There are many ways to see that $0.999\ldots=1$ over the Reals (or over $\Bbb Q$ or $\Bbb C$ for that matter) like "Is it true that 0.99999…=1?", and the reasoning is easy enough:

If $x=0.\bar9$ then $10x=9.\bar{9}$ and consequenty, $9x=9.0$.

Some time ago I watched a math video which explained that the equality is not true everywhere, without mentioning what would break down, and no specific example was given, but what can go wrong, what are explicit examples?

What comes to mind are fields like the $p$-adics, the Superreals and Hyperreals. While I can follow their basic definitions or popular introductions, I have absolutely no intuition, let alone knowledge, what makes them different or not.

To make sense of $$s_n:=9\sum_{k=1}^n 10^{-k}$$

it's enough to have the arithmetic of a field. To make sense of

$$0.\bar 9 := \lim_{n\to\infty} s_n$$

dunno what's needed here. A metric perhaps like $\operatorname{abs}:x\mapsto |x|$. The sequence should converge and the concept of convergence should make sense of course, and the limit should be $1$. Topologically closed is maybe too strong, but if the field is also a complete metric space that should suffice?

What I don't know is whether the above structures match (except for $\bar{\Bbb Q}_p$, which is however becond by comprehension / intuition).

And suppose the series $s_n$ from above converges to an element $0.\bar 9$ of the domain. Does this imply that $0.\bar 9 -1$ is an infinitesimal?

Maybe someone can iterate on these structures or on other domains that are worth talking about. And is it a requirement that $10^{-n}\to0$? Or may it be the case that $10^{-n}$ need not to become arbitrarily small and $0.\bar9$ still makes sense? What about Duals and split-complex?

**Note: My questions are not about uniqueness of representation.**

## Best Answer

I think this might help answer some of the spirit of the question, I try to give some of the reasoning in the p-adic cases which ends up having the sequence not converge in two different "flavors". I think learning more about the definition of a limit and Cauchy sequences might help you here.

Ostrowski's theorem says that every absolute value we can put on the rationals is either the real, a p-adic, or trivial absolute value. This is not entirely comprehensive, but a good start for seeing where your sequence can be seen to be valid. From here we can build up the real or any of the p-adic fields by completion with respect to this absolute value.

Specifically in the p-adic cases, we define the p-adic absolute value on rational numbers by pulling out the largest power of the prime. More precisely for $a,b, n \in \mathbb{Z}$ with $a,b$ not divisible by a prime number $p$, we can represent any rational number $x=p^n\frac{a}{b}$ and define its p-adic absolute value to be:

$$\left|p^n\frac{a}{b}\right|_p = \frac{1}{p^n}$$

Here if we take $.999...$ as a limit of the sequence $\frac{9}{10}, \frac{9}{10}+\frac{9}{100}, \frac{9}{10}+\frac{9}{100}+\frac{9}{1000}, ... $ we end up in either one of two situations.

In the 2-adics or the 5-adics, the sequence can't converge because we are adding on larger and larger terms to our sequence since their absolute values diverge as $n \to \infty$: $\left|\frac{9}{10^n}\right|_2 = 2^n$ and $\left|\frac{9}{10^n}\right|_5 = 5^n$

In the other p-adic fields we have that it doesn't converge because the sequence of terms never decays to $0$ but is still bounded above and infinitely "oscillates" around, very similar to Grandi's series. To directly see this, we can simplify the geometric series of partial sums to $9\frac{(10^{-1})^n-1}{10^{-1}-1}$ and focus on the nonconstant term for which infinitely many values, $(10^{-1})^n-1 \not \equiv 0 \mod p$.