We can show that the repeating decimal .999… is equal to 1.0 with only a couple of legal moves:
If x = .999…, then 10x = 9.999…
Now subtract x from 10x:
You get 9x = 9.0 and thus x = 1
So the repeating decimal .999… equals 1.0!
Is there some number that’s larger than .999… but smaller than 1.0?
Such a number would involve an infinitesimal, meaning a literally infinitely small mathematical entity.
— From Everything and More by David Foster Wallace