Cantor's intersection theorem: Difference between revisions

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A proof goes as follows. Since the diameters tend to zero, the diameter of the intersection of the ''C''_''n'' is zero, so it is either empty or consists of a single point. So it is sufficient to show that it is not empty. Pick an element ''x''_''n'' of ''C''_''n'' for each ''n''. Since the diameter of ''C''_''n'' tends to zero and the ''C''_''n'' are nested, the ''x''_''n'' form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point ''x''. Since each ''C''_''n'' is closed, and ''x'' is a limit of a sequence in ''C''_''n'', ''x'' must lie in ''C''_''n''. This is true for every ''n'', and therefore the intersection of the ''C''_''n'' must contain ''x''.

A converse to this theorem is also true: if ''X'' is a metric space with the property that the intersection of any nested family of nonemptynon-empty closed subsets whose diameters tend to zero is non-empty, then ''X'' is a complete metric space. (To prove this, let ''x''_''n'' be a Cauchy sequence in ''X'', and let ''C''_''n'' be the closure of the tail of this sequence.)

== References ==