# Proof

We want to show that the set $$I:=\{(a_n)_{n\in\mathbb N}~|~a_n\in\mathbb Q,\lim a_n=0\}$$ of all rational sequences convergent to $$0$$ is an ideal of the commutative unit ring of all rational Cauchy sequences $$(M , + , \cdot)$$, formally $$I\lhd R$$.

We will do so by proving all properties of an ideal:

### $$(2)$$ We want to show that $$(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.$$

It has been shown that given a rational Cauchy sequence $$(c_n)_{n\in\mathbb N}$$, the members $$c_n$$ are bounded, i.e. there exists a positive constant $$c > 0, c\in\mathbb Q$$, such that $$|c_n|\le c$$ for all $$n\in\mathbb N$$. By hypothesis, $$(a_n)\in I$$ is a rational sequence convergent to $$0$$. Thus, we can choose any arbitrarily small $$\epsilon\in\mathbb Q,\epsilon > 0$$, and conclude that there exists an $$N(\epsilon/c)\in\mathbb N$$ such that for all $$n > N(\epsilon/c)$$ we have the estimation $|c_n\cdot a_n|=|c_n|\cdot|a_n|\le c\cdot \frac\epsilon c=\epsilon.$ This shows that a right-product of a rational Cauchy sequence $$(c_n)_{n\in\mathbb N}$$ and a rational sequence $$(a_n)_{n\in\mathbb N}$$ convergent to $$0$$ is a sequence $$(c_n\cdot a_n)_{n\in\mathbb N}$$, which also converges to $$0$$, or equivalently an element of $$I$$.

### $$(3)$$ We want to show that $$(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (a_n\cdot c_n)_{n\in\mathbb N}\in I.$$

This can be concluded immediately from $$(2)$$, since $$(M, + ,\cdot)$$ is a commutative ring by hypothesis.

This demonstrates altogether that $$I$$ is an ideal of $$(M, + ,\cdot)$$.

Github: ### References

#### Bibliography

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