TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Cauchy criterion for improper integrals

Statement

The integral $\int_a^\infty f$ converges if and only if, for every $\varepsilon>0$ , there is $A>a$ such that $A\le u<v\quad\Longrightarrow\quad\left|\int_u^v f\right|<\varepsilon.$ At a finite right singular endpoint $b$ , replace the condition by $b-\delta<u<v<b$ ; at a finite left endpoint use $a<u<v<a+\delta$ ; at $-\infty$ use $u<v\le-A$ . In each case all displayed proper integrals must exist.

Facts & Assumptions

Given: A locally Riemann-integrable $f$ on the relevant one-ended interval.

[L1]

Adjacent proper integrals add (For $a<c<b$ : $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$ , and then $\int_a^b f = \int_a^c f + \int_c^b f$ ; with the oriented form for arbitrary $a,b,c$ ).

[L2]

A convergent real sequence is Cauchy, and every Cauchy real sequence converges (Every convergent sequence is Cauchy, The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges).

[L3]

The Archimedean property supplies integer truncations beyond every real bound and reciprocal truncations inside every positive neighborhood (Every complete ordered field is Archimedean, For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ).

Proof

technique · direct

1.1

Put $F(R)=\int_a^R f$ . If $F(R)$ has a finite limit as $R\to\infty$ , then $|F(v)-F(u)|<\varepsilon$ for all sufficiently large $u,v$ . By [L1], this difference is $\int_u^v f$ , proving necessity.

1.2

Conversely, the tail condition and [L3] make the sequence $F(n)$ Cauchy, hence convergent to some $L$ by [L2]. Given $\varepsilon>0$ , choose a large integer $n$ for which $|F(n)-L|<\varepsilon/2$ and the tail condition is below $\varepsilon/2$ . For every real $R\ge n$ , [L1] gives $F(R)-L=(F(R)-F(n))+(F(n)-L)$ , so $F(R)\to L$ .

L1L2L3

2.1

For a finite endpoint use the reciprocal sequence $a+1/n$ or $b-1/n$ furnished by [L3]; the identical Cauchy argument applies. Reversing the real line gives the $-\infty$ form.

L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 101 results over 28 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

William F. Trench, Introduction to Real Analysis, Section 3.4 (standard reference, not scraped)