Real Analysis Foundations

          Module 1: Introduction to Real Analysis and Fundamental Concepts

Section 1: The Nature of Mathematics and Real Analysis
- Lesson 1: What is Real Analysis?
  - Module 1, Section 1, Lesson 1: An overview of real analysis, its scope, and its significance in mathematics.
- Lesson 2: The Importance of Rigor
  - Module 1, Section 1, Lesson 2: Discussing why rigor is essential in mathematics and how it builds a solid foundation for advanced topics.

Section 2: The Set of Real Numbers
- Lesson 1: Properties of Real Numbers
  - Module 1, Section 2, Lesson 1: Exploring the defining characteristics of real numbers including order and density.
- Lesson 2: The Completeness Axiom
  - Module 1, Section 2, Lesson 2: Understanding the completeness property of real numbers and its role in analysis.

Section 3: Basics of Mathematical Proofs
- Lesson 1: Logic and Proof Methods
  - Module 1, Section 3, Lesson 1: Introducing logical reasoning and various proof techniques used in analysis.
- Lesson 2: Direct Proof and Counterexamples
  - Module 1, Section 3, Lesson 2: Learning how to construct direct proofs and identify counterexamples to mathematical statements.

Section 4: The Language of Mathematics
- Lesson 1: Definitions and Theorems
  - Module 1, Section 4, Lesson 1: Emphasizing the importance of precise definitions and clear statements of theorems.
- Lesson 2: Mathematical Notation and Symbols
  - Module 1, Section 4, Lesson 2: Reviewing common mathematical symbols and notation critical for reading and writing proofs.

Section 5: Overview of Course Themes
- Lesson 1: Introduction to Limits, Continuity, and Convergence
  - Module 1, Section 5, Lesson 1: Providing a preview of the key themes and ideas that will be explored throughout the course.
- Lesson 2: The Course Roadmap
  - Module 1, Section 5, Lesson 2: Outlining the structure of the course and what to expect in subsequent modules.

Module 2: Limits and Sequences

Section 1: Understanding Limits
- Lesson 1: Intuitive Notions of Limits
  - Module 2, Section 1, Lesson 1: Discussing the conceptual idea of a limit through everyday examples.
- Lesson 2: The Formal Definition of a Limit
  - Module 2, Section 1, Lesson 2: Introducing the epsilon definition of a limit and its precise meaning.

Section 2: The Algebra of Limits
- Lesson 1: Limit Laws
  - Module 2, Section 2, Lesson 1: Learning the fundamental laws governing limits and how they are applied.
- Lesson 2: Evaluating Basic Limits
  - Module 2, Section 2, Lesson 2: Practical techniques for computing limits of elementary functions.

Section 3: Sequences in Real Analysis
- Lesson 1: Definition of a Sequence
  - Module 2, Section 3, Lesson 1: Understanding what sequences are and how they are used in analysis.
- Lesson 2: Convergence of Sequences
  - Module 2, Section 3, Lesson 2: Examining the concept of convergence with examples of convergent and divergent sequences.

Section 4: Limit Theorems for Sequences
- Lesson 1: Monotone Convergence Theorem
  - Module 2, Section 4, Lesson 1: Exploring the monotone convergence theorem and its applications.
- Lesson 2: Bounded Sequences
  - Module 2, Section 4, Lesson 2: Understanding how boundedness interacts with convergence in sequences.

Section 5: Introduction to Infinite Sequences
- Lesson 1: Subsequences and Their Importance
  - Module 2, Section 5, Lesson 1: Discussing the concept of subsequences and why they are useful in analysis.
- Lesson 2: Accumulation Points
  - Module 2, Section 5, Lesson 2: Defining accumulation points and studying their role in the convergence of sequences.

Module 3: Continuity of Functions

Section 1: Fundamental Concepts of Continuity
- Lesson 1: The Definition of Continuity
  - Module 3, Section 1, Lesson 1: Introducing what it means for a function to be continuous.
- Lesson 2: Intuitive Examples of Continuous Functions
  - Module 3, Section 1, Lesson 2: Providing simple, real-life examples to illustrate continuity.

Section 2: The Epsilon-Delta Definition of Continuity
- Lesson 1: Understanding the Epsilon-Delta Framework
  - Module 3, Section 2, Lesson 1: Breaking down the epsilon-delta definition with clear explanations.
- Lesson 2: Constructing Epsilon-Delta Proofs
  - Module 3, Section 2, Lesson 2: Step-by-step methods for proving the continuity of functions using epsilon-delta arguments.

