Can someone help me in this exercise?
Show that $Z_{p}=\varprojlim_{n\in \mathbb{N}}\mathbb{Z}/p^{n}\mathbb{Z}$ can also be identified with the set of power series
$$Z_{p}= \{ b=\sum_{n=0}^{\infty}b_{n}p^{n} \ | \ b_{n}\in \mathbb{N}, \ 0\leq b_{n}\leq p \}$$
Explain how the addition and multiplication of series is carried out under this identification. How is $\mathbb{Z}$ embedded in $Z_{p}$ under this identification?
Thank you!
This is a classical exercise on inverse limits. Recall that $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$ is a subgroup of the product $\prod_{n\in \mathbb{N}}\mathbb{Z}/p^n \mathbb{Z}$. Specifically, $$ \varprojlim \mathbb{Z}/p^n\mathbb{Z} = \{(a_n)_n \mid a_n \equiv a_m (\text{mod }p^n) \text{ for } n\geq m \} $$ (check this using the UMP of the inverse limit)
Thus, for every sequence $(a_n)_n$ in $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$ we have: $a_1 \in \mathbb{Z}/p\mathbb{Z}$, then $a_1 \equiv a_2 (\text{ mod }p)$; thus, $a_2 = a_1 + k_1p$ for $k \in \mathbb{Z}/p\mathbb{Z}$. Analogously, $a_3 = a_1 + k_1p + k_2p^2$. We have then, $a_n = a_1 + k_1p + \ldots + k_{n-1}p^{n-1}$. Therefore, the sequence $(a_n)_n$ is just sequence of partial sums of elements in $Z_p$. This is the identification you're looking for.