The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges

Every Cauchy sequence of reals converges to a real (Limits and Cauchy sequences of reals).

More carefully, this is a statement about the axioms: in a complete ordered field, that is in an ordered field with the least-upper-bound property (Complete ordered field (least-upper-bound property)), every Cauchy sequence converges. The proof below uses nothing about $\mathbb{R}$ except that property, through Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence.

This library already knows the conclusion by a different route. It is proved on the Cauchy-construction page, where $\mathbb{R}$ is built out of Cauchy sequences of rationals and completeness is read off the construction. That proof is about a particular construction; this one is about the axioms, and it is what tells us the statement holds in any complete ordered field, however it was obtained.

• The three steps are exactly the three lemmas, and each is sharp. A Cauchy sequence is bounded (Every Cauchy sequence of reals is bounded); a bounded sequence has a convergent subsequence (Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence); a Cauchy sequence with a convergent subsequence converges (A Cauchy sequence with a convergent subsequence converges, to that subsequence’s limit). Dropping the Cauchy hypothesis at the last step breaks the chain, since a bounded sequence need not converge (FALSE: every bounded sequence converges).

• Where completeness enters. Only in the middle step, and there only through A monotone sequence converges if and only if it is bounded inside the proof of Bolzano-Weierstrass. The first and third steps hold in any ordered field. That localisation is the reason for the page order.

• The converse needs an extra hypothesis. Cauchy completeness alone does not imply the least-upper-bound property; it does so together with the Archimedean property, and there are Cauchy complete non-Archimedean ordered fields that are not Dedekind complete. This library does not prove that here; the equivalences between the forms of completeness are the subject of a later page, and Two independent proofs that $\mathbb{R}$ is Cauchy complete, and why the library records both states precisely what is and is not established now.

• The name. "Cauchy criterion" is the useful reading: the theorem lets one prove convergence without producing the limit, which is what makes it the standard tool for series and for uniform convergence later on.

• The construction-side proof of the same sentence is The reals are complete, and Two independent proofs that $\mathbb{R}$ is Cauchy complete, and why the library records both sets out why this library keeps both. Neither proof uses the other, and nothing above depends on that item.