I can't understand this property, left unproved by my textbook as a trivial fact: let $K$ be a valued field, with valuation $\left|\phantom{x}\right|:K\longrightarrow\mathbb{R}$, let $\{a_n\}$ be a Cauchy sequence in $K$. Then $\{\left|a_n\right|\}$ is a Cauchy sequence in $\mathbb{R}$.
I think it suffices to prove that $\left| \left|a_n\right|-\left|a_m\right|\right|\leq\left|a_n-a_m\right|$, but I have no ideas.
You are absolutely right, and this equation is the reverse triangle inequality:
From $|x+y|\leq|x|+|y|$ we obtain by setting $x:=a-b$ and $y:=b$ $$ |a|\leq|a-b|+|b| $$ and hence $$ |a|-|b|\leq|a-b|. $$ Analogously we get by setting $x=a$ and $y:=b-a$ $$ |b|-|a|\leq|b-a|=|a-b|, $$ and, in combination, we get the desired inequality.