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Proposition: How Convergence Preserves the Order Relation of Sequence Members

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 \(\lim_{n\rightarrow\infty} b_n=b\). Using the definition of the order relation for real numbers "\(\le\)" and "\( < \)" the following holds:

1. From \(a_n \le b_n\) for all \(n\in\mathbb N\) it follows that \(a \le b\).
2. From \(a_n \le b_n\) for all \(n\in\mathbb N\) it follows that \(a \le b\).

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