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An important property of the real numbers is that every set of real numbers has a supremum. This is sometimes called the supremum axiom and expresses the completeness of the real numbers.
Examples:
Note that the supremum of S does not have to belong to S (like in these examples). If the supremum value belongs to the set then we can say there is a largest element in the set.
In general, in order to show that sup(S) ≤ A, one only has to show that x ≤ A for all x in S. Showing that sup(S) ≥ A is a bit harder: for any ε > 0, you have to exhibit an element x in S with x ≥ A  ε.
In functional analysis, one often considers the supremum norm of a bounded function f : X > R (or C); it is defined as
See also: infimum or greatest lower bound, limit superior.
One can define suprema for subsets S of arbitrary partially ordered sets (P, <=) as follows:
In an arbitrary partially ordered set, there may exist subsets which don't have a supremum. In a lattice every nonempty finite subset has a supremum, and in a complete lattice every subset has a supremum.
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