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Proposition: Closed n-Dimensional Cuboids Are Compact

Let $a_\nu,b_\nu\in\mathbb R$ be real numbers with $a_\nu \le b_\nu$ for $\nu=1,\ldots,n$. The closed $n$-dimensional cuboid $$Q:=\{(x_1,\ldots,x_n)\in\mathbb R^n:\quad a_\nu \le x_\nu \le b_\nu\}$$ is a compact subset of the $n$-dimensional metric space of real numbers $\mathbb R^n$.

Table of Contents

Proofs: 1 Corollaries: 1

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Proofs: 1 2

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References

Bibliography

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