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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:
 \((I, +)\) is a subgroup of \((M , +)\).
 If we multiply a rational Cauchy sequence by a rational sequence convergent to \(0\) from the right, the product is convergent to \(0\), formally \[(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.\]
 If we multiply a rational Cauchy sequence by a rational sequence convergent to \(0\) from the right, the product is convergent to \(0\), formally \[(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.\]
Table of Contents
Proofs: 1
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Definitions: 1
Proofs: 2 3
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References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013