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Proof

(related to Theorem: Heine-Borel Theorem)

Let \(A\subset\mathbb R^n\) be a subset \(A\) of the $n$-dimensional metric space of real numbers $\mathbb R^n$.

"$\Rightarrow$"

• Let $A$ be compact.
• Since \(\mathbb R^n\) is a metric space, $A\subset X$ is a compact subset of a metric space.
• According to the corresponding proposition, all compact subsets of metric spaces are closed and bounded.
• Thus, also \(A\) must be closed and bounded.

"$\Leftarrow$"

• Let $A$ be closed and bounded.
• Because $A$ is bounded, it is contained in a sufficiently large closed $n$-dimensional cuboid $Q$.
• According to the corresponding proposition, all closed $n$-dimensional cuboids are compact.
• Thus, $Q$ must be compact.
• Because $A\subset Q$ by construction, it is a closed subset of a compact set.
• According to the corresponding proposition, all closed subsets of compact sets are compact.
• Thus, $A$ must be compact.
  ∎

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References

Bibliography

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