Graduate Completeness of the formal power series and valued fields

[aalma]

TL;DR

Trying to understand the completeness of K[[x]] and how to show that every cauchy sequence in it has a limit in K[[x]].

I had difficulty showing this no matter what I tried in (a) I am not getting it. Here for p(t) in K[[t]] : $|p|=e^{-v(p)}$ where v(p) is the minimal index with a non-zero coiefficient.
I said that p_i is a cauchy sequence so, for every epsilon>0 there exists a natural N such that for all i,j>N we have
$|p_i(t)-p_j(t)|<epsilon$, which is equivalent to that
$v(p_i(t)-p_j(t))>e^{-epsilon}$.
But could not see how it helps here!.

Any clarifications would be great

 

[Infrared]

In order for $||p_i-p_j||$ to be smaller than $\varepsilon=e^{-n}$ (for large $i,j$) it must be the case that the first nonzero term of $p_i-p_j$ has degree larger than $n$, i.e. the coefficients of $1,t,...,t^n$ are the same for $p_i$ and $p_j.$ So, for any fixed degree $k$, the coefficient of $t^k$ in your sequence is eventually constant and the limit of your Cauchy sequence is just the power series whose coefficient of $t^k$ is this element of $K$.

 

[aalma]

Thanks!
Here is what I did in a.
can you give direction for b? I think it would be similar to a but could not see how..