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  "body": "\n## Statement\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] in which every nonempty [[chain|0S8FE7ZMFM]] has a [[supremum|0R3H5PANRQ]].\nLet $f \\colon P \\to P$ be a [[monotone|0A7KV2WB7A]] map, and let $a \\in P$ satisfy $a \\le f(a)$.\nThen $f$ has a fixed point $p$ with $a \\le p$.\nMoreover, $p$ can be chosen as the least element of the set $\\set{x \\in P : f(x) = x \\text{ and } a \\le x}$.\n\n## Remarks\n\nProof idea: define $x_0 = a$, $x_{\\alpha + 1} = f(x_\\alpha)$, and $x_\\lambda = \\sup_{\\alpha \u003c \\lambda} x_\\alpha$ at limit ordinals $\\lambda$.\nBy transfinite induction, $(x_\\alpha)$ is increasing and each $x_\\alpha$ satisfies $x_\\alpha \\le f(x_\\alpha)$, so the suprema exist.\nThe sequence cannot be strictly increasing for all ordinals, since $P$ is a set, so $x_{\\alpha + 1} = x_\\alpha$ for some $\\alpha$, and $p = x_\\alpha$ is a fixed point.\nIf $q$ is a fixed point with $a \\le q$, induction gives $x_\\alpha \\le q$ for all $\\alpha$, so $p$ is the least such fixed point.\nThe theorem contains the existence part of the [[Knaster–Tarski theorem|RHXP3HA64M]]: in a complete lattice, take $a$ to be the least element.\nThe hypothesis $a \\le f(a)$ is needed.\nOn the two-element antichain $\\set{a, b}$, every nonempty chain is a single point, and the swap of $a$ and $b$ is monotone and has no fixed point.\n\n## Explanations\n\n### L1 human\n\nIf a monotone rule moves some point $a$ upward, keep applying it.\nThe points climb, and whenever the climb runs through infinitely many steps, jump to the top of what has been reached.\nBecause every chain has a top, the climb never gets stuck, and since there are only so many points it must stop at a point that the rule fixes.\nThis fixed point is the lowest one above $a$.\n\n### L2 human\n\nThe theorem weakens the completeness hypothesis of the Knaster–Tarski theorem to chain-completeness, at the price of needing a starting point $a$ with $a \\le f(a)$.\nIn a chain-complete poset with a least element, every monotone map has a least fixed point.\nCompared with the [[Bourbaki–Witt theorem|9EABA9ZDZS]], it asks for monotonicity instead of $x \\le f(x)$ for all $x$, and in return it gives a least fixed point above $a$.\n",
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