# Let S be a nonempty set in R which is bounded above and T is the set of all upper bounds for S. Show that sup S = inf T .

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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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.