## Contents - Part 1

I. Notions from Set Theory 1 ( ita - chapter 1 )

operations on sets 4
____ : The Compliment of a Set
X 12 : The Compliment of a Union
E 06 : The Compliment of an Intersection
____ : The Difference of two Sets

functions 8
____ : The General Definition of Function
____ : The Restriction of a Function
____ : Inverse Functions
____ : Image of a Set Under a Function
____ : Inverse Image of a Set Under a Function

finite and infinite sets 10
____ : Finite and Infinite Sets
X 12 : The Image of a Union is the Union of Images
X 12 : The Inverse Image of a Union
X 12 : The Inverse Image of an Image
X 12 : The Image of an Inverse Image

II. The Real Number System 15 ( ita - chapter 2 )

____ : Summary Page for Upper and Lower Bounds
____ : Bigger Numbers have Bigger Square Roots

III. Metric Spaces 33 ( ita - chapter 3 )

definition of a metric space 34
D 34 : Definition of a Metric Space
E 34 : The Real Numbers as a Metric Space
E 34 : Euclidean n-Space
D 36 : Subspaces in a Metric Space
P 37 : The Difference of Two Sides of a Triangle is Less than the Third Side

open and closed sets 37
D 37 : Open and Closed Balls in a Metric Space
D 38 : Intervals of Real Numbers
D 39 : Open Sets in a Metric Space
P 40 : An Open Ball is an Open Set
D 40 : Closed Sets in a Metric Space
P 41 : A Closed Ball is a Closed Set
____ : Examples with Real Numbers
P 39 : Combinations of Open Sets
P 41 : Combinations of Closed Sets
E 42 : Half Spaces in Euclidean n-Space
E 43 : Intervals in Euclidean n-Space
D 43 : Bounded Sets
P 44 : Extrema for Nonempty Bounded Closed Sets of Real Numbers
X 61 : Bounded Open Sets of Real Numbers

convergent sequences 44
D 45 : Convergence of a Sequence
E 46 : The Constant Sequence
P 46 : The Limit of a Sequence is Unique
D 46 : Subsequences
P 46 : Convergence of Subsequences
D 47 : Bounded Sequences
E 47 : Convergent Sequences are Bounded
T 47 : Convergent Sequences and Closed Sets
P 48 : Limit of a Product Sequence
P 48 : Limits of Sum, Difference, and Quotient Sequences
D 50 : Monotonic Sequences
P 50 : Bounded Monotonic Sequences are Convergent

completeness 51
D 51 : Cauchy Sequences
P 51 : Convergent Sequences are Cauchy
P 52 : Cauchy Sequences are Bounded
P 52 : Convergence of a Cauchy Sequence with a Convergent Subsequence
D 52 : Completeness
P 52 : The Completeness of a Closed Subset of a Complete Metric Space
T 53 : As a Metric Space, the Real Numbers are Complete

compactness 54
D 54 : Compactness
P 54 : A Compact Subset of a Metric Space is Bounded
D 55 : Cluster Points
____ : Cluster Points in Open Sets of Real Numbers
T 56 : Existence of a Cluster Point in Infinite Subsets of a Compact Metric Space
C 56 : Convergence of a Subsequence in a Compact Metric Space
C 56 : A Compact Metric Space is Complete
C 56 : A Compact Subset of a Metric Space is Closed
X 64 : [a,b] is Compact
X 65 : More on Compactness

connectedness 59
D 59 : Connectedness
P 60 : A Criterion for a Set of Real Numbers to be Not Connected
T 60 : Intervals of Real Numbers are Connected

IV. Continuous Functions 67 ( ita - chapter 4 )

definition of continuity 68
D 68 : Continuity at a Point
D 68 : Continuity on a Set
____ : Example: f(x,y) = y
E 69 : A Constant Function
E 70 : The Restriction of a Continuous Function is Continuous
P 70 : Inverse Images of Open Sets Under Continuous Functions are Open

continuity and limits 72
E & D 72 : Limit of a Function at a Point
E 74 : The Connection Between Continuity and Limits
P 74 : Continuous Functions and Convergent Sequences

continuity of rational operations 75
P 75 : Continuity of Rational Operations
C 76 : Limits in Rational Operations

continuous functions on a compact metric space 78
T 78 : The Continuous Image of a Compact Metric Space is Compact
C 78 : Max & Min - Continuous Real-valued Functions on Compact Metric Spaces
D 80 : Uniform Continuity
E 80 : Uniform Continuity Implies Continuity
T 80 : Continuous Functions on Compact Metric Spaces are Uniformly Continuous

continuous functions on a connected metric space 82
T 82 : The Continuous Image of a Connected Set is Connected
C 82 : The Intermediate Value Theorem

sequences of functions 83
D 83 : Convervent Sequences of Functions
D 85 : Uniformly Convervent Sequences of Functions
P 86 : Cauchy Criterion for Sequences of Functions
T 87 : Continuity of the Limit of a Uniformly Convergent Sequence of Continuous Functions
L 87 : Continuity of f,g: E to E' Implies Continuity of d'( f(p), g(p) )
T 90 : A Complete Metric Space of Continuous Functions

V. Differentiation 97 ( ita - chapter 5 )

definition of derivative 98
D 098 : Differentiability at a Point
P 099 : Differentiability Implies Continuity
D 100 : Differentiability on a Set
_____ : The Restriction of a Differentiable Function is Differentiable

rules of differentiation 100
E 100 : The Derivative of a Constant Function is Zero
E 100 : The Derivative of the Identity Function is One
P 101 : Rules of Differentiation
_____ : The Derivative of a Product of n Functions

the mean value theorem 103
P 103 : The Derivative at Extrema is Zero
L 104 : Rolle's Theorem
T 105 : The Mean Value Theorem

VI. Riemann Integration 111 ( ita - chapter 6 )

definitions and examples 112
D 112 : The Riemann Sum
D 112 : The Riemann Integral
E 114 : Integral of a Constant Function
E 114 : Integral of a Function with Only One Non-zero Point
E 114 : Integral of a Unit Step

linearity and order properties of the integral 116
P 116 : Integrals of Sums and Constant Multiples of Functions
P 117 : The Integral of a Non-negative Function is Non-negative
C 117 : Integration of an Inequality
C 118 : Bounds for an Integral

existence of the integral 118
L 118 : Integrability
D 119 : Step Functions
L 119 : Step Functions are Integrable
P 120 : General Condition for Riemann Integrability
T 123 : Continuous Functions are Riemann Integrable

the fundamental theorem of calculus 123
P 123 : Subintervals in Riemann Integration
D 125 : Negation of a Riemann Integral
C 125 : Integration from a to b to c - Regardless of Order
T 126 : The Fundamental Theorem of Calculus
C 127 : The Integral in Terms of its Antiderivative

VII. Interchange of Limit Operations 137 ( ita - chapter 7 )

differentiation under the integral sign 159
D 159 : The Partial Derivative

VIII. The Method of Successive Approximations 169 ( ita - chapter 8 )

the fixed point theorem 170
T 170 : The Fixed Point Theorem

uniqueness of the real number system 29
_____ : b-mal Expansions of Real Numbers
_____ : Every Infinite b-mal Represents a Real Number
_____ : Periodic Infinite b-mals
Discussion 29 : Uniqueness of the Real Number System

the simplest case of the implicit function theorem 173
T 174 : The Simplest Case of The Implicit Function Theorem
C 176 : The Inverse Function Theorem

existence and uniqueness theorems for ordinary differential equations 177
T 178 : dy/dx = f(x,y)