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.

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

Github: ### References

#### Bibliography

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