(related to Theorem: Completeness Principle for Complex Numbers)

- Let \((c_n)_{n\in\mathbb N}\) be a complex Cauchy sequence and let \(c_n=a_n+ib_n\) for all \(n\in\mathbb N\).
- From the proposition about complex Cauchy sequences versus real Cauchy sequences, it follows that \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) are real Cauchy sequences.
- According to the completeness principle for real numbers, they are convergent real sequences.
- From the proposition about convergent complex sequences versus convergent real sequences it follows that \((c_n)_{n\in\mathbb N}\) is a convergent complex sequence.∎

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