Proposition: Product of Convegent Complex Sequences

Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be complex sequences convergent to the limits \(\lim_{n\rightarrow\infty} a_n=a\) and \(\lim_{n\rightarrow\infty} b_n=b\). Consider the complex sequence \((c_n)_{n\in\mathbb N}\) with \(c_n:=a_n \cdot b_n\). Then \((c_n)_{n\in\mathbb N}\) is also convergent and its limit equals \(\lim_{n\rightarrow\infty} c_n=a \cdot b\).

This proposition can be expressed in the short form:

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

Proofs: 1

Proofs: 1 2

Thank you to the contributors under CC BY-SA 4.0!




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