Solution
= Solution
The set $P_x=\{y\in P:x\leq y\}$ is nonempty because it contains $x$. Every nonempty chain $C\subseteq P_x$ has an upper bound $u$ in $P$ by hypothesis. Since every member of $C$ is at least $x$, one has $x\leq u$, so $u\in P_x$. <Zorn lemma> therefore gives a maximal element $\sigma$ of $P_x$. If $\sigma\leq z$ in $P$, then $z\in P_x$, and maximality in $P_x$ forces $z=\sigma$. Thus $\sigma$ is maximal in $P$ and satisfies $x\leq\sigma$.