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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 .
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 <).
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 top example shows a case where z is much less than the sum x + y of the other two sides, and the bottom example shows a case where the side z is only slightly less than x + y. In mathematics , the triangle inequality states that for any triangle , the sum of the lengths of any two sides must be greater than or equal to the length of the ...
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
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 ...