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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:

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 \[(c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.\]

Table of Contents

Proofs: 1

Mentioned in:

Definitions: 1
Proofs: 2 3

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References

Bibliography

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