You are confusing the set of converging numbers 0,9 0,99 0,999 ... and the number it is converging to, aka 0,999... which is the same as 1. T

You are confusing the set of converging numbers 0,9 0,99 0,999 ... and the number it is converging to, aka 0,999... which is the same as 1. That number (written as 0,999... or 1) is NOT in the set. Another way to convince yourself: take 1. Divide by 9, you get 0,111... . Now multiply it back by 9. You get 0,999..., but this must be the same as 9. Also, 0,9999... is not a regular decimal number. It is a writing CONVENTION. This is in fact exactly the same as the fraction 9 / 9, which is exactly 1.