(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 $\mathbb R^n$.

- 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.

- 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.∎

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