Equivalence of alternative Definitions of $\mathbb{Q}_p$ In the literature there are three common definitions for the p-adic integers $\mathbb{Z}_p$ which are isomorphic. I refer to the definitions as inverse limit $\underset{\leftarrow}{\lim}{\mathbb{Z}}\diagup{p^n}$, completion of $\mathbb{Z}$ via $p$-adic metric and through power series/$p$-adic expansions.
Then $\mathbb{Q}_p$ is respectively defined as quotient field of the inverse limit, completion of $\mathbb{Q}$ w.r.t. $p$-adic metric and field of laurent series.
However I can't really find any sources giving the Isomorphisms between the different definitions of $\mathbb{Q}_p$. I suspect one can use the isomorphisms for $\mathbb{Z}_p$ and somehow expand them to $\mathbb{Q}_p$ but it's not clear to me how it is done. When trying myself especially the power series version gives me a lot of trouble. Does anyone know the proofs and wants to outline them for me or have some source in which this matter is proven?
I would be sincerely grateful.
 A: Two definitions of the topological ring $\mathbb Z_p$:
(1) The topological ring $\varprojlim \mathbb Z/p^n\mathbb Z$.
(2) The metric space completion of the ring $\mathbb Z$ with respect to the $p$-adic metric.
To prove this, one needs to show that for the ring $\mathbb Z_p$ in the sense of (2), every element can be expressed as
$$x = \sum\limits_{n=0}^{\infty} a_n p^n$$
for unique elements $a_n \in \{0, 1, ... , p-1\}, n = 0, 1, 2, ...$  An isomorphism of this ring with $\varprojlim \mathbb Z/p^n\mathbb Z$ is given by sending such $x$ to the sequence
$$x \mapsto (a_0, a_0 + a_1p, a_0 + a_1p + a_2p^2, ...).$$
Having established the equivalence of these, one can then define $\mathbb Q_p$ as:
(1) The quotient field of $\mathbb Z_p$, giving it the direct limit topology as the union of increasing additive subgroups $p^{-n}\mathbb Z_p, n = 1, 2, ...$.  Each group $p^{-n}\mathbb Z_p$ is given the topology  induced from the additive isomorphism $p^{-n}\mathbb Z_p \rightarrow \mathbb Z_p, x \mapsto p^nx$.  Not difficult to define the $p$-adic metric on this field and show it is complete.
(2) The metric space completion of $\mathbb Q$ with respect to the $p$-adic metric.
To do this, one should show that every element of $\mathbb Q_p$ in the sense of (2) can be expressed as
$$x = \sum\limits_{n=m} a_n p^n \tag{1}$$
for unique $m \in \mathbb Z, a_n \in \{0, 1, ... , p-1\}, a_m \neq 0$.
