Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 16H / b

The set $P_x=\{y\in P:x\le y\}$ 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 $x\le u$, so $u\in P_x$. Zorn lemma therefore gives a maximal element $\sigma$ of $P_x$. If $\sigma \le z$ in $P$, then $z\in P_x$, and maximality in $P_x$ forces $z=\sigma$. Thus $\sigma$ is maximal in $P$ and satisfies $x\le \sigma$.