Proof
(related to Theorem: Theorem of BolzanoWeierstrass)
 Let $X$ be a metric space.
 Let \(A\subset X\) be a compact subset.
 Let $(x_n)_{n\in\mathbb N}$ be a sequence of points \(x_n\in A\).
 We will prove by contradiction that $(x_n)_{n\in\mathbb N}$ contains a subsequence $(x_{n_k})_{k\in\mathbb N}$, which converges against some point \(a\in A\).
 Assume, there is no such subsequence.
 Then for every $x\in A$ we can choose arbitrary small \(\epsilon_x > 0\) such that the open ball $B(x,\epsilon_x)$ contains only finitely many sequence members. (Otherwise we could construct a subsequence convergent against $x$, since we can choose \(\epsilon_x\) arbitrarily small).
 Note that $(B(x,\epsilon_x))_{x\in A}$ is an open cover of $A$, i.e. $$ A\subset \bigcup_{x\in A} B(x,\epsilon_x).$$
 Since $A$ is compact, there are finitely many points \(x_1,\ldots,x_n\in A\) forming with their open balls a finite open subcover of $A$, formally
$$ A\subset \bigcup_{k=1}^n B(x_k,\epsilon_{x_k}).$$
 But then, the sequence $(x_n)_{n\in\mathbb N}$ would contain only finitely many points, which is a contradiction.
 Therefore, the assumption that there is no convergent subsequence $(x_{n_k})_{k\in\mathbb N}$ of $(x_n)_{n\in\mathbb N}$ is wrong.
∎
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References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984