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A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
The hyperbolic bound [7] is a tighter sufficient condition for schedulability than the one presented by Liu and Layland: = (+), where U i is the CPU utilization for each task. It is the tightest upper bound that can be found using only the individual task utilization factors.
If (,) is a partially ordered set, such that each pair of elements in has a meet, then indeed = if and only if , since in the latter case indeed is a lower bound of , and since is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original ...
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
By the least-upper-bound property of real numbers, = {} exists and . Now, for every ε > 0 {\displaystyle \varepsilon >0} , there exists N {\displaystyle N} such that c ≥ a N > c − ε {\displaystyle c\geq a_{N}>c-\varepsilon } , since otherwise c − ε {\displaystyle c-\varepsilon } is a strictly smaller upper bound of { a n ...
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The algebra of all subsets of an infinite set that are finite or have finite complement is a Boolean algebra but is not complete. The algebra of all measurable subsets of a measure space is a ℵ 1-complete Boolean algebra, but is not usually complete.