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

Proposition: Quotient of Convergent Real Sequences

Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be convergent real sequences with the limits \(\lim_{n\rightarrow\infty} a_n=a\) and \(\lim_{n\rightarrow\infty} b_n=b\) with \(b\neq 0\). Then there is a some index \(N\in\mathbb N\) such that \(b_n\neq 0\) for all \(n \ge N\) and the real sequence \((c_n)_{n\in\mathbb N,~n\ge N}\) with \(c_n:=a_n/b_n\) is convergent to the number \(a/b\).

Notes

• This proposition can be expressed in the short form: \[\lim_{n\rightarrow\infty,~n\ge N} \frac{a_n}{b_n}=\frac{\lim_{n\rightarrow\infty,~n\ge N} a_n}{\lim_{n\rightarrow\infty,~n\ge N} b_n}.\]
• 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

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References

Bibliography

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