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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