We present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.

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# Borel’s Proof of the Heine-Borel Theorem

Posted byCalvin Wooyoung ChinPosted inNotesTags:Analysis, Compactness, Heine-Borel Theorem, Set Theory, Topology, Transfinite Induction

We present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.

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