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 \(a\):

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

Proofs: 1

Lemmas: 1
Proofs: 2 3 4 5 6

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