In Serre's course in arithmetic, $p$-adic integers are defined as follows: let $A_n:=\mathbb{Z}/p^n\mathbb{Z}$. There are natural (ring) homomorphisms $\phi_n:A_n\rightarrow A_{n-1}$. The ring of $p$-adic integers is the inverse limit of the system $(A_n,\phi_n)$. Then the author says
By definition, a $p$-adic integer is a sequence $(\cdots, x_3,x_2,x_1)$ with $x_i\in A_i$ and $\phi_n(x_n)=x_{n-1}$.
On the other hand, the formally written expression for a $p$-adic integer is of the form $a_0 + a_1 p + a_2p^2 + \cdots $ with $0\le a_i<p$. Here, $a_i$'s are independent, whereas, in the definition of Serre, the coordinates $x_i$ are successively dependent through homomorphisms $\phi_i$'s.
I do not understand how to write a formally written $p$-adic integer as a sequence expressed by Serre?