Necessity of Archimedean property in construction of Reals? I am currently self-studying the construction of reals as equivalence classes of rationals. In it, I have read that the Archimedean property is a necessary assumption we have to make to construct $\mathbb{R}$. However, from what I have studied, I haven't been able to find out where this has been necessary. 
As far as I can see, we can prove that Cauchy Completeness $\implies$ Least Upper Bound Property (for example, like in this Wikipedia entry) without the Archimedean Property, which should make Cauchy Completeness equivalent to the Least Upper Bound Property. So, am I missing out on something, or is the Archimedean property necessary for a different reason?
 A: I think that you are missing nothing. The Archimedian property is a consequence of the least upper bound property. So, if you somehow define the reals in some way which allows you to prove that the least upper bound property holds, then automatically the Archimedian property will hold too.
A: I assume you are constructing the set $\mathbb {R} $ from $\mathbb {Q} $ and each element of $\mathbb{R} $ is an equivalence class of Cauchy sequences of rationals.
When this is done you should know that $\mathbb{Q} $ possesses Archimedean property ie if $a, b$ are rationals with $a>0$ then there is a positive integer $n$ such that $na>b$. And this is a trivial property of set $\mathbb {Q} $ and it remains trivial even in $\mathbb {R} $ when we construct it from $\mathbb {Q} $.
Only when the set of real numbers is presented to us axiomatically this Archimedean property acquires some sort of a non-trivial nature because then it can not be proved without the use of completeness axiom.
My own confusion regarding this was sorted out long back in this thread on mathoverflow.
