Originally posted by Bishopcrw
Personally I like asking the question,
What number is between the two numbers 1 and .999....?
This number by definition would be the difference (subtraction).
The difference you speak of is smaller than all positive numbers.
It is not a negative number. Let's call the difference X.
So we have
1) 0=X or 0 is smaller than X
2) X is smaller than p, for all positive numbers p.
Let's assume X is NOT equal to zero. From 1) we infer then that 0 is smaller than X
Make P = 0.5 * X.
0 is smaller than P clearly.
But now we have a contradiction with 2)!
Our assumption must then be false.
In words this means that the difference between 1 and 0.999... is zero, in other words the two numbers are equal.