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If the supremum of exists, it is unique, and if b is an upper bound of , then the supremum of is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB). [1] The infimum is, in a precise sense, dual to the concept of a
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
Exactly in the same way one defines the essential infimum as the supremum of the essential lower bound s, that is, = {: ({: <}) =} if the set of essential lower bounds is nonempty, and as otherwise; again there is an alternative expression as = {: ()} (with this being if the set is empty).
The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finite infimum. The dual notion is called supremum. Interval. For two elements a, b of a partially ordered set P, the interval [a,b] is the subset {x in P | a ≤ x ≤ b} of P. If a ≤ b does not hold the interval will be empty. Interval ...
If has an upper bound, i.e. there exists a number , such that for all , one can call the number = the supremum of , if the number s {\displaystyle s} is an upper bound of A {\displaystyle A} , meaning ∀ x ∈ A : x ≤ s {\displaystyle \forall x\in A:\;x\leq s}
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.
For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the integers or of the real numbers, etc.), and so is 4.On the other hand, 6 is not a lower bound for S since it is not smaller than every element in S.
The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.