integer function

Set $c_{0} = [x]$, and $c_{n} = [10^{n}(x-c_{0})-10^{n-1}c_{1}-10^{n-2}c_{2}- \cdots - 10c_{n-1}]$ for $n = 1,2,3, \ldots$. Verify the decimal representation of $x$ is $x = c_{0} + 0 \cdot c_{1}c_{2}c_{3} \ldots$ and that this construction excludes the possibility of an infinite string of $9$'s.

So we want to show that $x = [x] + 0 \cdot c_{1}c_{2}c_{3}$. Now $x-1 < [x] \leq x$. Then what?