Introductory Analysis-An Inquiry Approach

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Prerequisites
	P1. Exploring Mathematical Statements
		P1.1 What is a mathematical statement?
		P1.2 Basic set theory
		P1.3 Quantifiers, both existential and universal
		P1.4 Implication: the heart of a “provable” mathematical statement
		P1.5 Negations
		P1.6 Statements related to implication
	P2. Proving Mathematical Statements
		P2.1 Using definitions
		P2.2 Proving a basic statement with an existential quantifier
		P2.3 Proving a basic statement with a universal quantifier
		P2.4 Proving an implication directly
		P2.5 Proof by contrapositive
		P2.6 Proof involving cases
		P2.7 Proof by contradiction
		P2.8 Proof by induction
		P2.9 Proving that one of two (or one of several) conclusions is true
	P3. Preliminary Content
		P3.1 Relations and equivalence
		P3.2 Functions
		P3.3 Inequalities and epsilons
Main Content
	1. Properties of R
		1.1 Preliminary work
		1.2 Main Theorems
		1.3 Follow-up work
	2. Accumulation Points and Closed Sets
		2.1 Preliminary work
		2.2 Main Theorems
		2.3 Follow-up work
	3. Open Sets and Open Covers
		3.1 Preliminary work
		3.2 Main Theorems
		3.3 Follow-up work
	4. Sequences and Convergence
		4.1 Preliminary work
		4.2 Main Theorems
		4.3 Follow-up work
	5. Subsequences and Cauchy Sequences
		5.1 Preliminary Work
		5.2 Main Theorems
		5.3 Follow-up Work
	6. Functions, Limits, and Continuity
		6.1 Preliminary Work
		6.2 Main Theorems
		6.3 Follow-up Work
	7. Connected Sets and the Intermediate Value Theorem
		7.1 Preliminary Work
		7.2 Main Theorems
		7.3 Follow-up Work
	8. Compact Sets
		8.1 Preliminary Work
		8.2 Main Theorems
		8.3 Follow-up Work
	9. Uniform Continuity
		9.1 Preliminary Work
		9.2 Main Theorems
		9.3 Follow-up Work
	10. Introduction to the Derivative
		10.1 Preliminary Work
		10.2 Main Theorems
		10.3 Follow-up Work
	11. The Extreme and Mean Value Theorems
		11.1 Preliminary Work
		11.2 Main Theorems
		11.3 Follow-up Work
	12. The Definite Integral: Part I
		12.1 Preliminary Work
		12.2 Main Theorems
		12.3 Follow-up Work
	13. The Definite Integral: Part II
		13.1 Preliminary Work
		13.2 Main Theorems
		13.3 Follow-up Work
	14. The Fundamental Theorem(s) of Calculus
		14.1 Preliminary Work
		14.2 Main Theorems
		14.3 Follow-up Work
	15. Series
		15.1 Preliminary work
		15.2 Main Theorems
		15.3 Follow-up work
Extended Explorations
	E1. Function Approximation
		E1.1 Taylor Polynomials and Taylor\'s Theorem
		E1.2 Interpolation
		E1.3 Divided Differences
		E1.4 A Hybrid Approach
	E2. Power Series
		E2.1 Introduction to Power Series
		E2.2 Differentiation of a Power Series
		E2.3 Taylor Series
	E3. Sequences and Series of Functions
		E3.1 Pointwise Convergence
		E3.2 Uniform Convergence and Uniformly Cauchy Sequences of Functions
		E3.3 Consequences of Uniform Convergence
	E4. Metric Spaces
		E4.1 What is a Metric Space? Examples
		E4.2 Metric Space Completeness
		E4.3 Metric Space Compactness
	E5. Iterated Functions and Fixed Point Theorems
		E5.1 Iterative Maps and Fixed Points
		E5.2 Contraction Mappings
		E5.3 Newton\'s Method
Appendix
	A. Brief Summary of Ordered Field Properties
Bibliography
Index