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.
