# Proof

Let $$(M , + , \cdot)$$ be the unit ring of all rational Cauchy sequences. We want to show that the set $$I:=\{(a_n)_{n\in\mathbb N}~|~a_n\in\mathbb Q,\lim a_n=0\}$$, i.e. the set of all rational sequences convergent to $$0$$ is a subset of $$M$$, formally $$I\subseteq R$$. We will demonstrate that any rational sequence $$(a_n)_{n\in\mathbb N}$$ with $$\lim a_n=0$$ is also a rational Cauchy sequence, formally $(a_n)_{n\in\mathbb N}\in I\Longrightarrow (a_n)_{n\in\mathbb N}\in M.$

Following the definition of convergence in the metric space $$(\mathbb Q,|~|)$$, for any arbitrarily small $$\epsilon/2$$, $$\epsilon\in\mathbb Q$$, there exists an $$N(\epsilon/2)\in\mathbb N$$ such that for all $$n , m > N(\epsilon/2)$$, we have $|a_n| < \frac \epsilon2\quad\text{and}\quad|a_m| < \frac \epsilon2.$ Thus, we can estimate (using the triangle inequality, see third property of the metric $$|\cdot|$$),
$|a_n-a_m|\le |a_n|+|a_m| < \frac\epsilon2 + \frac\epsilon2 = \epsilon.$ This demonstrates, that $$(a_n)_{n\in\mathbb N}$$ with $$\lim a_n=0$$ is also a rational Cauchy sequence, or that $$I\subseteq M$$, as required.

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

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