# Proof

(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 sequence1 $$( c_{k}^{(m)}-c_{k}^{(n)})_{k\in\mathbb N}$$ 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 $$\mathbb R$$.

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

#### Footnotes

1. Please note that the difference of two Cauchy sequences is also a Cauchy sequence.