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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence and let $L$ be a real number.\nThen $(a_n)$ converges to $L$, in the sense of the [[limit of a sequence|0XKWSSY8P2]], if and only if the following holds.\nFor each real number $\\varepsilon \u003e 0$ there is a natural number $N$ such that $|a_n - L| \u003c \\varepsilon$ for all $n \\ge N$.\n\n## Remarks\n\nThe proof uses two facts. Each interval $(L - \\varepsilon, L + \\varepsilon)$ is open and contains $L$. Each open interval that contains $L$ contains such an interval.\nIn a metric space $(X, d)$ the same statement holds with $d(a_n, L)$ in place of $|a_n - L|$.\nThe number $N$ depends on $\\varepsilon$. A smaller $\\varepsilon$ in general needs a larger $N$.\n\n## Explanations\n\n### L1 human\n\nThe number $\\varepsilon$ is the allowed error, and $N$ is the point after which the error is always smaller than $\\varepsilon$.\nFor $a_n = 1/(n+1)$ and $L = 0$, the choice $N \u003e 1/\\varepsilon$ works.\n",
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  "msc": [
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  "name": "Epsilon-N definition of the limit of a sequence",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/09G/09G6KHS6GY.md",
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