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In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
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.
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.
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 ...
13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on ...
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 ...
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 ...
This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers ) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least upper bound property (see below).