Let S and T be nonempty subsets of R with the following property: s ≤ t for all s ∈ S and t ∈ T.
(a) Show that S is bounded above and T is bounded below.
(b) Prove supS ≤ inf T .
(c) Given an example of such sets S and T where S ∩ T is nonempty.
(d) Give an example of sets S and T where supS = infT and S ∩T is the empty set. In your explanation, make sure you justify why sup S = inf T .
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