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A converse to this theorem is also true: if <math>X</math> is a metric space with the property that the intersection of any nested family of non-empty closed subsets whose diameters tend to zero is non-empty, then <math>X</math> is a complete metric space. (To prove this, let <math>(x_k)_{k \geq 1}</math> be a Cauchy sequence in <math>X</math>, and let <math>C_k</math> be the closure of the tail <math>(x_j)_{j \geq k}</math> of this sequence.)
== See also ==
* [[Kuratowski's intersection theorem]]
== References ==
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