Prove that $\Bbb R$ satisfies Nested Interval Property (NIP) implies it satisfies Axiom of Completeness (AoC) as well.
I tried solving this problem hereby:
We consider $S\subset \mathbb R$ such that $S\neq\emptyset$ and $S$ is bounded above.
We need to show that, $\sup S$ exists in $\Bbb R.$
As $S$ is bounded then, $\exists M\in\Bbb R$ such that $s\leq M$$,\forall s\in S.$ We assume, $\exists s_0\in S$ as $S$ is nonempty and $m_0=M.$ We now bustect the interval $I_0=[s_0,M_0]$ at $\frac{s_0+m_0}{2}$. We write, $s_1=\frac{s_0+m_0}{2}$ if $\frac{s_0+m_0}{2}\in S$ and $m_1=m_0.$ If $\frac{s_0+m_0}{2}\notin S$ we write, $m_1=\frac{s_0+m_0}{2}$ and $s_1=s_0.$
In either cases, we note that, $I_1=[s_1,m_1]\subset I_0$ and $$|I_1|=m_1-s_1\leq \frac{1}{2}(m_0-s_0)=|I_0|.$$ We again bisect $I_1$ and continue this process to obtain a sequence of closed intervals $I_n,n\in\Bbb N\cup \{0\}=N'.$
By NIP, we have, $\exists$ a unique $c\in \cap I_n.$ We claim, $\sup S=c.$
If $s\in S$ and $\exists n\in N'$ such that $s\leq s_n$ then, $s\leq s_n\leq c.$
If $s\gt s_n,\forall n'\in N',$ then, $s_n\leq s\leq m_n,\forall n\in N'\implies c=s,$ as $c$ is unique by NIP.
Thus, $c$ is an upper bound of $S.$
If, $\exists M'\in\Bbb R$ such that $s\leq M'$ we claim, $c\leq M'.$ For if, $c\gt M'\geq s,\forall s\in S$ then, $$s_n\leq M'\lt c\leq m_n\implies M'\in \cap I_n\implies c=M',$$ a contradiction. So, $c\leq M'\implies c=\sup S.$
This completes the proof.
However, I still have some doubts regarding my solution. This is because, I am doubtful with the statement :
If $s\gt s_n,\forall n'\in N',$ then, $s_n\leq s\leq m_n,\forall n\in N'\implies c=s,$ as $c$ is unique by NIP.
in the above solution. For it may, happen that $\exists n\in N'$ such that $s\gt m_n$ if ($m_n\notin S$ by definition of $m_i's$ and,) $m_n$ is not an upper bound of $S.$ I think this is the point where the logic of my proof breaks down or rather becomes awry.
I want to know if this is an issue then how can we fix this? Else, if that statement above is correct, then how is that justified ?