A way to illustrate 0.999…=1 occurred to me the other day that I haven’t seen before, although I’m sure it’s not original.
Namely, the purported difference between them, if there was one, would be 0.000…1. But that “…” represents infinitely many zeroes! No matter how long you wait for it, the 1 “at the end” never comes: there is simply no end to the zeroes.
Contrast with other numbers that are intractable in seemingly the same way, e.g. π. If you start writing down 4−π, you get non-zero digits right away: 0.8584073…. There is a material difference between 4 and π.
But with 1−0.999… you can write down digits from now till all eternity and you will never get anything more than a string of zeroes. No matter how far down the rabbit hole you chase that seeming difference, you’ll never find it.
Because: it isn’t there. There is no difference. They are the same.