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Prove the Dedekind Cut Theorem

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Prove this theorem....Suppose that A and B are non empty sets such that AUB=R(real numbers) and if 'a' is an element of the set A and 'b' is an element of the set B then
a<b, and there exists a "cut point" c such that if x<c<y then x is an element of the set A and y is an element of the set B.

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The solution of the Posting 161244

It is given that (real number)
If , then .
There exists a cut point "c" such that .

Then by the property of Dedekind's Theorem, either belongs to or to .

Let be the real number defined by the cut of rational numbers such that and consists of
all rational numbers belonging to and respectively. Then either has a ...

Solution Summary

The Dedekind cut theorem is proven.

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