applicability: $\mathbb {Q, R}$

Definition: Real Cauchy Sequence

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

Corollaries: 1
Proofs: 2 3 4 5 6 7 8
Propositions: 9 10 11
Theorems: 12

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


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