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A question to Sequence.

  1. Mar 2, 2008 #1
    There is a theorem: If {En} is a sequence of closed, nonempty and bounded sets in a complete metric space X, if En[tex]\supset[/tex]En+1, and if lim diam En = 0, then [tex]\cap[/tex]En consists exactly one point.

    And what I'm asking is that, if either the sets were not closed or X was not a complete space (but not both), and all other condictions are still satisfied, then what will follow? And if I let X be the rational set, for instance, what will I get. And could you explain it?

    Last edited: Mar 2, 2008
  2. jcsd
  3. Mar 3, 2008 #2


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    Do you have the proof for this theorem? Then you could just scan through it and scrutinize each step to see which assumption(s) are used.
  4. Mar 3, 2008 #3


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    Hi, Ka Yan!

    You should be able to find a simple example of open sets (on a plane, say) whose intersection is empty.

    … there you go! :smile:

    (and: hint: are the rationals complete? if not, why not?)
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