# Proof

Note that since the rational zero $$0$$ exists , it is also true that the rational sequence $$(0)_{n\in\mathbb N}$$ exists. We have to show that $$(0)_{n\in\mathbb N}$$ is a rational Cauchy sequence. But this is trivially the case, since for any rational $$\epsilon > 0$$ we have

$|0_i-0_j|=0 < \epsilon\quad\quad\text{ for all }i,j\ge 1.$

It remains to be shown that the rational Cauchy sequence $$(0)_{n\in\mathbb N}$$ is neutral with respect to the addition of rational Cauchy sequences. Let $$(x_n)_{n\in\mathbb N}$$ be a rational Cauchy sequence. Because $$0$$ is neutral with respect to the addition of rational numbers, it follows that

$\begin{array}{ccll} (x_n)_{n\in\mathbb N}+(0)_{n\in\mathbb N}&=&(x_n+0)_{n\in\mathbb N}&\text{by definition of adding rational Cauchy sequences}\\ &=&(x_n)_{n\in\mathbb N}&\text{because }0\text{ is neutral with respect to the addition of rational numbers}\\ \end{array}$

It remains to be shown that also the equation $$(0)_{n\in\mathbb N}+(x_n)_{n\in\mathbb N}=(x_n)_{n\in\mathbb N}$$ holds for all rational Cauchy sequences $$(x_n)_{n\in\mathbb N}$$. It follows immediately from the commutativity of adding rational Cauchy sequenses.

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