The Cut Property is defined as follows:
If $A$ and $B$ are nonempty, disjoint sets with $A\cup B=\mathbb{R}$ and $a<b$ for all $a\in A$ and $b\in B$, then there exists $c\in\mathbb{R}$ such that $x\leq c$ whenever $x\in A$ and $x\geq c$ whenever $x\in B$.
One of the exercises in the book asks how the axiom of completeness implies this, and then how the cut theorem implies the axiom of completeness. I understand the forward direction, but need help on the reverse.
Here is what I have attempted thus far:
Assuming that $\mathbb{R}$ has the cut property, let $A\subset\mathbb{R}$ be nonempty and bounded above. Take some element $a_0\in A$, and define the set $A'=A\cup (-\infty, a_0)$. Define $B\subset\mathbb{R}$ by $B=\lbrace b\in\mathbb{R} | b\text{ is an upper bound of the set $A$}\rbrace$. Then, $A'\cup B=\mathbb{R}$, so there must exist some $c\in\mathbb{R}$ such that $a\leq c$ for any $a\in A$, and $c\leq b$ for any $b\in B$. Then, as $b$ is any upper bound of the set $A$, $c$ is the least upper bound for $A$, which completes the proof.
Here are my questions: I didn't really cover questions about infinity, and I am wondering if it was necessary to extend the set $A$ to $-\infty$ in order to satisfy the requirements of the cut property. Also, can I really say that $A'\cup B=\mathbb{R}$? How do I know that there are no gaps in between?