quality:correction
Metric space

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

for all points A, B, and C in M. That is, the distance from A to C is at most as large as the sum of the distance from A to B and the distance from B to C.

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 x and x such that d (x, x) < ε /2 and d (x, x) < ε /2, where ε > 0 is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, d (x, x) ≤ d (x, x) + d (x, x) < ε /2 + ε /2 = ε, so that the sequence { x } is a Cauchy sequence, by definition.

This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via d (u, v) ≔ ‖ u − v ‖, with u − v being the vector pointing from point v to u.