Why is the complement of the thick Cantor set dense in [0,1]?

[hj2000]

Let C be the thick Cantor set. let {a_n} be a sequence of positive numbers.
In the construction of the thick Cantor set, at the n-th stage we remove the middle a_n part of each interval (instead of the middle third as in the ordinary Cantor set).

I actually wanted to show that [0,1]-C is dense (where C is the thick Cantor set). How do I show it?

 

[EnumaElish]

My guess is you need to show C is nowhere dense. See http://en.wikipedia.org/wiki/Nowhere_dense

 

[jostpuur]

I would prove the following. Let $x\in C$ be arbitrary. It suffices to find a sequence $x_n\in [0,1]-C$ so that $x_n\to x$. Do you know how to identify points of Cantor's set with sequences of 0 and 1? I mean sequences like 0011... 1010... 1101... and so on.