  Shelah showed that the existence of free subsets over internally approachable
subalgebras follows from the failure of the PCF conjecture on intervals of
regular cardinals. We show that a stronger property called the Approachable
Bounded Subset Property can be forced from the assumption of a cardinal
$\lambda$ for which the set of Mitchell orders $\{ o(\mu) \mid \mu < \lambda\}$
is unbounded in $\lambda$.
  Furthermore, we study the related notion of continuous tree-like scales, and
show that such scales must exist on all products in canonical inner models. We
use this result, together with a covering-type argument, to show that the large
cardinal hypothesis from the forcing part is optimal.
