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?
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.