## Syllabus

Certainly, here’s a sample syllabus for an introductory course in Real Analysis 1. Please note that this is a general example, and you should adapt it to meet the specific requirements and standards of your educational institution:

**Course Title: Introduction to Real Analysis 1**

**Course Description:** Real Analysis 1 is the first part of a two-semester course that provides a rigorous introduction to the foundations of real analysis, focusing on the theory of real numbers, sequences, and continuity.

**Course Objectives:** By the end of this course, students should be able to:

- Develop a deep understanding of the real number system and its properties.
- Apply mathematical proof techniques to analyze and prove properties of real numbers and sequences.
- Understand and work with limits and continuity of functions.
- Gain proficiency in mathematical writing and rigorous mathematical thinking.
- Apply real analysis concepts to solve mathematical problems and explore real-world applications.

**Textbook:**

- [Insert Real Analysis 1 Textbook Title and Author(s)]

**Materials:**

- Notebook or binder for class notes and assignments.
- Graphing calculator (if required by the school or teacher).
- Pencils, erasers, and a ruler.

**Grading:**

- Homework and Class Participation: XX%
- Quizzes and Assignments: XX%
- Midterm Examinations: XX%
- Final Examination: XX%

**Course Outline:**

**Unit 1: Introduction to Real Analysis**

- The importance of real analysis.
- Axiomatic development of the real number system.
- The least upper bound property.
- Completeness of the real numbers.

**Unit 2: Sequences and Convergence**

- Convergence and divergence of sequences.
- Limit theorems for sequences.
- Subsequences and the Bolzano-Weierstrass theorem.

**Unit 3: Continuity**

- Definition of continuity and properties of continuous functions.
- Intermediate Value Theorem.
- Uniform continuity.

**Unit 4: Limits and Continuity of Functions**

- Limits of functions.
- Continuity and differentiability.
- The derivative and the mean value theorem.

**Unit 5: Review and Final Exam Preparation**

**Note:** This is a general example of a syllabus for an introductory Real Analysis 1 course. It’s important to adapt it to meet the specific needs and standards of your educational institution. The percentages for grading, the textbook, and materials may vary. Additionally, consult with your institution’s curriculum guidelines and any state or district standards that may apply. Real Analysis 1 is typically a college-level course, so the level of rigor and depth may vary based on the institution’s curriculum.

## Free textbooks

Real Analysis is a rigorous mathematical subject, and finding completely free textbooks for it can be a bit challenging. However, there are some open educational resources (OER) and free online textbooks that cover the basics of Real Analysis. Here are a few options:

**“Elementary Real Analysis” by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner**:- This textbook is available for free online and covers the fundamentals of Real Analysis. It’s suitable for beginners and includes exercises with solutions.
- Website: http://classicalrealanalysis.info/

**“Real Variables with Basic Metric Space Topology” by Robert B. Ash**:- This book is freely available online and provides an introduction to Real Analysis along with some basic topology concepts. It includes exercises and solutions.
- Website: https://www.math.uiuc.edu/~r-ash/Real-Variables/

**“Introduction to Real Analysis” by William F. Trench**:- This free textbook offers a comprehensive introduction to Real Analysis. It includes clear explanations and exercises with solutions.
- Website: https://digitalcommons.trinity.edu/mono/7/

**“Principles of Mathematical Analysis” by Walter Rudin**:- While not free, “Principles of Mathematical Analysis” by Walter Rudin is a classic textbook in Real Analysis. Some universities provide free access to this book through their libraries or websites.

**OpenStax Calculus Volume 1**:- OpenStax offers a free calculus textbook designed for college students. While it’s not specifically a Real Analysis textbook, it covers many topics relevant to Real Analysis, including limits and continuity.
- Website: https://openstax.org/details/books/calculus-volume-1

**MIT OpenCourseWare (OCW)**:- MIT OCW provides free access to course materials from Massachusetts Institute of Technology (MIT) courses, including Real Analysis. You can find lecture notes, assignments, and exams related to Real Analysis.
- Website: https://ocw.mit.edu/index.htm

These resources should provide you with a good foundation in Real Analysis. Be sure to explore them and choose the one that aligns best with your learning style and course objectives.

## Free MOOCs

As of my last knowledge update in January 2022, finding a completely free Massive Open Online Course (MOOC) specifically focused on Real Analysis might be challenging due to the advanced nature of the subject. However, you can explore MOOC platforms for related courses in advanced mathematics and analysis. While these courses may not cover the entire Real Analysis curriculum, they can still be valuable for gaining mathematical insight. Keep in mind that some MOOCs offer free auditing options, while others may charge for certificates. Here are some platforms where you can explore relevant courses:

**Coursera**:- Coursera offers advanced mathematics courses from universities and institutions. You can audit many of these courses for free, but fees may apply if you want certificates or additional features.
- Website: https://www.coursera.org/

**edX**:- edX provides courses in advanced mathematics, including topics related to analysis. You can audit these courses for free, but certificates may have associated fees.
- Website: https://www.edx.org/

**MIT OpenCourseWare (OCW)**:- MIT OCW offers free access to course materials from Massachusetts Institute of Technology (MIT) courses, including those related to Real Analysis. While you won’t receive a certificate, you can access lecture notes, assignments, and exams.
- Website: https://ocw.mit.edu/index.htm

**Khan Academy**:- Khan Academy offers a wide range of free math courses and tutorials that cover advanced topics, including calculus and mathematical analysis.
- Website: https://www.khanacademy.org/

**Stanford Online**:- Stanford University offers some free online courses in advanced mathematics and related fields. Check their website for current offerings.
- Website: https://online.stanford.edu/courses

While these resources may not be specifically tailored to Real Analysis, they cover advanced mathematical topics that are foundational to the subject. Be sure to explore the course descriptions and offerings on each platform to find courses that align with your learning objectives. Additionally, verify the availability of free auditing options for the courses you are interested in.