The p-adic completion is a complete valued field

By the specific construction fixed in [L3] and supplied by [L1], an element of ${\mathbb{Q}}_p$ is represented by a $d_p$-Cauchy sequence $(x_n)$ in $\mathbb{Q}$, and the distance between classes is ${\widehat{d}}_p([x],[y])={\lim }_n\mathrm{\mid }{x}_n-y_n{\mathrm{\mid }}_p\mathrm{.}$ Define $[x]+[y]:=[x_n+y_n],[x][y]:=[x_n{y}_n],\mathrm{\mid }[x]{\mathrm{\mid }}_p:={\lim }_n\mathrm{\mid }{x}_n{\mathrm{\mid }}_p\mathrm{.}$

The nonarchimedean inequality in [L2] shows that sums and differences of Cauchy sequences are Cauchy. A Cauchy sequence in a nonarchimedean metric is bounded, so products of Cauchy sequences are again Cauchy, and equivalent representatives give equivalent sums and products because $x_n{y}_n-x_n^{\mathrm{\prime }}{y}_n^{\mathrm{\prime }}=x_n(y_n-y_n^{\mathrm{\prime }})+y_n^{\mathrm{\prime }}(x_n-x_n^{\mathrm{\prime }})\mathrm{.}$ Passing to the limit through [L2] proves representative independence of the extended absolute value as well.

Let $[x]\in {\mathbb{Q}}_p$ be nonzero. Then $\mathrm{\mid }[x]{\mathrm{\mid }}_p>0$, and step 1.1 says $\mathrm{\mid }{x}_n{\mathrm{\mid }}_p\to \mathrm{\mid }[x]{\mathrm{\mid }}_p$ in $\mathbb{R}$. So there are a real constant $c>0$ and an index $N$ with $\mathrm{\mid }{x}_n{\mathrm{\mid }}_p\ge c$ for all $n\ge N$. In particular $x_n\mathrm{\ne }0$ eventually. For $n,m\ge N$, $\mathrm{\mid }{x}_n^{-1}-x_m^{-1}{\mathrm{\mid }}_p=\frac{\mathrm{\mid }{x}_n-x_m{\mathrm{\mid }}_p}{\mathrm{\mid }{x}_n{\mathrm{\mid }}_p\mathrm{\mid }{x}_m{\mathrm{\mid }}_p}\le c^{-2}\mathrm{\mid }{x}_n-x_m{\mathrm{\mid }}_p,$ and the right-hand side tends to $0$ because $(x_n)$ is Cauchy. Thus $(x_n^{-1})$ is eventually defined and Cauchy, so every nonzero class has an inverse.

Multiplicativity and the strong triangle inequality on ${\mathbb{Q}}_p$ follow by taking limits of the corresponding rational identities from [L2]. Completeness is already part of [L1]. Thus ${\mathbb{Q}}_p$ is a complete nonarchimedean valued field.