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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[preordered set|0VABYWJWH9]] and let $C \\subseteq P$ be a [[subset|0BJR9QR08E]].\nThe subset $C$ is *cofinal* in $P$ when for every $x \\in P$ there is $c \\in C$ with $x \\le c$.\n\n## Remarks\n\nExample: the set $\\N$ is cofinal in $\\R$ with the usual order.\nCheck: for $x \\in \\R$, the integer $\\ceil{\\abs{x}}$ is in $\\N$ and is at least $x$.\n\nNon-example: the set $(-\\infty, 0]$ is not cofinal in $\\R$, since no element of it is at least $1$.\n\nIf $P$ has a greatest element $m$, then a subset is cofinal exactly when it contains some $c$ with $m \\le c$.\nIn a partial order, this means that the subset contains $m$.\nBeing cofinal is transitive: a cofinal subset of a cofinal subset of $P$ is cofinal in $P$.\nThe dual notion is a coinitial subset, in which every element of $P$ lies above some element of the subset.\n",
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