◀ ▲ ▶Branches / Analysis / Theorem: Every Bounded Real Sequence has a Convergent Subsequence

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

Theorem: Every Bounded Real Sequence has a Convergent Subsequence

Every bounded real sequence \((a_n)_{n\in\mathbb N}\) has a convergent subsequence \((a_{n_k})_{k\in\mathbb N}\) (i.e. at least one accumulation point).

Notes

• The theorem is named after Bernard Bolzano and Karl-Theodor Weierstrass.
• The theorem can be also formulated for general metric spaces.

Table of Contents

Proofs: 1

Mentioned in:

Definitions: 1
Examples: 2
Proofs: 3 4 5 6
Propositions: 7 8
Theorems: 9

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References

Bibliography

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