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?