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==Topological Statement==
'''Theorem.''' ''Let <math>S</math> be a [[Topological Space|topological space]]. A decreasing nested sequence of non-empty compact, closed subsets of <math>S</math> has a non-empty intersection. In other words, supposing <math>(C_k)</math> is a sequence of non-empty compact, closed subsets of S satisfying''
:<math>C_0 \supset C_1 \supset \cdots \supset C_n \supset C_{n+1} \supset \cdots, </math>
''it follows that''
:<math>\left(\bigcap_{k = 0}^\infty C_k\right) \neq \emptyset. </math>
The closedness condition may be omitted in situations where every compact subset of ''<math>S''</math> is closed, for example when ''<math>S''</math> is [[Hausdorff space|Hausdorff]].
'''Proof.''' Assume, by way of contradiction, that <math>{\bigcaptextstyle \bigcap_{k = 0}^\infty C_k}=\emptyset</math>. For each ''<math>k''</math>, let <math>U_k=C_0\setminus C_k</math>. Since <math>{\bigcuptextstyle \bigcup_{k = 0}^\infty U_k}=C_0\setminus\left({\bigcaptextstyle \bigcap_{k = 0}^\infty C_k}\right)</math> and <math>{\bigcaptextstyle \bigcap_{k = 0}^\infty C_k}=\emptyset</math>, we have <math>{\bigcuptextstyle \bigcup_{k = 0}^\infty U_k}=C_0</math>. Since the <math>C_k</math> are closed relative to ''<math>S''</math> and therefore, also closed relative to <math>C_0</math>, the <math>U_k</math>, their set complements in <math>C_0</math>, are open relative to <math>C_0</math>.
Since <math>C_0\subset S</math> is compact and <math>(\{U_k) \vert k \geq 0\}</math> is an open cover (on <math>C_0</math>) of <math>C_0</math>, a finite cover <math>\{U_{k_1}, U_{k_2}, \ldots, U_{k_m}\}</math> can be extracted. Let <math>M=\max_{1\leq i\leq m} {k_i}</math>. Then <math>{\bigcuptextstyle \bigcup_{i = 1}^m U_{k_i}}=U_M</math> because <math>U_1\subset U_2\subset\cdots\subset U_n\subset U_{n+1}\cdots</math>, by the nesting hypothesis for the collection <math> (C_k)._{k \geq 0}</math>. Consequently, <math>C_0={\bigcuptextstyle \bigcup_{i = 1}^m U_{k_i}} = U_M</math>. But then <math>C_M=C_0\setminus U_M=\emptyset</math>, a contradiction. [[Q.E.D.|∎]]
==Statement for Real Numbers==
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