Real Numbers: Lecture 4 (LUB Property)

We introduce the notion of LUB property of R. We then show that R enjoys the LUB property iff it enjoys the GLB property. We offer two proofs. The second one is very much enjoyed by students whenever we teach it. We end with the Archimedean property of R.
There is a MISTAKE during 13:12--13:50, We should show -lambda is less than or equal to alpha. You can easily do it.



0:00 Introduction
1:42 Review of LUB and GLB
2:38 LUB property of R
7:26 GLB property of R
9:07 simple observation for LUB and GLB of a set
14:04 a=LUB(A) iff -a =GLB(-A)
14:26 proof-1 (of a=LUB(A) iff -a =GLB(-A))
17:53 Proof-2
27:17 lub property of the set of Rational numbers
31:24 Archimedean property -1
40:03 Z is not bounded above in R
44:48 Archimedean property -2
47:41 Equivalence relation between Archimedean property -1 and Archimedean property -2
50:08 Easy consequences of Archimedean property.