Theorem: HeineBorel Theorem
Let \(A\subset\mathbb R^n\) be a subset \(A\) of the $n$dimensional metric space of real numbers $\mathbb R^n$. $A$ is compact if and only if $A$ is closed and bounded.
This theorem was first proven by Eduard Heine (1821  1881) and Émile Borel (1871  1956)
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References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984