did not read the whole document. just the proof of theorem 1 and it has a minor error. it is not obvious that C(n) != x. that is because it is not true. to see that, consider a map C(n) such that

C(n) = 0.10000000...

that is: c_1(n) = 1 and c_i(n) = 0 for all other i.

If you let \overline{c}_1(n) = 0 and c_i(n) = 9 for all other i, then we have that c_i(n) != \overline{c}(c)_i(n) for all i but C(n) = x.

This is a common mistake people make. To fix it, one needs to be more careful when defining \overline{c}_i.