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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 1}$ be a [[real sequence|0CTV783HYT]] that is subadditive, that is, $a_{m+n} \\le a_m + a_n$ for all integers $m, n \\ge 1$.\nThen the [[limit|05M5SWSZZD]] of $a_n / n$ as $n \\to \\infty$ exists in $[-\\infty, \\infty)$, and\n$$\\lim_{n \\to \\infty} \\frac{a_n}{n} = \\inf_{n \\ge 1} \\frac{a_n}{n},$$\nwhere $\\inf$ is the [[infimum|04KFTA34C5]].\n\n## Remarks\n\nFix $m \\ge 1$ and write each $n$ as $n = qm + r$ with integers $q \\ge 0$ and $1 \\le r \\le m$.\nSubadditivity gives $a_n \\le q a_m + a_r$, so $a_n / n \\le (q m / n)(a_m / m) + a_r / n$.\nAs $n \\to \\infty$ the last term tends to $0$ and $qm/n$ tends to $1$, so $\\limsup a_n / n \\le a_m / m$.\nTaking the infimum over $m$ gives $\\limsup a_n/n \\le \\inf_m a_m/m \\le \\liminf a_n/n$, which proves the claim.\nThe limit can be $-\\infty$: the sequence $a_n = -n^2$ is subadditive, since $-(m+n)^2 \\le -m^2 - n^2$, and $a_n / n = -n$.\nA typical use: if $c_n \u003e 0$ and $c_{m+n} \\le c_m c_n$ for all $m, n$, then $a_n = \\log c_n$ is subadditive, so $c_n^{1/n}$ converges to $\\inf_n c_n^{1/n}$.\nFor a bounded linear operator $A$ on a Banach space with no power equal to $0$, this applies to $c_n = \\norm{A^n}$.\nThe same lemma is often called the subadditivity lemma.\n\n## Explanations\n\n### L1 human\n\nA sequence is subadditive when the value at $m + n$ is at most the sum of the values at $m$ and at $n$.\nFekete's lemma says that the average growth $a_n / n$ then settles down.\nIts limit is the smallest of all the averages $a_n / n$.\n\n### L2 human\n\nThe lemma turns a crude inequality into the existence of a growth rate, without any formula for the terms.\nIt is used to define constants such as the connective constant of self-avoiding walks, entropies, and the spectral radius $\\lim_n \\norm{A^n}^{1/n}$.\nKingman's subadditive ergodic theorem is a random version of the same idea.\nFor a superadditive sequence, with $a_{m+n} \\ge a_m + a_n$, apply the lemma to $-a_n$: then $a_n / n$ tends to $\\sup_n a_n / n$.\n",
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