Positive even integers dense in the p-adic integers

Question

On page 22 of the book p-adic Numbers, p-adic Analysis, and Zeta-Functions by Koblitz, it says that the set $S$ of positive even integers is "dense" in $\mathbb{Z}_p$ when $p>2$.

This seems very strange to me. Take for example $3\in \mathbb{Z}_3$. How is this the limit of a sequence of positive even integers?

Answer 1

For the general case:

Let $\sum_{j\ge0}a_jp^j\in \Bbb Z_p$. Let $b_n=\sum_{j=0}^na_jp^j$. Consider the sequence given by $$ c_n = \begin{cases}b_n & b_n\textrm{ even} \\ b_n + p^{n+1} & b_n\textrm{ odd} \end{cases} $$ .