*".9 is not 1; neither is .999, nor .9999999999. In fact if you stop the *

expansion of 9s at any finite point, the fraction you have (like .9999

= 9999/10000) is never equal to 1. But each time you add a 9, the

error is less. In fact, with each 9, the error is ten times smaller.

You can show (using calculus or other methods) that with a large

enough number of 9s in the expansion, you can get arbitrarily close to

1, and here's the key:

THERE IS NO OTHER NUMBER THAT THE SEQUENCE GETS ARBITRARILY CLOSE TO.

Thus, if you are going to assign a value to .9999... (going on

forever), **the only sensible value is 1.**"