Yeah, that's the difference here, which maybe wasn't clear. The ellipsis after a number in math means "going on to infinity" so 0.99999999... means nines going on to infinity, that is, you never run out of digits. That is emphatically NOT the same as 0.99999999. Which means any kind of numerical example given with a finite number of nines isn't talking about the same thing.

Anyway, I looked it up in my analysis book last night, and you can prove pretty straightfowardly (if you've read the first three chapters of the book) that both decimal representations (that is, 0.999999.... and 1.000000...) are, indeed, EQUAL to the number 1.