The Riemann Rearrangement Theorem // I can make this sum anything I want

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Here we investigate the Riemann Rearrangement Theorem from the 19th century. Analysis was changed forever after this discovery.

//Books

A great companion text to Rudin is Maxwell Rosenlicht's Introduction to Analysis. The explanations are a lot clearer and the book is under $20.

//Reference for Cantor’s Work

//Exercises
- Show that the elements of the Cantor set can be written in base-3 as 0.a_1 a_2 a_3 … where each a_i is either 0 or 2.

//Watch Next

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0:00 Introduction
0:40 Anticipating Topology
1:48 Proof of Part A
3:53 Can a sequence converge to more than one point?
5:17 Proof of Part B
8:21 Using a Theorem Backwards
8:47 Functional Analysis Example
10:28 What about a PROOF?
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I remember this theorem, stuck out to me because of how interesting it is. Fascinating video, Real Analysis has so much motivation and counterintuitive results like these ones really help highlight why all the formalism is necessary since not everything always works as expected. Loved the video 👍

Cyclonus-fcxx
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do you have any plan on elucidating about dedekind cuts (rudin's chapter 1's appendix)? Thanks in advanced 😁

spiderjerusalem
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I'm totally not a mathematician, but that's probably the most mind blowing thing I've seen, since being exposed to Gödel's incompleteness theorems. 😲

PhillipRhodes
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You need to catalog your entire math library and release it to us! Curious minds want to know 🙂

yareps
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I’m currently working on the Riemann hypothesis and anything in relation to it is much appreciated. I just finished your video on the polylogarithmic hardy space, which is stunning because I have novel research of number fields which connects the finite difference method with a plethora of old and new functions and tricks! 🎉 everyday I feel one step closer also to finding a groundbreaking revelation, thanks for all your hard work the quality doesn’t go unnoticed

ryanhinojosa
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can u suggest one book for abstract algebra for undergrad

solaris
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This vids are cool bits of math while i do apostle analysis 😄

richsmith