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.

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

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