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  2. Least-upper-bound property - Wikipedia

    en.wikipedia.org/wiki/Least-upper-bound_property

    has an upper bound in Q, but does not have a least upper bound in Q (since the square root of two is irrational). The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.

  3. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    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.

  4. Completeness of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Completeness_of_the_real...

    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.

  5. Construction of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Construction_of_the_real...

    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 ...

  6. Completeness (order theory) - Wikipedia

    en.wikipedia.org/wiki/Completeness_(order_theory)

    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 ...

  7. Upper and lower bounds - Wikipedia

    en.wikipedia.org/wiki/Upper_and_lower_bounds

    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 ...

  8. Archimedean property - Wikipedia

    en.wikipedia.org/wiki/Archimedean_property

    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.

  9. Inequality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Inequality_(mathematics)

    Least-upper-bound property: Every non-empty subset of P with an upper bound has a least upper bound (supremum) in P. Ordered fields. If (F, +, ×) is a field ...