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  "body": "\n## Example\n\nFor each integer $n \\ge 1$, let $a_n = n^{1/n}$ be the positive $n$-th root of $n$.\nThen $a_n \\to 1$ as $n \\to \\infty$.\nFor the proof, write $a_n = 1 + h_n$ with $h_n \\ge 0$.\nFor $n \\ge 2$, the binomial theorem gives $n = (1 + h_n)^n \\ge \\binom{n}{2} h_n^2 = \\frac{n(n-1)}{2} h_n^2$.\nThus $0 \\le h_n \\le \\sqrt{2/(n-1)}$, and the [[squeeze theorem|0XMF00T4XQ]] gives $h_n \\to 0$.\n\n## Remarks\n\nThe first terms are $1, \\sqrt{2} \\approx 1.414, 3^{1/3} \\approx 1.442, 4^{1/4} \\approx 1.414$. The largest term is at $n = 3$.\nIn the same way, $c^{1/n} \\to 1$ for each real $c \u003e 0$.\nMathlib proves the version for a real variable, $x^{1/x} \\to 1$ as $x \\to \\infty$, in the lemma `tendsto_rpow_div`.\n\n## Explanations\n\n### L1 human\n\nThe number $n$ grows, but the root index $n$ grows too, and the root wins.\nFor example, $1000^{1/1000}$ is about $1.0069$, very close to $1$.\n",
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