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Every nonempty subset of the real numbers that is bounded below has a real infimum.
If is nonempty and bounded below, then its greatest lower bound exists in .
This is the "lower" counterpart to the least upper bound theorem and follows immediately by applying the supremum property to .
A bounded below set is a subset for which there exists such that for every ; such an is called a lower bound of .
Least Upper Bound Theorem: If is nonempty and bounded above, then exists in .
A supremum of a nonempty set that is bounded above is a real number such that:
The supremum is the “least upper bound” and may exist even when has no maximum. The completeness axiom asserts that every nonempty bounded-above set of real numbers has a supremum.