Every finite subset of a totally ordered set such as the reals has a maximum. However, many infinite sets do not. The integers, have no maximum, since for any we can find such that . (Taking works nicely.)
A more subtle example of this phenomenon is the set . While this set has a least upper bound 1, it has no maximum.
The previous example suggests the following formulation: if is a set contained in some larger ordered set with the least upper bound property, then has a maximum if and only if the least upper bound of is a member of .
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