I think people who disagree that 0.(9) = 1 on the grounds that infinity does not exist make the same logical fallacy as the ancient philosopher Zenon who described the so called Zenon's paradox:

"Achilles is much faster than a tortoise. He can travel a certain distance x in 9/10’s of a minute. The tortoise can cover the same distance in 9 minutes, meaning that the tortoise is 10 times slower than Achilles or Achilles is 10 times faster than the tortoise, whichever you prefer.

Now, suppose that the tortoise is that particular distance x in front of Achilles. If someone was to ask when Achilles would gain upon the tortoise, we would say:

-If the time it takes Achilles to overtake the tortoise is t, then he would travel a distance which is t/(9/10 min) compared to the distance x in the problem. The tortoise would cover a distance which is only t/(9 min) compared to the distance x given in the problem. But, if Achilles overtook the tortoise, this would mean that

t*x/(9/10 min) = x + t*x/(9 min)

10*t*x/(9 min) = x + t*x/(9 min) / * (9 min) <- Least common multiple of all the denominators

10*t*x = (9 min)*x + t*x / : x <- common multple

10*t = 9 min + x

9*t = 9 min / : 9

t = 1 min

Piece of cake! But, not quite. What if the guy who was solving the problem was ‘narcberry’? He would reason in the following manner:

- If Achilles is to gain on the tortoise he would first have to cross the distance that was initially between them. We know that he can cross this distance in 9/10 minute. However, during this time the tortoise would move a small distance, actually 1/10 (because it would cover 10 times smaller distance than Achilles did) of the initial distance, so Achilles still did not gain upon the tortoise. In order to gain upon the tortoise, Achilles needs 1/10*(9/10) = 9/10^{2} minute to cross this distance. But, the tortoise would move an even smaller distance during this time, actually, (1/10)^{2} of the initial distance. This is enough for him to “prove” that, since in every step the tortoise moves away by an ever decreasing, but still nonzero distance, Achilles would never gain upon the tortoise. Never meaning it would take him an infinite amount of time! But, little does ‘narcberry’ know that:

9/10 min + 9/10^{2} min + 9/10^{3} min + … + 9/10^{n} min + … =

= (9/10)/(1 – 1/10) min = 1 min

The sum of infinite terms on the left is called an infinite series. The Ancient Greeks, just as ‘narcberry’ had trouble acknowledging the fact that a sum of infinite terms can itself be finite. Notice that I intentionally chose the numbers so that the infinite series on the left exactly represents the number 0.(9) = 0.99….