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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.\nAn element $u \\in P$ is an upper bound of $S$ when $s \\le u$ for each $s \\in S$.\nThe set $S$ is bounded above when it has an upper bound.\n\n## Remarks\n\nAn upper bound need not be in $S$. The number $1$ is an upper bound of $[0, 1)$ in $\\mathbb{R}$, and so is $5$.\nA lower bound is defined in the same way with $u \\le s$.\nThe least upper bound of $S$, when it exists, is the supremum of $S$.\n\n## Explanations\n\n### L1 human\n\nAn upper bound is a ceiling for the set: no element of the set is above it.\nThe set $\\{1, 1/2, 1/3, \\dots\\}$ has upper bounds $1$, $2$, $100$ and many more.\n",
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