Intro to Real Analysis: How to think of real numbers

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 - https://amzn.to/3NFYiAC Tom Apostol - Mathematical Analysis 2nd Edition - https://amzn.to/3mwxFlC Walter Rudin - Principles of Mathematical Analysis 3rd Edition - https://amzn.to/3MDHUis //Exercises - Finish showing that the Cauchy sequence definition of real numbers provides a field. //Watch Next The Real Analysis Survival Guide https://youtu.be/v5rD0B-zfXw The Analyticity of the Laplace transform https://youtu.be/FIMkbFQL6XM Introduction to Control Theory https://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: https://www.epidemicsound.com/referral/644nao //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://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