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  "body": "\n## Definition\n\nLet $(a_n)$ be a [[real sequence|0CTV783HYT]].\nThe sequence is bounded above if there is a real number $M$ with $a_n \\le M$ for all $n \\in \\mathbb{N}$.\nThe sequence is bounded below if there is a real number $m$ with $a_n \\ge m$ for all $n \\in \\mathbb{N}$.\nThe sequence is bounded if it is bounded above and bounded below.\nEquivalently, $(a_n)$ is bounded if there is a real number $K \\ge 0$ with $|a_n| \\le K$ for all $n \\in \\mathbb{N}$.\n\n## Remarks\n\nThe sequence $a_n = (-1)^n$ is bounded, with $K = 1$.\nThe sequence $a_n = n$ is bounded below by $0$, but it is not bounded above.\nA sequence is a function on $\\mathbb{N}$, so a bounded sequence is a bounded function on $\\mathbb{N}$.\nEach [[convergent sequence|0AZ3R52593]] is bounded, as the node [[convergent sequences are bounded|0S4JF619V7]] states.\n\n## Explanations\n\n### L1 human\n\nA sequence is bounded when all its terms stay inside one fixed interval $[-K, K]$.\nThe terms can move up and down, but they never escape from that interval.\n",
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