How to Prove Existence of Integer n in Dedekind Cut Using Archimedean Property?

[Shaji D R]

Let α be a Dedekind Cut. w a positive rational.How to prove that there exists a integer n such that nw is a member of α and (n+1)w is not a member of α, using Archemedian propoerty of Q.

Suppose p is a member of α. we can find n such that nw < p < (n+1)w. So nw is
a member of α. Further I am not able to proceed.

Please help me.

 

[Shaji D R]

OK. Looks like I got the answer. It was a trivial case. If there is no such n then nw is a member of alpha for every n by induction. Since aplpha is not equal to Q and Q is archemedian it is not possible.If q is not a member
of alpha then q > any member of alpha and nw > q for some n.And nw cannot be a member of alpha.

Sporry for asking such a simple question.