In practice, in p-adic analysis, when referring to a sequence of numbers it is common to use the terminology "converges to 0". However, isn't this terminology, technically, incorrect by the definition of the p-adic order since we extend the domain of the p-adic order to 0 by defining v(0)=∞ so that |0|=0? In consideration of the definition of the p-adic norm, |x|=p^(-v(x)), should the correct terminology not be, in fact, "diverges to 0", analogous to the notion of an infinite product diverging to 0 (inversely, of course)?
I admit that I may be nitpicking here, and it is very possible that I am simply incorrect or overthinking. But, this terminology does imply subtle differences. In particular, while in real analysis, it is very natural to define an elementary function by a sum or product, in p-adic analysis, the terminology seems to suggest that it is more natural to define an elementary function by a product, for example, the Artin-Hasse exponential function.
EDIT: I would like to point out, before others do, that we also technically extend the real absolute value by defining |0|=0 also. However, this actually reinforces my original post rather than contradicting it. What this says, or at least, what I infer from this, is that it is technically always more correct to use the terminology "diverges to 0" as opposed to "converges to 0". It is only in the real case, in other words, the case for the p-adic absolute value || for arbitrary p, under the limit as p approaches the least infinite ordinal, that the distinction between the two notions becomes ambiguous. The ambiguity arises because the real line is connected, so that, in special circumstances like the one considered (where the extension is in the topological boundary of the domain), extension of the domain of definition of a function becomes continuation, while restricting our consideration to only positive numbers in the domain, the absolute value becomes the identity map.
EDIT 2: I expected this to be obvious from the question, but do note that the question is in fact a question of consistency. Although, at face value it may appear to be somewhat shallow, it is actually very deep, in the sense that it has very deep consequences for an individual's conceptualization of the p-adic numbers, and in particular how the aforementioned individual uses them as a mathematical tool.
I do apologize for not writing in latex, by the way, I currently only have my surface with me, and doing so on it is an extremely tedious task.
For convergence, we need a topology, here obtained from a metric, here obtained from a norm, here defined in terms of the order: $|x|_p=\begin{cases}p^{-v(x)}&x\ne0\\0&x=0\end{cases}$. After this, convergence has its standard meaning from topology and $0$ is really "very near" to $p^{1000}$.