What is $\displaystyle\lim_{n\to\infty} p^{-n}$ in $\mathbb{Q}_p$?

Question

What is $\displaystyle\lim_{n\to\infty} p^{-n}$ in $\mathbb{Q}_p$?

I'm thinking it should be infinity or undefined as its $\frac{1}{0}$ or looking another way, it's an infinite distance from $0$ by $\lvert \cdot\rvert_p$.

But I have a hint of doubt because it's $0.000..._p=0$

Answer 1

Yes, $|p^{-n}|_p=p^n$ so the sequence is clearly unbounded in $\mathbb{Q}_p$ and has no limit.

Saying the limit is $0.000\cdots_p\,$ in $\mathbb{Q}_p$ is like saying $\lim\limits_{n\to\infty}10^n\,$ is $\cdots0000\,$ in $\mathbb{R}$.