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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:

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

Table of Contents

Proofs: 1

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Proofs: 1 2

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References

Bibliography

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