So just to verify, you believe the rest of my proof, assuming this:

Now that, I believe, is a matter of definition. And it's a good point to disagree on. Let me explain my point of view on this...Code:`0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )`

I believe the definition of a limit is hidden inside the definition of "0.9999...". Okay, let's consider a decimal point followed by a billion 9s. That may be quite close to "0.9999..." for our considerations, but it's still some distance away from "0.9999...", namely all the nines after the billion 9s. So we can add ANOTHER 9. We get closer to the number, but we're still not there.

So we add more and more 9s, and we get closer and closer, but never actually get there. Or: as the number of 9s goes to infinity we get closer and closer to "0.9999...". That's exactly what I'm saying here:

But at the very least, even if you take another definition, to me this seems to prove that there is no number between "0.9999..." and 1: if there were, the limit there would be less than 1!Code:`0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )`

And one can always find a number between two numbers that aren't equal: their average, for example.

I think the real problem is that "0.9999..." doesn't actually exist in real numbers (and thus takes the properties of the closest number). I mean, a decimal point followed by a billion 9s exists and may be quite close. But two billion 9s is even closer, but still not there. But there isn't an actual real number with an INFINITE amount of 9s. It can get very, very high, as close to infinity as you want but never actually infinity.

That doesn't mean we can't say "0.9999..." is a number, and check what the results are. The properties that follow are exactly that of 1. And two numbers with the same properties, at least in real numbers, are equal.