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 index^{1} \(N(\epsilon)\in\mathbb N\) 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
\(N(\epsilon)\) means that the natural number \(N\) depends only on the real number \(\epsilon\). ↩