Question on complete discrete valuation field Let $K$ ba a complete discrete valuation field. Let $A$ be a complete set of representatives of the residue field (The ring od integers over the maximum ideal) containing $0$. Then how to show the standard fact that any element $x$ in $K$ can be uniquely written in the form $\sum_{i=m}^{\infty}a_it^i$, where $t$ is the uniformising parameter of $K$, and $a_i\in A.$
 A: Let $x \in K$ and, say, $m=v(x)$ where $v$ stands for the valuation, then $x=t^mu$ where $u$ is a unity in the ring of the valuation $\cal O$. Take $a_0$ the unique representant of $u$ in $A$, thus $u = a_0+tb_1$ for some $b_1 \in \cal O$ and $b_1$ is uniquely determined by this equation, proceeding in the same way we can find others representants $a_1,\dots,a_n \in A$ such that $u=a_0 + a_1t + a_2t^2 +\dots+a_nt^n + t^{n+1}b_{n+1}$ for some $b_{n+1} \in \cal O$. The point is that the $a_i$ are all uniquely determined by this equation, so the representant $a_{n+1}$ of $b_{n+1}$ is uniquely determined by $u$ and satisfy $b_{n+1} = a_{n+1} + b_{n+2}t$. Plug in the equation above and proceed to obtain an infinite series, $u=\sum a_it^i$, this converges since the term $t^{n+1}b_{n+1}$ tends to $0$. Now plug it into $x=ut^m$.
A: For any given $n>0$ you can write
$$
x=\sum_{i=0}^{n-1}a_it^i+y_n
$$
where $y_n\in{\cal O}t^n$ and note that $a_i$ does not depend on $n>i$. This can be achieved recursively on $n$ considering quotients $\cal O/\cal O t^n$ for bigger and bigger $n$. Then you conclude observing that $\bigcap_{n>0}{\cal O}t^n=(0)$, i.e. the sequence $y_n$ has limit $0$.
