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 $\displaystyle u_0 \in S$ and $\displaystyle N_0$ be an upper bound of S.

Let $\displaystyle v_{0} = \frac {u_0 + N_0 }{2} $.

If $\displaystyle v_0 $ is an upper bound, let $\displaystyle U_1 = v_0 $ and $\displaystyle u_1 = x_0 $

If $\displaystyle v_0 $ is not an upper bound, then let $\displaystyle N_1 = N_0 $ and $\displaystyle u_1 > v_0 \ \ \ u_1 \in S$

Repeat the process to obtain sequence $\displaystyle N_n$ and $\displaystyle u_n$, we have to show that they both converge to the LUB (S).

To be honest, I'm a bit lost on how to generate $\displaystyle N_n$ and $\displaystyle u_n$, so if $\displaystyle v_0$ is not an upper bound, that means $\displaystyle N_0$ is still the least upper bound since $\displaystyle v_0 \in S $, right? Suppose it is not true, then we let $\displaystyle v_1$ be something lower, or closer to the sequence, but how should the sequences $\displaystyle N_n$ and $\displaystyle u_n$ look like?

Thank you very much!