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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a partially ordered set and let $S$ be a subset of $P$.\nAn element $M$ of $P$ is an upper bound of $S$ when $s \\le M$ for each $s$ in $S$.\nThe set $S$ is bounded above when it has at least one upper bound in $P$.\nSimilarly, $m$ is a lower bound of $S$ when $m \\le s$ for each $s$ in $S$.\nThe set $S$ is bounded below when it has a lower bound.\n\n## Remarks\n\nIn $\\mathbb{R}$, the interval $[0, 1)$ is bounded above. Each number $M \\ge 1$ is an upper bound.\nThe set $\\mathbb{N}$ is not bounded above in $\\mathbb{R}$, by the [[Archimedean property|0A74SX55QP]].\nAn upper bound of $S$ does not need to be an element of $S$.\n\n## Explanations\n\n### L1 human\n\nA set is bounded above when it does not go up forever: some number is a ceiling for all its elements.\nThe set $\\{1, 1/2, 1/3, \\dots\\}$ is bounded above by $1$, and also by $2$ and by $100$.\n",
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