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thenthanable
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Homework Statement
Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s [itex]\in[/itex] S and t [itex]\in[/itex] T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T.
The Attempt at a Solution
I tried a proof by contradiction, but I started with the assumption that the preposition was true (not sure if that is OK). I haven't had a formal introduction into proof-writing. I'm 3 weeks into my calculus course and that was one of the exercises given in the TB.
The preposition states that S has a sup β, and T has an inf β, where β is a unique real number.
Suppose that β does not exist. Thus S does not have a supremum, and is not bounded above. T does not have an infimum, and is not bounded below. Thus both S and T are the set of real numbers.
Therefore, there exists an so and a to such that So > To.
I would really appreciate comments because like I said I'm completely new to this stuff. :(