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**Danneedshelp** Let $\displaystyle C$ represent the Cantor set.

Q: Show that there exists $\displaystyle x_{1},y_{1}\in{C}$ satisfying $\displaystyle x_{1}+y_{1}=s\in[0,2]$.

I found some work that goes like this...

$\displaystyle C_{1}=[0,1/3]\cup[2/3,1]$. Then we have, $\displaystyle \color{blue}[0,1/3]+[0,1/3]=[0,1/3]\cup[0,2/3]+[2/3,1]=[2/3,4/3]\cup[2/3,1]

+[2/3,1]$. So, $\displaystyle C_{1}+C_{1}=[0, 2/3]\cup[2/3, 4/3]\cup[4/3, 2]=[0, 2]$. Thus, for all $\displaystyle s\in[0, 2]$, we can find an $\displaystyle x_{1} , y_{1}$ $\displaystyle C_{1}$ satisfying $\displaystyle x_{1} + y_{1} = s$

I am not sure that I follow exactly what is being done. Can someone help me to understand the work above. Do I think of these intervals as only lengths?