  Given a unital inductive limit of C*-algebras for which each C*-algebra of
the inductive sequence comes equipped with a compact quantum metric of Rieffel,
we produce sufficient conditions to build a compact quantum metric on the
inductive limit from the quantum metrics on the inductive sequence by utilizing
the completeness of the dual Gromov-Hausdorff propinquity of Latremoliere on
compact quantum metric spaces. This allows us to place new quantum metrics on
all unital AF algebras that extend our previous work with Latremoliere on
unital AF algebras with faithful tracial state. As a consequence, we produce a
continuous image of the entire Fell topology on the ideal space of any unital
AF algebra in the dual Gromov-Hausdorff propinquity topology.
