Cofinal subset

Remarks

Example: the set $\N$ is cofinal in $\R$ with the usual order. Check: for $x \in \R$, the integer $\ceil{\abs{x}}$ is in $\N$ and is at least $x$.

Non-example: the set $(-\infty, 0]$ is not cofinal in $\R$, since no element of it is at least $1$.

If $P$ has a greatest element $m$, then a subset is cofinal exactly when it contains some $c$ with $m \le c$. In a partial order, this means that the subset contains $m$. Being cofinal is transitive: a cofinal subset of a cofinal subset of $P$ is cofinal in $P$. The dual notion is a coinitial subset, in which every element of $P$ lies above some element of the subset.