Sir, I have a question. In 3:21, definition4. is "A is bounded above means that it has at least one upper bound". Then, in 4:59, the L.U.B principle says that (A is bounded above and A is not empty) can imply A has a least upper bound. I wonder if A is an empty set, whether A has an upper bound; if it does, then why this upper bound is not the least upper bound?
Just curious, would one proceed on to prove this theorem by making use of the method in which the real numbers are constructed. In other words, would one proceed to prove this theorem with the idea of Dedekind Cuts?
There are at least two standard way to construct the reals: as Dedekind cuts, and as the completion of the rationals (using an equivalence relation of Cauchy sequences). Whichever one you use, you can prove this property from it.
Sorry. I read it too quickly. The correct definition is with a non-strict inequality. The definitions of upper and lower bound are the same, but reversing the direction of the inequality.
Thank you sir...God bless you and ur family ..started loving maths
Great video
Sir, I have a question. In 3:21, definition4. is "A is bounded above means that it has at least one upper bound". Then, in 4:59, the L.U.B principle says that (A is bounded above and A is not empty) can imply A has a least upper bound. I wonder if A is an empty set, whether A has an upper bound; if it does, then why this upper bound is not the least upper bound?
Use the definition of upper bound. Can you tell me what are all the upper bounds of the empty set?
@@mat1378 All right, I understand my question now :(, and I suddenly realized that you did discuss it in the lecture...
@@mat1378 Thank you!
@@mat1378 Thank you!
Just curious, would one proceed on to prove this theorem by making use of the method in which the real numbers are constructed. In other words, would one proceed to prove this theorem with the idea of Dedekind Cuts?
There are at least two standard way to construct the reals: as Dedekind cuts, and as the completion of the rationals (using an equivalence relation of Cauchy sequences). Whichever one you use, you can prove this property from it.
While defining lower bound, it'll become x > c right? Not >=
Yes
@@mat1378 May I ask why? cuz when checking answer mathworld.wolfram.com/GreatestLowerBound.html writes c
Sorry. I read it too quickly. The correct definition is with a non-strict inequality. The definitions of upper and lower bound are the same, but reversing the direction of the inequality.
Wow