Skip to main content Contents Prev Up Next \(\DeclareMathOperator{\RE}{Re}
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Section 9.1 Results from real analysis
Theorem 9.1.1 . Heine-Borel.
A closed and bounded set in \(\R^n\) or \(\C^n\) is compact.
Theorem 9.1.2 .
Let \(\mathcal X, \mathcal R\) be metric spaces. Let \(S \subseteq \mathcal X\text{.}\) If \(f: S \to \mathcal R\) is continuous, then \(S\) compact implies that \(f(S)\) is compact.
Theorem 9.1.3 .
Suppose that \(K\) is a non-empty compact set in a metric space \(\mathcal X\) and that \(f:K \to \R\) is continuous. Then \(f\) attains its infimum and supremum on \(K\) - that is, \(f\) has both a minimum and maximum on \(K\text{.}\)
