 By applying the Cauchy criterion for the convergence of a complex series, we can also prove this fact as a simple implication of the triangle inequality.By the Cauchy criterion, formula_46 converges if and only if for any formula_47, there exists formula_48 such that formula_49 for any formula_50.But the triangle inequality implies that formula_51, so that formula_52 for any formula_50, which is exactly the Cauchy criterion for formula_54.