Completeness of the real numbers
The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom.An example is the subset of rational numbers This set has an upper bound.In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom.(In this real number line, this sequence converges to pi.)The rational number line does not satisfy the nested interval theorem.In other words, nested intervals theorem by itself is weaker than other forms of completeness, although taken together with Archimedean property, it is equivalent to the others.In weaker foundations such as in constructive analysis where the law of the excluded middle does not hold, the full form of the least upper bound property fails for the Dedekind reals, while the open induction property remains true in most models (following from Brouwer's bar theorem) and is strong enough to give short proofs of key theorems.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.The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.(The definition of continuity does not depend on any form of completeness, so there is no circularity: what is meant is that the intermediate value theorem and the least upper bound property are equivalent statements.)