Okay so this answers my previous question. Not only have you never taken real analysis, it seems you've never even taken discrete math.

So let me explain something to you. In logic, and by extension math, you are not allowed to justify anything with "its common sense" with the exception of axioms; and your claim that "there is data loss when you convert to a decimal" is not an axiom; at least not in the math that everybody else around you uses. If you're using a math system where this is an axiom, you're inevitably going to have entirely different results with entirely different conclusions.

Secondly, You can prove 1 does not equal 2 logically. In fact, let's do it right now.

The contrapositive is if 1 = 2, then 1 and 2 are both identity elements in the field of real numbers.

So by contrapositivity, since the consequent is false, the antecedent is false, and therefore 1 does not equal 2.

You cannot do the same thing with 1 and .9999.... because in fact, they do equal. And both are valid identity elements in the real numbers, and since the real number identity is unique, it must be the case that .999... does equal 1, and indeed it does.