I have no idea what any of these mean. I know that "0.999..." means the decimal number whose only digits are "9". But I don't know what you mean by "0...9". How many digits does that "..." skip over? It can't be an infinite number since there is no "last" digit in that case.

"ten times"? Do you mean 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9+ 0.9= 10(0.9)= 9? I don't know what you mean by 1- y because I don't know what you mean by "0.0...1" which is apparently how you have defined y0

x=0.9.. / lets *10

0.9..+0.9..+0.9.. ten times =(1-y)+(1-y).. ten times SO

And so nothing beyond here makes sense.

Should I conclude that you do not understand how "decimal notation" works?10x= 10-10y / -0.9...

10x-x=(10-10y)-(1-y)

9x=9-9y

9x=9-9y

x=(9-9y)/9 and this is NOT 9 !

x=9(1-y)/9

x=0.9....

This last one clearly shows that0.9...9 is not some "magic" number its number like anyother. This 0.00...1 is more "special" but is stillordinary number.

If you really want a proof that 0.999...= 1, do this: 0.999...= 0.9+ 0.09+ 0.009+ ...= 0.9+ 0.9(.1)+ 0.9(.01)+ 0.9(.001)+ ... In other words, it is ageometricseries, of the form with a= 0.9 and r= 0.1. It is easy to show that the sum of such a geometric series is which, here, is .