Theorem of Bolzano-Weierstrass

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.

1.Proof: (related to "Theorem of Bolzano-Weierstrass")

Edit or Add Notation Axiomatic Method

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Bibliography (further reading)

[582] Forster Otto: “Analysis 2, Differentialrechnung im $\mathbb R^n$, Gewöhnliche Differentialgleichungen”, Vieweg Studium, 1984