Every finite subset of the reals (or any other totally ordered set) has a minimum. However, many infinite subsets do not. The integers, have no minimum, 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 greatest lower bound 0, it has no minimum.
The previous example suggests the following formulation: if is a set contained in some larger ordered set with the greatest lower bound property, then has a minimum if and only if the greatest lower bound of is a member of .
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