## Ultrametric

An ultrametric is a Metric which satisfies the following strengthened version of the Triangle Inequality,

for all . At least two of , , and are the same.

Let be a Set, and let (where N is the Set of Natural Numbers) denote the collection of sequences of elements of (i.e., all the possible sequences , , , ...). For sequences , , let be the number of initial places where the sequences agree, i.e., , , ..., , but . Take if . Then defining gives an ultrametric.

The p-adic Number metric is another example of an ultrametric.