 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.This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via , with being the vector pointing from point to .