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  "body": "\n## Statement\n\nLet $f, g \\colon [a, \\infty) \\to \\mathbb{R}$ be Riemann integrable on each $[a, b]$, with $0 \\le f(x) \\le g(x)$ for all $x \\ge a$.\nIf $\\int_a^\\infty g(x) \\, dx$ converges, then $\\int_a^\\infty f(x) \\, dx$ converges, and\n$$\\int_a^\\infty f(x) \\, dx \\le \\int_a^\\infty g(x) \\, dx.$$\nIf $\\int_a^\\infty f(x) \\, dx$ diverges, then $\\int_a^\\infty g(x) \\, dx$ diverges.\n\n## Remarks\n\nThe function $b \\mapsto \\int_a^b f$ is increasing, and it is bounded by $\\int_a^\\infty g$ by [[monotonicity|0MBWQKM8QR]]. A [[bounded monotone function has a limit|0KJXHN0TW0]].\nFor example, $0 \\le e^{-x^2} \\le e^{-x}$ for $x \\ge 1$. So $\\int_1^\\infty e^{-x^2} \\, dx$ converges.\nThe same test holds for integrals that are improper at a finite endpoint.\n\n## Explanations\n\n### L1 human\n\nIf a larger function has finite area, a smaller nonnegative function also has finite area.\nThis lets us prove convergence without computing the integral.\n",
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