Prove Sums of Cantor Sets in [0,2]

[kathrynag]

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) Let's 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.

 

[kathrynag]

I also considered writing x as a^i/3^i where a is in {0,1} but not sure what else to do

 

[HallsofIvy]

I would guess from your title that C is the Cantor trinary set but it would have been better if you had said so. And I have no idea what C1, C2, or Cn is.

 

[kathrynag]

I'm assuming C is the Cantor set since the problem is dealing with Cantor sets, but the problem doesn't explicitly say what C is.

 

[kathrynag]

I say earlier in my book some talk about cantor sets with C1=C0\(1/3,2/3)=[0,1/3]U[2/3,1]
C2=([0,1/9)U[2/3,1/3])U([2/3,7/9]U[8/9,1])

 

[kathrynag]

Cantor set C=intersection(C_n)