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Proof

(related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)

This follows immediately from the corresponding rule, how convergence preserves the order relation of sequence members, if we define the constant sequences

• \((u_n)_{n\in\mathbb N}\) with \(u_n:=U\) for and \(n\in\mathbb N\) and the upper bound \(U\) and
• \((l_n)_{n\in\mathbb N}\) with \(l_n:=L\) for and \(n\in\mathbb N\) and the lower bound \(L\) and consider the order relation

\[l_n\le a_n\le u_n.\]

For the limits \(\lim_{n\rightarrow\infty} a_n=a\), \(\lim_{n\rightarrow\infty} l_n=L\) and \(\lim_{n\rightarrow\infty} u_n=U\) it follows then that

\[L\le a\le U.\]

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References

Bibliography

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