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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
Hence, it is the supremum of the limit points. The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of ...
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.
Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima.
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.
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).
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It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets of =, where each is in some . This is only true if convergence is determined by the discrete metric (that is, x n → x {\displaystyle x_{n}\to x} if there is N {\displaystyle N} such that x n = x {\displaystyle x ...