Exercise 9.2.3.12 ($\operatorname{Ind}$-Completion of Partially Ordered Sets). Let $(A, \leq )$ be a (small) partially ordered set. Recall that an ideal of $A$ is a subset $J \subseteq A$ satisfying the following conditions:
The set $J$ is downward-closed: that is, for elements $a \leq b$ of $A$, if $b$ belongs to $J$, then $a$ also belongs to $J$.
The set $J$ is directed: that is, every finite subset of $J$ has an upper bound (which is also contained in $J$).
Let $\widehat{A}$ denote the collection of all ideals $J \subseteq A$, which we regard as partially ordered by inclusion. Show that the nondecreasing function
exhibits the $\infty $-category $\operatorname{N}_{\bullet }( \widehat{A} )$ as an $\operatorname{Ind}$-completion of $\operatorname{N}_{\bullet }(A)$.