The most familiar example is the completeness of the real numbers.

The completeness of the real numbers is one of the defining properties of the real number system.

The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction.

It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom.

The sequential completeness of the real numbers (every bounded increasing sequence of real numbers has a limit).

Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.

The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological.

This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.

These last two properties, together with a lemma used in the proof of the Bolzano-Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano-Weierstrass theorem and the Heine-Borel theorem.