# Real Analysis : Bounded Sets

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Let S be a bounded nonempty set and let S^2 = {s^2 : s E S}. Show that sup S^2 = max((sup S)^2, (inf S)^2).

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Problem: Suppose is a bounded nonempty set and let . Show that

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Before the proof, we need the following definitions and theorems.

Definition 1: A nonempty set is bounded if there exists some , such that for any , we have .

Definition 2: A nonempty set is upper bounded if there exists some , such that for any , we have .

Definition 3: A nonempty set is lower bounded if there exists some , such that for any , we have

Definition 4: For any nonempty set , is defined as the smallest upper bound of and is defined as the greatest lower bound of . In another word, if is an upper bound of and is the lower bound of , ...

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Bounded Sets are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.