 Cauchy formulated such a condition by requiring formula_5 to be infinitesimal for every pair of infinite "m", "n".If formula_6 is a sequence in the set formula_7, then a "modulus of Cauchy convergence" for the sequence is a function formula_8 from the set of natural numbers to itself, such that formula_9.Any sequence with a modulus of Cauchy convergence is a Cauchy sequence.