Definition: Complex Cauchy Sequence

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

Proofs: 1 2 3
Propositions: 4 5
Theorems: 6

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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 real number \(\epsilon\).