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Proposition: Addition of Rational Cauchy Sequences

Let \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\) be rational Cauchy sequences. The addition of rational Cauchy Sequences is defined as: \[\begin{array}{cclc} (x_n)_{n\in\mathbb N}+(y_n)_{n\in\mathbb N}&:=&(x_n+y_n)_{n\in\mathbb N}\\ \end{array}\] where the result \((x_n+y_n)_{n\in\mathbb N}\) is also a rational Cauchy sequence, called the sum of the rational Cauchy sequences \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\). The sum exists and is well-defined.

Table of Contents

Proofs: 1

1. Proposition: Addition of Rational Cauchy Sequences Is Associative
2. Proposition: Addition of Rational Cauchy Sequences Is Commutative
3. Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences)
4. Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition
5. Proposition: Addition of Rational Cauchy Sequences Is Cancellative

Mentioned in:

Proofs: 1 2 3 4 5 6 7 8 9 10 11
Propositions: 12 13 14 15 16 17

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References

Bibliography

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