content: The one that I posted is the real one. The fake one (intentionally) makes a lot of mistakes. For example, it is true that x was 1 all along. But that doesn't mean that you're presupposing that x=1. x works simply as a pretend name for whatever 0.999... denotes, so when you say at the beginning that x=0.999... you're basically just saying that 0.999... is identical to itself. This is obviously true for any number whatsover, so you're not assuming anything about 0.999.. that isn't true of any other number. Every other step of the proof proceeds according to universal algebra laws. Hence the proof is valid. (If your not convinced, check the following: you can carry out the whole proof by writing 0.999...=0.999... at the beginning and repeating the whole proof by writing 0.999... instead of x. No need to use x at all.) Or, if you want a more pedantic explanation, what the proof demonstrates is this: For every number x, if x=0.999..., then x=1. So in particular, we have that if 0.999...=0.999..., then 0.999...=1. But obviously 0.999...=0.999... so 0.999...=1.