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Cantor Ternary Set : Countable or Not?

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How do I proof of a Cantor ternary set and how to identify whether its countable or not?

(See attached file for full problem description with equation)

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Consider the set C all elements of R that have the form

Where each αi is either 0 or 2. Prove that in fact S is the Cantor ternary set. Given that C is the Cantor set, explain why it is now obvious that the Cantor set is uncountable.
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Countability of Cantor Ternary Sets is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Recall, the Cantor ternary set is constructed inductively as the intersection of sets A_n, each of which is a union of 2^(n) disjoint closed intervals of length (1/3)^n. Each A_(n+1) is constructed by removing open middle thirds from each closed interval in A_n.

We can construct a map f from any ternary expansion sum a_i (1/3)^i, a_i = 0,2 to the cantor set by using ...

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