How to self-study real analysis - a step-by-step guide
+ a cute problem about infinite series!
What is real analysis?
I remember hearing a mathematician say the word “analysis” in my first year of undergrad, and I thought they were talking about data analysis or statistics.
Not true! When a mathematician says analysis, what they mean is the rigorous study of calculus.
Real analysis is one of the core branches of pure math — and one that is notoriously confusing to learn.
In this article, I’ll give a roadmap to self-study real analysis. I’ll include some textbooks that I found helpful as well as some online video tutorials.
How do I learn real analysis?
Book Recommendations
Understanding Analysis by Stephen Abbott. Oh my goodness. I cannot recommend this book enough. If you are struggling to understand “delta-epsilon” proofs, please just read this book. It was a lifesaver when I was learning this in undergrad.
Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings. To quote the preface: “Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy.” I really enjoyed this book and I hope you do too.
Video Recommendations
One of my favorite video lectures about analysis is the following playlist by Francis Su. The content is crystal clear and presented in a very beautiful way! I’ve put the first video below to whet your appetite:
The MAT137 YouTube series. These are the official class videos from the MAT137 (Analysis with proofs) course at the University of Toronto. I’m a huge fan of these videos: they’re all in 5 minute chunks which makes them easy to digest.
Challenge Problem
In the spirit of calculus, here is a problem about infinite sums.
Show that
(You have probably learned in a calculus course that this sum is equal to e, which is approximately 2.718…. But that's cheating! Use "low-tech" methods.)
Got a solution to the challenge problem? Submit it here.
Solution from last week
Last week’s problem was:
Given a positive integer n, find distinct positive integers x and y such that
Wow — I’m blown away by the response to last week’s question! There were a record number of submissions for this problem, each with different and creative solutions.
Special shoutout to Mark Spindler (Maryland), Shawn Varghese (New Jersey), Ben Elkins (Evanston, IL), Vipin Bhat (Pennsylvania), David Galarza (Bogotá), Spencer Wadsworth (Idaho), Olivier Massicot (Champaign, IL), Yirmi Levitas (Safed, Israel), Cameron Montag (Jersey City), Jorge Solloa (Mexico City), Ankit Agarwal (San Francisco), Vinicius Brunelli (Brazil), Michael Vincent (Washington), Javier Clavijo (Buenos Aires, Argentina), Tyler Blom (Wisconsin), Ritoprovo Roy, (Budapest), Mehmet Şamil Çelik (Türkiye), and Vyom (Manipur) for submitting solutions to last week’s problem.
See here for the solution to last week’s problem. Thank you to Mark Spindler from Maryland for submitting this very pretty solution.
Thanks for reading and happy learning! Until next time,
Adithya