Search results
Results From The WOW.Com Content Network
More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
Then has an upper bound (, for example, or ) but no least upper bound in : If we suppose is the least upper bound, a contradiction is immediately deduced because between any two reals and (including and ) there exists some rational , which itself would have to be the least upper bound (if >) or a member of greater than (if <).
The least-upper-bound property states that every nonempty subset of real numbers having an upper bound (or bounded above) must have a least upper bound (or supremum) in the set of real numbers. The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers
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.
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 ...
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 ...
Proof of the Extreme Value Theorem. By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a point d in [a, b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M – 1/n is not an upper ...