Mathematical Economics IV

Course Overview

This is an elective course designed for students pursuing advanced studies in economic sciences. The course provides a rigorous foundation in mathematical analysis, covering key concepts essential for formal reasoning and further applications in economics.

Students are expected to have prior knowledge of basic calculus and linear algebra. Before delving into analysis, the course begins with an introduction to logic and proof techniques to ensure a strong mathematical foundation.

Course Outline

Lecture 1: Set Theory, Logic & Proofs

1.1 Propositional and Predicate Logic
1.2 Methods of Proof (Direct, Indirect, Contradiction, Induction)
1.3 Set Operations and Properties
1.4 Relations and Functions
1.5 Cardinality and Countability

Lecture 2: The Real Number System

2.1 Axioms of the Real Number System
2.2 Completeness Axiom and the Least Upper Bound Property
2.3 Archimedean Property and Density of Rational Numbers
2.4 Intervals and Order Properties

Lecture 3: Basic Topology

3.1 Open and Closed Sets in R\mathbb{R}R
3.2 Limit Points and Closure
3.3 Compactness (Heine-Borel Theorem)
3.4 Connectedness and Intermediate Value Theorem

Lecture 4: Sequences

4.1 Convergence and Limits of Sequences
4.2 Cauchy Sequences and Completeness
4.3 Subsequences and Bolzano-Weierstrass Theorem
4.4 Monotone Sequences and Limit Theorems

Lecture 5: Continuous Functions

5.1 Definition and Properties of Continuity
5.2 Limits of Functions and the ϵ\epsilonϵ-δ\deltaδ Definition
5.3 Uniform Continuity and Compactness
5.4 The Extreme Value and Intermediate Value Theorems
5.5 Differentiability and Relationship with Continuity

Reading Materials

Prescribed Textbook:

• Walter Rudin (1976). Principles of Mathematical Analysis (Baby Rudin). 3rd Edition, McGraw-Hill. [Chapters 1–6]

Recommended Readings:

Logic & Proofs:

• Richard Hammack (2013). Book of Proof. 3rd Edition, Open-Sourced Textbook.

Mathematical Analysis:

• Stephen Abbott (2015). Understanding Analysis. 2nd Edition, Springer. (For additional intuition and examples)

• Terence Tao (2006). Analysis I. Hindustan Book Agency. (For deeper insights and alternative perspectives)