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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 *subadditive* if\n$$f(x + y) \\le f(x) + f(y) \\quad \\text{for all } x, y \\in A.$$\n\n## Remarks\n\nExample: $f(x) = \\sqrt{x}$ on $[0, \\infty)$ is subadditive.\nCheck: both sides of $\\sqrt{x + y} \\le \\sqrt{x} + \\sqrt{y}$ are nonnegative, and squaring gives $x + y \\le x + y + 2\\sqrt{xy}$, which is true.\n\nNon-example: $f(x) = x^2$ on $[0, \\infty)$ is not subadditive, since $f(2) = 4 \u003e 2 = f(1) + f(1)$.\n\nThe defining condition is a [[functional inequality|YQB2B0EXY8]].\nA subadditive sequence is a subadditive function on the positive integers, $a_{m+n} \\le a_m + a_n$.\nEvery concave function $f \\colon [0, \\infty) \\to \\R$ with $f(0) \\ge 0$ is subadditive.\n",
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