This course is a natural continuation of our Foundations of Analysis course. A rigorous understanding of real numbers and sequence limits enables us to study key concepts in Real Analysis: convergence of series, subsets of the real line, continuity of functions.
We have structured the course to provide you with opportunities to collaboratively prove all theorems with your peers. Several days before each class, you will receive lecture notes containing definitions, examples, and propositions that we need to prove. In class, we will discuss your approaches, comments, and any questions you may have.
Prerequisites
Concepts studied in the Foundations of Analysis course.
Curriculum
- Real exponentiation.
- Sequences, defined by recurrence relations.
- Infinite series. Absolute and conditional convergence.
- Cauchy Criterion for series. The harmonic series.
- The root and ratio tests.
- Subsets of the real line. Open and closed subsets.
- Closure. Adherent and limit points of a set.
- The Heine-Borel theorem for the line.
- Limiting values of functions. Left and right limits.
- Continuous functions. Singularities.
- Dirichlet’s and Thomae’s functions.
- The maximum principle.
- The intermediate value theorem.
- Limits at infinity.
- Compact sets. The Cantor set. The Heine-Borel covering theorem.
- Continuous functions of several variables
Grading & Assessment
There will be two classes each week: one lecture and one problem-solving class. The lecture is not a traditional presentation; it is a guided discussion of the pre‑read notes where we jointly prove the day’s theorems. We will use a collaborative online whiteboard (Miro) for each session where lecture notes or problem sets are uploaded in advance. Each week you should write solutions to at least four problems total across those boards. You will receive 1 point for each correct solution, up to a maximum of 4 points per week. Additional bonus points may be awarded for the shortest and most elegant solutions or for solutions to the most challenging exercises.
Both the midterm and final exams are oral, open-book, and open-notes. The midterm consists of 5 problems worth 3 points each, and the final exam consists of 5 problems worth 4 points each. You also receive 1 bonus point each time you are the first to correct the professor’s error during a lecture.
Grade = (regular points earned + bonus points) / (regular maximum possible points). Thus it is possible to score above 100%. To receive an A+, you need to earn at least 90% of regular points; an A requires 85%, and so on.
Textbooks
We recommend several textbooks below. Please select those that best meet your needs. You are not required to use the most recent editions; these are listed for your reference only.
- T. Tao, Analysis I, 4th edition, 2022.
- S. Abbott, Understanding Analysis, 2nd edition, 2016.
- W. Rudin, Principles of Mathematical Analysis, 3rd edition, 1976.
- V. Zorich, Mathematical Analysis I, 2nd edition, 2015.