[Math] How are the real numbers complete

order-theoryreal-analysis

The completeness axiom for the real numbers states that any subset of the reals that is bounded above has a supremum.

But if we take an example: Completeness of the real numbers – Least upper bound property

Consider a set of rational numbers $x$ such that $ x^{2}<2 $.
It's easy to see that it has an upper bound, but as the Wikipedia article says, it has no supremum since for any given number in the set you can find another number greater than that within the set.

But isn't this set a subset of the real numbers? How can it have no supremum?

Best Answer

The set of rational numbers $x$ such that $x^2 < 2,$ also written $\{x \in \mathbb Q\mid x^2 < 2\},$ has no supremum within the rational numbers. The supremum within the real numbers is just $\sqrt 2.$ It is true that $\sqrt 2 \not\in\{x \in \mathbb Q\mid x^2 < 2\},$ but so what? The supremum of a set does not have to be a member of the set.

The maximum value of a set must be in the set, but that is a different beast entirely (and indeed there is no such value in $\{x \in \mathbb Q\mid x^2 < 2\}$).