Ultrametric example Can anybody give an example for ultrametric space? i.e., in the metric space definition, instead of triangle inequality, we have strong triangle inequality, namely $d(x,y) \leq \max \left\{d(x,z),d(z,y)\right\}$ for all $x,y,z$.
A trivial example is the discrete metric. I want some non trivial example. 

First observe that any nonzero rational number in $\mathbb Q$ has a unique factorization into prime numbers (allowing negative exponents of course). Next, let $p$ be a prime and for any $\frac{a}{b}\in\mathbb Q$ define $v_p\left(\frac{a}{b}\right)$ to be the exponent of the prime $p$ in the factorization of $\frac{a}{b}$, and $v_p(0)=\infty$. Now let $\epsilon<1$ be positive, and define $d_p(r,s) = \epsilon^{v_p(r-s)}$. 
This is known as the $p$-adic norm on $\mathbb Q$, you can check it is an ultrametric.
