*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.

Theorem: X=0.

Proof:

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.

So X=0.

QED

In words this means that the difference between 1 and 0.999... is zero, in other words the two numbers are equal.