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Cantor set defined via sums, whaaaaa?

  1. Oct 11, 2007 #1
    Cantor set defined via sums, whaaaaa?!?

    problem 19 chapter 3 of Rudin. I'm totally lost, I've even done a project on the Cantor set before but I just don't know where to start here.

    Associate to each sequence a=(p_n) in which p_n is either 0 or 2, the real number

    x(a) = sum from 1 to infinity of (p_n)/(3^n).

    Prove that the set of all x(a) is the cantor middle-thirds set.

    Cheerio
     
  2. jcsd
  3. Oct 11, 2007 #2

    matt grime

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    What's the problem? Just consider the base three expansion of points in [0,1].
     
  4. Oct 11, 2007 #3

    D H

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    Represent the real numbers between 0 and 1 inclusive in trinary. It will help to represent 1 as 0.222...3. If you use this representation, when you construct Cantor's set recursively you will arrive at the set {x(a)}.
     
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