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 be any nonempty set. Define a metric by letting and be any number in the interval when .
And the problem is to show that this is an ultrametric, or that for all .
Here's my attempt at the proof:
First, since is a metric, we are given that . If , then . If and OR , then
The only remaining case is when .
I can't figure out the solution for this. It may be that I am taking the wrong approach. Any help is greatly appreciated.