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http://www.scribd.com/mobile/doc/76236535

page 49-50 Exercise 3.19

Let $A=\{0,2\}$ and $C$ be the Cantor Set. Define $x(\alpha) = \sum_{n=1}^\infty (\alpha_n / {3^n})$ for all $\alpha \in A^{\mathbb{N}}$.

Then $x$ is a well defined function.

I think the argument in the link assumed, without any notice, the existence of a sequence $\beta$ such that $x(\beta) = z$, for each $z\in C$.

Am I correct? Hence, the argument proves only ${ran} x \subset C$.

How do i prove that there exists a such sequence for each $z\in C$?

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    I am not sure if it's widely used to denote range set as $ran$.2012-10-07

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