truth Tables

            1.4.2Types of Proof

            1.4.3Quantifiers

            1.4.4Mathematical Induction

            1.4.5Negations

    1.5LIMIT POINTS AND CLOSURE

2.Some Basic Topological Properties of R^p

    2.1UNIONS AND INTERSECTIONS OF OPEN AND CLOSED SETS

    2.2COMPACTNESS

            2.2.1Definition

            2.2.2Nested Set Property

            2.2.3Definition

            2.2.5 Heine-Borel Theorem

    2.3SOME APPLICATIONS OF COMPACTNESS

            2.3.2Bolzano–Weierstrass Theorem

    2.4LEAST UPPER BOUNDS AND COMPLETENESS

            2.4.1Definitions

            2.4.4Definitions and Comments

3.Upper and Lower Limits of Sequences of Real Numbers

    3.1GENERALIZATION OF THE LIMIT CONCEPT

            3.1.1Definitions and Comments

    3.2SOME PROPERTIES OF UPPER AND LOWER LIMITS

            3.2.1Definitions and Comments

            3.2.4Remark

    3.3CONVERGENCE OF POWER SERIES

4.Continuous Functions

    4.1CONTINUITY: IDEAS, BASIC TERMINOLOGY, PROPERTIES

            4.1.1Definition

            4.1.4Definition

            4.1.7Definition

    4.2CONTINUITY AND COMPACTNESS

            4.2.3Definition

    4.3TYPES OF DISCONTINUITIES

            4.3.2Intermediate Value Theorem

    4.4THE CANTOR SET

            4.4.2Remarks

5.Differentiation

    5.1THE DERIVATIVE AND ITS BASIC PROPERTIES

            5.1.1Definition and Comments

            5.1.4Mean Value Theorem

            5.1.6Generalized Mean Value Theorem

    5.2ADDITIONAL PROPERTIES OF THE DERIVATIVE; SOME APPLICATIONS OF THE MEAN VALUE THEOREM

            5.2.1Intermediate Value Theorem for Derivatives

            5.2.3L’Hospital’s Rule

            5.2.4Taylor’s Formula with Remainder

6.Riemann-Stieltjes Integration

    6.1DEFINITION OF THE INTEGRAL

            6.1.1Definitions and Comments

            6.1.3Definition

    6.2PROPERTIES OF THE INTEGRAL

            6.2.3Evaluation Formula

            6.2.4Fundamental Theorem of Calculus

    6.3FUNCTIONS OF BOUNDED VARIATION

            6.3.1Definitions and Comments

    6.4SOME USEFUL INTEGRATION THEOREMS

            6.4.1Integration by Parts

            6.4.2Change of Variable Formula

            6.4.3Mean Value Theorem for Integrals

            6.4.4Upper Bounds on Integrals

            6.4.5Improper Integrals

7.Uniform Convergence and Applications

    7.1POINTWISE AND UNIFORM CONVERGENCE

            7.1.1Examples of Invalid Interchange of Operations

            7.1.2Definitions

            7.1.3Example

    7.2UNIFORM CONVERGENCE AND LIMIT OPERATIONS

            7.2.5Dini's Theorem

    7.3THE WEIERSTRASS M-TEST AND APPLICATIONS

            7.3.1Weierstrass M-Test

            7.3.2Example

            7.3.3An Everywhere Continuous, Nowhere Differentiate Function

    7.4 EQUICONTINU1TY AND THE ARZELA-ASCOU THEOREM

            7.4.3Definition

            7.4.4Arzela-Ascoli Theorem

    7.5THE WEIERSTRASS APPROXIMATION THEOREM

            7.5.3Weierstrass Approximation Theorem

8.Further Topological Results

    8.1THE EXTENSION PROBLEM

            8.1.3Tietze Extension Theorem

    8.2BAIRE CATEGORY THEOREM

            8.2.1Definitions and Comments

            8.2.2Baire Category Theorem

    8.3CONNECTEDNESS

            8.3.1Definitions

    8.4SEMICONTINUOUS FUNCTIONS

            8.4.1Definitions and Comments

9.Epilogue

    9.1SOME COMPACTNESS RESULTS

            9.1.1Definitions and Comments

    9.2REPLACING CANTORS NESTED SET PROPERTY

    9.3THE REAL NUMBERS REVISITED

Solutions to Problems