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Prove the Dedekind Cut Theorem

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 Dedekind cut theorem is proven.

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