therefore 10 c - c = 8.999...1
9 c = 8.999...1
That's the wrong concept of infinity. All digits are 9 and there is no last digit.
there are similar problems involving infinity, like how
1 + 2 + 3 ... = ?
2 + 4 + 6 ... = ?
even though both have the same number of terms, the second series is obviously twice that of the first series. however, two times infinity still equals infinity.
so it looks like that's the way we structure the number 'infinity'.
actually there are different kinds of infinity. for example there are more real numbers then there are whole numbers.
in fact you can easily prove that there are infinitely many types infinity.
but on the other hand from a mathematically useful point of view the amount of rational numbers is the same as whole numbers. the rational numbers are a "countable set", you can put them in order: (1, -1/2, 1/2, -1/3, 1/3, -2/3, 2/3, -1/4, 1/4, -3/4, 3/4, ...). you can define a distinct order where every rational number has its index. this would not work with the real numbers (can be proven easily too)
another question is if there is a cardinality step in between the whole and real numbers. no one has found one. but no one has proven that it does not exist either. the continuum hypothesis states there isn't one.