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**HallsofIvy** What is, or is not, an axiom depends on where you are starting. If you are defining the real numbers as a field itself, axiomatically, with no prior information, then you must take the "least upper bound" property as an axiom- it "defines" the real numbers

But you can also define the rational numbers first, axiomatically, then **construct** the real numbers system from the rational numbers. That is what Rudin does, using Dedekind cuts. That way, you can prove the "least upper bound property" from the definition of Dedeking cuts and the properties of the rational numbers.

You can also construct the real numbers using Cauchy sequences- say that two Cauchy sequences, $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$ are "equivalent" if and only if the sequence $\displaystyle \{a_n- b_n\}$ converges to 0 and then define the real numbers to be equivalence classes of such sequences. That makes it easy to prove the "Cauchy Criterion", that all, in the real numbers, all Cauchy sequences converge, which is equivalent to the least upper bound property (in the sense that given either one, you can prove the other).

A third way to construct the real numbers is to use "increasing sequences with upper bounds". We define "equivalence" for two such sequences in the same way and define the real numbers to be the equivalence classes. That makes it easy to prove "monotone convergence" which is equivalent to both "Cauchy Criterion" and "least upper bound property".