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  "body": "\n## Definition\n\nLet $(a_n)$ be a [[real sequence|0CTV783HYT]] and let $L$ be a real number.\nThe sequence $(a_n)$ converges to $L$ if the following holds for each open interval $U$ that contains $L$.\nThere is a natural number $N$ such that $a_n \\in U$ for all $n \\ge N$.\nThen $L$ is a limit of $(a_n)$, with the notation $\\lim_{n \\to \\infty} a_n = L$ or $a_n \\to L$ as $n \\to \\infty$.\n\n## Remarks\n\nThe [[epsilon-N definition of the limit of a sequence|09G6KHS6GY]] is an equivalent form, with intervals $(L - \\varepsilon, L + \\varepsilon)$.\nA sequence has at most one limit, by the [[uniqueness of limits|0ZXJ2455NJ]]. So the notation $\\lim_{n \\to \\infty} a_n$ is well defined.\nThe same definition works in each topological space, with open neighbourhoods of $L$ in place of open intervals.\nIn Mathlib, the limit is the statement `Filter.Tendsto a Filter.atTop (nhds L)`.\n\n## Explanations\n\n### L1 human\n\nThe sequence $(a_n)$ tends to $L$ when its terms come close to $L$ and stay close.\nHowever small a window around $L$ you choose, from some index on, all terms are inside the window.\nFor example, $1/(n+1)$ tends to $0$.\n",
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