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Why Calculus Works: A Beginner’s Tour of Real Analysis

SAT Score Range

1 session

+1

This series ended on July 16, 2025. All 1:1 and group chats related to this series are disabled 7 days after the last session.

About

Have you ever wondered why the rules of calculus work the way they do? In this session, we’ll explore the foundational ideas of real analysis that give calculus its rigour and precision. We’ll introduce concepts like the epsilon delta definition of a limit, what it truly means for a function to be continuous, and how to begin thinking in proofs. This session is designed for students who have studied or are currently studying calculus and are curious about the "why" behind the formulas. No experience with proof-writing is required, just curiosity and a willingness to think more deeply about familiar ideas.

Tutored by

Anna P 🇬🇧

Certified in 15 topics

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I'm an Anglo-Saxon. I'm British and American. I'm a high-functioning autistic. I love trains. I love Our Lord Jesus Christ and Our Lady, the Blessed Virgin Mary. I love mathematics and physics. I provide tutoring in Algebra II, Calculus, and Physics. I was a winner in the acsimc math competition and earned distinction in the most prestigious humanities research competition, the International John Locke Competition, which is jointly hosted by Oxford and Princeton. At the moment, I’m doing a stem research internship. I’m working on a research project in analytic number theory, alongside a historical research project on the English Civil War and the French Revolution. I know multiple programming languages. I’ve also completed a selective summer stem program at UC Berkeley. I have successfully taken a university chemistry course. I’ve earned 5’s on my AP exams and received multiple academic awards at school in a range of subjects. I served as vice president of my school’s engineering club, contributing to projects in electrical and mechanical engineering. I’m also developing a blog and an annotated bibliography. In addition, I earned distinction in a science fair competition a few years ago, and I look forward to continuing to build on these experiences. I moderate a forum dedicated to Catholic and medieval history, as well as European royalist and reactionary movements such as Jacobitism, Anglo-American loyalism, and French legitimism. I'm fluent in French. I founded and currently serve as president of my school’s French club. (Keep in mind that I’m trying to be as specific as possible without doxxing myself) Please feel free to reach out to me via Slack or the Schoolhouse chat. I would be delighted to see you in my classes : )

Sessions

✋ ATTENDANCE POLICY

If you sign up, please try your best to attend live or let me know ahead of time if you can’t make it. If you miss two sessions without notice, you may be removed from the series to make room for others.

SESSION 1

16

Jul

SESSION 1

Other Topics

Other Topics

Wed 4:15 AM - 5:15 AM UTCJul 16, 4:15 AM - 5:15 AM UTC

In this series, we’ll take a deeper look at the foundations of calculus through the lens of introductory real analysis. We’ll begin by understanding the real number system, including concepts like bounds, supremum and infimum, and why completeness matters. From there, we’ll build toward the formal ε δ definition of a limit, carefully unpacking what it means and how to apply it in simple proofs. We’ll then use this framework to define and explore continuity, examining functions that are continuous and those that are not, with intuitive and rigorous examples. If time permits, we may touch on the Intermediate Value Theorem, uniform continuity, and the idea of convergence of sequences and functions. Along the way, you’ll be introduced to proof-based mathematical thinking, including basic techniques like direct proofs and proof by contradiction, all in a supportive, beginner-friendly environment.

Public Discussion

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Jul 16

1 week

60 mins

/ session

SCHEDULE

Wednesday, Jul 16

4:15AM