# Least Upper Bound

Calculus Level 3

The greatest lower bound of an ordered set $$S \subseteq U$$ is the largest number $$x$$ in $$U$$ such that $$x \leq y$$ for all $$y$$ in $$S$$.

For example, let $$U$$ be the set of all real numbers, and let $$S = (0,1)$$. Then $$0$$ is the greatest lower bound of $$S$$. Notice how the greatest lower bound differs from the minima of a set in that the greatest lower bound of a set might not even belong to the set.

An ordered set $$X$$ is said to have the greatest-lower-bound property if the following holds true:

If $$E \subseteq X,$$ $$E$$ is not empty, and $$E$$ is bounded below, then $$E$$ has a greatest lower bound.

Which of the following ordered sets (under their usual order) does not have the greatest-lower-bound property?

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