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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 $b$ of $P$ is a supremum, or least upper bound, of $S$ when these two conditions hold:\n\n1. $b$ is an [[upper bound|0B5H2V4JW8]] of $S$;\n2. if $M$ is an upper bound of $S$, then $b \\le M$.\n\nThe notation is $b = \\sup S$.\n\n## Remarks\n\nA set has at most one supremum, see [[uniqueness of the supremum|04ETX30HT2]].\nIn $\\mathbb{R}$, $\\sup [0, 1) = 1$ and $\\sup [0, 1] = 1$. The first set does not contain its supremum.\nIn $\\mathbb{Q}$, the set $\\{q \\in \\mathbb{Q} : q^2 \u003c 2\\}$ is bounded above but has no supremum.\n\n## Explanations\n\n### L1 human\n\nThe supremum is the lowest possible ceiling for a set.\nFor the set $\\{0.9, 0.99, 0.999, \\dots\\}$, every number at least $1$ is a ceiling, and $1$ is the lowest of these, so the supremum is $1$.\n",
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