Lecture Notes

Lectures:

Lecture 1: Where Do We Start?
Lecture 2: The Real Numbers
Lecture 3: Countability
Lecture 4: Metric Spaces, Open Sets
Lecture 5: Closed Sets
Lecture 6: Compact Sets
Lecture 7: Compact Subsets of Euclidean Space
Lecture 8: Completeness
Lecture 9: Sequences and Series
Lecture 10: Continuity
Lecture 11: Continuity and Sets
Lecture 12: Continuity and Compactness
Lecture 13: First In-Class Test
Lecture 14: Differentiability
Lecture 15: Mean Value Theorem
Lecture 16: Riemann-Stieltjes Integral Defined
Lecture 17: Integrability of a Continuous Function
Lecture 18: Riemann-Stieltjes Integral
Lecture 19: Fundamental Theorem of Calculus
Lecture 20: Sequences of Functions
Lecture 21: Second In-Class Test
Lecture 22: Uniform Convergence
Lecture 23: Equicontinuity
Lecture 24: Power Series
Lecture 25: Fundamental Theorem of Algebra
Lecture 26: Final Review
Final Exam

 

Lecture Outlines

Lecture 1: Where Do We Start?

Reading: Rudin Pages 1-11
Problems: Rudin Chapter 1, Problems 1, 3, 5

1. First 'Proof' -- that there is no rational with square 2
2. Naive Set Theory
3. Fields, Rational Numbers
4. Ordering

Lecture 2: The Real Numbers

Reading: Rudin Pages 11-17
Problems: Rudin Chapter 1, Problems 8, 9, 10

1. Least Upper Bound Property
2. Archimedean Property of Real Numbers
3. Euclidean Spaces
4. Schwarz Inequality
5. Triangle Inequality
6. Complex Numbers

Lecture 3: Countability

Reading: Rudin Pages 24-30
Problems: Rudin Chapter 2, Problems 2, 3, 4

1. Maps, Surjectivity, Injectivity, Bijectivity
2. Equivalence of sets
3. Finite Sets, Countable Sets, Uncountable Sets, At-most-countable Sets, Infinite sets
4. Countability of the Integers (duh)
5. A Countable Union of Countable Sets is Countable
6. Cartesian Product of Two Countable Sets is Countable
7. Countability of the Rationals
8. The Noncountability of the Set of Sequences with Values in {0,1}

Lecture 4: Metric Spaces, Open Sets

Reading: Rudin Pages 31-35
Problems: Rudin Chapter 2, Problems 9a, 9b, 9c, 11

1. Metric Spaces, Definition and Examples -- Euclidean Metric, Discrete Metric and Supremum Metric
2. Open Balls in a Metric Space
3. Open Subsets of a Metric Space
4. Unions and Countable Intersections of Open Sets are Open
5. Open Balls are Open (duh)
6. Limit Points and Closed Sets

Lecture 5: Closed Sets

Reading: Rudin Pages 34-36
Problems: Rudin Chapter 2, Problems 10, 22, 23

Metric Spaces, Basic Theory

1. Complements of closed sets are open and vice versa
2. Closure of a set
3. Relatively open subsets
4. Compact sets are closed

Problem set 3 is due on the same day as lecture #6.

Lecture 6: Compact Sets

Reading: Rudin Pages 36-38
Problems: Rudin Chapter 2, Problems 12, 16, 25

1. Infinite Subsets of Compact Sets have Limit Points
2. Countable intersection property

Lecture 7: Compact Subsets of Euclidean Space

Reading: Rudin Pages 38-40
Problems: Rudin Chapter 2, Problems 24, 26, 29

1. Compactness of the unit cube
2. Heine-Borel theorem
3. Weierstrass's theorem
4. Connectedness of sets

Lecture 8: Completeness

Reading: Rudin Pages 42-43, 47-55
Problems: Rudin Chapter 2, Problems 19, 20, 21

Problem set 4 due at the end of class.

1. Sequential Compactness
2. Convergence of Sequences
3. Cauchy Sequences
4. Completeness
5. Completeness of Euclidean spaces

Lecture 9: Sequences and Series

Reading: Rudin Pages 55-69, 71-75
Problems: Rudin Chapter 3, Problems 2, 7, 12, 16

1. Completeness of compact spaces
2. Sequential compactness
3. Did not do series, root, ratio tests, absolute convergence

Lecture 10: Continuity

Reading: Rudin pages 83-86
Problems: Rudin Chapter 4, Problems 1, 4, 15

Problem set 5 is due on the same day as this lecture.

1. Limits of functions at a point
2. Continuity of functions at a point
3. Continuity of composites
4. Continuity of maps

Lecture 11: Continuity and Sets

Reading: Rudin pages 85-93
Problems: Rudin Chapter 4, Problems 1, 4, 15

1. Continuity and open sets
2. Continuity and closed sets
3. Continuity of components

Lecture 12: Continuity and Compactness

Reading: Rudin pages 89-93
Problems: -

1. Continuity and compactness
2. A continuous function on a compact set has a maximum
3. Continuity of connectedness
4. A continuous function on an interval takes intermediate values

Lecture 13: First In-Class Test Covering All Material In Lectures 1-10

Lecture 14: Differentiability

Reading: Rudin pages 103-107
Problems: -

1. Differentiability and the derivative
2. Differentiability implies continuity
3. Sums and products
4. Chain rule
5. Maxima and minima

Lecture 15: Mean Value Theorem

Reading: Rudin pages 107-110
Problems: -

1. Mean value theorems
2. Increasing and decreasing functions
3. l'Hopital's rule
4. Higher derivatives
5. Taylor's theorem

Lecture 16: Riemann-Stieltjes Integral Defined

Reading: Rudin pages 120-124
Problems: -

1. Upper and lower sums
2. Upper and lower integrals
3. Integrability
4. Refinement

Lecture 17: Integrability of a Continuous Function

Reading: Rudin pages 124-127
Problems: -

1. Integrability criterion
2. Continuous functions are Riemann integrable

Lecture 18: Riemann-Stieltjes Integral

Reading: Rudin pages 128-133
Problems: -

1. Riemann-Stieltjes Integral
2. Properties of the integral

Lecture 19: Fundamental Theorem of Calculus

Reading: Rudin pages 133-136
Problems: -

1. Integration by parts
2. FTC version 1
3. FTC version 2

Lecture 20: Sequences of Functions

Reading: Rudin pages 143-151
Problems: -

1. Pointwise convergence of sequences of functions
2. Uniform convergence
3. Cauchy criterion
4. Uniform convergence and continuity

Lecture 21: Second In-Class Test

Lecture 22: Uniform Convergence

Reading: Rudin pages 150-154
Problems: -

1. The metric space of bounded continuous functions on a metric space
2. Uniform convergence and integration
3. Uniform convergence and differentiation

Lecture 23: Equicontinuity

Reading: Rudin pages 154-161
Problems: -

1. Equicontinuity and compactness
2. Stone-Weierstrass theorem

Lecture 24: Power Series

Reading: Rudin pages 83-86
Problems: -

1. Convergent Taylor series
2. Analytic Functions

Lecture 25: Fundamental Theorem of Algebra

Reading: Rudin pages 180-185
Problems: -

1. Exponential, logarithm and trigonometric functions

Lecture 26: Final Review

Reading: -
Problems: -

1. Final Exam review
2. Indications of what we could have covered with more time
3. Structure of final exam
4. Relationship of this material to other mathematics courses

Final Exam