Theorem (Heine-Borel). \itseries Let I := [a, b] \subset \mathbb{R} be a closed interval and \mathcal{C} be a collection of open sets whose union \bigcup \mathcal{C} contains I. Then there is a finite subcollection \{S_1, S_2, \cdots, S_k\} of \mathcal{C} for some integer k such that the finite union \bigcup_{i=1}^k S_i still contains I.

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