**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 **:** d*y*/dx = f(*x*,*y*)