How can every $p$-adic integer be the limit of a sequence of non-negative integers?

Question

See Andrew Baker's P-adic Notes. Every element of $\mathbb{Z}_p = \{a \in \mathbb{Q}_p : |a|_p \leq 1 \}$ is a limit of a sequence of non-negative integers, with respect to the $|\cdot|_p$ norm. How is this possible?

Answer 1

Hint: You can think of $\mathbb{Z}_p$ as being sequences $\displaystyle (a_n)\in\prod_n (\mathbb{Z}/p^n\mathbb{Z})$ where $a_n\equiv a_m\text{ mod }p^m$ for $m\leqslant n$. The integers sit inside of here as the set of all such sequences that are eventually constant. Do you see how to approximate now? the ability to use non-negative integers comes from the fact that in every $\mathbb{Z}/p\mathbb{Z}$ the image of $x\in\mathbb{Z}_-$ is equal to the image of some $y\in\mathbb{Z}_{\geqslant 0}$.