Least Upper Bound Property Proof

Prove that an nonempty set S that is bounded above has a least upper bound.

Proof.

For this problem, we must use the following way to show the proof:

Let and be an upper bound of S.

Let .

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!