  Density functional theory together with the Kohn-Sham (KS) scheme represent
an efficient framework to recover the ground state density and energy of a
many-body quantum system from an auxiliary "non-interacting" system (one-body
with a local potential that is a density functional). However, to fully achieve
establishing the KS scheme, a general proof of the validity of the so-called
"non-interacting $v$-representability" conjecture is still required. In this
article, we propose such a proof. We reduce the demonstration to proving (1)
that the KS potential is differentiable for all non-interacting
$v$-representable densities and (2) that its derivative is bounded. Then, we
demonstrate points (1) and (2) applying static linear response to the
non-interacting system.
