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An ultrametric is a Metric which satisfies the following strengthened version of the Triangle Inequality,
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.
See also Metric, p-adic Number