◀ ▲ ▶Branches / Analysis / Proposition: Convergent Real Sequences are Bounded

applicability: $\mathbb {N, Z, Q, R, C}$

Proposition: Convergent Real Sequences are Bounded

Every convergent real sequence \((a_n)_{n\in\mathbb N}\) is bounded.

Notes

• In some literature, you will find the previous proposition that every monotonic bounded sequence is convergent formulated together with the following proposition that "the opposite" also is true, suggesting an equivalence. It is not really an equivalence since the first proposition is not valid for complex sequences, while the following one is. Complex numbers cannot be ordered. Thus, it does not make sense to talk about "monotonic" complex sequences. However, it does make sense to talk about bounded and convergent complex sequences.
• A generalization for complex numbers can be found here.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2
Propositions: 3 4

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References

Bibliography

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