Foundations of Analysis

This course is the first in a three-part series on Real Analysis: Foundations of Analysis, followed by Analysis I, and concluding with Analysis II: Differentiation and Integration. Our primary goal is to establish the rigorous foundations upon which all of analysis is built.

Pedagogically, the course is structured around active, collaborative engagement. Several days before each class, you will receive lecture notes that include precise definitions, illustrative examples, and a series of propositions and theorems to be proved. Class time will be devoted to discussing your ideas, refining your arguments, and addressing any questions that arise. In this way, you will develop not only a deep conceptual understanding but also the essential skill of constructing and communicating rigorous proofs.

Prerequisites

There are no formal prerequisites but familiarity with basic set notation and elementary mathematical logic would be helpful. We will review these concepts in optional, self‑paced Miro review boards before the semester, and again during the first classes.

Curriculum

1. Sets, functions, set operations (union, intersection, etc.), composition of functions. Image and preimage. Injectivity, surjectivity.
2. The Peano axioms. Addition and multiplication of natural numbers. The ordering of the natural numbers.
3. Well-ordering principle.
4. Construction of the integers as equivalence classes of pairs of natural numbers; addition, multiplication, ordering.
5. Fundamental Theorem of Arithmetic for natural numbers.
6. Construction of the rational numbers; addition, multiplication, ordering.
7. Cauchy sequences of rational numbers.
8. Definition of real numbers. Addition and multiplication.
9. The ordering of the real numbers.
10. The least upper bound property. The Archimedean property.
11. Definition of the limit of a sequence. Cauchy sequences. Completeness of the set of real numbers.
12. Algebraic properties of limits of sequences.
13. Real exponentiation.
14. Subsequences. Limit points of sequences.
15. Nested Intervals Property. The Bolzano–Weierstrass theorem.
16. Limit superior and limit inferior.

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.