From: TheSnail - PM

I'm a bit puzzled, though, by "until you have calculated all other transactions". What other transactions? The process of dividing by 9 leaving remainder 1 isn't going to change.

The 'other transactions' are that you started by showing 1/9= 0.111..
(which is only an approximation) and then showed:

9 * 1/9 = 9 * 0.1111....
1 = 0.9999....

It is in this second stage that you have overcomplicated things by replacing 1/9 by a close approximation of 1/9.
9* 1/9 = 9/1 * 1/9 = 9/9 =1

All you have shown is that an approximation of one ninth is very close to the value of One.