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  "body": "\n## Definition\n\nLet $(A, +)$ be a [[semigroup|0XMEM406JM]], for example $[0, \\infty)$, $\\R$ or the positive integers under addition.\nA [[function|0EPAT51DXQ]] $f \\colon A \\to \\R$ is *superadditive* if\n$$f(x + y) \\ge f(x) + f(y) \\quad \\text{for all } x, y \\in A.$$\n\n## Remarks\n\nExample: $f(x) = x^2$ on $[0, \\infty)$ is superadditive.\nCheck: $(x + y)^2 = x^2 + y^2 + 2xy \\ge x^2 + y^2$, since $xy \\ge 0$.\n\nNon-example: $f(x) = \\sqrt{x}$ on $[0, \\infty)$ is not superadditive, since $\\sqrt{2} \u003c 2 = \\sqrt{1} + \\sqrt{1}$.\n\nA function $f$ is superadditive if and only if $-f$ is [[subadditive|3X9DD9J3CV]].\nEvery convex function $f \\colon [0, \\infty) \\to \\R$ with $f(0) \\le 0$ is superadditive.\nFor a superadditive sequence, $a_{m+n} \\ge a_m + a_n$, the limit of $a_n / n$ exists in $(-\\infty, \\infty]$ and equals $\\sup_n a_n / n$.\n",
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