[thaumasiotes]

> To fix it, one needs to be more careful when defining \overline{c}_i.

I think the usual way to fix it is actually to be more careful when defining C(n) - that is, to specify that the value of C(n) is a sequence of digits, with the sequences in 1:1 correspondence with [0,1], not itself a value that belongs to the interval [0,1]. Then you can just say that trailing zeroes, or trailing nines, aren't allowed, giving you a unique representation for every value in [0,1]. At that point c-bar will work as defined.