I took some liberties with the word "the", but I'm pretty sure that you know that it's easy to construct an ordered sequence of all rational numbers using any of a number of standard diagonalization arguments. I'll skip the details you already know.
Eventually, yes, you're going to be able to poke a hole in my argument. It is definitely flawed. I don't know how long it'd take us to get there, but it doesn't really matter. But we're already at the point where this cannot be considered "basic", and that is my true point here.
Attempting to demonstrate that 0.999... = 1 while meticulously avoiding any rigorous definition of what 0.999... means is not very easy and will require you to fend off all sorts of potential jabs from various directions. It's much easier to just talk about infinite sums and be done with it.
