In most books, the LUB property of is treated as an axiom and then it is shown that it is equivalent to completeness. However, in baby Rudin, it appears that in defining as the set of Dedekind cuts, we prove that has the LUB property directly from the definition.
Likewise, in defining as the set of equivalence classes of Cauchy sequences (under the relation that two sequences are equivalent if and only if the sequence of differences is null), we show directly from the definition that is complete.
Here is what confuses me: Why is it considered an axiom if we can prove each of these (equivalent) properties from the equivalent definitions of ?
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, and are "equivalent" if and only if the sequence 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".
Now, we can easily show that is a Cauchy sequence, but we don't know that the limit actually exists without assuming that is complete (which unless taken as an axiom we do not know). Therefore, on second thought, I don't see how one can show that with defined in this way, it is complete (ie. every Cauchy sequence converges).
If we know that the metric I wrote above exists, then I agree that we can show that is complete. However, we don't know that such a metric exists without knowing is complete, so it seems completely hopeless to define the real numbers in this way. Do you have any other ideas on how to define a metric and show that the space is complete as you suggested?