complete ultrametric field

A field $K$ equipped with a non-archimedean valuation  $|\cdot |$  is called a non-archimedean field or also an ultrametric field, since the valuation the ultrametric  $d(x,y):=|x-y|$  of $K$.

Theorem.

Let $(K,d)$ be a complete (http://planetmath.org/Complete) ultrametric field.  A necessary and sufficient condition for the convergence of the series

$a_1+a_2+a_3+\mathrm{\dots }$(1)

in $K$ is that

$\underset{n\to \mathrm{\infty }}{lim}{a}_n=\mathrm{\hspace{0.33em}0}.$(2)

Proof.  Let $\epsilon$ be any positive number.  When (1) converges, it satisfies the Cauchy condition and therefore exists a number $m_{\epsilon }$ such that surely

for all  $m\geqq m_{\epsilon }$;  thus (2) is necessary.  On the contrary, suppose the validity of (2).  Now one may determine such a great number $n_{\epsilon }$ that

No matter how great is the natural number $n$, the ultrametric then guarantees the inequality

always when  $m\geqq n_{\epsilon }$.  Thus the partial sums of (1) form a Cauchy sequence, which converges in the complete field.  Hence the series (1) converges, and (2) is sufficient.