Thread: Cantor set question

Let represent the Cantor set.

Q: Show that there exists satisfying .

I found some work that goes like this...

. Then we have, . So, . Thus, for all , we can find an satisfying

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?

thanks

Originally Posted by Danneedshelp

Let represent the Cantor set.

Q: Show that there exists satisfying .

I found some work that goes like this...

. Then we have, . So, . Thus, for all , we can find an satisfying

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?

means the set of all numbers that are equal to an element of plus an element of . But . By adding two numbers from the interval [0,1/3] you can get anything in the interval [0,2/3]. By adding a number from the interval [0,1/3] and a number from the interval [2/3,1] you can get anything in the interval [2/3,4/3]. By adding two numbers from the interval [2/3,1] you can get anything in the interval [4/3,2]. Putting those facts together, you see that by adding two numbers from the set you can get anything in the interval [0,2]. That is what is meant by the line . (The previous line, which I have highlighted in blue, seems totally confused.) Therefore, for any , we can find with .

That is just the first stage of this proof. You need to go on to show that for each of the sets used in the construction of the Cantor set. So there exist with . Finally, you need to show that a subsequence of the sequence converges to a point x in the Cantor set. The corresponding subsequence of the s will then converge to a point y in the Cantor set, with x+y=s.