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- October 15th 2013, 02:29 PM #1
## Ultrametrics

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 .

or

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

- October 17th 2013, 12:48 PM #2

- October 17th 2013, 01:31 PM #3

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- October 17th 2013, 02:09 PM #4
## Re: Ultrametrics

I really appreciate your response. I'm just wondering how the definition of the metric implies that they are all equal... I looked at all of the properties of the metric that are in the definition I was given, and none of them outright imply this. The closest I can see is the property that for all , but that doesn't lead me straight to where you were.

- October 17th 2013, 02:21 PM #5

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- October 17th 2013, 02:23 PM #6

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- October 17th 2013, 03:09 PM #7

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- October 17th 2013, 09:59 PM #8