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Math Help - Cantor sets

  1. #1
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    Cantor sets

    I'm supposed to show that the sum C+C ={x+y,x,y in C}=[0,2]
    a) Show there exist x1,y1 in C1 for which x1+y1=s. Show in general for any arbitrary n in the naturals, we can always find xn, yn in Cn for which xn+yn=s.
    b) Keeping in mind that the sequences xn and yn do not necessarily converge show show that they never the less be used to produce the desired x and y in C satisfying x+y=s.


    a) Lets be in [0,2]
    C1=[0,1/3]U[2/3,1]
    That's about as far as I get and then I get stuck.
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  2. #2
    Senior Member Tinyboss's Avatar
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    It might help to think of C as the set of reals in [0,1] which can be expressed in base-3 with no 1's in the digit sequence. Which base-3 sequences can you get by adding sequences with no 1's?
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  3. #3
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    I can write it as a^i/3^i where a is in {0,1}.
    I'm still confused on how to use that to show x1+y1=s
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