Introduction To Real Analysis

1. Understand and apply the properties of real numbers.
2. Prove convergence and divergence of sequences.
3. Apply the Bolzano-Weierstrass Theorem.
4. Evaluate limits of functions.
5. Understand and apply the Cauchy criterion.
6. Analyze continuity and differentiability of functions.

13 weeks, about 52 hours in total. Yours will differ.

1. Week 1Module 1: Preliminaries

   • Unit 1: Sets and Functions · 4 hours

     Review set theory notations and definitions.. Practice identifying subsets, unions, intersections, and complements.. Work through examples of sets defined by properties..

2. Week 2Module 1: Preliminaries

   • Unit 2: Mathematical Induction · 4 hours

     Study the principle of mathematical induction.. Practice proving statements using mathematical induction.. Understand the well-ordering property of natural numbers..

3. Week 3Module 1: Preliminaries

   • Unit 3: Finite and Infinite Sets · 4 hours

     Learn the definitions of finite, infinite, denumerable, countable, and uncountable sets.. Understand and apply the uniqueness theorem.. Practice identifying countable and uncountable sets..

4. Week 4Module 2: The Real Numbers

   • Unit 1: The Algebraic and Order Properties of R · 4 hours

     Study the algebraic properties of real numbers.. Understand the order properties and trichotomy property.. Practice working with inequalities and proving basic theorems..

5. Week 5Module 2: The Real Numbers

   • Unit 2: Absolute Value and Real Line · 4 hours

     Understand the definition and properties of absolute value.. Apply the triangle inequality and its variations.. Practice solving inequalities involving absolute values..

6. Week 6Module 2: The Real Numbers

   • Unit 3: The Completeness Property of R · 4 hours

     Study the completeness property of real numbers.. Understand suprema and infima.. Apply the completeness property to prove fundamental results..

7. Week 7Module 2: The Real Numbers

   • Unit 4: Applications of the Supremum Property · 4 hours

     Apply the Archimedean property.. Understand the existence of square roots.. Study the density of rational numbers in R..

8. Week 8Module 2: The Real Numbers

   • Unit 5: Intervals · 4 hours

     Understand the nested interval property.. Study binary and decimal representations of real numbers.. Prove the uncountability of R..

9. Week 9Module 3: Sequences and Series

   • Unit 1: Sequences and Their Limits · 4 hours

     Learn the definition of a sequence and its limit.. Apply the uniqueness of limits theorem.. Practice proving convergence using the definition of limit..

10. Week 10Module 3: Sequences and Series

    • Unit 2: Limit Theorems · 4 hours

      Study the limit theorems for sequences.. Apply the squeeze theorem.. Understand the properties of bounded sequences..

11. Week 11Module 3: Sequences and Series

    • Unit 3: Monotone Sequences · 4 hours

      Understand monotone increasing and decreasing sequences.. Apply the monotone convergence theorem.. Study Euler's number..

12. Week 12Module 3: Sequences and Series

    • Unit 4: Subsequences and the Bolzano-Weierstrass Theorem · 4 hours

      Learn the definition of a subsequence.. Understand the Bolzano-Weierstrass theorem.. Apply divergence criteria..

13. Week 13Module 3: Sequences and Series

    • Unit 5: The Cauchy Criterion · 4 hours

      Understand the Cauchy criterion for convergence.. Apply the Cauchy criterion to determine convergence.. Study contractive sequences..