= Cauchy criterion for an improper integral
{c}
{title2=$\int_a^\infty f\text{ converges}\iff\forall\varepsilon>0\ \exists R\ \forall v>u>R:\ |\int_u^vf|<\varepsilon$}
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.
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