Cauchy completion builds real numbers from rational approximating sequences. It is the metric counterpart of Dedekind's order construction.
Cauchy sequences and null sequences
A sequence is Cauchy when its tail becomes internally indistinguishable, even before a limit is known.
Definition. A rational sequence \((q_n)\) is Cauchy if for every \(\varepsilon>0\) there is \(N\) such that \(|q_n-q_m|\lt \varepsilon\) for all \(m,n\ge N\). A null sequence satisfies \(q_n\to0\) in \(\mathbb Q\).
Lemma. Every Cauchy sequence of rationals is bounded.
Proof. Choose \(N\) such that \(|q_n-q_N|\lt 1\) for \(n\ge N\). The tail is bounded by \(|q_N|+1\), and the finitely many initial terms have a maximum absolute value.
Equivalence classes of approximations
Two rational Cauchy sequences represent the same real number when their difference vanishes.
Definition. Let \(\mathcal C\) be the set of rational Cauchy sequences and \(\mathcal N\) the set of rational null sequences. Define \((q_n)\sim(r_n)\) if \((q_n-r_n)\in\mathcal N\). The completion is \(\mathcal C/\mathcal N\).
Proposition. Coordinatewise addition and multiplication are well-defined on \(\mathcal C/\mathcal N\).
Proposition. Every nonzero class has a multiplicative inverse, and constant rational sequences embed \(\mathbb Q\) as a dense subfield of \(\mathcal C/\mathcal N\).
Proof. If \([q_n]\ne0\), then \((q_n)\) is not null. Cauchyness implies that there are \(\delta>0,N\) such that \(|q_n|\ge\delta\) for \(n\ge N\); otherwise a subsequence would tend to zero and force the whole Cauchy sequence to do so. Define \(r_n=1/q_n\) for \(n\ge N\), with arbitrary nonzero initial terms. Then \[ |r_n-r_m| =\frac{|q_n-q_m|}{|q_nq_m|} \le\delta^{-2}|q_n-q_m|, \] so \((r_n)\) is Cauchy and \([q_n][r_n]=[1]\). Equivalent representatives give equivalent reciprocal sequences by the same estimate.
The map \(q\mapsto[(q,q,\ldots)]\) is an injective field homomorphism. Given \(x=[q_n]\) and \(\varepsilon>0\), choose \(N\) so that \(|q_n-q_N|<\varepsilon\) for \(n\ge N\). Then the constant rational class \(q_N\) is within \(\varepsilon\) of \(x\), proving density.
Proof. Sums and products of Cauchy sequences are Cauchy; products use boundedness of Cauchy sequences. If \(q_n-r_n\to0\) and \(u_n-v_n\to0\), then \((q_n+u_n)-(r_n+v_n)\to0\), and \(q_nu_n-r_nv_n=q_n(u_n-v_n)+v_n(q_n-r_n)\to0\) by boundedness.
Order from eventual positivity
Positivity is defined by being eventually separated from zero.
Definition. A class \([q_n]\) is positive if there exist \(\delta>0\) and \(N\) such that \(q_n>\delta\) for all \(n\ge N\). Define \([q_n]\le[r_n]\) if \([r_n-q_n]\) is positive or zero.
Proposition. This order is total and compatible with the field operations.
Proof. The preceding inverse argument shows that a nonzero class has a representative eventually bounded away from zero. Cauchyness then prevents sign changes in its tail: terms of opposite signs would differ by at least \(2\delta\). Hence exactly one of \([q_n]\) and \([-q_n]\) is positive. This proves trichotomy. If \(q_n\le r_n\) eventually, then adding the same representative preserves the inequality; products of two eventually positive representatives are eventually positive. Independence from representatives follows because a null perturbation is eventually smaller than half the positive separation from zero.
Completeness of the completed field
The construction is designed so that Cauchy approximations converge.
Theorem. The metric field \(\mathcal C/\mathcal N\) is complete.
Proof. Let \((x_k)\) be Cauchy in the completion. Choose indices \(k_j\uparrow\infty\) such that \(|x_{k_j}-x_{k_{j+1}}|<2^{-j}\). By density of the embedded rationals, choose \(a_j\in\mathbb Q\) with \(|a_j-x_{k_j}|<2^{-j}\). For \(m>j\), \[ |a_m-a_j| \le2^{-m}+|x_{k_m}-x_{k_j}|+2^{-j}, \] and the middle term tends uniformly to zero as \(j\to\infty\), so \((a_j)\) is rational Cauchy. Let \(x=[a_j]\).
For fixed \(j\), the tail of the representative \((a_m)\) is close to \(a_j\) by the same estimate, which gives \(|x-x_{k_j}|\to0\). Thus the selected subsequence converges to \(x\). Since the original \((x_k)\) is Cauchy, for every \(\varepsilon>0\) all sufficiently late \(x_k\) are within \(\varepsilon/2\) of a late \(x_{k_j}\), and that subsequence term is within \(\varepsilon/2\) of \(x\). Hence \(x_k\to x\).
Equivalence with Dedekind reals
The metric and order repairs give the same real line.
Remark. A Cauchy class maps to the cut of rationals eventually below its representatives. Conversely, a cut is represented by any rational sequence increasing through the cut and approaching its boundary. These maps are inverse order-preserving field isomorphisms.