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  "body": "\n## Statement\n\nLet $H_n = \\sum_{k=1}^{n} \\frac{1}{k}$ be the [[harmonic numbers|0D2Y6G4C4S]].\nThen the limit $\\lim_{n \\to \\infty} (H_n - \\ln n)$ exists, and it lies in $[0, 1]$.\nIn particular, $H_n \\sim \\ln n$ as $n \\to \\infty$.\n\n## Remarks\n\nLet $a_n = H_n - \\ln n$. Since $\\ln(n + 1) - \\ln n = \\int_n^{n+1} \\frac{dt}{t}$ lies between $\\frac{1}{n+1}$ and $\\frac{1}{n}$, the sequence $a_n$ decreases. Also $a_n \\ge H_n - \\ln(n + 1) \\ge 0$.\nThe [[monotone convergence theorem|0FM6N4R0VB]] gives the limit. It is the [[Euler–Mascheroni constant|00C5P9JZBK]] $\\gamma$.\nThe symbol $\\sim$ is [[asymptotic equivalence|00SB5EA3VW]]. Mathlib states the limit as `Real.tendsto_harmonic_sub_log`.\n\n## Explanations\n\n### L1 human\n\nThe sum $1 + 1/2 + \\cdots + 1/n$ is close to the area under $1/x$ from $1$ to $n$, which is $\\ln n$.\nThe error tends to a constant.\n",
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