# Definition: Rational Cauchy Sequence

Let $$(\mathbb Q,|~|)$$ be the mectric space of all rational numbers, together with the distance defined by the absolute value "$$|~|$$", and let $$(a_n)_{n\in\mathbb N}$$ be a sequence of rational numbers in $$(\mathbb Q,|~|)$$. The sequence $$(a_n)$$ is called a rational Cauchy sequence, if for all rational numbers $$\epsilon > 0$$ there exists an index1 $$N(\epsilon)\in\mathbb N$$ with $|a_i-a_j| < \epsilon\quad\quad\text{ for all }i,j\ge N(\epsilon).$

Definitions: 1 2 3
Lemmas: 4 5 6
Parts: 7
Proofs: 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Propositions: 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983

#### Footnotes

1. $$N(\epsilon)$$ means that the natural number $$N$$ depends only on the rational number $$\epsilon$$.