Uploaded June 2022 | Updated September 2026, 15 hours ago
Mathematicians didn't feel that the decimal expansion of a number was quite rigorous enough in the 19th century, so they came up with some new ways to build them. Two prominent methods are Dedekind Cuts and Equivalence Classes of Cauchy sequences of rational numbers (or in other words, real numbers are convergent sequences of rationals).
//Books
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - amzn.to/3mwxFlC
Walter Rudin - Principles of Mathematical Analysis 3rd Edition - amzn.to/3MDHUis
//Exercises
- Finish showing that the Cauchy sequence definition of real numbers provides a field.
//Watch Next
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music Provided by Epidemic Sound
Light Up - Ben Elson
Morning Rises - Airae
Drop That - Reversus
Mumbai - Ooyy
All Parts Equal - Airae
Pandora's Box - Squid
We Can't Stop - Reversus
Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel:
epidemicsound.com/referral/644nao
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: amzn.to/3D3upUu
DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!
0:00 Introduction
0:23 What are real numbers?
1:49 1 is 0.9999
2:32 Two approaches to defining real numbers
4:00 An example through pi
6:32 Convergent sequences
9:35 Cauchy Sequences
13:21 Sums of Cauchy Sequences
16:40 Inequalities
Mathematicians didn't feel that the decimal expansion of a number was quite rigorous enough in the 19th century, so they came up with some new ways to build them. Two prominent methods are Dedekind Cuts and Equivalence Classes of Cauchy sequences of rational numbers (or in other words, real numbers are convergent sequences of rationals).
//Books
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - amzn.to/3mwxFlC
Walter Rudin - Principles of Mathematical Analysis 3rd Edition - amzn.to/3MDHUis
//Exercises
- Finish showing that the Cauchy sequence definition of real numbers provides a field.
//Watch Next
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764
//Music Provided by Epidemic Sound
Light Up - Ben Elson
Morning Rises - Airae
Drop That - Reversus
Mumbai - Ooyy
All Parts Equal - Airae
Pandora's Box - Squid
We Can't Stop - Reversus
Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel:
epidemicsound.com/referral/644nao
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: amzn.to/3D3upUu
DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!
0:00 Introduction
0:23 What are real numbers?
1:49 1 is 0.9999
2:32 Two approaches to defining real numbers
4:00 An example through pi
6:32 Convergent sequences
9:35 Cauchy Sequences
13:21 Sums of Cauchy Sequences
16:40 Inequalities
