There are no integers between two integers whose difference is 1. However there are always a real number between two different real numbers. That's how you tell numbers apart.

If 0.999... and 1 were different numbers I could subtract 0.999... from 1 and I would get a number. What is this number? It must exist if the two numbers were different.

Interesting how there's no such thing as a smallest number. I guess that Planck's constant and Planck-time and other Planck stuff do not apply to number theory :)