I'm reading a book about PDEs and it opens with a discussion of set theory and limit points and whatnot and this got me thinking about how we define distance (since it uses neighborhoods to talk about closed and open sets).
So we all agree that $\frac{3}{3}=1$, and $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{3}{3}=1$but $\frac{1}{3} = 0.333.....$ and so on to infinity, so the claim can be made that $0.333... + 0.333... + 0.333...=0.999...$
so we generally say that $0.999... = 1$
flipping this around, we can say $0.000...01 = 0$ where $0.000...01$ is an infinite number of zeroes with a 1 at the end. To me, to equate these two numbers is to say there is no distance between them.
If we add $0.000...01$ to both sides, we get $0.000...02 = 0.000...01$, but $0.000...01 = 0$ so we can then say that $0.000...02 = 0$ which reads to me that there is no distance between these two numbers, yet there technically is a number between them, so I feel like there has to be some distance there. So my question is, at what point do we define distance?
In reference to set theory, if we have the closed set of all numbers $x \le 0$, then $0$ is a limit point to a set, then technically $0.000...01$ would be a limit point and so would $0.000...02$, yet these are not within the set, so how is it closed?
This also shows me something else: if we carried on adding $0.000...01$ an infinite number of times to both sides, we would eventually reach $0.999...=1$, so we have "travelled" a total of one unit by adding $0$ an infinite amount of times. How does that work?
Maybe this is just a silly way that the concept of infinity works, but it just seems counterintuitive.