Triangle Inequality - Metric Space

Metric Space

In a metric space M with metric d, the triangle inequality is a requirement upon distance:

for all x, y, z in M. That is, the distance from x to z is at most as large as the sum of the distance from x to y and the distance from y to z.

The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any and such that and, where is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, so that the sequence is a Cauchy sequence, by definition.