TITLE: How to prove that the both definition of completeness of $\mathbb{R}$ are equivalent?
QUESTION [10 upvotes]: In the definition of completeness of a set, in particular $\mathbb{R}$, I have seen the following definitions:


Dedekind: Every non-empty bounded of subset has a least upper bound (with respect to the natural order).
Cauchy: Every Cauchy sequence converges.


However, how can one prove that both of these definitions are equivalent ?

REPLY [2 votes]: Indeed 1. implies 2., but the converse does not hold for ordered fields in general unless the Archimedean Property is added as an extra hypothesis. This is the assumption implicitly made in "proofs" of that implication for claiming that the sequence constructed is Cauchy: the Archimedean Property is required for moving from some $\varepsilon>0$ to a natural number $n$ such that $n>\dfrac{1}{\varepsilon}$.
The field of formal Laurent series is an example of an ordered field that is Cauchy-complete but not Dedekind-complete. See
James Propp, Real analysis in reverse, The American Mathematical Monthly
120:5 (2013), 392-408
for more details on this (and also other equivalences and non-equivalences of Dedekind-completeness).