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has an upper bound in Q, but does not have a least upper bound in Q ... [a, b] that takes negative values under f. Then b is an upper bound for S, ...
A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set. An example of a set that lacks the least-upper-bound property is Q , {\displaystyle \mathbb {Q} ,} the set of rational numbers.
The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers = {<}. This set has an upper bound. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be √2, but it does not exist in Q.
The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality. Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of f, if g(x) ≥ f (x) for each x in D.
Thus u is an upper bound for S. To see that it is a least upper bound, notice that the limit of (u n − l n) is 0, and so l = u. Now suppose b < u = l is a smaller upper bound for S. Since (l n) is monotonic increasing it is easy to see that b < l n for some n. But l n is not an upper bound for S and so neither is b. Hence u is a least upper ...
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.
The seldom-considered dual notion to a dcpo is the filtered-complete poset. Dcpos with a least element ("pointed dcpos") are one of the possible meanings of the phrase complete partial order (cpo). If every subset that has some upper bound has also a least upper bound, then the respective poset is called bounded complete. The term is used ...
On the other hand, / is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal between / and , and if / < / then is not infinitesimal. But 1 / ( 4 n ) < c / 2 {\displaystyle 1/(4n)<c/2} , so c / 2 {\displaystyle c/2} is not infinitesimal, and this is a contradiction.