An Excursion through Elementary Mathematics, Volume I: Real Numbers and Functions

Preface
Contents
1 The Set of Real Numbers
	1.1 Arithmetic in R
	Problems: Section 1.1
	1.2 The Order Relation in R
	Problems: Section 1.2
	1.3 Completeness of the Real Number System
	Problems: Section 1.3
	1.4 The Geometric Representation
	Problems: Section 1.4
2 Algebraic Identities, Equations and Systems
	2.1 Algebraic Identities
	Problems: Section 2.1
	2.2 The Modulus of a Real Number
	Problems: Section 2.2
	2.3 A First Look at Polynomial Equations
	Problems: Section 2.3
	2.4 Linear Systems and Elimination
	Problems: Section 2.4
	2.5 Miscellaneous
	Problems: Section 2.5
3 Elementary Sequences
	3.1 Progressions
	Problems: Section 3.1
	3.2 Linear Recurrences of Orders 2 and 3
	Problems: Section 3.2
	3.3 The  and  Notations
	Problems: Section 3.3
4 Induction and the Binomial Formula
	4.1 The Principle of Mathematical Induction
	Problems – Section 4.1
	4.2 Binomial Numbers
	Problems – Section 4.2
	4.3 The Binomial Formula
	Problems – Section 4.3
5 Elementary Inequalities
	5.1 The AM-GM Inequality
	Problems: Section 5.1
	5.2 Cauchy's Inequality
	Problems: Section 5.2
	5.3 More on Inequalities
	Problems: Section 5.3
6 The Concept of Function
	6.1 Definitions and Examples
	Problems: Section 6.1
	6.2 Monotonicity, Extrema and Image
	Problems: Section 6.2
	6.3 Composition of Functions
	Problems: Section 6.3
	6.4 Inversion of Functions
	Problems: Section 6.4
	6.5 Defining Functions Implicitly
	Problems: Section 6.5
	6.6 Graphs of Functions
	Problems: Section 6.6
	6.7 Trigonometric Functions
	Problems: Section 6.7
7 More on Real Numbers
	7.1 Supremum and Infimum
	Problems: Section 7.1
	7.2 Limits of Sequences
	Problems: Section 7.2
	7.3 Kronecker's Lemma
	Problems: Section 7.3
	7.4 Series of Real Numbers
	Problems: Section 7.4
8 Continuous Functions
	8.1 The Concept of Continuity
	Problems: Section 8.1
	8.2 Sequential Continuity
	Problems: Section 8.2
	8.3 The Intermediate Value Theorem
	Problems: Section 8.3
9 Limits and Derivatives
	9.1 Some Heuristics I
	9.2 Limits of Functions
	Problems: Section 9.2
	9.3 Basic Properties of Derivatives
	Problems: Section 9.3
	9.4 Computing Derivatives
	Problems: Section 9.4
	9.5 Rôlle's Theorem and Applications
	Problems: Section 9.5
	9.6 The First Variation of a Function
	Problems: Section 9.6
	9.7 The Second Variation of a Function
	Problems: Section 9.7
	9.8 Sketching Graphs
	Problems: Section 9.8
10 Riemann's Integral
	10.1 Some Heuristics II
	10.2 The Concept of Integral
	Problems: Section 10.2
	10.3 Riemann's Theorem and Some Remarks
	Problems: Section 10.3
	10.4 Operating with Integrable Functions
	Problems: Section 10.4
	10.5 The Fundamental Theorem of Calculus
	Problems: Section 10.5
	10.6 The Change of Variables Formula
	Problems: Section 10.6
	10.7 Logarithms and Exponentials
	Problems: Section 10.7
	10.8 Miscellaneous
	Problems: Section 10.8
	10.9 Improper Integration
	Problems: Section 10.9
	10.10 Two Important Applications
	Problems: Section 10.10
11 Series of Functions
	11.1 Taylor Series
	Problems: Section 11.1
	11.2 Series of Functions
	Problems: Section 11.2
	11.3 Power Series
	Problems: Section 11.3
	11.4 Some Applications
	Problems: Section 11.4
	11.5 A Glimpse on Analytic Functions
	Problems: Section 11.5
Bibliography
Appendix A
Glossary
Appendix B Hints and Solutions
	Section 1.1
	Section 1.2
	Section 1.3
	Section 2.1
	Section 2.2
	Section 2.3
	Section 2.4
	Sections 2.5
	Section 3.1
	Section 3.2
	Sections 3.3
	Section 4.1
	Section 4.2
	Section 4.3
	Section 5.1
	Section 5.2
	Section 5.3
	Section 6.1
	Section 6.2
	Section 6.3
	Section 6.4
	Section 6.5
	Section 6.6
	Section 6.7
	Section 7.1
	Section 7.2
	Section 7.3
	Section 7.4
	Section 8.1
	Section 8.2
	Section 8.3
	Section 9.2
	Section 9.3
	Section 9.4
	Section 9.5
	Section 9.6
	Section 9.7
	Section 9.8
	Section 10.2
	Section 10.3
	Section 10.4
	Section 10.5
	Section 10.6
	Section 10.7
	Section 10.8
	Section 10.9
	Section 10.10
	Section 11.1
	Section 11.2
	Section 11.3
	Section 11.4
	Section 11.5
Index