Proving the least upper bound property for real numbers

From the definition of the real number to the least upper bound property takes a good amount of work. In this video we go through the entire process of defining a least upper bound and proving that a certain construction actually gives us just that. We use the convergent sequence definition of Real Numbers that comes from Cantor. The Extended Version of this video can be found here:    • Least Upper Bound Property (Extended)   //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 Real Analysis: How to think of Real Numbers    • Intro to Real Analysis: How to think of re...   The Banach Fixed Point Theorem    • Fixed Point Iteration - Loki SLAPPED 16 ti...   The Real Analysis Survival Guide    • The Real Analysis Survival Guide   Control Theory: The Analyticity of the Laplace transform    • Morera's Theorem in Practice - Laplace Tra...   Introduction to Control Theory    • Introduction to Control Theory   //Music Provided by Epidemic Sound Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referra... //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 Start 0:55 Partial Orderings and Least Upper Bounds 2:16 Total Orderings and Real Numbers 4:03 The LUB Algorithm 5:16 Square Root of 2 7:33 Real Distances 8:36 Proving the LUB Algorithm Works 9:50 Constructing a LUB 12:22 To Be Continued...