Instructor: Eran Assaf
Course on canvas.dartmouth.edu.⇗
Course 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, continuous functions, integration theory, sequences and series of functions, and polynomial approximation. Some applications of the theory may be presented. MATH 63 presents similar material, but from a more sophisticated point of view. This course may not serve as an adequate prerequisite for either MATH 73 or 83. Students who contemplate taking one of these two advanced courses should consider taking MATH 63 instead of this course. Not open to students who have taken MATH 63.
Textbook
Russell A. Gordon, Real Analysis: A First Course, 2nd Edition
Syllabus
| Week | Date | Topics | Reading | Homework |
|---|---|---|---|---|
| 1 | Fri 21 Jun Download Fri 21 Jun | Course Introduction, Notation and Mathematical Logic. |
Appendix A Appendix B |
|
| 2 | Mon 24 Jun Download Mon 24 Jun |
What is a Real Number? Fields. |
1.1 |
Problem Set #1 |
| Wed 26 Jun Download Wed 26 Jun | Ordered Fields, Absolute Value. | 1.2 | ||
| Fri 28 Jun | No Lesson | |||
| 3 | Mon 1 Jul Download Mon 1 Jul | Supremum and Infimum of Sets, Completeness Axiom. | 1.3 | Problem Set #2 |
| Wed 3 Jul Download Wed 3 Jul | Completeness Axiom (cont'd), Archimedean Property of , Existence of roots in . | 1.3 | ||
| Fri 5 Jul Download Fri 5 Jul | Countable and Uncountable Sets. |
1.4 |
||
| 4 | Mon 8 Jul Download Mon 8 Jul | Countable and Uncountable Sets. (Cont'd) | 1.4 | Problem Set #3 |
| Wed 10 Jul Download Wed 10 Jul | Uncountable sets and Sequences | 2.1 | ||
| Fri 12 Jul Download Fri 12 Jul | Sequences and Convergence. | 2.1 | ||
| 5 | Mon 15 Jul | No Lesson | Problem Set #4 | |
| Wed 17 Jul Download Wed 17 Jul | Sequences and Convergence (Cont'd). | 2.1 | ||
|
Thu 18 Jul Download Thu 18 Jul (X-Hour) |
Unbounded and Monotone Sequences. | 2.2 | ||
| Cauchy Sequences, Cauchy Completeness of , Nested Intervals Theorem. | 2.2 | |||
| 6 | Mon 22 Jul Download Mon 22 Jul | Midterm Review. |
Practice - midterm from Winter 2024 |
|
| Wed 24 Jul | Midterm Exam (In-class Part). | |||
| Thu 25 Jul Download Thu 25 Jul (X-Hour) | Subsequences. | 2.3 | ||
| Fri 26 Jul Download Fri 26 Jul | Limit of a Function. | 3.1 | ||
| 7 | Mon 29 Jul Download Mon 29 Jul | Limit of a Function (cont'd). | 3.1 | |
| Wed 31 Jul Download Wed 31 Jul | One-sided Limits, Limits at Infinity, Infinite Limits. |
3.1 |
||
| Thu 1 Aug Download Thu 1 Aug (X-Hour) | Continuity. |
3.2 |
||
| Fri 2 Aug Download Fri 2 Aug | Continuity and Discontinuities. |
3.2 |
||
| 8 | Mon 5 Aug Download Mon 5 Aug | Intermediate & Extreme Value Theorems. | 3.3 | Problem Set #6 |
| Wed 7 Aug Download Wed 7 Aug | Uniform Continuity. | 3.4 | ||
| Fri 9 Aug Download Fri 9 Aug | The Derivative of a Function. | 4.1 | ||
| 9 | Mon 12 Aug Download Mon 12 Aug | Rolle's Theorem and the Mean Value Theorem. | 4.2 | Problem Set #7 |
| Wed 14 Aug Download Wed 14 Aug | The Riemann Integral. | 5.1 | ||
| Fri 16 Aug Download Fri 16 Aug | The Riemann Integral (cont'd). | 5.1 | ||
| 10 | Mon 19 Aug Download Mon 19 Aug | Conditions for Riemann Integrability. | 5.2 | Problem Set #8 |
| Wed 21 Aug Download Wed 21 Aug | The Fundamental Theorem of Calculus. | 5.3 | ||
| Thu 22 Aug Download Thu 22 Aug (X-hour) | Final Exam Review Session. | |||
| Sun 25 Aug | Final Exam | Practice Final - from Winter 2024 Download Practice Final - from Winter 2024 |