Proposition: Product of a Complex Number and a Convergent Complex Sequence

Let \((a_n)_{n\in\mathbb N}\) be a convergent complex sequence with \(\lim_{n\rightarrow\infty} a_n=a\) and let \(x\in\mathbb R\) be a complex number. Consider the complex sequence \((c_n)_{n\in\mathbb N}\) with \(c_n:=x a_n\). Then \((c_n)_{n\in\mathbb N}\) is also convergent and its limit equals \(\lim_{n\rightarrow\infty} c_n=xa\).

This proposition can be expressed in the short form:

\[\lim_{n\rightarrow\infty} (x \cdot a_n)=x \cdot \lim_{n\rightarrow\infty} a_n.\]

Proofs: 1

Proofs: 1

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