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.
How do they know that that's a real number?