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Theorem: Theorem of Bolzano-Weierstrass

Let $X$ be a metric space and let \(A\subset X\) be a compact subset. Let $(x_n)_{n\in\mathbb N}$ be a sequence of points \(x_n\in A\). Then $(x_n)_{n\in\mathbb N}$ contains a subsequence $(x_{n_k})_{k\in\mathbb N}$, which converges against some point \(a\in A\).

Notes

• This theorem is a generalization of the corresponding theorem for real sequences.
• The theorem is named after Bernard Bolzano and Karl-Theodor Weierstrass.

Table of Contents

Proofs: 1

Mentioned in:

Theorems: 1

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References

Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984