Let $L(E)$ be the set of lower bounds of a set $E$ and $(S, \le)$ a partially ordered set. For each $s \in S$, let
$$ \langle s] := \{x \in S \mid x \le s\} $$
and
$$ [s\rangle := \{x \in S \mid x \ge s\}. $$
I want to prove that for $E \subseteq S$, one has $L(E) = \langle s]$ for some $s \in S$ $\iff$ inf $E$ exists (i.e., $L(E) = \langle$inf $E]$).
I was wondering if the following is a valid argument: If $e =$ inf $E$ exists, then $L(E)$ contains no $\gamma > e$. Thus, every $\alpha$ $\le e$ is in $L(E)$, which means that, by definition of $\langle s]$, $L(E) = \langle e]$ = $\langle$inf $E]$.
Thanks in advance.
Hint: By definition of infimum, $\text{inf}(E)=\text{max}\{s\in S: \forall e\in E, s\le e\ \}=\text{max}(L(E)),$ if it exists. Saying $L(E)=\left<s\right]$ for some $s$ is just saying that $L(E)$ has a unique greatest element.