{
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  "body": "\n## Definition\n\nLet $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.\nFor each $n \\in \\mathbb{N}$, let $i_n = \\inf \\{a_k : k \\ge n\\}$.\nThe sequence $(i_n)$ is increasing and bounded above.\nThe limit inferior of $(a_n)$ is\n$$\\liminf_{n \\to \\infty} a_n = \\lim_{n \\to \\infty} i_n = \\sup_{n \\in \\mathbb{N}} \\inf_{k \\ge n} a_k.$$\n\n## Remarks\n\nThe limit exists by the [[monotone convergence theorem|0FM6N4R0VB]].\nIt satisfies $\\liminf a_n = -\\limsup (-a_n)$, where $\\limsup$ is the [[limit superior|0P8BA7T7S5]].\nAlways $\\inf_n a_n \\le \\liminf a_n \\le \\limsup a_n \\le \\sup_n a_n$.\nFor $a_n = (-1)^n$, the limit inferior is $-1$ and the limit superior is $1$.\n\n## Explanations\n\n### L1 human\n\nThe limit inferior is the lowest value that the sequence keeps coming back to, in the long run.\nEach number below it is undercut by only finitely many terms.\n",
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    "name": "Limit inferior",
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  "msc": [
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  "name": "Limit inferior",
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