Kerodon

Proof. Let $T \subseteq Q$ be the collection of all upper bounds for $S$: that is, the collection of all elements $t \in Q$ satisfying $s \leq t$ for each $s \in S$. We wish to show that $T$ has a smallest element. Our assumption that $(Q, \leq )$ is a complete lattice guarantees that $T$ has a greatest lower bound $t = \mathrm{inf}(T)$. Note that every element $s \in S$ is a lower bound for $T$, and therefore satisfies $s \leq t$. It follows that $t \in T$ is a least upper bound for $S$. $\square$