> If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious.
How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
They have to know that 0.999... means you never stop writing nines.
Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
> They have to know that 0.999... means you never stop writing nines.
How do they know that that's a real number?
They don't have to know that it's a real number. Knowing that you never stop writing nines is sufficient to perform the calculation.
You're asking them to perform subtraction. They probably know how to do that with real numbers, but problably not with much else. So they'll have to know that they're real numbers (or whatever numbers you are demanding that they be – you're still unclear on this point if it's not actually the reals).
Internally I'm saying they're real numbers. In what I say to the person trying to intuit that 0.999... = 1, I'm deliberately avoiding talking about number systems. I'm assuming this person thinks of numbers as sequences of digits, possibly with a decimal point.
I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...