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  "body": "\n## Statement\n\nLet $(a_n)_{n \\ge 1}$ be a real sequence that converges to a real number $L$.\nFor $n \\ge 1$, let $\\sigma_n = (a_1 + \\cdots + a_n)/n$ be its [[Cesàro mean|0P0DJHBGBB]].\nThen $\\sigma_n \\to L$ as $n \\to \\infty$.\n\n## Remarks\n\nFor the proof, let $\\varepsilon \u003e 0$ and take $N$ with $|a_k - L| \u003c \\varepsilon$ for $k \u003e N$. For $n \u003e N$,\n$$|\\sigma_n - L| \\le \\frac{1}{n} \\sum_{k=1}^{N} |a_k - L| + \\frac{n - N}{n} \\varepsilon.$$\nThe first term tends to $0$, so $|\\sigma_n - L| \u003c 2\\varepsilon$ for large $n$.\nThe converse fails. The means of $(-1)^n$ tend to $0$, but $(-1)^n$ diverges.\nThe statement is a special case of the [[Stolz–Cesàro theorem|0Q3AKZV74D]] with $b_n = n$.\n\n## Explanations\n\n### L1 human\n\nIf the terms settle near $L$, then the running average also settles near $L$.\nThe early terms that are far from $L$ count less and less, because we divide by $n$.\n",
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