Proposition: Sum of Convergent Complex Sequences

Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be convergent complex sequences 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 \(c_n:=a_n + b_n\). Then \((c_n)_{n\in\mathbb N}\) is also convergent and its limit equals \(\lim_{n\rightarrow\infty} c_n=a + b\). This proposition can be expressed in the short form:

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

Proofs: 1

Proofs: 1


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References

Bibliography

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