◀ ▲ ▶Branches / Analysis / Definition: Monotonic Sequences
applicability: $\mathbb {N, Z, Q, R}$
Definition: Monotonic Sequences
Based on the order relation for real numbers, a real sequences \((a_n)_{n\in\mathbb N}\) can be classified by the way the values \(a_n\) increase (decrease), depending on increasing index \(n\). The sequence \((a_n)_{n\in\mathbb N}\) is called:
- monotonically increasing, if for all \(n\in\mathbb N\) one has \(a_n \le a_{n+1}\),
- strictly monotonically increasing, if for all \(n\in\mathbb N\) one has \(a_n < a_{n+1}\),
- monotonically decreasing, if for all \(n\in\mathbb N\) one has \(a_n \ge a_{n+1}\),
- strictly monotonically decreasing, if for all \(n\in\mathbb N\) one has \(a_n > a_{n+1}\).
If \((a_n)_{n\in\mathbb N}\) is either (strictly) monotonically increasing or decreasing, it is called monotonic.
Table of Contents
- Proposition: Real Sequences Contain Monotonic Subsequences
Mentioned in:
Applications: 1
Definitions: 2
Lemmas: 3 4 5
Parts: 6
Proofs: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Propositions: 24 25 26 27 28
Theorems: 29
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
