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?
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.