Cauchy sequence
Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the well-ordering property of the natural numbers (let formula_20 be the smallest possible formula_21 in the definition of Cauchy sequence, taking formula_22 to be formula_23). The existence of a modulus also follows from the principle of dependent choice, which is a weak form of the axiom of choice, and it also follows from an even weaker condition called AC00. "Regular Cauchy sequences" are sequences with a given modulus of Cauchy convergence (usually formula_24 or formula_25). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice.