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

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