The supremum is the least upper bound of a set S, defined as a quantity M such that no member of the set exceeds M, but if ϵ is any positive quantity, however small, there is a member that exceeds M - ϵ. When it exists (which is not required by this definition, e.g., sup R does not exist), it is denoted sup_(x element S) x (or sometimes simply sup_S for short). The supremum is implemented in the Wolfram Language as MaxValue[f, constr, vars]. More formally, the supremum sup_(x element S) x for S a (nonempty) subset of the affinely extended real numbers R^_ = R union { ± ∞} is the smallest value y element R^_ such that for all x element S we have x<=y. Using this definition, sup_(x element S) x always exists and, in particular, sup R = ∞.
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