# Math 104

## Introduction to Analysis

Lectures MWF 11:10-12:00, 220 Wheeler Hall

Office hours F 16:00-18:00, 821 Evans Hall, and by appt.

#### Topics covered

• 08/30: Basic notation and terminology. Integers and rationals.
• 09/02: Ordered sets. Upper and lower bounds. Suprema and infima.
• 09/04: Fields.
• 09/06: Ordered fields.
• 09/09: The real number system.
• 09/11: Complex numbers.
• 09/13: Euclidean spaces.
• 09/16: Finite and infinite sets.
• 09/18: Basic set operations. Countable and uncountable sets.
• 09/20: Metric spaces. Open and closed sets.
• 09/23: Limit points. Interior and closure.
• 09/25: Relatively open and closed sets.
• 09/27: Compactness.
• 09/30: The Heine-Borel theorem and consequences.
• 10/02: Perfect sets.
• 10/04: The Cantor set and its variants.
• 10/07: Connected and disconnected sets.
• 10/09: Convergent and divergent sequences.
• 10/11: Basic limit results.
• 10/14: Subsequences.
• 10/16: Midterm.
• 10/18: Cauchy sequences. Monotone sequences.
• 10/21: Limit points. Liminf and limsup.
• 10/23: Series. Nonnegative series.
• 10/25: The dyadic trick. The number e.
• 10/28: The root and ratio tests.
• 10/30: Power series.
• 11/01: Summation by parts. Absolute convergence.
• 11/04: Addition and multiplication of series.
• 11/06: Rearrangements.
• 11/08: Limits of functions.
• 11/13: Continuous functions.
• 11/15: Continuity and compactness.
• 11/18: Continuity and connectedness. Discontinuities.
• 11/20: Monotonic functions. Infinite limits and limits at infinity.
• 11/22: The derivative of a real function. Mean value theorems.
• 11/25: Continuity of derivatives. L'Hospital rule.
• 11/27: Taylor's formula. Differentiation of vector-valued functions.
• 12/02: Riemann-Stieltjes integral: Riemann sums, refinements, and integrability.
• 12/04: Integrating continuous and monotone functions. Integrating against step functions.
• 12/06: The fundamental theorem of Calculus (various versions).
• 12/09: Properties of the integral. Some fun examples.

### Resources

• Books.
• Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill.
• Charles Pugh, Real Mathematical Analysis, Springer-Verlag (2002).

The first book will be the main textbook for the course.

The instructor welcomes cooperation among students and the use of books. However, handing in homework that makes use of other people's work (be it from a fellow student, a book or paper, or whatever) without explicit acknowledgement is considered academic misconduct.

### Assignments

Homework problems listed as Exercises are from the 3rd edition of Rudin.

• Homework assignment #1, due Sep 9th.
• Homework assignment #2, due Sep 16th: Exercises 7, 10, 12, 18 (pp.22-23).
• Homework assignment #3, due Sep 23rd: Exercises 3, 6, 7, 9 (p.43).
• Homework assignment #4, due Sep 30th: Exercises 10, 11, 13, 16 (p.44).
• Homework assignment #5, due Oct 7th: Exercises 18, 20, 23, 27 (p.45).
• Homework assignment #6, due Oct 14th: Exercises 3, 4, 5 (p.78).
• Homework assignment #7, due Oct 21st.
• Homework assignment #8, due Oct 28th.
• Homework assignment #9, due Nov 4th.
• Homework assignment #10, due Nov 13th: Exercises 6, 9, 12, 14 (p.78).
• Homework assignment #11, due Nov 18th: Exercises 1, 2, 3, 4 (p.98).
• Homework assignment #12, due Nov 25th: Exercises 10, 14, 18, 21 (p.98).
• Homework assignment #13, due Dec 2nd.
• Homework assignment #14, due Dec 9th.

### Exams

Here is a mock midterm.