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  "body": "\n## Statement\n\nLet $S$ be a nonempty subset of the real numbers that is bounded above, and let $b = \\sup S$ be its [[supremum|0BTMQ04556]].\nFor each real number $c \u003c b$, there is an element $s$ of $S$ with $c \u003c s \\le b$.\n\n## Remarks\n\nProof: if no such $s$ exists, then $c$ is an upper bound of $S$ that is less than $b$. This contradicts the definition of the supremum.\nAn equivalent form: for each $\\varepsilon \u003e 0$, there is $s$ in $S$ with $b - \\varepsilon \u003c s$.\nFor an infimum $a = \\inf S$ and each $c \u003e a$, there is $s$ in $S$ with $a \\le s \u003c c$.\n\n## Explanations\n\n### L1 human\n\nYou can find elements of a set as close to its supremum as you want.\nIf you lower the ceiling even a little, then some element of the set sticks out above it.\n",
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