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  "body": "\n## Statement\n\nLet $S$ be a nonempty subset of the real numbers that is [[bounded below|0B5H2V4JW8]].\nThen $S$ has an [[infimum|04KFTA34C5]] in $\\mathbb{R}$.\n\n## Remarks\n\nProof: the set $-S = \\{-s : s \\in S\\}$ is nonempty and bounded above. By the [[completeness axiom|0AXWET80YW]], it has a supremum $b$. Then $-b$ is the infimum of $S$.\nAnother proof: the set of lower bounds of $S$ is nonempty and bounded above by each element of $S$, and its supremum is $\\inf S$.\n\n## Explanations\n\n### L1 human\n\nIf a set of real numbers has a floor, then it has a highest floor.\nThis follows from the completeness axiom by a reflection of the number line through $0$, which turns floors into ceilings.\n",
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