An Infinite Decimal Is a Limit, Not a Nearly Finished Number
A decimal with infinitely many digits is understood through its finite approximations. The first few approximations to 0.999... are 0.9, 0.99, and 0.999. Each is less than 1, but after writing n nines, the gap is exactly 1 divided by 10 to the n power: 0.001 after three nines, for instance. As n grows without bound, that gap shrinks toward zero.
The infinite decimal denotes the number those approximations approach. Since their distance from 1 becomes arbitrarily small, the limit is 1—not a number just short of 1. A “sliver left over” would have to be a positive distance, but no positive distance remains in the limit.
Why the Final Digit Never Arrives
The intuition that the decimal is unfinished can be misleading: there is no final step at which another 9 gets added. The notation describes the whole infinite pattern at once. It does not describe a process that pauses after a very large, finite number of digits.
This is also why 0.999... and 1 are not two numbers squeezed infinitely close together. They are two decimal spellings of the same real number. The distinction matters whenever decimals are used to represent exact values: an endless expansion can be exact, even when every finite version of it is only an approximation.
