Real Analysis (WI24)

Course Info:

  • Lectures: Monday, Wednesday, Friday, block 10 (10:10 am - 11:15 am ET)
  • x-period: Thursday, block 10X (12:15 - 1:05 pm ET); generally won't be used unless announced
  • Room: Kemeny 242
  • Instructor: Alena Erchenko
  • Instructor office: Kemeny 331
  • E-mail: alena.erchenko@dartmouth.edu
  • Office hours: Monday, Wednesday 9-10am ET, Friday 11:30am-12:30pm, or by appt in my office Kemeny 331
  • Prerequisites: MATH 22 or 24, or MATH 13 and permission of the instructor
  • Recommended Texts:
  • Grading: Grades will be based on weekly homework (30%), weekly quizzes (35%), and final (35%). Your lowest homework and quiz scores will be dropped.

Homework:

Homework is due on Gradescope on the day indicated below before 11:59 pm ET. Please upload your solutions by following the instructions in this video.  Feel free to submit handwritten or work typeset in LaTeX (Overleaf is a great resource and tool for the latter). If you are skipping a problem, it is easier for the grader if you write "Skip" and upload that as your solution to the problem. You are more than welcome to work with others on the homework problems. However, the solutions should be written up on your own and reflect your understanding.

Due Date Homework Solutions
01/05 Homework 0 N/A
01/10 Homework 1 Solutions
01/17 Homework 2 Solutions (pdf) / Solutions (TeX)
01/24 Homework 3 Solutions (pdf) / Solutions (TeX)
01/31 Homework 4 Solutions (pdf) / Solutions (TeX)
02/07 Homework 5 Solutions (pdf) / Solutions (TeX)
02/14 Homework 6 Solutions (pdf) / Solutions (TeX)
02/21 Homework 7 Solutions (pdf) / Solutions (TeX)
02/28 Homework 8 Solutions (pdf) / Solutions (TeX)

Quizzes:
There will be weekly in-class quizzes on Fridays for 15-20 mins at the end of the class. The quizzes will be covering the material of the week before.

Date Quiz Solutions
01/12 Quiz 1 Solutions
01/19 Quiz 2 Solutions (pdf) / Solutions (TeX)
01/26 Quiz 3 Solutions (pdf) / Solutions (TeX)
02/02 Quiz 4 Solutions (pdf) / Solutions (TeX)
02/09 Quiz 5 Solutions (pdf) / Solutions (TeX)
02/16 Quiz 6 Solutions (pdf) / Solutions (TeX)
02/23 Quiz 7 Solutions (pdf) / Solutions (TeX)
03/01 Quiz 8 Solutions (pdf) / Solutions (TeX)

 

Schedule:

The specific topics covered in each class are tentative and will be updated as the semester progresses. Moreover, the notes posted for each lecture may contain slightly more content than the lecture and may bleed a bit into the next lecture. Let me know if you find any errors.

Date Topic References
01/03 Introduction. Basic Set Theory.

Lecture 1

(R) Chapter I
(L) Introduction,
Section 0.3
01/05 Bijections, invertible maps. Cardinality. Cantor's theorem.

Lecture 2

(L) Introduction, Section 0.3
01/08 Ordered fields

Lecture 3

(R) Chapter II, Section 1-2
(L) Section 1.1
01/10 Least upper bound. Existence of square roots

Lecture 4

(R) Chapter II, Section 3-4
(L) Section 1.2

01/11

(block 10X, 12:15pm - 1:05 pm ET)

 Metric spaces

Lecture 5

(R) Chapter III, Sections 1-2
(L) Section 7.1
01/12 Ball neighborhoods. Open sets

Lecture 6

(R) Chapter III, Section 2

(L) Section 7.2
01/15 No class -> Moved to X-hour on 01/11
01/17 Open and closed sets. 

Lecture 7

(R) Chapter III, Sections 2

(L) Section 7.2
01/19 Convergence. 

Lecture 8

(R) Chapter III, Section 3

(L) Section 7.3
01/22 Complete spaces. 

Lecture 9

(R) Chapter III, Section 4

(L) Section 7.4
01/24 Compact sets

Lecture 10

(R) Chapter III, Section 5

(L) Section 7.4.2
01/26 Equivalent definitions of compactness. 

Lecture 11

(R) Chapter III, Section 5

(L) Section 7.4.2
01/29 Heine-Borel. Connected sets

Lecture 12

(R) Chapter III, Section 6

(L) Sections 7.4.2 and 7.2.2
01/31 Continuous function. Functions on connected spaces.

Lecture 13

(R) Chapter IV, Sections 1, 5
(L) Sections 3.2, 7.5.3, 
02/02 Intermediate Value Theorem. Limits

Lecture 14

(R) Chapter IV, Section 2, 5
(L) Sections 3.2, 3.3, 7.5.1
02/05 Rational functions

Lecture 15

(R) Chapter IV, Section 3
(L) Sections 3.2, 7.5.2
02/07 Functions on a compact. Sequences of functions.

Lecture 16

(R) Chapter IV, Section 4

(L) Sections 7.5.2
02/09 Sequences of functions

Lecture 17

(R) Chapter IV, Section 6
(L) Section 6
02/12 Derivatives

Lecture 18

(R) Chapter V, Section 1-2

(L) Section 4.1
02/14 Derivatives (continue). Mean Value Theorem

Lecture 19

(R) Chapter V, Section 3

(L) Sections 4.1 and 4.2
02/15
(block 10X, 12:15pm - 1:05 pm ET)		 Taylor's Theorem Lecture 20
(R) Chapter V, Section 4
(L) Section 4.3
02/16 Riemann Integral

Lecture 21

(R) Chapter VI, Section 1
(L) Section 5.1
02/19 Properties of integral

Lecture 22

(R) Chapter VI, Sections 2-3

(L) Section 5.2
02/21 Integrable functions Lecture 23
02/23 Fundamental Theorem. Interchange of limit operations.

Lecture 24

(R) Chapter VI, Sections 4-5. Chapter VII, Section 1
02/26 Contractions. The fixed point theorem. Lecture 25
02/28 Application in dynamics Lecture 26
03/01 Application in dynamics Lecture 27
03/04 Review Lecture 28
03/08
(11:30am-2:30pm ET)		 Final exam 

 

Solutions of the Final Exam

Course Catalogue Description:

This course introduces the basic concepts of real-variable theory. Topics include real numbers and cardinality of sets, sequences and series of real numbers, metric spaces, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Students may not take both Mathematics 35 and 63 for credit.

Learning Outcomes:

By the end of this course, you should be able to:

  1. Understand the real number system, metric spaces, continuity, differentiability, and Riemann integration;
  2. Solve mathematical problems: utilize abstraction and think creatively; 
  3. Write clear mathematical proofs: recognize and construct mathematically rigorous arguments.

Additional Pages:

  1. Academic Honor Principle
  2. Expectations
  3. Student Accessibility Needs
  4. Mental Health
  5. Title IX
  6. Religious Observances
  7. Additional Support for your Learning

Additional Resources: 

  • It is highly recommended to solve as many exercises from the textbook or other resources as you have time for. This will be more useful than rereading your class notes or textbook as it will often force you to go back to look at the material anyway. The only way to truly learn mathematics is through practice. Feel free to ask about your solutions.

  • There are many other textbooks and online materials that you can consult. If you have questions on whether a particular one might be useful, do not hesitate to reach out.