How to prove: $|nx|\le |n|\cdot|x|$, for $x\in K$ and $n \in \mathbb Z$ ?
The absolute value here is a nonnegative function from a field $K$ to $\mathbb R$ and in the definition there's a point;
$|xy|=|x|\cdot |y|,\quad \forall x,y\in K$ ?
What is not working for my case ? Is $n$ not necessarily contained in $K$ ? Are $1$ and $-1$ always in $K$ ?
Can I prove it inductively using $|-1|=|1|=1$ (this is already known)