I've been asked to show that a particular metric is actually an ultrametric. I'm a little lost, and I'll post what I have.

Here's the metric we're supposed to look at:

Let $\displaystyle X$ be any nonempty set. Define a metric $\displaystyle \mu : X \times X \to \mathbb{R}$ by letting $\displaystyle \mu (x,x) = 0$ and $\displaystyle \mu (x,y) = \mu (y,z)$ be any number in the interval $\displaystyle [1,2]$ when $\displaystyle x \neq y$.

And the problem is to show that this is an ultrametric, or that $\displaystyle \mu (x,z) \leq Max\[\mu (x,y), \mu (y,z)\]$ for all $\displaystyle x,y,z \in X$.

Here's my attempt at the proof:

First, since $\displaystyle \mu$ is a metric, we are given that $\displaystyle \mu (x,z) \leq \mu (x,y) + \mu (y,z)$. If $\displaystyle x = z$, then $\displaystyle \mu (x,z) = 0 \leq Max\[\mu (x,y), \mu (y,z)\]$. If $\displaystyle x \neq z$ and $\displaystyle x = y$ OR $\displaystyle y = z$, then

$\displaystyle \mu (x,z) \leq \mu (x,y) + \mu (y,z) = \mu (x,y) + 0 = Max\[\mu (x,y),\mu (y,z)\]$

or

$\displaystyle \mu (x,z) \leq \mu (x,y) + \mu (y,z) = 0 + \mu (y,z) = Max\[\mu (x,y), \mu (y,z)\]$

The only remaining case is when $\displaystyle x\neq y\neq z$.

I can't figure out the solution for this. It may be that I am taking the wrong approach. Any help is greatly appreciated.