Cauchy criterion for an improper integral (${\int }_a^{\infty }f\text{ converges}⟺\forall \epsilon >0\exists R\forall v>u>R:|{\int }_u^{v}f|<\epsilon$)

For a locally integrable real or complex function, the displayed tail condition is equivalent to ordinary convergence of its improper integral at infinity. Apply the Cauchy criterion for a function limit to its truncated integrals, using completeness of the real or complex numbers. The necessity follows by subtracting two nearby-to-limit truncations; sufficiency follows by first taking a convergent sequence of truncations and then bounding all other late truncations against it. The analogous criterion applies at a finite singular endpoint. Bounding the modulus of every tail by its absolute-value integral proves that an absolutely convergent improper integral converges ordinarily.