Beta Enrollment Now Open

Some lessons are pending completion; however, we are creating and adding new content every week. Beta students have access to tutoring during weekly Zoom office hours.

Official Enrollment Opens September 2023


This course is for students who want to pursue higher studies in mathematics, machine learning, engineering, or economics. It is also ideal for anyone with a genuine interest in mathematics that wants to deepen their understanding.

  • Improve mathematical problem-solving skills

  • Formally analyze limits, sequences and series

  • Identify and prove properties hold for functions

  • Understand why derivative and integral results in calculus work

Course Outline

    1. Course Overview

    2. Practice Quiz

    3. Notation

    4. Induction

    5. Assignment: Induction

    6. Archimedean Property

    7. Quiz: Archimedean Property

    8. Infimums + Supremums

    9. Rationals and Density

    10. Office Hours

    1. Metric Spaces

    2. Quiz: Metrics

    3. Open and Closed Sets

    4. Quiz: Open and Closed Sets

    5. Assignment: Open and Closed Sets

    6. Compact Sets

    7. Quiz: Compact Sets

    8. Survey: Topology

    9. Midterm Exam 1

    1. Intro to Sequences and Limits

    2. Algebraic Properties of Limits

    3. Cauchy Sequences

    4. Divergent Sequences

    5. Monotone Sequences

    6. Subsequences

    7. Liminf and Limsup

    8. Series and Convergence Tests

    1. Limits of Functions

    2. Quiz: Limits of Functions

    3. Continuous Functions

    4. Assigment: Continuous Functions

    5. Intermediate Value Theorem

    6. Extreme Value Theorem

    7. Uniform Continuity

    8. Midterm Exam 2

    1. Lesson - Derivative Definition

    2. Lesson - Mean Value Theorem

    1. Lesson - Riemann Integral

    2. Lesson - Darboux Sums

    3. Final Exam

Course Highlights

  • Lessons for each major topic
  • Graphics for example proofs
  • Ability to ask questions

Analysis Taught via Graphics

Abstract definitions and theorems can be hard to digest. Our course makes concepts concrete by visualizing definitions and proof arguments.

Focus on Proof Techniques

Analysis courses usually focus on key theorems and too often leave details "as an exercise for the reader." However, students often experience the most headache in how to write proofs for homework and exams. Thus, we focus on supplying students with high-level outlines and walk-through proof examples.

Meet Your Instructors

Eric Erdős

Founder / Instructor

Our founder Eric loves learning and sharing new ideas. He has degrees in computer science, math, and physics. Eric spent several years tutoring and teaching for many STEM courses both through university and privately, including teaching in the math department at UCLA. During off time, he loves running and surfing.

Daisy Dirac


Daisy brings the cheerful spirit to the team. She applies the same gentle and warm teaching style as a mother reading children's books. In her off time, Daisy enjoys exploring coffee shops and practicing yoga and pilates.

Student Reviews

5 star rating


Yan Yang



Read Less


Your course at your pace, priced at the cost of a textbook rental.

  • $39.00 / month with 3 day free trial

    Course Subscription


Why Typal Academy?

We are unique in focusing our lessons on showing you how to write clear proofs and how to reason about each concept with intuitive diagrams. See below for how our course compares to alternatives.

Completion Certificate

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  • Will this help me with my class at university?

    Yes! This is precisely our goal, to help students better understand the concepts and how to apply them. We expect many students to be simultaneously enrolled in university courses (e.g. MATH 131A at UCLA).

  • What does it mean to be a beta student?

    Course memberships are $39/month for the course content; the ability to join Zoom office hours usually costs $30/month. However, current beta users have early access to Typal Academy while content is still undergoing much growth. To remedy any gaps in lesson plans, all beta students are welcome to join office hours with their membership.

  • What is so different about adding graphics?

    Real analysis concepts are typically geometric in nature, and seeing canonical examples can go a long ways in developing intuition. Real analysis textbooks often either do not contain diagrams or very few (that are of low quality).

  • Do I need to be enrolled in a university?

    Not at all! We aim to empower aspiring scholars to learn effectively without forking out money to expensive universities. Some members even find this resource useful for preparing to go back to school for a graduate degree.

  • What are the lessons like?

    Each lesson includes key definitions and theorems along with several walkthrough examples that include both proofs and illustrative diagrams. These no-fluff lessons are written with a lighter tone than most textbooks, with sidebars from instructors giving intuition for the examples. Some lessons also include video mini-lectures.

  • How long will this course take?

    The content for this course is typically studied over the course of 10 weeks. Yet, you can adjust that timeline as needed to suit your needs.

  • What's the cost?

    Access to the course contents costs $39 per month. All current users (i.e. beta users) can also attend weekly office hours with Eric on Sunday evenings.

  • What do I do if I have a question?

    Each lesson and quiz has an associated discussion tab where you are welcome to ask clarifying questions to the community. We update lessons to better clarify material based on community comments. You are also welcome to ask for more examples on a particular topic!

  • Do I need a textbook?

    Nope! The content in our online course is self-contained. If you also want to complement your analysis studies with a textbook, we certainly encourage this!