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Proof

(related to Corollary: Closed Real Intervals Are Compact)

• According to the respective proposition, closed $n$-dimensional cuboids of \(\mathbb R^n\) are compact.
• A closed real interval $[a,b]\subset\mathbb R$ is a special case of an $n$-dimensional closed cuboid for \(n=1\).
• Thus, it follows from the general case that it is a compact subset of the metric space of real numbers \(\mathbb R\).
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References

Bibliography

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