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There are several ways to construct real numbers, such as the Dedekind cut, monotone bounded sequences and Cauchy sequences. It is obvious that the former two involves the order of the rational numbers, but whether the order is needed in constructing the real numbers via the Cauchy sequences is not clear at the first glance. However, there are many people claimed that it is because the Cauchy sequence doesn't require the order of the set, we can generalize this concept, using Cauchy sequence to complete the incomplete metric space. But I'm not quite convinced, so I ask here. Although the way to defining real numbers by Cauchy sequences requires only the rational numbers to be a metric space, in order to discuss $\forall \varepsilon\in\mathbb{Q}^{+},~ \exists N\in\mathbb{N},~ \forall m,n\in\mathbb{N},~m,n>N\Rightarrow d(x_m,x_n)<\varepsilon$, how did we define the metric here? It's $d(x,y)=|x-y|$ for any $x,y\in\mathbb{Q}$, and the absolute function is defined as the following way:

$|a|:= \begin{cases} a~~~~~~~,a\geq 0\\ -a~~~~,a<0 \end{cases}$

So there is the situation to compare a number with $0$, hence the order relation involved. Therefore, it seems the Cauchy sequences way also requires the order relation.

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You are right. To build the reals by completing the rationals you need the metric, and the metric does depends on the order relation.

You may be surprised to know that there are other completions of the rationals that come from different metrics. Those are defined using prime factoring in the integers, so don't depend on the order. You can read about them in wikipedia: https://en.wikipedia.org/wiki/P-adic_number

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