◀ ▲ ▶Branches / Analysis / Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
applicability: $\mathbb {N, Z, Q, R}$
Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
Let \((a_n)_{n\in\mathbb N}\) be a real sequence and let \(D\) be its carrier set.
1. \((a_n)_{n\in\mathbb N}\) is called bounded above, if \(D\) is bounded above.
1. \((a_n)_{n\in\mathbb N}\) is called bounded below, if \(D\) is bounded below.
1. \((a_n)_{n\in\mathbb N}\) is called bounded, if \(D\) is bounded.
Examples
 The alternating sequence is bounded above by $1$ and bounded below by $1.$
 The sequence $\left(\frac{2n+1}{n+1}\right)_{n\in\mathbb N}=\left(2\frac{1}{n+1}\right)_{n\in\mathbb N}$ is bounded above by $2$ and bounded below by $1.$
 The sequence $(n)_{n\in\mathbb N}$ is bounded below by $0$ and unbounded above.
Mentioned in:
Applications: 1
Corollaries: 2
Definitions: 3 4 5 6
Explanations: 7
Parts: 8
Proofs: 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Propositions: 28 29 30 31 32 33 34
Theorems: 35 36
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983