Completeness of the formal power series and valued fields

[aalma]

TL;DR Summary

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..