It works as a demonstration fine. The problem from the point of view of pure mathematics is that it is an infinite process. Now, when you represent a number like, oh, 0.7687326 to one less decimal place the convention normally used is that you round the penultimate digit up if it is 5 or more, and down if it less than five (at least, that's the normal convention - there are weirdnesses like banker's rounding, but we will ignore that!), this would be0.768733.

Mow, if we have 0.99999999... to any *finite* number of places, and you wish to stop writing the places out, then the rounding rules take you to 1.00000000.... which does equal 1, but not in an interesting way. Conversely, if you use the other common convention and truncate, you have 1=0.9999 (say), which isn't true. Either way, you don't really fully express the concept. And if you want to 'go on forever' you are on treacherous ground: infinite series are vicious beasts.

By way of illustration, and for a spot of light-hearted maths (!!!!), think about the series

S = 1-1+1-1+1...

What does it add up to? Well, lets bracket things this way

S = (1-1) + (1-1) + (1-1) ... ===> S = 0

But noting that a-b is the same as a+(-b), a different bracketing gives
S = 1 + (-1+1) + (-1+1) + (-1+1) ... ===> S = 1

Or even
S = 1 - (1-1+1-1+1-1..) ... ===> S = 1 - S ===> S = 0.5

It's problems like that which made mathematicians very wary of infinite series and the need to treat 'em with kid gloves.