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In [[real analysis]], a branch of mathematics, '''Cantor's intersection theorem''', named after [[Georg Cantor]], is a theorem related to [[compact set]]s in '''R''', the set of [[real number]]s. It states that a decreasing nested [[sequence]] of non-empty, [[closed set|closed]] and [[bounded set|bounded]] subsets of '''R''' has nonempty intersection. In other words, supposing {''C''<sub>''k''</sub>} is a sequence of non-empty, closed and bounded sets satisfying
:<math>C_0 \supseteq C_1 \supseteq \cdots C_k \supseteq C_{k+1} \cdots, \, </math>
it follows that
:<math>\left(\bigcap_{k} C_k\right) \neq \emptyset. \, </math>
The result is typically used as a lemma in proving the [[Heine–Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. Conversely, if the Heine–Borel theorem is known, then it can be restated as: a decreasing nested sequence of non-empty, compact subsets of '''R''' has nonempty intersection.
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