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Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members
Let \((a_n)_{n\in\mathbb N}\) be a convergent real sequences with the limit \(\lim_{n\rightarrow\infty} a_n=a\) and let \(L\in\mathbb R\) and \(U\in\mathbb R\) be any given lower and upper bounds of its sequence members:
\[L\le a_n\le U\quad\quad\forall n\in\mathbb N.\]
Then \(L\) and \(U\) are also the bounds for the limit \(a\):
\[L\le a\le U.\]
Table of Contents
Proofs: 1
Mentioned in:
Lemmas: 1
Proofs: 2 3 4 5 6
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983