Section 3: Properties of Continuous Functions
- Lesson 1: The Intermediate Value Theorem
  - Module 3, Section 3, Lesson 1: Explaining the Intermediate Value Theorem and its implications.
- Lesson 2: The Extreme Value Theorem
  - Module 3, Section 3, Lesson 2: Understanding how continuous functions attain maximum and minimum values on closed intervals.

Section 4: Exploring Discontinuities
- Lesson 1: Types of Discontinuities
  - Module 3, Section 4, Lesson 1: Identifying and classifying different types of discontinuities.
- Lesson 2: Analyzing Discontinuous Functions
  - Module 3, Section 4, Lesson 2: Techniques to study functions that have points of discontinuity.

Section 5: Continuity in Practice
- Lesson 1: Graphical Analysis of Continuity
  - Module 3, Section 5, Lesson 1: Using graphs to recognize continuous and discontinuous behavior in functions.
- Lesson 2: Applications of Continuity in Real World Problems
  - Module 3, Section 5, Lesson 2: Discussing how the concept of continuity applies to real-world scenarios and problem-solving.

Module 4: Convergence and Series

Section 1: Introduction to Convergence
- Lesson 1: The Concept of Convergence
  - Module 4, Section 1, Lesson 1: Defining convergence in the context of sequences and functions.
- Lesson 2: Recognizing Convergence in Examples
  - Module 4, Section 1, Lesson 2: Analyzing basic examples to distinguish convergent and divergent behavior.

Section 2: Tests for Convergence
- Lesson 1: The Comparison Test
  - Module 4, Section 2, Lesson 1: Learning how to use the comparison test to examine the convergence of series.
- Lesson 2: Cauchy's Convergence Criterion
  - Module 4, Section 2, Lesson 2: Introducing Cauchy’s criterion as a fundamental tool in evaluating convergence.

Section 3: Introduction to Infinite Series
- Lesson 1: Definitions and Key Concepts of Series
  - Module 4, Section 3, Lesson 1: Explaining the basic definition of an infinite series and how it relates to sequences.
- Lesson 2: Convergence of Series
  - Module 4, Section 3, Lesson 2: Studying conditions under which infinite series converge.

Section 4: Basics of Power Series
- Lesson 1: Understanding the Radius of Convergence
  - Module 4, Section 4, Lesson 1: Explaining how to determine the radius within which a power series converges.
- Lesson 2: Examples of Power Series
  - Module 4, Section 4, Lesson 2: Working through examples to solidify understanding of power series convergence.

Section 5: Practical Evaluation of Series
- Lesson 1: Techniques for Series Approximation
  - Module 4, Section 5, Lesson 1: Methods used to approximate the sum of a convergent series.
- Lesson 2: Error Estimation in Series
  - Module 4, Section 5, Lesson 2: Learning how to estimate the error when approximating series sums.

Module 5: Techniques for Rigorous Analysis

Section 1: Structure of Mathematical Proofs
- Lesson 1: Basics of Proof Writing
  - Module 5, Section 1, Lesson 1: Outlining the structure of clear and effective mathematical proofs.
- Lesson 2: Common Proof Techniques
  - Module 5, Section 1, Lesson 2: A review of various proof strategies including direct proof and induction.

Section 2: Proof by Contradiction and Contrapositive
- Lesson 1: Applying Proof by Contradiction
  - Module 5, Section 2, Lesson 1: Understanding the logic behind proof by contradiction through examples.
- Lesson 2: Using the Contrapositive Method
  - Module 5, Section 2, Lesson 2: Learning how to rewrite statements in their contrapositive form to prove assertions.

Section 3: Constructing and Analyzing Counterexamples
- Lesson 1: The Role of Counterexamples in Analysis
  - Module 5, Section 3, Lesson 1: Learning why finding counterexamples is essential in testing mathematical statements.
- Lesson 2: Techniques for Discovering Counterexamples
  - Module 5, Section 3, Lesson 2: Practical strategies to construct counterexamples in various contexts.

Section 4: Proofs Involving Limits and Continuity
- Lesson 1: Writing Proofs for Limits
  - Module 5, Section 4, Lesson 1: Detailed methods for constructing epsilon-delta proofs related to limits.
- Lesson 2: Proving Continuity Rigorously
  - Module 5, Section 4, Lesson 2: Step-by-step approaches to prove the continuity of functions using rigorous logic.

Section 5: Synthesis and Review
- Lesson 1: Review of Key Concepts in Real Analysis
  - Module 5, Section 5, Lesson 1: Summarizing the main ideas from the course to reinforce understanding.
- Lesson 2: Preparing for Further Study in Analysis
  - Module 5, Section 5, Lesson 2: Discussing strategies and next steps for students interested in deepening their study of higher mathematics.