Let $p$ be prime. I am tasked to prove that the sequence $10^{-n}$ does not converge in $\mathbb{Q}_{p}$ for any $p$ where $\mathbb{Q}_{p}$ is the set of p-adic numbers.
For $p=2$ or $5$, we see that $|10^{-n}|_{p}$ goes to infinity so it can't be Cauchy. For $p\neq 2$ or $5$ i'm really confused here. We want to compute $|10^{-n}-10^{-m}|_{p}$. With the nonarchimedian norm we have $ |10^{-n-m}-1|_{p}\leq 1$. I don't really know where to proceed from here. Any hint would be very nice. Thank you.
Hint: Let $f(x)=x/10$. Then if there is a limit $x_{\infty}$ to this sequence, $x_0=1, x_{n+1}=f(x_n)$ then the limit must satisfy $f(x_{\infty})=x_{\infty}$.