OK, this has actually been a can of mass-internet-anger worms for years so I probably shouldn’t start up this debate. But general consensus is that the proof is true. It is really a matter of notation rather than a matter of number theory, because of the fact that the notational concept of “recurring” means “repeat to infinity”, and infinity is a strange and often misunderstood concept. Personally though it seems clear to me that 0.999… is simply another way of writing 1, and similarly 8.999… is another way of writing 9. Numerically they are the same, they only differ in how they are written.

By the way, to specifically address your point: your statement is true, and I presume that you derived by simply multiplying 0.999… by 9. If you keep multiplying each digit by 9 and adding it to the result, it will endlessly add 9s to the end of the number, seemingly forever.

Now consider this:

x = 0.999…
10x = 9.999…
10x – x = 9
9x = 9

This part is the key, so it is unfortunate that it is missing from rotatebilly’s post. You can see from the LHS that this is correct and should also be identical to your result.