I was given "this" Dedekind Cut in undergrad. real analysis class
- a$\lt$b, $\forall$ $a\in A, \forall b\in B$
$\to \exists \alpha$ such that, $a\le \alpha \le b $
And I was told this "is" the "Dedekind cut"
But from my memories and from research(not much, mostly wikipedia and here), i think we need
- if $ a \in A , \exists a' \in $ A s.t. $a\lt a'$
to ensure there exist no largest element in A
Plus, I'm not sure about number 2, their union being set of real number.
Anyway, I had a problem proving "Dedekind cut" $\Rightarrow$Existence of Sup(S)
Which definition of $\alpha$=Sup(S) given (S: set)
$\alpha \in$Upperbound(S) (set of upperbounds of S)
$\forall \epsilon \gt$0, $\exists x \in $S s.t. $\alpha - \epsilon \lt x \leq \alpha$
(I prefer "if x$\lt \alpha \to$ x$\notin$Upperbound(S)" though)
if 4. is included in definition, then I have it. (even with not-rigorously given conditions)
1.Are those $3$ sufficient for proving existence of Sup($S$)?
2.If given "Dedekind cut" was wrong definition and therefore we can't prove existence of Sup($S$), then is it possible to prove that it is impossible to prove?
3.Or I just want clear evidence that given definition is wrong in logical way
I uploaded my answer to first question and second question.
So I hope someone can check if it's right.
And i wrote there, but I still want to know if it is right to omit "no maximum" statement, and the difference between omitted and not omitted Dedekind cut.