The Cauchy condition for an infinite series Σ a_n on a compact set is:

For all ε > 0, there exists an N such that for all m, n that satisfies m ≥ n > N,

| a_n + a_n+1 + … + a_m| < ε

The series is said to be uniformly Cauchy if it satisfies the Cauchy condition.

A series of complex numbers is convergent if and only if it is uniformly Cauchy.