Let $P$ be a dcpo, $Idl(P)$ be the set of ideals ordered by inclusion, and $f : Idl(P) \to P$ be the function defined by $f(I) = \bigvee I$.
Question: How to show that $f$ is Scott-continuous?
My attempt: Let $D$ be a directed set of ideals with a supremum $\hat I$. One needs to show that $\bigvee \hat I$ is the supremum of the set $\{\bigvee I \; \mid \; I \in D \}$. It is easy to show that $\bigvee \hat I$ is an upper bound, but I cannot figure out how to show that it is the least one.
You already concluded that $\bigvee \hat I$ is an upper bound of $\{\bigvee I : I \in D\}$.
Let $p$ be an arbitrary upper bound of $\{\bigvee I : I \in D\}$.
Then ${\downarrow}p$, the ideal generated by $p$, can easily be proven to be an upper bound of $D$, since $I \subseteq {\downarrow}p$ for each $I \in D$. Thus $\hat I \subseteq {\downarrow}p$.