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Let S be a nonempty set in R which is bounded above and T is the set of all upper bounds for S , T ={ x is in R : x is an upperbound for S } . Show that sup S = inf T .

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As I mentioned to you on your last question, to get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, some would consider your post rude because it is a command ("Show"), not a request for help, so please consider rewriting it. –  Zev Chonoles Mar 10 '13 at 7:28

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It is the definition of the dedekind completeness of $\mathbb{R}$ also known as the Dedekind cut. Not that the supremum of a set is the lowest upper bound, and the infimum is the tallest lower bound.

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