Least Upper Bound Property Proof
Prove that an nonempty set S that is bounded above has a least upper bound.
For this problem, we must use the following way to show the proof:
Let and be an upper bound of S.
If is an upper bound, let and
If is not an upper bound, then let and
Repeat the process to obtain sequence and , we have to show that they both converge to the LUB (S).
To be honest, I'm a bit lost on how to generate and , so if is not an upper bound, that means is still the least upper bound since , right? Suppose it is not true, then we let be something lower, or closer to the sequence, but how should the sequences and look like?
Thank you very much!