Cantor set defined via sums, whaaaaa?

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SUMMARY

The Cantor set can be defined through the sum of a specific sequence, where each element p_n is either 0 or 2. The formula x(a) = Σ (p_n)/(3^n) from n=1 to infinity generates the Cantor middle-thirds set. To understand this, one must consider the base three expansion of real numbers within the interval [0,1], including the representation of 1 as 0.222...3. This approach clarifies how the recursive construction of the Cantor set aligns with the defined sequence.

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Cantor set defined via sums, whaaaaa?!?

problem 19 chapter 3 of Rudin. I'm totally lost, I've even done a project on the Cantor set before but I just don't know where to start here.

Associate to each sequence a=(p_n) in which p_n is either 0 or 2, the real number

x(a) = sum from 1 to infinity of (p_n)/(3^n).

Prove that the set of all x(a) is the cantor middle-thirds set.

Cheerio
 
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What's the problem? Just consider the base three expansion of points in [0,1].
 
Represent the real numbers between 0 and 1 inclusive in trinary. It will help to represent 1 as 0.222...3. If you use this representation, when you construct Cantor's set recursively you will arrive at the set {x(a)}.
 

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