In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
If it does, it is a minimum or least element of S.
The number 1 is a lower bound, but not the greatest lower bound, and hence not the infimum. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a maximal element. The distinction between "minimal" and "least" is only possible when the given order is not a total one.
On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. In a totally ordered set, like the real numbers, the concepts are the same. For example, it applies for real functions, and, since these can be considered special cases of functions, for real n-tuples and sequences of real numbers.
Existence and uniqueness[ edit ] Infima and suprema do not necessarily exist.
For example, consider the set of negative real numbers excluding zero. In this case, it is also called the minimum of the set. For subsets of the real numbers, another kind of duality holds: Another example is the hyperreals ; there is no least upper bound of the set of positive infinitesimals.
Hence, 0 is the least upper bound of the negative reals, so the supremum is 0. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound.
The inferior and superior limits agree if and only if the sequence is convergent i. However, if an infimum or supremum does exist, it is unique.
Minimal upper bounds[ edit ] Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. More generally, if a set has a smallest element, then the smallest element is the infimum for the set.
If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S or does not exist.
This set has a supremum but no greatest element. The least-upper-bound property is an indicator of the suprema. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
The two red curves approach the limit superior and limit inferior of xn, shown as dashed black lines. Relation to maximum and minimum elements[ edit ] The infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S.
If an ordered set S has the property that every nonempty subset of S having an upper bound also has a least upper bound, then S is said to have the least-upper-bound property.Supremum and infimum We start with a straightforward definition similar to many others in this course.
Read the definitions carefully, and note the use of ⩽ and ⩾ here rather than. The supremum A non-empty set is bounded from above if there exists such that The number is called an upper bound of. If a set is bounded from above, then it has infinitely many upper bounds, because every number greater then the upper bound is also an upper bound.
Among all the [ ]. In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is the.
If the max exists, then it is the supremum. If the min exists, then it is the infimum.
For an infinite set, it can happen that the max does not exist but the supremum does exist, and/or that the min does not exist but the infimum does exist. Supremum of a sequence of functions. Ask Question. up vote 4 down vote favorite. 1. Browse other questions tagged real-analysis supremum-and-infimum or ask your own question.
3 years, 4 months ago. viewed. 4, times. active. 3 years, 4 months ago. Related. 4. The infimum and supremum are concepts in mathematical analysis that generalize the notions of minimum and maximum of finite sets. They are extensively used in real analysis, including the axiomatic construction of the real numbers and the formal definition of the Riemann integral.
The limits of the infimum and supremum of parts of sequences of real numbers are used in some convergence tests.Download