In Abbot's "Understanding Analysis," the Axiom of Completeness is stated as "every nonempty set of real numbers that is bounded above has a least upper bound." He then gives $S = \{r \in \mathbb{Q}| r^2 < 2\}$ as an example of a set of rational numbers such that the Axiom of Completeness isn't valid. But since there exists a rational number in $S$ arbitrarily close to $\sqrt2$, isn't $\sqrt2$ a supremum of $S$ ? Since a supremum of a set doesn't have to be part of the set, why does the supremum of $S$ have to be a rational number for the Axiom to be valid?
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