In the book I'm following the order of a formal real power series $a=\sum_{n=1}^{\infty} a_nx^n$ is defined as: $$o(a)=\begin{cases} \infty, \, \, a=0\\ \min\{n:a_n \neq 0\}, \, \, a \neq 0 \end{cases}$$ I'm asked to prove that $$d(a,b)=2^{-o(a-b)}$$ is a metric in the space of formal power series. I've managed to prove that $d(a,b)=0$ iff $a=b$ and $d(a,b)=d(b,a)$ easily, but now I'm struggling with the triangle inequality.
How should I procede?
A rather big hint is that $d$ is actually an ultrametric meaning
$$d(a,c)\le \max(d(a,b),d(b,c))$$
This implies the triangle inequality, and is true of your metric.