The fractional part of x is what you have left after you subtract the largest integer k ≤ x. Hence the fractional part is always less than 1. Since 0.999... =

The fractional part of x is what you have left after you subtract the largest integer k ≤ x. Hence the fractional part is always less than 1. Since 0.999... = 1, it is not less than 1, so cannot be the fractional part of any number.