Question Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s and t , then s<t. Prove that there is a unique real number such that every real number less than is in S and every real number greater than is in T. (A decomposition of the reals into two sets with these properties is a Dedekind cut. This is known as Dedekind's theorem.)
I'm not really sure how to approach the problem.First of all, from what i understand if we have an open interval of real numbers, in this case (s,t), there are many rational and irrational numbers between s and t ; because the rational and irrationals are dense in the reals.
So what's all this talk about been unique ? What's going on ?
Secondly, the question does not specify if is in any of the sets. I used that assumption in my won answer.
What I did: ( I doubt this is correct)
The inequality s < t s&t would mean that the smallest member in t is an upper bound of S.
Taking the smallest real number in T, say , the following is true if T is bounded.
This implies that
Because s<t is an upper bound of S and possible inf S.
Which implies supS =
So this means that
When i do it this way it seems more correct because sup S and inf T is unique because they are bounded and according to the theorem i have here every non emepty set of real numbers that is bounded from above or below, has a unique inf or sup.