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  "body": "\n## Definition\n\nLet $A$ be a set of positive integers.\nFor each integer $n \\ge 1$, let $A(n)$ be the number of elements $a \\in A$ with $a \\le n$.\nIf the [[limit|05M5SWSZZD]]\n$$d(A) = \\lim_{n \\to \\infty} \\frac{A(n)}{n}$$\nexists, then $d(A)$ is the *natural density* (or *asymptotic density*) of $A$.\nThe *upper density* $\\overline{d}(A)$ and the *lower density* $\\underline{d}(A)$ are the limit superior and the [[limit inferior|033RZ32T4Q]] of $A(n)/n$; they always exist.\n\n## Remarks\n\nThe natural density exists if and only if $\\overline{d}(A) = \\underline{d}(A)$, and then all three numbers are equal.\n\nExample: for the multiples of a positive integer $m$, $A(n) = \\floor{n/m}$, and $n/m - 1 \u003c A(n) \\le n/m$; so $d(A) = 1/m$.\n\nNon-example: let $A$ be the set of positive integers whose first decimal digit is $1$.\nFor $n = 2 \\cdot 10^k - 1$ one has $A(n) = 1 + 10 + \\dots + 10^k = (10^{k+1} - 1)/9$, so $A(n)/n \\to 5/9$.\nFor $n = 10^{k+1} - 1$ the count $A(n)$ is the same, so $A(n)/n \\to 1/9$.\nHence the limit does not exist; here $\\overline{d}(A) = 5/9$ and $\\underline{d}(A) = 1/9$.\n\nIf $d(A)$ and $d(B)$ exist and $A \\cap B$ is empty, then $d(A \\cup B) = d(A) + d(B)$.\nNatural density is not the same as [[Schnirelmann density|QHH8RY9XCZ]], which is an infimum over all $n$ and is $0$ for every set not containing $1$.\n",
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