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 norm metric is another example of an ultrametric.