I am confused. The question is as follows:

Let $(S, \preceq$), be a sup-dense subset of $(T, \preceq)$, both partially ordered sets. In other words, for every $t \in T$, there exists a subset $E \subseteq S$ such that $t = \sup\_T E$. Show that the canonical "inclusion" (an order embedding such that $\iota(s) = s \forall s \in S$) is inf-continuous, or in other words, that for every subset $E \subseteq S$ such that $\inf\_S E$ exists, $\inf\_S E = \inf\_T E$.

I can't figure out how to prove that $\inf\_T E$ exists in the first place. How might I go about doing that?