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

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1 session

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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 23 topics

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I'm Anglo-Germanic. I love trains. I love mathematics and physics. I provide tutoring in Algebra II, Calculus, and Physics. I was a winner in the acsimc math competition and got a Very High Commendation and a scholarship in the most prestigious humanities research competition, the International John Locke Competition. 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. 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. 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 ancient 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. I have won top awards twice in the National French Competition. (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

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SESSION 1

16

Jul

SESSION 1

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