Cantor set defined via sums, whaaaaa?

[SiddharthM]

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

 

[matt grime]

What's the problem? Just consider the base three expansion of points in [0,1].

 

[D H]

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)}.