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applicability: $\mathbb {N, Z, Q, R, C}$

Proposition: Product of Convegent Real Sequences

Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be convergent real sequences to the limits \(\lim_{n\rightarrow\infty} a_n=a\) and \(\lim_{n\rightarrow\infty} b_n=b\). Consider the real 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 \(\lim_{n\rightarrow\infty} c_n=a \cdot b\).

Notes

• This proposition can be expressed in the short form: \[\lim_{n\rightarrow\infty} (a_n \cdot b_n)=\lim_{n\rightarrow\infty} a_n \cdot \lim_{n\rightarrow\infty} b_n.\]
• The proposition's proof can be transferred also to sequences of other kinds than real numbers, for example, the complex numbers.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3

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References

Bibliography

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