This question was left as an exercise in a seminar of p-adic integers.
The issue is that I have not studied any algebraic number theory and hence I am not able to think about how to prove this particular result. But I want to learn basics of p-adic number theory and hence want to know how to solve this particular problem.
Question: In $\mathbb{Q}_p$ , show that $(x_n)$ is Cauchy sequence iff $|x_{n+1} - x_{n}|_p \to 0$.
I found a solution here:If $(x_n)_{n\geq 1}$, $x_n \in \mathbb{Q}$ is p-adic Cauchy, show ord$_p(x_n)$ eventually constant.
But in the answer I am having these questions and the OP of question is not seen since atleast 1 month ago and OP of answer has deleted the account:
Assuming $ (x_n)$ is cauchy , then in the answer I have question that :(i) why can't the sequence $(x_n)$converge to 0?
(ii) what is non-acrhemedian property and how does it implies that $|x_m|= |x_n|$? Rest of the answer is clear to me.
Conversly, I am not able to prove that $(x_n)$ is cauchy assuming that $|x_{n+1} - x_{n}|_p| \to 0$ ?
Can you please help me with this?
Thanks!