# Lemma: Convergent Rational Sequences With Limit $$0$$ Are an Ideal Of the Ring of Rational Cauchy Sequences

Let $$(M , + , \cdot)$$ be the commutative unit ring of all rational Cauchy sequences. 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 $$M$$, formally $$I\lhd R$$, which is equivalent to the following properties:

1. $$(I, +)$$ is a subgroup of $$(M , +)$$.
2. If we multiply a rational Cauchy sequence by a rational sequence convergent to $$0$$ from the right, the product is convergent to $$0$$, formally $(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.$
3. If we multiply a rational Cauchy sequence by a rational sequence convergent to $$0$$ from the right, the product is convergent to $$0$$, formally $(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.$

Proofs: 1

Definitions: 1
Proofs: 2 3

Github: ### References

#### Bibliography

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