Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences

Let \((c_n)_{n\in\mathbb N}\) be a complex sequence \((c_n)_{n\in\mathbb N}\) is a convergent complex sequence, if and only if the real sequences of the real parts \((\Re(c_n))_{n\in\mathbb N}\) and imaginary parts \((\Im(c_n))_{n\in\mathbb N}\) are convergent real sequences. In case of convergence, the following equation is fulfilled:

\[\lim_{n\to\infty}c_n=\lim_{n\to\infty}\Re(c_n) + i\cdot \lim_{n\to\infty}\Im(c_n).\]

Proofs: 1 Corollaries: 1

Proofs: 1 2 3

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