this post was submitted on 17 Jul 2023
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No, there is no difference. Infitesimal or otherwise. They are the same number, able to be shown mathematically in a number of ways.
Yes, thats what we're saying. No one said it's an infinitesimally small difference as in hyperbolically its there but really small. Like literally, if you start with 0.9 = 1-0.1, 0.99 = 1-0.01, 0.9... n nines ...9 = 1-0.1^n. You'll start to approach one, and the difference with one would be 0.1^n correct? So if you make that difference infinitely small (infinite: to an infinite extent or amount): lim n -> inf of 0.1^n = 0. And therefore 0.999... = lim n -> inf of 1-0.1^n = 1-0 = 1.
I think it's a good way to rationalize, why 0.999... is THE SAME as 1. The more 9s you add, the smaller the difference, at infinite nines, you'll have an infinitely small difference which is the same as no difference at all. It's the literal proof, idk how to make it more clear. I think you're confusing infinitely and infinitesimally which are not at all the same.
Technically you're both right as there are no infinitesimals in the real number system, which is also one of the easiest ways to explain why this is true.