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  "body": "\n## Statement\n\nLet $a$ be a real number, let $n$ be a natural number, and let $f \\colon [a, a + n] \\to \\mathbb{R}$ be a decreasing function.\nThen\n$$\\sum_{i=1}^{n} f(a + i) \\le \\int_a^{a+n} f(x)\\, dx \\le \\sum_{i=0}^{n-1} f(a + i).$$\n\n## Remarks\n\nOn each interval $[a + i, a + i + 1]$ we have $f(a + i + 1) \\le f(x) \\le f(a + i)$. Integrate and add over $i$.\nThe integral exists because monotone functions are [[Riemann integrable|04YKAYDQPW]].\nWith $f(x) = 1/x$ and $a = 1$ it gives $\\ln(n + 1) \\le H_n \\le 1 + \\ln n$ for the [[harmonic numbers|0D2Y6G4C4S]].\nMathlib states the first inequality as `AntitoneOn.sum_le_integral`.\n\n## Explanations\n\n### L1 human\n\nThe sum of the values at the right ends of unit intervals is an area of rectangles below the graph of a decreasing function.\nThe sum of the values at the left ends is an area of rectangles above it.\n",
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