(related to Theorem: Completeness Principle for Real Numbers)
By assumption, \((x_n)_{n\in\mathbb N}\) is a real Cauchy sequence, i.e. a Cauchy sequence in the metric space \((\mathbb R,|~|)\). We have to show that it converges against a limit \(a\in\mathbb R\).
By definition of real Cauchy sequences, for every \(\epsilon\in\mathbb R\), \(\epsilon > 0\) there is a \(N(\epsilon)\in\mathbb N\) such that for all \(m,n > N(\epsilon)\) we have the estimation
\[|x_m - x_n| < \epsilon\quad\quad ( * )\]
First, we observe that by the definition of real numbers, each \(n\)-th sequence member \(x_n\) can be represented by the residue class
\[x_n=(c_{k}^{(n)})_{k\in\mathbb N} + I,\]
i.e. of the class of all rational Cauchy sequences, the difference of which converges to \(0\). Correspondingly, also the real number \(\epsilon > 0\) can be represented as its residue class:
\[\epsilon=(\epsilon_k)_{k\in\mathbb N} + I.\]
Moreover, we can assume without loss of generality that \(\epsilon\) is an arbitrarily small (but fixed) rational number. In this case, we can set
\[\epsilon=(\epsilon)_{k\in\mathbb N} + I.\]
In fact, the magnitude of \(\epsilon\) does not depend on the index \(k\).
Second, we observe that by the definition of the order relation for real numbers the estimation \( ( * ) \) means that
\[|x_m - x_n| < \epsilon \Longleftrightarrow \exists q\in\mathbb Q, q > 0, N(q)\in\mathbb N: \epsilon - |c_{k}^{(m)}-c_{k}^{(n)}| > q\quad\quad\forall k\in\mathbb N, k > N(q). \] In particular, for all pairs of natural numbers \(n,m > N(\epsilon)\) there exists an index \(N(q)\), such that \(|c_{k}^{(m)}-c_{k}^{(n)}| < \epsilon\) for all \(k > N(q)\). Because \(N(q)\) depends only on the pair \(m,n\) and this pair only depends on \(\epsilon\), we can set \(N(q):=N(n,m)=M(\epsilon)\) and conclude that for all \(k > M(\epsilon)\), we have \(|c_{k}^{(m)}-c_{k}^{(n)}| < \epsilon\).
This means that the rational Cauchy sequence^{1} \(( c_{k}^{(m)}-c_{k}^{(n)})_{k\in\mathbb N}\) converges to \(0\), or that \(( c_{k}^{(m)}-c_{k}^{(n)})_{k\in\mathbb N}\in I\). But this means that both rational Cauchy sequences \(( c_{k}^{(m)})_{k\in\mathbb N}\) and \((c_{k}^{(n)})_{k\in\mathbb N}\) are the representatives of the same residue class, or real number \(a\in\mathbb R\) with
\[a=( c_{k}^{(m)})_{k\in\mathbb N} + I=(c_{k}^{(n)})_{k\in\mathbb N} + I.\]
Therefore, we get together with the triangle inequality
\[|c^{(m)}_k - a + a - c^{(n)}_k | \le |c^{(m)}_k - a| + |a - c^{(n)}_k | < 2\cdot\epsilon,\quad\quad \forall k > M(\epsilon),\]
or that
\[| x_n - a | < 2\cdot\epsilon,\quad\quad \forall n >\max(M(\epsilon),N(\epsilon)).\]
Since \(\epsilon\) is arbitrarily small, it completes the proof that the real Cauchy sequence \((x_n)_{n\in\mathbb N}\) is convergent in \(\mathbb R\).
Please note that the difference of two Cauchy sequences is also a Cauchy sequence. ↩