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Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum
Let $A$ be a nonempty compact subset of a the real numbers $\mathbb R$. Then $$\max(A)\in A,~\min(A)\in A,$$ where $\max(A)$ and $\min(A)$ denote the maximum and minimum of \(A\), respectively.
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Proofs: 1
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References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984