Dunno if it helps your understanding, but 0.99 is closer to 1 than 0.9 is, so for each 9 you add, you get closer to 1. If you have an infinite amount of 9's then you are getting infinitely close to 1. Or if you want it in math terms:
[tex].9 = \frac{9}{10^1}[/tex]
[tex].09 = \frac{9}{10^2}[/tex]
[tex].009 = \frac{9}{10^3}[/tex]
etc.
[tex].9 + .09 + .009 = \frac{9}{10^1} + \frac{9}{10^2} + \frac{9}{10^3} = .999[/tex]
So:
[tex]\sum_{n=1}^{\infty} \frac{9}{10^n} = 0.\overline{999} = 1[/tex]
You see that as [tex]n \to \infty[/tex] (analogous to "you are adding more 9's on the end") you are getting closer to 1.