In standard mathematics there is no such thing as an infinitely small number. All nonzero standard real numbers have some finite separation from zero. The diff

In standard mathematics there is no such thing as an infinitely small number. All nonzero standard real numbers have some finite separation from zero. The difference between 0.9 recurring and 1 would have to be infinitely small, as you note. So if it's a number, it can only be zero. It can't be any nonzero number because none of them are infinitely small. You may not like it intuitively, but this way of thinking is perfectly rigorous and logical and works very well for all practical calculations. Admitting infinitely small numbers makes math way more complicated (nonstandard analysis) and under the hood it ultimately falls back on the standard model anyway.