This is a follow up question to one I posted earlier. I'm trying to decide that if for $(S,\preceq)$ a partially ordered set and $E\subseteq S$, one has $L(E)=\langle s]$ for some $s\in S$ iff $\inf E$ exists, and in particular, $L(E)=\langle\inf E]$.
I'm simply curious about showing that $L(E)\subseteq\langle\inf E]$ when $\inf E$ exists. Taking some $x\in L(E)$, if $x$ is comparable to $\inf E$, then $x\preceq\inf E$, and so $x\in\langle\inf E]$. But what if $x$ and $\inf E$ are not comparable? Is it still possible to show such an inclusion?
Edit: $\langle s]$ is the set {$x\in S \ | \ x\preceq s$}.