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On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element. However, the definition of maximal and minimal elements is more general. In particular, a set can have many ...
The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm.
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
This concept is also called supremum or join, and for a set S one writes sup(S) or for its least upper bound. Conversely, the greatest lower bound is known as infimum or meet and denoted inf(S) or . These concepts play an important role in many applications of order theory.
A set with upper bounds and its least upper bound. In mathematics, particularly in order theory, an upper bound or majorant [1] of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S.
As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function does at points (that is, the image of ), but rather by asking for the set of points where equals a specific value (that is, the preimage of under ).