Is 0.9 repeating = 1 disproved by asymptotes? I'm discussing proofs that 0.9 repeating equals 1 with some friends, and they use asymptotes to disprove this. One says if we had the function $y=x/0.000\ldots1$ (and he's only using that impossible number for theoretical purposes), the slope of the asymptote would be as close to undefined as you can get, but the value would be 0.9 repeating (if you let 1 represent infinity). 

I don't know that much about asymptotes yet, but I'm comparing it to an infinite series, because he says that the function's graph forever approaches the number that is the value of the asymptote, but effectively never reaches it. The infinite series $1/2 + 1/4 + 1/8+\cdots$ forever approaches $1$, and you could say that it effectively never does, but even still the answer is exactly $1$. Is there a better argument against the asymptote argument?

EDIT: I know about the many valid proofs for this, but my problem is that they all refuse to accept those proofs. They're throwing invalid arguments my way, and since they reject mine, I'm stuck having to disprove theirs. I know those arguments can be disproved, just not how, and my question is how to disprove this one. Just clarifying.
$1/2 + 1/4 + 1/8 + \ldots$ is just a number. It doesn't "approach" anything: it simply is $1$.

It is the sequence of partial sums that approaches $1$: that is, the list of numbers


$1/2 = 1/2$
$3/4 = 1/2 + 1/4$
$7/8 = 1/2 + 1/4+ 1/8$
$15/16 = 1/2 + 1/4+ 1/8 + 1/16$
...


is a sequence of numbers whose limit is $1$.

$1/2 + 1/4 + 1/8 + \ldots$ is not this sequence: it is the number that is equal to the limit of this sequence.
$0.9$ repeating is equivalent to $1$ because the difference of them is $0.0000...1$ or $0.1$ to the power of infinity, but if there is an infinite amount of zeros there's no room for the one. You might say that it will always have enough room for one because it's infinite, but if you can add a one you can add a zero, making the number invalid for the difference, so if there can't be a one at the end it means it can't exist, therefore saying that there is no difference between $0.9$ repeating and $1$ because the difference between them cannot exist.